Robotics and automation an introduction-.pdf

Robotics
Matjaž Mihelj · Tadej Bajd
Aleš Ude · Jadran Lenarčič
Aleš Stanovnik · Marko Munih
Jure Rejc · Sebastjan Šlajpah
Second Edition

Matjaž Mihelj • Tadej Bajd • Aleš Ude
Jadran Lenarčič • Aleš Stanovnik
Marko Munih • Jure Rejc • Sebastjan Šlajpah
Robotics
Second Edition
123

Matjaž Mihelj
Faculty of Electrical Engineering
University of Ljubljana
Ljubljana, Slovenia
Tadej Bajd
Ljubljana, Slovenia
Aleš Ude
Department of Automatics,
Biocybernetics and Robotics
Jožef Stefan Institute
Ljubljana, Slovenia
Jadran Lenarčič
Jožef Stefan Institute
Ljubljana, Slovenia
Aleš Stanovnik
Ljubljana, Slovenia
Marko Munih
Ljubljana, Slovenia
Jure Rejc
Ljubljana, Slovenia
Sebastjan Šlajpah
Ljubljana, Slovenia
ISBN 978-3-319-72910-7 ISBN 978-3-319-72911-4 (eBook)
https://doi.org/10.1007/978-3-319-72911-4
Library of Congress Control Number: 2018946678
1st edition: © Springer Science+Business Media B.V. 2010
2nd edition: © Springer International Publishing AG, part of Springer Nature 2019
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Preface
It is perhaps difficult to agree on what a robot is, but most people working in
robotics would probably quote the “Father of Robotics”, Joseph F. Engelberger
(1925–2015), a pioneer in industrial robotics, stating “I can’t define a robot, but I
know one when I see one”.
The word robot does not originate from a scientific or engineering vocabulary,
but was first used in the Czech drama “R.U.R.” (Rossum’s Universal Robots) by
Karel Čapek, that was first played in Prague in 1921. The word itself was invented
by his brother Josef. In the drama the robot is an artificial human being which is a
brilliant worker, deprived of all “unnecessary qualities”, such as emotions, cre-
ativity, and the capacity for feeling pain. In the prologue of the drama the following
definition of robots is given: “Robots are not people (Roboti nejsou lidé). They are
mechanically more perfect than we are, they have an astounding intellectual
capacity, but they have no soul. The creation of an engineer is technically more
refined than the product of nature”.
The book Robotics evolved through decades of teaching robotics at the Faculty
of Electrical Engineering, University of Ljubljana, Slovenia, where the first text-
book on industrial robotics was published in 1980 (A. Kralj and T. Bajd,
“Industrijska robotika”). The way of presenting this rather demanding subject was
successfully tested with several generations of undergraduate students.
The second edition of the book continues the legacy of the first edition that won
the Outstanding Academic Title distinction from the library magazine CHOICE in
2011. The major feature of the book remains its simplicity. The introductory
chapter now comprehensively covers different robot classes with the main focus on
industrial robots. The position, orientation, and displacement of an object are
described by homogenous transformation matrices. These matrices, which are the
basis for any analysis of robot mechanisms, are introduced through simple geo-
metrical reasoning. Geometrical models of the robot mechanism are explained with
the help of an original, user-friendly vector description. With the world of the
roboticist being six-dimensional, orientation of robot end effectors received more
attention in this edition.
v

Robot kinematics and dynamics are introduced via a mechanism with only two
rotational degrees of freedom, which is however an important part of the most
popular industrial robot structures. The presentation of robot dynamics is based on
only the knowledge of Newton’s law and was additionally simpliﬁed for easier
understanding of this relatively complex matter. The workspace plays an important
role in selecting a robot appropriate for the planned task. The kinematics of parallel
robots is signiﬁcantly different from the kinematics of serial manipulators and
merits additional attention.
Robot sensors presented in this edition are relevant not only for industrial
manipulators, but also for complex systems such as humanoid robots. Robot vision
has an increasingly important role in industrial applications and robot trajectory
planning is a prerequisite for successful robot control. Basic control schemes,
resulting in either the desired end-point trajectory or in the force between the robot
and its environment, are explained. Robot environments are illustrated by product
assembly processes, where robots are a part of a production line or operate as
completely independent units. Robot grippers, tools, and feeding devices are also
described.
With the factory floor becoming ever more complex, interaction between
humans and robots will be inevitable. Collaborative robots are designed for safe
human-robot interaction. Flexibility of production can be further increased with the
use of wheeled mobile robots. A glimpse into the future, when humans and robots
will be companions, is presented in the chapter on humanoid robotics, the com-
plexity of which requires more advanced knowledge of mathematics. The chapter
on standardization and measurement of accuracy and repeatability is of interest for
users of industrial robots.
The book requires a minimal advanced knowledge of mathematics and physics.
It is therefore appropriate for introductory courses in robotics at engineering fac-
ulties (electrical, mechanical, computer, civil). It could also be of interest for
engineers who had not studied robotics, but who have encountered robots in the
working environment and wish to acquire some basic knowledge in a simple and
fast manner.
Ljubljana, Slovenia Matjaž Mihelj
April 2018 Tadej Bajd
vi Preface

Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Robot Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Industrial Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Homogenous Transformation Matrices . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Translational Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Rotational Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Pose and Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Geometrical Robot Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Geometric Description of the Robot Mechanism . . . . . . . . . . . . . . . 27
3.1 Vector Parameters of a Kinematic Pair . . . . . . . . . . . . . . . . . . . 27
3.2 Vector Parameters of the Mechanism . . . . . . . . . . . . . . . . . . . . 31
4 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Two-Segment Robot Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3 Workspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6 Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.1 Characteristics of Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Kinematic Arrangements of Parallel Robots . . . . . . . . . . . . . . . 73
6.3 Modelling and Design of Parallel Robots . . . . . . . . . . . . . . . . . 78
7 Robot Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.1 Principles of Sensing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.2 Sensors of Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.2.1 Placing of Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.2.2 Potentiometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
vii

7.2.3 Optical Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.2.4 Magnetic Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2.5 Tachometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2.6 Inertial Measurement Unit. . . . . . . . . . . . . . . . . . . . . . 94
7.3 Contact Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.3.1 Tactile Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.3.2 Limit Switch and Bumper . . . . . . . . . . . . . . . . . . . . . . 98
7.3.3 Force and Torque Sensor . . . . . . . . . . . . . . . . . . . . . . 98
7.3.4 Joint Torque Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.4 Proximity and Ranging Sensors . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4.1 Ultrasonic Rangefinder . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4.2 Laser Rangefinder and Laser Scanner . . . . . . . . . . . . . 103
8 Robot Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.1 System Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.2 Forward Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.3 Backward Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.3.1 Single Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.3.2 Stereo Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.4 Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.5 Object Pose from Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.5.1 Camera Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.5.2 Object Pose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9 Trajectory Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.1 Interpolation of the Trajectory Between Two Points . . . . . . . . . 123
9.2 Interpolation by Use of via Points . . . . . . . . . . . . . . . . . . . . . . 126
10 Robot Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
10.1 Control of the Robot in Internal Coordinates . . . . . . . . . . . . . . 134
10.1.1 PD Control of Position . . . . . . . . . . . . . . . . . . . . . . . . 135
10.1.2 PD Control of Position with Gravity Compensation . . . 136
10.1.3 Control of the Robot Based on Inverse Dynamics . . . . 137
10.2 Control of the Robot in External Coordinates . . . . . . . . . . . . . . 141
10.2.1 Control Based on the Transposed Jacobian Matrix . . . . 142
10.2.2 Control Based on the Inverse Jacobian Matrix . . . . . . . 143
10.2.3 PD Control of Position with Gravity Compensation . . . 144
10.2.4 Control of the Robot Based on Inverse Dynamics . . . . 144
10.3 Control of the Contact Force . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10.3.1 Linearization of a Robot System Through Inverse
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
10.3.2 Force Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
viii Contents

11 Robot Environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
11.1 Robot Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
11.2 Robot Peripherals in Assembly Processes . . . . . . . . . . . . . . . . . 158
11.2.1 Assembly Production Line Conﬁgurations . . . . . . . . . . 158
11.3 Feeding Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
11.4 Conveyors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
11.5 Robot Grippers and Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
12 Collaborative Robots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
12.1 Collaborative Industrial Robot System . . . . . . . . . . . . . . . . . . . 173
12.2 Collaborative Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
12.3 Collaborative Operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
12.3.1 Safety-Rated Monitored Stop . . . . . . . . . . . . . . . . . . . 178
12.3.2 Hand Guiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
12.3.3 Speed and Separation Monitoring . . . . . . . . . . . . . . . . 180
12.3.4 Power and Force Limiting . . . . . . . . . . . . . . . . . . . . . 181
12.4 Collaborative Robot Grippers . . . . . . . . . . . . . . . . . . . . . . . . . 184
12.5 Applications of Collaborative Robotic System . . . . . . . . . . . . . 185
13 Mobile Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
13.1 Mobile Robot Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
13.2 Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
13.2.1 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
13.2.2 Path Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
13.2.3 Path Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
14 Humanoid Robotics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
14.1 Biped Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
14.1.1 Zero-Moment Point . . . . . . . . . . . . . . . . . . . . . . . . . . 211
14.1.2 Generation of Walking Patterns . . . . . . . . . . . . . . . . . . 213
14.2 Imitation Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
14.2.1 Observation of Human Motion and Its Transfer
to Humanoid Robot Motion . . . . . . . . . . . . . . . . . . . . 217
14.2.2 Dynamic Movement Primitives . . . . . . . . . . . . . . . . . . 221
14.2.3 Convergence Properties of Linear Dynamic
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
14.2.4 Dynamic Movement Primitives for Point-to-Point
Movements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
14.2.5 Estimation of DMP Parameters from a Single
Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
14.2.6 Modulation of DMPs . . . . . . . . . . . . . . . . . . . . . . . . . 227
15 Accuracy and Repeatability of Industrial Manipulators . . . . . . . . . 231
Derivation of the Acceleration in Circular Motion . . . . . . . . . . . . . . . . . . 243
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Contents ix

Chapter 1
Introduction
Today’s robotics can be described as a science dealing with intelligent movement of
various robot mechanisms which can be classified in the following four groups: robot
manipulators, robot vehicles, man-robot systems and biologically inspired robots
(Fig.1.1). The most frequently encountered robot manipulators are serial robot mech-
anisms. The robot manipulator is represented by a serial chain of rigid bodies, called
robot segments, connected by joints. Serial robot manipulators will be described in
more details in the next section of this chapter. Parallel robots are of considerable
interest both in science and in industry. With these, the robot base and platform
are connected to each other with parallel segments, called legs. The segments are
equipped with translational actuators, while the joints at the base and platform are
passive. Parallel robots are predominantly used for pick-and-place tasks. They are
characterized by high accelerations, repeatability, and accuracy. As the robot ma-
nipulators replace the human operator at various production jobs, their size is often
similar to that of a human arm. Manufacturers can also provide robot manipulators
which are up to ten times larger, capable of manipulating complete car bodies. By
contrast in the areas of biotechnology and new materials micro- and nanorobots are
used. Nanorobots enable pushing, pulling, pick-and-place manipulations, orienting,
bending, and grooving on the scale of molecules and particles. The most widespread
nanomanipulator is based on the principle of atomic force microscope. The actuator
of this nanomanipulator is a piezoelectric crystal, the movement of which is assessed
by the use of a laser source and photocell.
Autonomous robot vehicles are found on land, in the water and in the air. The
land-based mobile robots are most often applied in man-made environments, such
as apartments, hospitals, department stores, or museums, but can increasingly be
found on highways and even pathless grounds. Most mobile robots are nevertheless
used on flat ground with movement enabled by wheels, with three wheels providing
the necessary stability. Often the wheels are specially designed to enable omnidi-
rectional movements. Robot vehicles can be found as vacuum cleaners, autonomous
lawn mowers, intelligent guides through department stores or museums, attendants
© Springer International Publishing AG, part of Springer Nature 2019
M. Mihelj et al., Robotics, https://doi.org/10.1007/978-3-319-72911-4_1
1

2 1 Introduction
ROBOTS
ROBOT
MANIPULATORS
ROBOT
VEHICLES
MAN-ROBOT
SYSTEMS
BIOLOGICALLY
INSPIRED ROBOTS
SERIAL
PARALLEL
MICRO
AND NANO
LAND
WATER
AIR
HAPTIC
ROBOTICS
TELEMANI-
PULATORS
EXOSKELETONS
HUMANOIDS
ROBOTS FROM
ANIMAL WORLD
Fig. 1.1 Classification of robots
in clinical centers, space rovers, or autonomous cars. Students can enjoy learning
in various competitions, for example football or rescue games, based on the use
of small mobile robots. Among the aerial vehicles, the most popular appear to be
small quadrocopters. They have a very simple mechanical structure what makes them
comparatively inexpensive. Quadrocopters fly using four rotors and are equipped
with sensors such as gyroscopes, accelerometers, and cameras, and mostly used for
surveillance purposes. Larger autonomous versions are used for military reconnais-
sance missions. Water-based robots either float on the surface or operate under water.
The underwater versions can have the shape of smaller autonomous submarines. They
can often be equipped with a robotic arm and used in ocean research, sea floor or ship
wrecks observation or as attendants on oil platforms. Autonomous floating robots
are used for marine ecological assesments.
New knowledge in the area of robot control is strongly influencing the develop-
ment of man-robot systems, such as haptic robots, telemanipulators, and exoskele-
tons. The use of haptic robots is related to virtual environments which are usually
displayed on the computer screens. Early virtual environments provided sight and
sound to the observer, but no sense of touch. Haptic robots provide the user with the
feeling of touch, limited motion, compliance, friction, and texture in virtual environ-
ment. Haptic robots play an important role in rehabilitation robotics, where small
haptic robots are used for the assessment and evaluation of movements of the upper
extremities in paralyzed persons. Stronger haptic systems can hold the wrist of a
paralyzed person and guide the arm end-point along the desired path which is shown
to the subject in a virtual environment presented on the computer screen. The haptic
robot exerts two types of the forces to the subject’s wrist. When the patient is unable
to perform a movement along the path shown to him in the virtual environment, the
robot pushes the wrist along the required trajectory and helps the patient to accom-
plish the task. The robot is helping only to the extent necessary for the patient to
reach the goal point. When the patient’s paralyzed extremity travels away from the

1 Introduction 3
planned curve, the robot pushes the wrist to the vicinity of the required trajectory.
Telemanipulators are robots which are controlled by a human operator when there is
a barrier between the telemanipulator and the human operator. The barrier between
the operator and working environment is usually either distance (e.g. outer space) or
dangerousness (e.g. inside a nuclear plant). Telemanipulators are also entering the
medical world, being used in surgery (telemedicine). Exoskeletons are active mech-
anisms which are attached to human upper or lower extremities. They are mainly
used for rehabilitation purposes. Lower limb exoskeletons may increase the strength
of healthy persons or enable the retraining of paralyzed persons in walking. In com-
parison with haptic rehabilitation robots, exoskeletons for upper extremities exert
forces to all segments of paralyzed arm.
Biologically inspired robots can be divided into humanoid robots and the robots
from the animal world. Examples from the animal world are various types of robotic
snakes, fish, quadrupeds, six- or eight leg walking robots. Humanoid robots are by
far the most advanced robot systems in the group of the biologically inspired robots.
They are designed to live and work in a human environment. The most noticeable
property of humanoid robots is their ability of bipedal walking. They walk either
with statically stable or dynamically stable gait, they can balance while standing on
a single leg, they move in accordance with human co-worker, they can even run. The
current problems in humanoid robotics are related to artificial vision, perception and
analysis of environment, natural language processing, human interaction, cognitive
systems, machine learning and behaviors. Some robots also learn from experience
replicating natural processes such as trial-and-error and learning by doing, in the
same way a small child learns. In this way the humanoid robot gains a certain degree
of autonomy which further means that humanoid robots can behave in some situations
in a way that is unpredictable to their human designers. Humanoid robots are coming
into our homes and are becoming our partners. They may soon be companions to the
elderly and children, assistants to nurses, physicians, firemen, and workers. The need
is arising to embody ethics into a robot, which is refered to as robo-ethics. Robo-
ethics is an applied ethics whose objective is to develop scientific/cultural/technical
tools that can be shared by different social groups and beliefs. These tools aim to
promote and encourage the development of robotics for the advancement of human
society and individuals, and to help preventing its misuse against humankind. In 1942
the outstanding novelist Isaac Asimov formulated his famous three laws of robotics.
Later, in 1983, he added the fourth law, known as the zeroth law: No robot may harm
humanity or through inaction, allow humanity to come in harm. The new generation
of humanoid robots will be partners that coexist with humans assisting them both
physically and psychologically and will contribute to the realization of a safe and
peaceful society. They will be potentially more ethical than humans.

4 1 Introduction
arm
gripper
wrist
1
2
3
Fig. 1.2 Robot manipulator
1.1 Robot Manipulator
Today the most useful and efficient robotic systems are the industrial robot manipu-
lators which can replace the human workers in difficult or monotonous jobs, or where
a human would otherwise be faced with hazardous conditions. The robot manipu-
lator consists of a robot arm, wrist, and gripper (Fig.1.2). The robot arm is a serial
chain of three rigid segments which are relatively long and provide positioning of
the gripper in the workspace. Neighboring segments of a robot arm are connected
through a robot joint, which is (Fig.1.3) either translational (prismatic) or rotational
(revolute). The rotational joint has the form of a hinge and limits the motion of two
neighbor segments to rotation around the joint axis. The relative position is given by
the angle of rotation around the joint axis. In robotics the joint angles are denoted by
the Greek letter ϑ. In the simplified diagrams the rotational joint is represented by a
cylinder. The translational joint restricts the movement of two neighboring segments
to translation. The relative position between two segments is measured as a distance.
The symbol of the translational joint is a prism, while the distance is denoted by the
letter d. Robot joints are powered by either electric or hydraulic motors. The sensors
in the joints are measuring the angle or distance, velocity, and torque.
The robot wrist usually consists of three rotational joints. The task of the robot
wrist is to enable the required orientation of the object grasped by the robot gripper.
The two- or multi-fingered robot gripper is placed at the robot endpoint. Different
tools, to enable drilling, spray painting, or welding devices, can be also attached to
the endpoint. Industrial robot manipulators usually allow mobility in six degrees of

1.1 Robot Manipulator 5
angle of rotation
joint axis joint axis
distance
d
ϑ
Fig. 1.3 Rotational (left) and translational (right) robot joint
3 translations 2 rotations 1 rotation
POSITION ORIENTATION
POSE
Fig. 1.4 Degrees of freedom of a rigid body
freedom, meaning that the robotic mechanism has six joints and also six actuators.
In this way the robot arm can position an object to an arbitrary place in the robot
workspace, while the gripper can rotate the object about all three axes of a rectangular
coordinate frame.
In order to clarify the term degree of freedom, let us first consider a rigid body
which usually represents the object manipulated by the industrial robot. The simplest
rigid body consists of three mass particles (Fig.1.4). A single mass particle has three
degrees of freedom, described by three displacements along the axes of a rectangular
frame. The displacement along a line is called translation. We add another mass
particle to the first one in such a way that there is constant distance between them.
The second particle is restricted to move on the surface of a sphere surrounding the
first particle. Its position on the sphere can be described by two circles reminding
us of meridians and latitudes on a globe. The displacement along a circular line is

6 1 Introduction
called rotation. The third mass particle is added in such a way that the distances with
respect to the first two particles are kept constant. In this way the third particle may
move along the circle, a kind of equator, around the axis determined by the first two
particles. A rigid body therefore has six degrees of freedom: three translations and
three rotations. The first three degrees of freedom describe the position of the body,
while the other three degrees of freedom determine its orientation. The term pose
is used to include both position and orientation. It is often said that while the world
surrounding us is three-dimensional, the world of a roboticist is six-dimensional.
Modern industrial robot manipulators are reprogrammable and multipurpose. In
modern industrial production, it is no longer economical to hold large stocks of either
materials or products. This is known as: “Just in time” production. As a consequence,
it may happen that different types of a certain product find themselves on the same
production line during the same day. This problem, which is most inconvenient for
fixed automation devices, can be efficiently resolved by using reprogrammable in-
dustrial robotic manipulators. Reprogrammable robots allow us to switch from the
production of one type of product to another type by touching a push-button. Fur-
thermore, the robot manipulator is a multipurpose mechanism. The robot mechanism
is a crude imitation of the human arm. In the same way as we use our arm for both
precise and heavy work, we can apply the same robot manipulator to different tasks.
This is even more important in view of the economic life span of an industrial robot,
which is rather long (12–16 years). It could therefore happen that a robot manipulator
acquired for welding purposes, could be reassigned to a pick and place task. Robot
arms have another important property, namely, the axes of two neighboring joints
are either parallel or perpendicular. As the robot arm has only three degrees of free-
dom, there exist a limited number of possible structures of robot arms. Among them
the most frequently used are anthropomorphic and the so-called SCARA (Selective
Compliant Articulated Robot for Assembly) robot arm. Anthropomorphic type of
robot arm (Fig.1.5), has all three joints of the rotational type, and as such it resem-
bles the human arm to the largest extent. The second joint axis is perpendicular to
the first one, while the third joint axis is parallel to the second one. The workspace of
the anthropomorphic robot arm, encompassing all the points that can be reached by
the robot endpoint, has a spherical shape. The SCARA robot arm appeared relatively
late in the development of industrial robotics (Fig.1.6) and is predominantly used
for industrial assembly processes. Two of the joints are rotational and one is transla-
tional. The axes of all three joints are parallel. The workspace of SCARA robot arm
is of the cylindrical type. In the market we can also find three other commercially
available structures of the robot arms: cylindrical, Cartesian, and to a lesser extent
spherical.
1.2 Industrial Robotics
Today’s industry cannot be imagined any longer without industrial robotic manipu-
lators, which can be divided into three different groups. In the first group we classify
the industrial robots which have the role of master in a robot cell. A robot cell usually

1.2 Industrial Robotics 7
RRR
Fig. 1.5 Antropomorphic robot arm
RRT
Fig. 1.6 SCARA robot arm
compromises one or more robots, workstations, storage buffers, transport systems
and numerically controlled machines. In the second group there are the robots which
are slaves within the robot cell. In the third group we include the industrial robots
which are used in special applications (Fig.1.7).
Robot masters in a robot cell, can be found in the following production processes:
welding, painting, coating, and sealing, machining, and assembly. Robot welding
(spot, arc, laser) represents the most frequent robot applications. It is characterized
by speed, precision, and accuracy. Robot welding is specially economic when per-
formed in three shifts. Today we encounter the largest number of welding robots
in the car industry. There, the ratio of human workers and robots is 6:1. Industrial
robots are often used in aggressive or dangerous environments, such as spray paint-
ing. Robotic spray painting represents a saving of material together with a higher
quality of painted surfaces. Where toxic environment exist, the social motivation for
introduction of robots can outweigh economic factors. In machining applications the
robot typically holds either a workpiece or a powered spindle and performs drilling,
grinding, deburring or other similar applications. Robot manipulators are increas-
ingly entering the area of industrial assembly, where component parts are assembled
into a functional systems. The electronic and electromechanical industries represents

8 1 Introduction
INDUSTRIAL ROBOTICS
ROBOT MASTER
IN WORKCELL
ROBOT SLAVE
IN WORKCELL
ROBOT IN SPECIAL
APPLICATIONS
WELDING
PAINTING, COATING,
SEALING
MACHINING
ASSEMBLY
WORKPIECE AND
MATERIAL HANDLING
PALLETIZING AND
PARTS FEEDING
DIE CASTING
FLEXIBLE FIXTURING
QUALITY ASSURANCE
INSPECTION, TESTING
MAINTENANCE
AND REPAIR
FOOD INDUSTRY
TEXTILE AND
CLOTHING INDUSTRY
CONSTRUCTION
Fig. 1.7 Classification of industrial robots
an important area for the application of assembly robots. There are also attractive
assembly operations in the automotive industry, where one robot dispenses adhesive
on the windshield glass, while another robot holds the windshield and inserts it into
the opening on the vehicle body.
The robot plays the role of a slave in the following industrial applications: work-
piece and material handling, palletizing and parts feeding, die casting, and flexible
fixturing. In this situation the role of a master can be given to a numerically con-
trolled machine in the robot cell. Pick and place robots represent the most common
use of robots in material handling, where tasks are often tedious or repetitive and
potentially hazardous (e.g. press loading). Often the industrial robots are used in the
tasks when they execute point to point movements. Such examples are encountered
in palletizing i.e. arranging of workpieces or products for the purpose of packaging
or handing them over to a machine. Robot palletizing is especially appropriate and
welcome when heavy objects are considered (e.g. barrels in a brewery). Die cast-
ing operations are hot, dirty, and hazardous, providing an unpleasant environment
for human workers. With robot handling, the die cast parts are precisely oriented
in the die casting machine. The effectiveness of a robot cell can be upgraded by

1.2 Industrial Robotics 9
using of flexible fixturing systems. The flexibility of a robot cell is achieved via
servodriven programmable positioners, allowing the manufacturing process to be
performed faster and more dexterously.
Special applications of the industrial robots are the following: quality assurance,
inspection, and testing, maintenance and repair, robots in food, textile and clothing
industry, and in construction. Quality assurance, inspection, and testing are often
applied in the electronic industry, where electric parameters (e.g. voltage, current,
resistance) are tested during assembly of electronic circuits. In this situation the
robot performs the necessary measurements on the object (dimensional, electric),
while grasping and placing it into a new position. In robot maintenance and repair
teleoperated and autonomous robots are used for variety of applications in nuclear
industry, highways, railways, power lines maintenance, and aircraft servicing. Robots
are also entering the food industry, where in addition to handling and packaging
applications in food processing, they are used for the tasks such as food preparation
or even decorating chocolates. The textile and clothing industry presents unique
problems because of the limp nature of the workpieces, making handling of textiles or
similar materials extremely complicated. Many different types of construction robots
have been developed all over the world, however very few have been commercialized.
The key challenges of the present-day robotics are human-robot interaction and
human-robot collaboration. The development of the so-called soft robotics enables
humans and robots to interact and collaborate in industrial environments, in service
and everyday settings. When developing collaborative robots, or shortly co-bots,
the safety of human-robot interaction must be ensured. Analysis of human injuries
caused by blunt or sharp tool impacts was therefore necessary as the first step in
collaborative robots research. Based on numerous studies of human-robot collisions,
the safe robot velocities were determined for given robot inertial properties. Safe
human-robot interaction is further guaranteed by novel control schemes which mea-
sure the torque in each robot joint, detecting the slightest contacts between the robot
and the human operator and instantly stopping the robot. The prerequisite for the
efficient torque control is an extremely detailed model of the dynamics of the robot.
To make the robot manipulator compliant, when in contact with human operator, a
biologically inspired approach is also used. Storing the energy in the spring elements
in the robot arm joints makes the motion control efficient and natural. Complex co-
bots, often applied as multi-arm robot systems, cannot be programmed in the same
way as ordinary industrial robot manipulators. Cognitive robotics approaches based
on artificial intelligence techniques must be introduced, such as imitation learning,
learning from demonstrations, reinforcement learning, or learning from rewards. In
this way co-bots are able to perform tasks in unknown and unstructured environ-
ments. Special attention must also be devoted to robot hands. In collaborating with
human operator, the robot hand must be humanoid in order to be able to operate
tools and equipment designed for the human hand. Also, the robot hand must mea-
sure the forces exerted to provide a gentle grip. The today’s industrial robots are
for safety reasons still working behind the fences. Fenceless industrial soft robotics
has the potential to open novel unforeseen applications, leading to more flexible and
cost-effective automation.

Chapter 2
Homogenous Transformation Matrices
2.1 Translational Transformation
As stated previously robots have either translational or rotational joints. To describe
the degree of displacement in a joint we need a unified mathematical description of
translational and rotational displacements. The translational displacement d, given
by the vector
d = ai + bj + ck, (2.1)
can be described also by the following homogenous transformation matrix H
H = Trans(a, b, c) =
⎡
⎢
⎢
⎣
1 0 0 a
0 1 0 b
0 0 1 c
0 0 0 1
⎤
⎥
⎥
⎦ . (2.2)
When using homogenous transformation matrices an arbitrary vector has the follow-
ing 4 × 1 form
q =
⎡
⎢
⎢
⎣
x
y
z
1
⎤
⎥
⎥
⎦ =

x y z 1
T
. (2.3)
A translational displacement of vector q for a distance d is obtained by multiplying
the vector q with the matrix H
v =
⎡
⎢
⎢
⎣
1 0 0 a
0 1 0 b
0 0 1 c
0 0 0 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
x
y
z
1
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎣
x + a
y + b
z + c
1
⎤
⎥
⎥
⎦ . (2.4)
11

12 2 Homogenous transformation matrices
The translation, which is presented by multiplication with a homogenous matrix, is
equivalent to the sum of vectors q and d
v = q + d = (xi + yj + zk) + (ai + bj + ck) = (x + a)i + (y + b)j + (z + c)k.
(2.5)
In a simple example, the vector 1i + 2j + 3k is translationally displaced for the
distance 2i − 5j + 4k
v =
⎡
⎢
⎢
⎣
1 0 0 2
0 1 0 −5
0 0 1 4
0 0 0 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
1
2
3
1
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎣
3
−3
7
1
⎤
⎥
⎥
⎦ .
The same result is obtained by adding the two vectors.
2.2 Rotational Transformation
Rotational displacements will be described in a right-handed rectangular coordinate
frame, where the rotations around the three axes, as shown in Fig.2.1, are considered
as positive. Positive rotations around the selected axis are counter-clockwise when
looking from the positive end of the axis towards the origin O of the frame x–y–z.
The positive rotation can be described also by the so called right hand rule, where the
thumb is directed along the axis towards its positive end, while the fingers show the
Fig. 2.1 Right-hand rectangular frame with positive rotations

2.2 Rotational Transformation 13
Fig. 2.2 Rotation around x axis
positive direction of the rotational displacement. The direction of running of athletes
in a stadium is also an example of a positive rotation.
Let us first take a closer look at the rotation around the x axis. The coordinate
frame x
–y
–z
shown in Fig.2.2 was obtained by rotating the reference frame x–y–z
in the positive direction around the x axis for the angle α. The axes x and x
are
collinear.
The rotational displacement is also described by a homogenous transformation
matrix. The first three rows of the transformation matrix correspond to the x, y, and
z axes of the reference frame, while the first three columns refer to the x
, y
, and z
axes of the rotated frame. The upper left nine elements of the matrix H represent the
3 × 3 rotation matrix. The elements of the rotation matrix are cosines of the angles
between the axes given by the corresponding column and row
Rot(x, α) =
x
y
z
⎡
⎢
⎢
⎣
cos 0◦
cos 90◦
cos 90◦
0
cos 90◦
cos α cos(90◦
+ α) 0
cos 90◦
cos(90◦
− α) cos α 0
0 0 0 1
⎤
⎥
⎥
⎦
x
y
z
=
⎡
⎢
⎢
⎣
1 0 0 0
0 cos α − sin α 0
0 sin α cos α 0
0 0 0 1
⎤
⎥
⎥
⎦
.
(2.6)
The angle between the x
and the x axes is 0◦
, therefore we have cos 0◦
in the
intersection of the x
column and the x row. The angle between the x
and the y axes

Fig. 2.3 Rotation around y axis
is 90◦
, we put cos 90◦
in the corresponding intersection. The angle between the y
and the y axes is α, the corresponding matrix element is cos α.
To become more familiar with rotation matrices, we shall derive the matrix
describing a rotation around the y axis by using Fig.2.3. The collinear axes are
y and y
y = y
. (2.7)
By considering the similarity of triangles in Fig.2.3, it is not difficult to derive the
following two equations
x = x
cos β + z
sin β
z = −x
sin β + z
cos β. (2.8)
All three Eqs.(2.7) and (2.8) can be rewritten in the matrix form

2.2 Rotational Transformation 15
Rot(y, β) =
x
y
z
⎡
⎢
⎢
⎣
cos β 0 sin β 0
0 1 0 0
− sin β 0 cos β 0
0 0 0 1
⎤
⎥
⎥
⎦
x
y
z
. (2.9)
The rotation around the z axis is described by the following homogenous trans-
formation matrix
Rot(z, γ ) =
⎡
⎢
⎢
⎣
cos γ − sin γ 0 0
sin γ cos γ 0 0
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦ . (2.10)
In a simple numerical example we wish to determine the vector w, which is
obtained by rotating the vector u = 14i + 6j + 0k for 90◦
in the counter clockwise
(i.e., positive) direction around the z axis. As cos 90◦
= 0 and sin 90◦
= 1, it is not
difficult to determine the matrix describing Rot(z, 90◦
) and multiplying it by the
vector u
w =
⎡
⎢
⎢
⎣
0 −1 0 0
1 0 0 0
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
14
6
0
1
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎣
−6
14
0
1
⎤
⎥
⎥
⎦ .
The graphical presentation of rotating the vector u around the z axis is shown in
Fig.2.4.
Fig. 2.4 Example of rotational transformation

2.3 Pose and Displacement
In the previous section we have learned how a point is translated or rotated around
the axes of the cartesian frame. In continuation we shall be interested in displace-
ments of objects. We can always attach a coordinate frame to a rigid object under
consideration. In this section we shall deal with the pose and the displacement of
rectangular frames. Here we see that a homogenous transformation matrix describes
either the pose of a frame with respect to a reference frame, or it represents the dis-
placement of a frame into a new pose. In the first case the upper left 3 × 3 matrix
represents the orientation of the object, while the right-hand 3 × 1 column describes
its position (e.g., the position of its center of mass). The last row of the homogenous
transformation matrix will be always represented by [0 0 0 1]. In the case of object
displacement, the upper left matrix corresponds to rotation and the right-hand col-
umn corresponds to translation of the object. We shall examine both cases through
simple examples. Let us first clear up the meaning of the homogenous transformation
matrix describing the pose of an arbitrary frame with respect to the reference frame.
Let us consider the following product of homogenous matrices which gives a new
homogenous transformation matrix H
H = Trans(8, −6, 14)Rot(y, 90◦
)Rot(z, 90◦
)
=
⎡
⎢
⎢
⎣
1 0 0 8
0 1 0 −6
0 0 1 14
0 0 0 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
0 0 1 0
0 1 0 0
−1 0 0 0
0 0 0 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
0 −1 0 0
1 0 0 0
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦
=
⎡
⎢
⎢
⎣
0 0 1 8
1 0 0 −6
0 1 0 14
0 0 0 1
⎤
⎥
⎥
⎦ .
(2.11)
When defining the homogenous matrix representing rotation, we learned that the first
three columns describe the rotation of the frame x
–y
–z
with respect to the reference
frame x–y–z
x
y
z
⎡
⎢
⎢
⎣
0 0 1 8
1 0 0 −6
0 1 0 14
0 0 0 1
⎤
⎥
⎥
⎦
x
y
z
.
(2.12)
The fourth column represents the position of the origin of the frame x
–y
–z
with respect to the reference frame x–y–z. With this knowledge we can represent
graphically the frame x
–y
–z
described by the homogenous transformation matrix
(2.11), relative to the reference frame x–y–z (Fig.2.5). The x
axis points in the

2.3 Pose and Displacement 17
Fig. 2.5 The pose of an arbitrary frame x–y–z with respect to the reference frame x–y–z
Fig. 2.6 Displacement of the reference frame into a new pose (from right to left). The origins O1,
O2 and O are in the same point
direction of y axis of the reference frame, the y
axis is in the direction of the z axis,
and the z
axis is in the x direction.
To convince ourselves of the correctness of the frame drawn in Fig.2.6, we shall
check the displacements included in Eq.(2.11). The reference frame is first translated
into the point (8, −6, 14), afterwards it is rotated for 90◦
around the new y axis and
finally it is rotated for 90◦
around the newest z axis (Fig.2.6). The three displacements
of the reference frame result in the same final pose as shown in Fig.2.5.
In continuation of this chapter we wish to elucidate the second meaning of the
homogenous transformation matrix, i.e., a displacement of an object or coordinate
frame into a new pose (Fig.2.7). First, we wish to rotate the coordinate frame x–y–z
for 90◦
in the counter-clockwise direction around the z axis. This can be achieved by
the following post-multiplication of the matrix H describing the initial pose of the

coordinate frame x–y–z
H1 = H · Rot(z, 90◦
). (2.13)
The displacement resulted in a new pose of the object and new frame x
–y
–z
shown
in Fig.2.7. We shall displace this new frame for −1 along the x
axis, 3 units along
y
axis and −3 along z
axis
H2 = H1 · Trans(−1, 3, −3). (2.14)
After translation a new pose of the object is obtained together with a new frame
x
–y
–z
. This frame will be finally rotated for 90◦
around the y
axis in the positive
direction
H3 = H2 · Rot(y
, 90◦
). (2.15)
The Eqs.(2.13), (2.14), and (2.15) can be successively inserted one into another
H3 = H · Rot(z, 90◦
) · Trans(−1, 3, −3) · Rot(y
, 90◦
) = H · D. (2.16)
In Eq.(2.16), the matrix H represents the initial pose of the frame, H3 is the final
pose, while D represents the displacement
D = Rot(z, 90◦
) · Trans(−1, 3, −3) · Rot(y
, 90◦
)
=
⎡
⎢
⎢
⎣
0 −1 0 0
1 0 0 0
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
1 0 0 −1
0 1 0 3
0 0 1 −3
0 0 0 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
0 0 1 0
0 1 0 0
−1 0 0 0
0 0 0 1
⎤
⎥
⎥
⎦
=
⎡
⎢
⎢
⎣
0 −1 0 −3
0 0 1 −1
−1 0 0 −3
0 0 0 1
⎤
⎥
⎥
⎦ .
(2.17)
Finally, we shall perform the post-multiplication describing the new relative pose of
the object

2.3 Pose and Displacement 19
Fig. 2.7 Displacement of the object into a new pose
H3 = H · D =
⎡
⎢
⎢
⎣
1 0 0 2
0 0 −1 −1
0 1 0 2
0 0 0 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
0 −1 0 −3
0 0 1 −1
−1 0 0 −3
0 0 0 1
⎤
⎥
⎥
⎦
=
x
y
z
⎡
⎢
⎢
⎣
0 −1 0 −1
1 0 0 2
0 0 1 1
0 0 0 1
⎤
⎥
⎥
⎦
x0
y0
z0
.
(2.18)
As in the previous example we shall graphically verify the correctness of the
matrix (2.18). The three displacements of the frame x–y–z: rotation for 90◦
in counter-
clockwise direction around the z axis, translation for −1 along the x
axis, 3 units
along y
axis and −3 along z
axis, and rotation for 90◦
around y
axis in the positive
direction are shown in Fig.2.7. The result is the final pose of the object x
, y
, z
.
The x
axis points in the positive direction of the y0 axis, y
points in the negative
direction of x0 axis and z
points in the positive direction of z0 axis of the reference
frame. The directions of the axes of the final frame correspond to the first three
columns of the matrix H3. There is also agreement between the position of the origin
of the final frame in Fig.2.7 and the fourth column of the matrix H3.
2.4 Geometrical Robot Model
Our final goal is the geometrical model of a robot manipulator. A geometrical robot
model is given by the description of the pose of the last segment of the robot (end-
effector) expressed in the reference (base) frame. The knowledge how to describe the

Fig. 2.8 Mechanical assembly
pose of an object using homogenous transformation matrices will be first applied to
the process of assembly. For this purpose, a mechanical assembly consisting of four
blocks, such as presented in Fig.2.8, will be considered. A plate with dimensions (5 ×
15 × 1) is placed over a block (5 × 4 × 10). Another plate (8 × 4 × 1) is positioned
perpendicularly to the first one, holding another small block (1 × 1 × 5).
A frame is attached to each of the four blocks as shown in Fig.2.8. Our task will be
to calculate the pose of the frame x3–y3–z3 with respect to the reference frame x0–y0–
z0. In the last chapter we learned that the pose of a displaced frame can be expressed
with respect to the reference frame using the homogenous transformation matrix H.
The pose of the frame x1–y1–z1 with respect to the frame x0–y0–z0 will be denoted
by 0
H1. In the same way 1
H2 represents the pose of frame x2–y2–z2 with respect to
x1–y1–z1 and 2
H3 the pose of x3–y3–z3 with regard to frame x2–y2–z2. We learned
also that the successive displacements are expressed by post-multiplications (suc-
cessive multiplications from left to right) of homogenous transformation matrices.
The assembly process can be described by post-multiplication of the corresponding
matrices. The pose of the fourth block can be written with respect to the first one by
the following matrix
0
H3 = 0
H1
1
H2
2
H3. (2.19)
The blocks were positioned perpendicularly one to another. In this way it is not
necessary to calculate the sines and cosines of the rotation angles. The matrices can
be determined directly from Fig.2.8. The x axis of frame x1–y1–z1 points in negative
direction of the y axis in the frame x0–y0–z0. The y axis of frame x1–y1–z1 points in

2.4 Geometrical Robot Model 21
negative direction of the z axis in the frame x0–y0–z0. The z axis of the frame x1–y1–
z1 has the same direction as x axis of the frame x0–y0–z0. The described geometrical
properties of the assembly structure are written into the first three columns of the
homogenous matrix. The position of the origin of the frame x1–y1–z1 with respect
to the frame x0–y0–z0 is written into the fourth column
O1
x y z
0
H1 =
⎡
⎢
⎢
⎣
0 0 1 0
−1 0 0 6
0 −1 0 11
0 0 0 1
⎤
⎥
⎥
⎦
x
y
z
⎫
⎬
⎭
O0
.
(2.20)
In the same way the other two matrices are determined
1
H2 =
⎡
⎢
⎢
⎣
1 0 0 11
0 0 1 −1
0 −1 0 8
0 0 0 1
⎤
⎥
⎥
⎦ (2.21)
2
H3 =
⎡
⎢
⎢
⎣
1 0 0 3
0 −1 0 1
0 0 −1 6
0 0 0 1
⎤
⎥
⎥
⎦ . (2.22)
The position and orientation of the fourth block with respect to the first one is given
by the 0
H3 matrix which is obtained by successive multiplication of the matrices
(2.20), (2.21) and (2.22)
0
H3 =
⎡
⎢
⎢
⎣
0 1 0 7
−1 0 0 −8
0 0 1 6
0 0 0 1
⎤
⎥
⎥
⎦ . (2.23)
The fourth column of the matrix 0
H3 [7, −8, 6, 1]T
represents the position of the
origin of the frame x3–y3–z3 with respect to the reference frame x0–y0–z0. The
accuracy of the fourth column can be checked from Fig.2.8. The rotational part of
the matrix 0
H3 represents the orientation of the frame x3–y3–z3 with respect to the
reference frame x0–y0–z0.
Now let us imagine that the first horizontal plate rotates with respect to the first
vertical block around axis 1 for angle ϑ1. The second plate also rotates around the
vertical axis 2 for angle ϑ2. The last block is elongated for distance d3 along the third

Fig. 2.9 Displacements of the mechanical assembly
Fig. 2.10 SCARA robot manipulator in an arbitrary pose
axis. In this way we obtained a robot manipulator, of the SCARA type as mentioned
in the introductory chapter.
Our goal is to develop a geometrical model of the SCARA robot. Blocks and plates
from Fig.2.9 will be replaced by symbols for rotational and translational joints that
we know from the introduction (Fig.2.10).
The first vertical segment with the length l1 starts from the base (where the robot
is attached to the ground) and is terminated by the first rotational joint. The second
segment with length l2 is horizontal and rotates around the first segment. The rotation
in the first joint is denoted by the angle ϑ1. The third segment with the length l3 is also

Fig. 2.11 The SCARA robot manipulator in the initial pose
horizontal and rotates around the vertical axis at the end of the second segment. The
angle is denoted as ϑ2. There is a translational joint at the end of the third segment.
It enables the robot end-effector to approach the working plane where the robot task
takes place. The translational joint is displaced from zero initial length to the length
described by the variable d3.
The robot mechanism is first brought to the initial pose which is also called “home
position”. In the initial pose two neighboring segments must be either parallel or
perpendicular. The translational joints are in their initial position di = 0. The initial
pose of the SCARA manipulator is shown in Fig.2.11.
First, the coordinate frames must be drawn into the SCARA robot presented in
Fig.2.11. The first (reference) coordinate frame x0–y0–z0 is placed onto the base
of the robot. In the last chapter we shall learn that robot standards require the z0
axis to point perpendicularly out from the base. In this case it is aligned with the
first segment. The other two axes are selected in such a way that robot segments are
parallel to one of the axes of the reference coordinate frame, when the robot is in its
initial home position. In this case we align the y0 axis with the segments l2 and l3.
The coordinate frame must be right handed. The rest of the frames are placed into
the robot joints. The origins of the frames are drawn in the center of each joint. One

of the frame axes must be aligned with the joint axis. The simplest way to calculate
the geometrical model of a robot is to make all the frames in the robot joints parallel
to the reference frame (Fig.2.11).
The geometrical model of a robot describes the pose of the frame attached to the
end-effector with respect to the reference frame on the robot base. Similarly, as in the
case of the mechanical assembly, we shall obtain the geometrical model by successive
multiplication (post-multiplication) of homogenous transformation matrices. The
main difference between the mechanical assembly and the robot manipulator is the
displacements of the robot joints. For this purpose, each matrix i−1
Hi describing the
pose of a segment will be followed by a matrix Di representing the displacement of
either the translational or the rotational joint. Our SCARA robot has three joints. The
pose of the end frame x3–y3–z3 with respect to the base frame x0–y0–z0 is expressed
by the following postmultiplication of three pairs of homogenous transformation
matrices
0
H3 = (0
H1D1) · (1
H2D2) · (2
H3D3). (2.24)
In Eq.(2.24), the matrices 0
H1, 1
H2, and 2
H3 describe the pose of each joint frame
with respect to the preceding frame in the same way as in the case of assembly of
the blocs. From Fig.2.11 it is evident that the D1 matrix represents a rotation around
the positive z1 axis. The following product of two matrices describes the pose and
the displacement in the first joint
0
H1D1 =
⎡
⎢
⎢
⎣
1 0 0 0
0 1 0 0
0 0 1 l1
0 0 0 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
c1 −s1 0 0
s1 c1 0 0
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎣
c1 −s1 0 0
s1 c1 0 0
0 0 1 l1
0 0 0 1
⎤
⎥
⎥
⎦ .
In the above matrices the following shorter notation was used sin ϑ1 = s1 and
cos ϑ1 = c1.
In the second joint there is a rotation around the z2 axis
1
H2D2 =
⎡
⎢
⎢
⎣
1 0 0 0
0 1 0 l2
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
c2 −s2 0 0
s2 c2 0 0
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎣
c2 −s2 0 0
s2 c2 0 l2
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦ .
In the last joint there is translation along the z3 axis
2
H3D3 =
⎡
⎢
⎢
⎣
1 0 0 0
0 1 0 l3
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
1 0 0 0
0 1 0 0
0 0 1 −d3
0 0 0 1
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎣
1 0 0 0
0 1 0 l3
0 0 1 −d3
0 0 0 1
⎤
⎥
⎥
⎦ .

The geometrical model of the SCARA robot manipulator is obtained by post-
multiplication of the three matrices derived above
0
H3 =
⎡
⎢
⎢
⎣
c12 −s12 0 −l3s12 − l2s1
s12 c12 0 l3c12 + l2c1
0 0 1 l1 − d3
0 0 0 1
⎤
⎥
⎥
⎦ .
When multiplying the three matrices the following abbreviation was introduced
c12 = cos(ϑ1 + ϑ2) = c1c2 − s1s2 and s12 = sin(ϑ1 + ϑ2) = s1c2 + c1s2.

Chapter 3
Geometric Description of the Robot
Mechanism
The geometric description of the robot mechanism is based on the usage of trans-
lational and rotational homogenous transformation matrices. A coordinate frame is
attached to the robot base and to each segment of the mechanism, as shown in Fig.3.1.
Then, the corresponding transformation matrices between the consecutive frames are
determined. A vector expressed in one of the frames can be transformed into another
frame by successive multiplication of intermediate transformation matrices.
Vector a in Fig.3.1 is expressed relative to the coordinate frame x3–y3–z3, while
vector b is given in the frame x0–y0–z0 belonging to the robot base. A mathemati-
cal relationship between the two vectors is obtained by the following homogenous
transformation

b
1

= 0
H1
1
H2
2
H3

a
1

. (3.1)
3.1 Vector Parameters of a Kinematic Pair
Vector parameters will be used for the geometric description of a robot mechanism.
For simplicity we shall limit our consideration to the mechanisms with either par-
allel or perpendicular consecutive joint axes. Such mechanisms are by far the most
frequent in industrial robotics.
In Fig.3.2, a kinematic pair is shown consisting of two consecutive segments of
a robot mechanism, segment i − 1 and segment i. The two segments are connected
by the joint i including both translation and rotation. The relative pose of the joint is
determined by the segment vector bi−1 and unit joint vector ei , as shown in Fig.3.2.
The segment i can be translated with respect to the segment i − 1 along the vector
ei for the distance di and can be rotated around ei for the angle ϑi . The coordinate
27

28 3 Geometric Description of the Robot Mechanism
Fig. 3.1 Robot mechanism with coordinate frames attached to its segments
Fig. 3.2 Vector parameters of a kinematic pair
frame xi –yi –zi is attached to the segment i, while the frame xi−1–yi−1–zi−1 belongs
to the segment i − 1.
The coordinate frame xi –yi –zi is placed into the axis of the joint i in such a way
that it is parallel to the previous frame xi−1–yi−1–zi−1 when the kinematic pair is in
its initial pose (both joint variables are zero ϑi = 0 and di = 0).
The geometric relations and the relative displacement of two neighboring seg-
ments of a robot mechanism are determined by the following parameters:
ei —unit vector describing either the axis of rotation or direction of translation
in the joint i and is expressed as one of the axes of the frame xi –yi –zi . Its
components are the following
ei =
⎡
⎣
1
0
0
⎤
⎦ or
⎡
⎣
0
1
0
⎤
⎦ or
⎡
⎣
0
0
1
⎤
⎦ ;

3.1 Vector Parameters of a Kinematic Pair 29
bi−1 —segment vector describing the segment i − 1 expressed in the frame xi−1–
yi−1–zi−1. Its components are the following
bi−1 =
⎡
⎣
bi−1,x
bi−1,y
bi−1,z
⎤
⎦ ;
ϑi —rotational variable representing the angle measured around the ei axis in the
plane which is perpendicular to ei (the angle is zero when the kinematic pair is
in the initial position);
di —translational variable representing the distance measured along the direction
of ei (the distance equals zero when the kinematic pair is in the initial position).
If the joint is only rotational (Fig.3.3 above), the joint variable is represented by
the angle ϑi , while di = 0. When the robot mechanism is in its initial pose, the joint
angle equals zero ϑi = 0 and the coordinate frames xi –yi –zi and xi−1–yi−1–zi−1
are parallel. If the joint is only translational (Fig.3.3 below), the joint variable is di ,
Fig. 3.3 Vector parameters of a kinematic pair

while ϑi = 0. When the joint is in its initial position, then di = 0. In this case the
coordinate frames xi –yi –zi and xi−1–yi−1–zi−1 are parallel irrespective of the value
of the translational variable di .
By changing the value of the rotational joint variable ϑi , the coordinate frame xi –
yi –zi is rotated together with the segment i with respect to the preceding segment
i − 1 and the corresponding frame xi−1–yi−1–zi−1. By changing the translational
variable di , the displacement is translational, where only the distance between the
two neighboring frames is changing.
The transformation between the coordinate frames xi−1–yi−1–zi−1 and xi –yi –
zi is determined by the homogenous transformation matrix taking one of the three
possible forms regarding the direction of the joint vector ei . When the unit vector ei
is parallel to the xi axis, there is
i−1
Hi =
⎡
⎢
⎢
⎣
1 0 0 di + bi−1,x
0 cos ϑi − sin ϑi bi−1,y
0 sin ϑi cos ϑi bi−1,z
0 0 0 1
⎤
⎥
⎥
⎦ , (3.2)
when ei is parallel to the yi axis, we have the following transformation matrix
i−1
Hi =
⎡
⎢
⎢
⎣
cos ϑi 0 sin ϑi bi−1,x
0 1 0 di + bi−1,y
− sin ϑi 0 cos ϑi bi−1,z
0 0 0 1
⎤
⎥
⎥
⎦ (3.3)
When ei is parallel to the zi axis, the matrix has the following form
i−1
Hi =
⎡
⎢
⎢
⎣
cos ϑi − sin ϑi 0 bi−1,x
sin ϑi cos ϑi 0 bi−1,y
0 0 1 di + bi−1,z
0 0 0 1
⎤
⎥
⎥
⎦ . (3.4)
In the initial pose the coordinate frames xi−1–yi−1–zi−1 and xi –yi –zi are parallel
(ϑi = 0 and di = 0) and displaced only for the vector bi−1
i−1
Hi =
⎡
⎢
⎢
⎣
1 0 0 bi−1,x
0 1 0 bi−1,y
0 0 1 bi−1,z
0 0 0 1
⎤
⎥
⎥
⎦ . (3.5)

3.2 Vector Parameters of the Mechanism 31
3.2 Vector Parameters of the Mechanism
The vector parameters of a robot mechanism are determined in the following four
steps:
step 1 —the robot mechanism is placed into the desired initial (reference) pose. The
joint axes must be parallel to one of the axes of the reference coordinate frame
x0–y0–z0 attached to the robot base. In the reference pose all values of joint
variables equal zero, ϑi = 0 and di = 0, i = 1, 2, . . . , n;
step 2 —the centers of the joints i = 1, 2, . . . , n are selected. The center of joint i
can be anywhere along the corresponding joint axis. A local coordinate frame
xi –yi –zi is placed into the joint center in such a way that its axes are parallel to
the axes of the reference frame x0–y0–z0. The local coordinate frame xi –yi –zi
is displaced together with the segment i;
step 3 —the unit joint vector ei is allocated to each joint axis i = 1, 2, . . . , n. It
is directed along one of the axes of the coordinate frame xi –yi –zi . In the
direction of this vector the translational variable di is measured, while the
rotational variable ϑi is assessed around the joint vector ei ;
step 4 —the segment vectors bi−1 are drawn between the origins of the frames xi –
yi –zi , i = 1, 2, . . . , n. The segment vector bn connects the origin of the frame
xn–yn–zn with the robot end-point.
Sometimes an additional coordinate frame is positioned in the reference point of a
gripper and denoted as xn+1–yn+1–zn+1. There exists no degree of freedom between
the frames xn–yn–zn and xn+1–yn+1–zn+1, as both frames are attached to the same
segment. The transformation between them is therefore constant.
The approach to geometric modeling of robot mechanisms will be illustrated by
an example of a robot mechanism with four degrees of freedom shown in Fig.3.4.
The selected initial pose of the mechanism together with the marked positions of the
joint centers is presented in Fig.3.5. The corresponding vector parameters and joint
variables are gathered in Table3.1.
The rotational variables ϑ1, ϑ2 and ϑ4 are measured in the planes perpendicular
to the joint axes e1, e2 and e4, while the translational variable di is measured along
the axis e3. Their values are zero when the robot mechanism is in its initial pose. In
Fig.3.6 the robot manipulator is shown in a pose where all four variables are positive
and nonzero. The variable ϑ1 represents the angle between the initial and momentary
y1 axis, the variable ϑ2 the angle between the initial and momentary z2 axis, variable
d3 is the distance between the initial and actual position of the x3 axis, while ϑ4
represents the angle between the initial and momentary x4 axis.
The selected vector parameters of the robot mechanism are inserted into the
homogenous transformation matrices (3.2)–(3.4)
0
H1 =
⎡
⎢
⎢
⎣
c1 −s1 0 0
s1 c1 0 0
0 0 1 h0
0 0 0 1
⎤
⎥
⎥
⎦ ,

Fig. 3.4 Robot mechanism with four degrees of freedom
1
H2 =
⎡
⎢
⎢
⎣
1 0 0 0
0 c2 −s2 l1
0 s2 c2 h1
0 0 0 1
⎤
⎥
⎥
⎦ ,
2
H3 =
⎡
⎢
⎢
⎣
1 0 0 0
0 1 0 d3 + l2
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦ ,
3
H4 =
⎡
⎢
⎢
⎣
c4 −s4 0 0
s4 c4 0 l3
0 0 1 −h3
0 0 0 1
⎤
⎥
⎥
⎦ .

Fig. 3.5 Positioning of the coordinate frames for the robot mechanism with four degrees of freedom
An additional homogenous matrix describes the position of the gripper reference
point where the coordinate frame x5–y5–z5 can be allocated
4
H5 =
⎡
⎢
⎢
⎣
1 0 0 0
0 1 0 l4
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦ .

Table 3.1 Vector parameters and joint variables for the robot mechanism in Fig.3.5
i 1 2 3 4
ϑi ϑ1 ϑ2 0 ϑ4
di 0 0 d3 0
i 1 2 3 4
0 1 0 0
ei 0 0 1 0
1 0 0 1
i 1 2 3 4 5
0 0 0 0 0
bi−1 0 l1 l2 l3 l4
h0 h1 0 −h3 0
Fig. 3.6 Determining the rotational and translational variables for the robot mechanism with four
degrees of freedom

Fig. 3.7 The SCARA robot manipulator in the initial pose
This last matrix is constant as the frames x4–y4–z4 and x5–y5–z5 are parallel and
displaced for the distance l4. Usually this additional frame is not even attached to the
robot mechanism, as the position and orientation of the gripper can be described in
the frame x4–y4–z4.
When determining the initial (home) pose of the robot mechanism we must take
care that the joint axes are parallel to one of the axes of the reference coordinate frame.
The initial pose should be selected in such a way that it is simple and easy to examine,
that it corresponds well to the anticipated robot tasks and that it minimizes the number
of required mathematical operations included in the transformation matrices.
As another example we shall consider the SCARA robot manipulator whose geo-
metric model was developed already in the previous chapter and is shown in Fig.2.10.
The robot mechanism should be first positioned into the initial pose in such a way
that the joint axes are parallel to one of the axes of the reference frame x0–y0–z0.
In this way the two neighboring segments are either parallel or perpendicular. The
translational joint must be in its initial position (d3 = 0). The SCARA robot in the
selected initial pose is shown in Fig.3.7.
The joint coordinate frames xi –yi –zi are all parallel to the reference frame. There-
fore, we shall draw only the reference frame and have the dots indicate the joint
centers. In the centers of both rotational joints, unit vectors e1 and e2 are placed

Table 3.2 Vector parameters and joint variables for the SCARA robot manipulator
i 1 2 3 4
ϑi ϑ1 ϑ2 0 ϑ4
di 0 0 d3 0
i 1 2 3 4
0 1 0 0
ei 0 0 1 0
1 0 0 1
i 1 2 3 4 5
0 0 0 0 0
bi−1 0 l1 l2 l3 l4
h0 h1 0 −h3 0
along the joint axes. The rotation around the e1 vector is described by the variable
ϑ1, while ϑ2 represents the angle about the e2 vector. Vector e3 is placed along the
translational axis of the third joint. Its translation variable is described by d3. The
first joint is connected to the robot base by the vector b0. Vector b1 connects the first
and the second joint and vector b2 the second and the third joint. The variables and
vectors are gathered in the three tables (Table3.2).
In our case all ei vectors are parallel to the z0 axis, the homogenous transformation
matrices are therefore written according Eq.(3.4). Similar matrices are obtained for
both rotational joints.
0
H1 =
⎡
⎢
⎢
⎣
c1 −s1 0 0
s1 c1 0 0
0 0 1 l1
0 0 0 1
⎤
⎥
⎥
⎦ .
1
H2 =
⎡
⎢
⎢
⎣
c2 −s2 0 0
s2 c2 0 l2
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦ .

For the translational joint, ϑ3 = 0 must be inserted into Eq.(3.4), giving
2
H3 =
⎡
⎢
⎢
⎣
1 0 0 0
0 1 0 l3
0 0 1 −d3
0 0 0 1
⎤
⎥
⎥
⎦ .
With postmultiplication of all three matrices the geometric model of the SCARA
robot is obtained
0
H3 = 0
H1
1
H2
2
H3 =
⎡
⎢
⎢
⎣
c12 −s12 0 −l3s12 − l2s1
s12 c12 0 l3c12 + l2c1
0 0 1 l1 − d3
0 0 0 1
⎤
⎥
⎥
⎦ .
We obtained the same result as in previous chapter, however in a much simpler
and more clearer way.

Chapter 4
Orientation
We often describe our environment as a three-dimensional world. The world of the
roboticist is, however, six-dimensional. He must not only consider the position of an
object, but also its orientation. When a robot gripper or end-effector approaches an
object to be grasped, the space angles between the gripper and the object are of the
utmost importance.
Six parameters are required to completely describe the position and orientation
of an object in a space. Three parameters refer to the position and the other three to
the orientation of the object. There are three possible ways how to mathematically
describe the orientation of the object. The first possibility is a rotation/orientation
matrix consisting of nine elements. The matrix represents a redundant description of
the orientation. A non-redundant description is given by RPY or Euler angles. In both
cases we have three angles. The RPY angles are defined about the axes of a fixed
coordinate frame, while the Euler angles describe the orientation about a relative
coordinate frame. The third possible description of the orientation is enabled by four
parameters of quaternion.
In the second chapter we already became acquainted with rotation matrices around
x, y, and z axis of a rectangular frame. We found them useful when developing the
geometrical model of a robot mechanism. It is not difficult to understand that there
exists also a matrix describing the rotation around an arbitrary axis. This can be
expressed in the following form
0
R1 =
⎡
⎣
1
i0
i 1
j0
i 1
k0
i
1
i0
j 1
j0
j 1
k0
j
1
i0
k 1
j0
k 1
k0
k
⎤
⎦ . (4.1)
The matrix of the dimension 3 × 3 does not only represent the rotation, but also the
orientation of the frame x1–y1–z1 with respect to the frame x0–y0–z0, as it can be
seen from Fig.4.1. The reference frame x0–y0–z0 is described by the unit vectors
39

40 4 Orientation
x1
x0
0i
1i
y1
y0
1j
0j
z1
z0
1k
0k
Fig. 4.1 Orientation of the coordinate frame x1–y1–z1 with respect to the reference coordinate
frame x0–y0–z0
0
i, 0
j, and 0
k and the rotated frame x1–y1–z1 with the unit vectors 1
i, 1
j, and 1
k.
Both coordinate frames coincide in the same origin. As we are dealing with the unit
vectors, the elements of the rotation/orientation matrix are simply the cosines of the
angles appertaining to each pair of the axes.
Let us consider the example from Fig.4.2 and calculate the matrix representing
the orientation of the frame x1–y1–z1, which is rotated for the angle +ϑ with respect
to the frame x0–y0–z0.
We are dealing with the following non-zero products of the unit vectors
0
i1
i = 1,
0
j1
j = cos ϑ,
0
k1
k = cos ϑ,
0
j1
k = − sin ϑ,
0
k1
j = sin ϑ.
(4.2)
The matrix describing the orientation of the frame x1–y1–z1 with respect to x0–
y0–z0 is therefore
Rx =
⎡
⎣
1 0 0
0 cϑ −sϑ
0 sϑ cϑ
⎤
⎦ (4.3)

4 Orientation 41
x0,x1
0i,1 i
y1
y0
1j
0j
z1
z0
1k 0k
ϑ
ϑ
Fig. 4.2 Two coordinate frames rotated about the x0 axis
x0
y0
z0
a n
s
Fig. 4.3 Orientation of robot gripper
The matrix (4.3) can be interpreted also as the rotation matrix around the x axis that
we already know as part of the homogeneous matrix (2.6) from the second chapter.
The notion of orientation is in robotics mostly related to the orientation of the
robot gripper. A coordinate frame with three unit vectors n, s, and a, describing the
orientation of the gripper, is placed between two fingers of a simple robot gripper
(Fig.4.3).

42 4 Orientation
The z axis vector lays in the direction of the approach of the gripper to the object.
It is therefore denoted by vector a (approach). Vector, which is aligned with y axis,
describes the direction of sliding of the fingers and is denoted as s (slide). The third
vector completes the right-handed coordinate frame and is called normal. This can
be shown as n = s × a. The matrix describing the orientation of the gripper with
respect to the reference frame x0–y0–z0 has the following form
R =
⎡
⎣
nx sx ax
ny sy ay
nz sz az
⎤
⎦ . (4.4)
The element nx of the matrix (4.3) denotes the projection of the unit vector n on
the x0 axis of the reference frame. It equals the cosine of the angle between the axes
x and x0 and has the same meaning as the element 1
i0
i of the rotation/orientation
matrix (4.1). The same is valid for the eight other elements of the orientation matrix
R (4.3).
To describe the orientation of an object we do not need nine elements of the
matrix. The left column vector is the cross product of vectors s and a. The vectors s
and a are unit vectors which are perpendicular with respect to each other, so that we
have
s · s = 1,
a · a = 1,
s · a = 0.
(4.5)
Three elements are therefore sufficient to describe the orientation. This orientation
is often described by the following sequence of rotations
R - roll - about z axis,
P - pitch - about y axis,
Y - yaw - about x axis.
This description is mostly used when describing the orientation of a ship or air-
plane. Let us imagine that the airplane flies along z axis and that the coordinate frame
is positioned into the center of the airplane. Then, R represents the rotation ϕ about
z axis, P refers to the rotation ϑ about y axis and Y to the rotation ψ about x axis,
as shown in Fig.4.4.
The use of the RPY angles for a robot gripper is shown in Fig.4.5. As it can
be realized from Figs.4.4 and 4.5, the RPY orientation is defined with respect to
a fixed coordinate frame. When developing the geometrical model of the SCARA
robot manipulator in the second chapter, we were postmultiplying the homogenous
transformation matrices describing the rotation (or translation) of each particular
joint. The position and orientation of each joint frame was defined with respect to
the preceding frame, appertaining to the joint axis which is not fixed. In this case, as

4 Orientation 43
x
y
z
Y,ψ
P,ϑ
R,ϕ
Fig. 4.4 RPY angles for the case of an airplane
x y
z
R,ϕ
P,ϑ
Y,ψ
Fig. 4.5 RPY angles for the case of robot gripper
we have seen, we are multiplying the matrices from left to right. When we are dealing
with consecutive rotations about the axes of the same coordinate frame, we make use
of the premultiplication of the rotation matrices. In other words, the multiplications
are performed in the reverse order from right to left.

44 4 Orientation
We start with the rotation ϕ about z axis, continue with rotation ϑ around y axis
and finish with the rotation ψ about x axis. The reverse order of rotations is also
evident from the naming of RPY angles. The orientation matrix, which belongs to
RPY angles, is obtained by the following multiplication of the rotation matrices
R(ϕ, ϑ, ψ) = Rot(z, ϕ)Rot(y, ϑ)Rot(x, ψ) =
=
⎡
⎣
cϕ −sϕ 0
sϕ cϕ 0
0 0 1
⎤
⎦
⎡
⎣
cϑ 0 sϑ
0 1 0
−sϑ 0 cϑ
⎤
⎦
⎡
⎣
1 0 0
0 cψ −sψ
0 sψ cψ
⎤
⎦ =
=
⎡
⎣
cϕcϑ cϕsϑsψ − sϕcψ cϕsϑcψ + sϕsψ
sϕsϑ sϕsϑsψ + cϕcψ sϕsϑcψ − cϕcψ
−sϑ cϑsψ cϑcψ
⎤
⎦ .
(4.6)
Equation (4.6) calculates the rotation matrix from the corresponding RPY angles.
We learned that rotation and orientation can be described either by rotation matri-
ces or by RPY angles. In the first case we need 9 parameters, while only 3 parameters
are required in the latter case. While matrices are convenient for computations, they
do not however, provide a fast and clear image of, for example, the orientation of
a robot gripper within a space. RPY and Euler angles do nicely present the orien-
tation of a gripper, but they are not appropriate for calculations. In this chapter we
shall learn that quaternions are appropriate for either calculation or description of
orientation.
The quaternions represent extension of the complex numbers
z = a + ib, (4.7)
where i means the square root of −1, therefore i2
= −1. The complex numbers can be
geometrically presented in a plane by introducing a rectangular frame with e (real)
and m (imaginary) axis. When going from plane into space, two unit vectors j and k
must be added to already existing i. The following equality i2
= j2
= k2
= ijk = −1
is also valid. The quaternion has the following form
q = q0 + q1i + q2j + q3k. (4.8)
In the Eq.(4.8) qi are real numbers, while i, j, and k correspond to the unit vectors
along the axes of the rectangular coordinate frame.
When describing the orientation by the RPY angles, the multiplications of the
rotation matrices were needed. In a similar way we need to multiply the quaternions
pq = (p0 + p1i + p2j + p3k)(q0 + q1i + q2j + q3k). (4.9)

4 Orientation 45
Table 4.1 Rules for quaternion multiplications
* 1 i j k
1 1 i j k
i i −1 k −j
j j −k −1 i
k k j −i −1
The multiplication of quaternions is not commutative. When multiplying two quater-
nions we shall make use of the Table4.1. Let us multiply two quaternions
(2 + 3i − j + 5k)(3 − 4i + 2j + k) =
= 6 + 9i − 3j + 15k−
− 8i − 12i2
+ 4ji − 20ki+
+ 4j + 6ij − 2j2
+ 10kj+
+ 2k + 3ik − jk + 5k2
=
= 6 + 9i − 3j + 15k−
− 8i + 12 − 4k − 20j+
+ 4j + 6k + 2 − 10i+
+ 2k − 3j − i − 5 =
= 15 − 10i − 22j + 19k.
(4.10)
The following expression of a quaternion is specially appropriate to describe the
orientation in the space
q = cos
ϑ
2
+ sin
ϑ
2
s. (4.11)
In the Eq.(4.11) s is a unit vector aligned with the rotation axis, while ϑ is the angle of
rotation. The orientation quaternion can be obtained from the RPY angles. Rotation
R is described by the quaternion
qzϕ = cos
ϕ
2
+ sin
ϕ
2
k. (4.12)
The following quaternion belongs to the rotation P
qyϑ = cos
ϑ
2
+ sin
ϑ
2
j, (4.13)

46 4 Orientation
while rotation Y can be written as follows
qxψ = cos
ψ
2
+ sin
ψ
2
i. (4.14)
After multiplying the above three quaternions (4.12–4.14), the resulting orienta-
tion quaternion is obtained
q(ϕ, ϑ, ψ) = qzϕqyϑ qxψ . (4.15)
Let us illustrate the three descriptions of the orientation, i.e. RPY angles, rotation
matrix, and quaternions, by an example of description of gripper orientation. To make
the example clear and simple, the plane of the two-finger gripper will be placed into
the x0–y0 plane of the reference frame (Fig.4.6). The RPY angles can be read from
the Fig.4.6. The rotations around z and y axis equal zero. The rotation for −60◦
around the x axis can be seen from the Fig.4.6. The orientation of the gripper can
be, therefore, described by the following set of RPY angles
ϕ = 0, ϑ = 0, ψ = −60◦
. (4.16)
From the Fig.4.6 we can read also the angles between the axes of the reference and
gripper coordinate frame. Their cosines represent the orientation/rotation matrix R
nx = cos 0◦
, sx = cos 90◦
, ax = cos 90◦
,
ny = cos 90◦
, sy = cos 60◦
, ay = cos 30◦
,
nz = cos 0◦
, sz = cos 150◦
, az = cos 60◦
.
(4.17)
x0
y0
z0
a
n
s
ψ = −60◦
Fig. 4.6 Orientation of robot gripper

4 Orientation 47
The matrix R can be calculated also by inserting the known RPY angles into the
Eq.(4.6)
R =
⎡
⎣
1 0 0
0 0.5 0.866
0 −0.866 0.5
⎤
⎦ . (4.18)
In this way the correctness of our reading of the angles from the Fig.4.6 was
tested. We shall calculate the orientation quaternion by inserting the RPY angles into
the Eqs.(4.12–4.14)
qzϕ = 1 + 0k,
qyϑ = 1 + 0j,
qxψ = 0.866 − 0.5i.
(4.19)
The orientation quaternion is obtained after multiplying the three above quaternions
(4.15)
q0 = 0.866, q1 = −0.5, q2 = 0, q3 = 0. (4.20)
The Eqs.(4.16), (4.18) and (4.20) demonstrate three different descriptions of the
same gripper orientation.

Chapter 5
Two-Segment Robot Manipulator
5.1 Kinematics
Kinematics is part of classic mechanics that study motion without considering the
forces which are responsible for this motion. Motion is in general described by
trajectories, velocities and accelerations. In robotics we are mainly interested in tra-
jectories and velocities, as both can be measured by the joint sensors. In robot joints,
the trajectories are measured either as the angle in a rotational joint or as the distance
in a translational joint. The joint variables are also called internal coordinates. When
planning and programming a robot task the trajectory of the robot end-point is of
utmost importance. The position and orientation of the end-effector are described by
external coordinates. Computation of external variables from the internal variables,
and vice versa, is the central problem of robot kinematics.
In this chapter we shall limit our interest to a planar two-segment robot manip-
ulator with two rotational joints (Fig.5.1). According to the definition given in the
introductory chapter, such a mechanism can hardly be called a robot. Nevertheless,
this mechanism is an important part of the SCARA and anthropomorphic robot struc-
tures and will allow us to study several characteristic properties of the motion of robot
mechanisms.
There is a distinction between direct and inverse kinematics. Direct kinematics
in the case of a two-segment robot represents the calculation of the position of
the robot end-point from the known joint angles. Inverse kinematics calculates the
joint variables from the known position of the robot end-point. Direct kinematics
represents the simpler problem, as we have a single solution for the position of the
robot end-point. The solutions of inverse kinematics depend largely on the structure
of the robot manipulator. We often deal with several solutions for the joint variables
resulting in the same position of the robot end-point, while in some cases an analytic
solution of inverse kinematics does not exist.
Kinematic analysis includes also the relations between the velocity of the robot
end-point and the velocities of individual joints. We shall find that inverse kinematics
for velocities is simpler than inverse kinematics for trajectories. We shall first find the
49

50 5 Two-segment robot manipulator
ϑ1
p1,l1
ϑ2
p2,l2
x
y
z
x
Fig. 5.1 Planar two-segment robot manipulator
solution of direct kinematics for the trajectories. By differentiation we then obtain the
equations describing direct kinematics for the velocities. By simple matrix inversion
the inverse kinematics for velocities can be computed. Let us now consider the planar
two-segment robot manipulator shown in Fig.5.1.
The axis of rotation of the first joint is presented by the vertical z axis pointing
out of the plane. Vector p1 is directed along the first segment
p1 = l1

cos ϑ1
sin ϑ1

. (5.1)
Vector p2 is along with the second segment. Its components can be read from Fig.5.1
p2 = l2

cos(ϑ1 + ϑ2)
sin(ϑ1 + ϑ2)

. (5.2)
Vector x connects the origin of the coordinate frame with the robot end-point
x = p1 + p2. (5.3)
So we have for the position of the robot end-point
x =

x
y

=

l1 cos ϑ1 + l2 cos(ϑ1 + ϑ2)
l1 sin ϑ1 + l2 sin(ϑ1 + ϑ2)

. (5.4)
By defining the vector of joint angles
q =

ϑ1 ϑ2
T
, (5.5)

5.1 Kinematics 51
the Eq.(5.4) can be written in the following shorter form
x = k(q), (5.6)
where k(·) represents the equations of direct kinematics.
The relation between the velocities of the robot end-point and joint velocities is
obtained by differentiation. The coordinates of the end-point are functions of the
joint angles, which in turn are functions of time
x = x(ϑ1(t), ϑ2(t)) (5.7)
y = y(ϑ1(t), ϑ2(t)).
By calculating the time derivatives of Eq.(5.7) and arranging them into matrix form,
we can write
ẋ
ẏ

=

∂x
∂ϑ1
∂x
∂ϑ2
∂y
∂ϑ1
∂y
∂ϑ2

ϑ̇1
ϑ̇2

. (5.8)
For our two-segment robot manipulator we obtain the following expression

ẋ
ẏ

=

−l1s1 − l2s12 −l2s12
l1c1 + l2c12 l2c12

ϑ̇1
ϑ̇2

. (5.9)
The matrix, which is in our case of the second order, is called the Jacobian matrix
J(q). The relation (5.9) can be written in short form as
ẋ = J(q)q̇. (5.10)
In this way the problems of direct kinematics for trajectories and velocities are solved.
When solving the inverse kinematics, we calculate the joint angles from the known
position of the robot end-point. Figure5.2 shows only those parameters of the two-
segment robot mechanism which are relevant for the calculation of the ϑ2 angle. The
cosine rule is used
x2
+ y2
= l2
1 + l2
2 − 2l1l2 cos(180◦
− ϑ2), (5.11)
where − cos(180◦
− ϑ2) = cos(ϑ2). The angle of the second segment of the two-
segment manipulator is calculated as the inverse trigonometric function
ϑ2 = arccos
x2
+ y2
− l2
1 − l2
2
2l1l2
. (5.12)

ϑ2
l1
l2
x2 +y2
Fig. 5.2 Calculation of the ϑ2 angle
ϑ1
ϑ2
l1
l2
x x
y
y
α1
α2
Fig. 5.3 Calculation of the ϑ1 angle
The angle of the first segment is calculated with the aid of Fig.5.3. It is obtained
as the difference of angles α1 and α2
ϑ1 = α1 − α2.
The angle α1 is obtained from the right-angle triangle made of horizontal x and
vertical y coordinates of the robot end-point. The angle α2 is obtained by elongating
the triangle of Fig.5.2 into the right-angle triangle, as shown in Fig.5.3. Again we
make use of the inverse trigonometric functions
ϑ1 = arctan
y
x
− arctan
l2 sin ϑ2
l1 + l2 cos ϑ2
. (5.13)
When calculating the ϑ2 angle, we have two solutions, elbow-up and elbow-
down, as shown in Fig.5.4. A degenerate solution is represented by the end-point

5.1 Kinematics 53
elbow-up
elbow-down
x
y
Fig. 5.4 Two solutions of inverse kinematics
position x = y = 0 when both segments are of equal length l1 = l2. In this case
arctan y
x
is not defined. When the angle ϑ2 = 180◦
, the base of the simple two-
segment mechanism can be reached at an arbitrary angle ϑ1. However, when a point
(x, y) lies outside of the manipulator workspace, the problem of inverse kinematics
cannot be solved.
The relation between the joint velocities and the velocity of the end-point is
obtained by inverting the Jacobian matrix J(q)
q̇ = J−1
(q)ẋ. (5.14)
The matrices of order 2 × 2 can be inverted as follows
A =

a b
c d

A−1
=
1
ad − cb

d −b
−c a

.
For our two-segment manipulator we can write

ϑ̇1
ϑ̇2

=
1
l1l2 s2

l2 c12 l2 s12
−l1 c1 − l2 c12 −l1 s1 − l2 s12

ẋ
ẏ

. (5.15)
In general examples of robot manipulators, it is not necessary that the Jacobian matrix
has the quadratic form. In this case, the so called pseudoinverse matrix (JJT
)−1
is
calculated. For a robot with six degrees of freedom the Jacobian matrix is quadratic,
however after inverting, it becomes rather impractical. When the manipulator is close

to singular poses (e.g., when the angle ϑ2 is close to zero for the simple two-segment
robot), the inverse Jacobian matrix is ill defined. We shall make use of the Jacobian
matrix when studying robot control.
5.2 Statics
After the end of the robot kinematics section let us make a short leap to robot statics.
Let us suppose that the end-point of the two-segment robot manipulator bumped
into an obstacle (Fig.5.5). In this way the robot is producing a force against the
obstacle. The horizontal component of the force acts in the positive direction of the
x axis, while the vertical component is directed along the y axis. The force against
the obstacle is produced by the motors in the robot joints. The motor of the first joint
is producing the torque M1, while M2 is the torque in the second joint.
The positive directions of both joint torques are counter-clockwise. As the robot
is not moving, the sum of the external torques equals zero. This means that the torque
M1 in the first joint is equal to the torque of the external force or it is equal to the
torque that the manipulator exerts on the obstacle
M1 = −Fx y + Fy x. (5.16)
The end-point coordinates x and y, calculated by Eq.(5.4), are inserted into Eq.(5.16)
M1 = −Fx (l1 sin ϑ1 + l2 sin(ϑ1 + ϑ2)) + Fy(l1 cos ϑ1 + l2 cos(ϑ1 + ϑ2)). (5.17)
ϑ1
ϑ2
M1
M2
l1
l2
Fx
Fy
f
x x
y
y
Fig. 5.5 Two-segment robot manipulator in contact with the environment

5.2 Statics 55
In a similar way the torque in the second joint is determined
M2 = −Fxl2 sin(ϑ1 + ϑ2) + Fyl2 cos(ϑ1 + ϑ2). (5.18)
Equations (5.17) and (5.18) can be written in matrix form

M1
M2

=

−l1 s1 − l2 s12 l1 c1 + l2 c12
−l2 s12 l2 c12

Fx
Fy

. (5.19)
The matrix in Eq.(5.19) is a transposed Jacobian matrix. The transposed matrix of
order 2 × 2 has the following form
A =

a b
c d

AT
=

a c
b d

.
In this way we obtained an important relation between the joint torques and the forces
at the robot end-effector
τ = JT
(q)f, (5.20)
where
τ =

M1
M2

f =

Fx
Fy

.
Equation (5.20) describes the robot statics. It will be used in the control of a robot
which is in contact with the environment.
5.3 Workspace
The robot workspace consists of all points that can be reached by the robot end-point.
It plays an important role when selecting an industrial robot for an anticipated task.
It is our aim to describe an approach to determine the workspace of a chosen robot.
We shall again consider the example of the simple planar two-segment robot with
rotational joints. Our study of the robot workspace will therefore take place in a plane
andweshallinfactdealwithaworkingsurface.Regardlessoftheconstraintsimposed
by the plane we shall become aware of the most important characteristic properties
of the robot workspaces. Industrial robots usually have the ability to rotate around
the first vertical joint axis. We shall therefore rotate the working surface around the
vertical axis of the reference coordinate frame and thus obtain an idea of the realistic
three-dimensional robot workspaces.
Let us consider the planar two-segment robot manipulator as shown in Fig.5.6.
The rotational degrees of freedom are denoted as ϑ1 and ϑ2. The lengths of the
segments l1 and l2 will be considered equal. The coordinates of the robot end-point
can be expressed as in (5.4) with the following two equations:

ϑ1
ϑ2
l1
l2
x
y
Fig. 5.6 Two-segment robot manipulator
x = l1 cos ϑ1 + l2 cos(ϑ1 + ϑ2)
y = l1 sin ϑ1 + l2 sin(ϑ1 + ϑ2).
(5.21)
If Eqs.(5.21) are first squared and then summed, the equations of a circle are obtained
(x − l1 cos ϑ1)2
+ (y − l1 sin ϑ1)2
= l2
2
x2
+ y2
= l2
1 + l2
2 + 2l1l2 cos ϑ2.
(5.22)
The first equation depends only on the angle ϑ1, while only ϑ2 appears in the second
equation. The mesh of the circles plotted for different values ϑ1 and ϑ2 is shown
in Fig.5.7. The first equation describes the circles which are in Fig.5.7 denoted as
ϑ1 = 0◦
, 30◦
, 60◦
, 90◦
, 120◦
, 150◦
, and 180◦
. Their radii are equal to the length of the
second segment l2, the centers of the circles depend on the angle ϑ1 and travel along a
circle with the center in the origin of the coordinate frame and with the radius l1. The
circles of the second equation have all their centers in the origin of the coordinate
frame, while their radii depend on the lengths of both segments and the angle ϑ2
between them.
The mesh in Fig.5.7 serves for a simple graphical presentation of the working
surface of a two-segment robot. It is not difficult to determine the working surface for
the case when ϑ1 and ϑ2 vary in the full range from 0◦
to 360◦
. For the two-segment
manipulator with equal lengths of both segments this is simply a circle with the radius
l1 + l2. Much more irregular shapes of workspaces are obtained when the range of
motion of the robot joints is constrained, as is usually the case. Part of the working
surface where ϑ1 changes from 0◦
to 60◦
and ϑ2 from 60◦
to 120◦
is displayed as
hatched in Fig.5.7.
When plotting the working surfaces of the two-segment manipulator we assumed
that the lengths of both segments are equal. This assumption will be now supported by

5.3 Workspace 57
ϑ1
ϑ2
ϑ1
=
0
◦
ϑ2
=
0
◦
ϑ2
=
30
◦
ϑ2
=
60
◦
ϑ
2
=
9
0
◦
ϑ2
=
120
◦
ϑ2
=
150
◦
ϑ1 = 180◦
ϑ
1
=
1
5
0
◦
ϑ
1
=
1
2
0
◦
ϑ
1
=
3
0
◦
ϑ
1
=
9
0
◦
ϑ
1
=
6
0
◦
l1
l2
x
y
Fig. 5.7 Workspace of a planar two-segment robot manipulator (l1 = l2, 0◦ ≤ ϑ1 ≤ 180◦, 0◦ ≤
ϑ2 ≤ 180◦)
an adequate proof. It is not difficult to realize that the segments of industrial SCARA
and anthropomorphic robots are of equal length. Let us consider a two-segment robot,
where the second segment is shorter than the first one, while the angles ϑ1 and ϑ2
vary from 0◦
to 360◦
(Fig.5.8). The working area of such a manipulator is a ring with
inner radius Ri = l1 − l2 and outer radius Ro = l1 + l2. It is our aim to find the ratio
of the segments lengths l1 and l2 resulting in the largest working area at constant
sum of lengths of both segments Ro. The working area of the described two-segment
robot manipulator is
A = π R2
o − π R2
i . (5.23)
By inserting the expression for the inner radius in Eq.(5.23)
R2
i = (l1 − l2)2
= (2l1 − Ro)2
(5.24)
we can write
A = π R2
o − π(2l1 − Ro)2
. (5.25)

Ro Ri
l1
l1
l2
l2
Fig. 5.8 Working area of two-segment manipulator with the second segment shorter
For maximum area, the derivative with respect to segment length l1 should be equal
zero
∂ A
∂l1
= 2π(2l1 − Ro) = 0. (5.26)
The solution is
l1 =
Ro
2
, (5.27)
giving
l1 = l2. (5.28)
The largest working area of the two-segment mechanism occurs for equal lengths of
both segments.
The area of the working surface depends on the segment lengths l1 and l2 and
on the minimal and maximal values of the angles ϑ1 and ϑ2. When changing the
ratios l1/l2 we can obtain various shapes of the robot working surface. The area of
a such working surface is always equal to the one shown in Fig.5.9. In this Figure
Δϑ1 refers to the difference between the maximal and minimal joint angle value
Δϑ1 = (ϑ1max
− ϑ1min
). The area of the working surface is the area of a ring segment
A =
Δϑ1π
360
(r2
1 − r2
2 ) (5.29)
for Δϑ1 given in angular degrees.

5.3 Workspace 59
Δϑ1
r1 r2
Fig. 5.9 Working surface of a two-segment manipulator
In Eq.(5.29), the radii r1 and r2 are obtained by the cosine rule
r1 =

l2
1 + l2
2 + 2l1l2 cos ϑ2min
r2 =

l2
1 + l2
2 + 2l1l2 cos ϑ2max
. (5.30)
The area of the working surface is, in the same way as its shape, dependent on the ratio
l2/l1 and on the constraints in the joint angles. The angle ϑ1 determines the position
of the working surface with respect to the reference frame and has no influence on
its shape. Let us examine the influence of the second angle ϑ2 on the area of the
working surface. We shall assume that l1 = l2 = 1 and ϑ1 change from 30◦
to 60◦
.
For equal ranges of the angle ϑ2 (30◦
) and for different values of ϑ2max
and ϑ2min
we
obtain different values of the working areas
0◦
≤ ϑ2 ≤ 30◦
A = 0.07
30◦
≤ ϑ2 ≤ 60◦
A = 0.19
60◦
≤ ϑ2 ≤ 90◦
A = 0.26
90◦
≤ ϑ2 ≤ 120◦
A = 0.26
120◦
≤ ϑ2 ≤ 150◦
A = 0.19
150◦
≤ ϑ2 ≤ 180◦
A = 0.07.
Until now, under the term workspace we were considering the so called reachable
robot workspace. This includes all the points in the robot surroundings that can be
reached by the robot end-point. Often this so-called dexterous workspace is of greater
importance. The dexterous workspace comprises all the points that can be reached
with any arbitrary orientation of the robot end-effector. This workspace is always

1 2 3 4
Fig. 5.10 Reachable and dexterous workspace of a two-segment manipulator with end-effector
smaller than the reachable workspace. The dexterous workspace is larger when the
last segment (end-effector) is shorter. The reachable and the dexterous workspaces
of a two-segment robot with the end-effector are shown in Fig.5.10. The second and
the third circle are obtained when the robot end-effector is oriented towards the area
constrained by the two circles. These two circles represent the limits of the dexterous
workspace. The first and the fourth circle constrain the reachable workspace. The
points between the first and the second and the third and the fourth circle cannot be
reached with an arbitrary orientation of the end-effector.
For robots having more than three joints, the described graphical approach is not
appropriate. In that case we make use of numerical methods and computer algorithms.
5.4 Dynamics
For illustration purposes, we shall study the planar, two-segment robot manipulator
as shown in Fig.5.11. The segments of length l1 and l2 may move in the vertical x–y
plane, their positions being described by angles with respect to the horizontal (x)
axis; ϑ1 and ϑ = ϑ1 + ϑ2. Actuators at the joints provide torques M1 and M2, whose
positive direction is defined by increasing angles, i.e., along the positive direction of
the z axis of our reference coordinate frame.
We now approximate the segments by point masses m1 and m2 at the midpoints
of rigid, but otherwise massless rods (see Fig.5.12). Let r1 denote the position of
point mass m1 with respect to the first joint, which is at the origin of our coordinate
frame. Let r2 denote the position of point mass m2 with respect to the second joint,
which is at the junction of the two segments.

5.4 Dynamics 61
x
y
z M1
M2
l1
l2
ϑ1
ϑ1
ϑ2
Fig. 5.11 Parameters of the planar, two-segment robot manipulator, which moves in the vertical
x–y plane
x
y
z
m1
m2
ϑ1
ϑ2
r
r1
r2
F1
F2
Fig. 5.12 Planar robot manipulator approximated by point masses m1 and m2

The point masses m1 and m2 are acted upon by the forces that are transmitted by
the massless rods, as well as by the force of gravity. Newton’s law claims that the
vector sum of all the forces acting on a particle is equal to the product of the particle
mass and it’s acceleration. Therefore,
F1 = m1a1 and F2 = m2a2, (5.31)
where F1 and F2 represent the sums of all forces (i.e., the force of the rod and the force
of gravity), acting on each of the point masses m1 and m2, while a1 and a2 are their
accelerations with respect to the origin of the coordinate frame. So, a calculation of
the accelerations amounts to the determination of the forces on the two “particles”.
The position of m1 with respect to the reference frame origin, is given by r1,
while the position of m2 is given by r = 2r1 + r2 (see Fig.5.12). The corresponding
accelerations are therefore a1 = r̈1 and a2 = r̈, where the two dots above the vector
symbol denote second derivatives with respect to time. Therefore,
a1 = r̈1 and a2 = r̈ = 2r̈1 + r̈2. (5.32)
Now, r1 and r2 represent rigid rods, so their lengths are fixed. Therefore, these
vectors can only rotate. Let us remind ourselves of basic physics, which says that a
rotating vector describes a particle in circular motion. Such motion may have two
components of acceleration (Fig.5.13, see also Appendix ??). The first component
is the radial or centripetal acceleration ar , which is directed towards the center of
rotation. It is due to the change only of the direction of velocity and is thus present
also in uniform circular motion. It is given by the expression
ar = −ω2
r, (5.33)
where ω is the angular velocity ω = θ̇. The second component is the tangential
acceleration, which is directed along the tangent to the circle (Fig.5.13). It is due to
the change of the magnitude of velocity and is present only in circular motion with
angular acceleration α = θ̈. It is given by
at = α × r, (5.34)
where α is the vector of angular acceleration, which is perpendicular to the plane
of motion, i.e., it is along the z axis of our reference coordinate frame. The total
acceleration is obviously
a = ar + at = −ω2
r + α × r. (5.35)
Let us now calculate the second derivatives with respect to time of the vectors r1
and r2. As noted above, each of these derivatives has two components corresponding
to the radial and to the tangential acceleration. So
r̈1 = −ω2
1r1 + α1 × r1 and r̈2 = −ω2
2r2 + α2 × r2. (5.36)

5.4 Dynamics 63
x
y
α
θ
r
m
ar
at
Fig. 5.13 A rotating vector r of fixed length describes a particle in circular motion
The magnitude of the angular velocity ω1 and the vector of angular acceleration α1
of the first segment are
ω1 = ϑ̇1 and α1 = ϑ̈1k, (5.37)
where k is the unit vector along the z axis. The angular velocity ω2 and the angular
acceleration α2 of the second segment are
ω2 = ϑ̇ = ϑ̇1 + ϑ̇2 and α2 = ϑ̈k = (ϑ̈1 + ϑ̈2)k. (5.38)
Here we used ϑ = ϑ1 + ϑ2 (see Figs.5.11 and 5.12). The second derivatives of
vectors r1 and r2 may be written as
r̈1 = −ω2
1r1 + α1 × r1 = −ϑ̇2
1 r1 + ϑ̈1k × r1 (5.39)
and
r̈2 = −ω2
2r2 + α2 × r2 = −ϑ̇2
r2 + ϑ̈k × r2 =
= −(ϑ̇1 + ϑ̇2)2
r2 + (ϑ̈1 + ϑ̈2)k × r2.
(5.40)
We may now use these expressions to calculate the accelerations of the two point
masses m1 and m2 corresponding to our two-segment robot. The acceleration a1 of
m1 is
a1 = r̈1 = −ϑ̇2
1 r1 + ϑ̈1(k × r1). (5.41)

The acceleration a2 of m2 is
a2 = r̈ = 2r̈1 + r̈2 =
= −2ϑ̇2
1 r1 + 2ϑ̈1(k × r1) − (ϑ̇1 + ϑ̇2)2
r2 + (ϑ̈1 + ϑ̈2)(k × r2).
(5.42)
From these accelerations we get the total forces acting on particles m1 and m2
F1 = m1a1 and F2 = m2a2. (5.43)
We can now calculate the torques of these forces with respect to the coordinate frame
origin
τ1 = r1 × F1 = r1 × m1a1 and τ2 = r × F2 = (2r1 + r2) × m2a2. (5.44)
Inserting expressions for a1 and a2 as derived above, reminding ourselves of the
double vector product [a × (b × c) = b(a · c) − c(a · b)], and by patiently doing
the lengthy algebra, we obtain
τ1 =m1r2
1 ϑ̈1k
and
τ2 =[ϑ̈1(4m2r2
1 + 4m2r1r2 cos ϑ2 + m2r2
2 )+
+ ϑ̈2(m2r2
2 + 2m2r1r2 cos ϑ2)−
− ϑ̇1ϑ̇24m2r1r2 sin ϑ2 − ϑ̇2
2 2m2r1r2 sin ϑ2]k.
(5.45)
The sum of both torques on the two “particles” of our system is obviously τ =
τ1 + τ2.
On the other hand, we may consider our two-segment system consisting of two
point masses and two massless rods from a different viewpoint. As a consequence of
Newton’s third law (To every action there is an equal but opposite reaction), we have
a theorem stating that internal torques in a system cancel out, so that only torques of
external forces are relevant. The torques of external forces on our robot system are
the torques of gravity and the torque exerted by the base on which the robot stands.
The torque of the base is equal to the torque M1 of the actuator in the first joint. The
sum of these torques of external forces (base + gravity) must be equal to τ1 + τ2
(derived above), as both results represent two ways of viewing the total torque on
the same system. So
M1 + r1 × m1g + r × m2g = τ1 + τ2. (5.46)
With r = 2r1 + r2 we have the torque of the actuator in the first joint
M1 = τ1 + τ2 − r1 × m1g − (2r1 + r2) × m2g. (5.47)

5.4 Dynamics 65
Remembering that g points vertically downward (i.e., along −y) and inserting the
above results for τ1 and τ2, we finally get
M1 = ϑ̈1(m1r2
1 + m2r2
2 + 4m2r2
1 + 4m2r1r2 cos ϑ2)+
+ ϑ̈2(m2r2
2 + 2m2r1r2 cos ϑ2)−
− ϑ̇1ϑ̇24m2r1r2 sin ϑ2 − ϑ̇2
2 2m2r1r2 sin ϑ2+
+ m1gr1 cos ϑ1 + 2m2gr1 cos ϑ1 + m2gr2 cos(ϑ1 + ϑ2).
(5.48)
In order to obtain the torque M2 of the actuator in the second joint, we will first
consider the total force F2 acting on the point mass m2. The force F2 is a sum of two
contributions. One is the force of gravity m2g, the other is the force F
2 exerted on
m2 by the massless and rigid rod of the second segment. So
F2 = F
2 + m2g. (5.49)
To this equation we apply a vector product of r2 from the left and obtain
r2 × F2 = r2 × F
2 + r2 × m2g. (5.50)
The first term on the right-hand side is the vector product of r2 with the force F
2
exerted on m2 by the massless and rigid rod. This term is equal to the torque M2
of the actuator in the second joint. (Note that the rod may also exert a force on m2
directed along the rod, but the vector product of that component with r2 vanishes).
We therefore obtain
M2 = r2 × F2 − r2 × m2g. (5.51)
Substituting m2a2 for F2 and the expression derived previously for a2, leads to
M2 = ϑ̈1(m2r2
2 + 2m2r1r2 cos ϑ2) + ϑ̈2m2r2
2 +
+ ϑ̇2
1 2m2r1r2 sin ϑ2 + m2r2g cos(ϑ1 + ϑ2).
(5.52)
The expressions for M1 (5.48) and M2 (5.52) seem relatively complicated, so
let us investigate some simple and familiar cases. First, assume ϑ1 = −90◦
and no
torque in the second joint M2 = 0 (Fig.5.14 left). The equation for M2 reduces to
ϑ̈2m2r2
2 = −m2gr2 sin ϑ2. (5.53)
This is the equation of a simple pendulum with mass m2, moment of inertia m2r2
2 =
J2, which is rotating around the second joint with angular acceleration ϑ̈2 (Fig.5.14
left). The left-hand side is thus J2α2 and on the right-hand side we have the torque
due to gravity. So, this is an example of the simple equation M = Jα, to which our

x
x
y
y
m1
m1
m2
m2
r1
r1
r1
r1
r2
r2
ϑ1 = −90◦
ϑ2
ϑ1
Fig. 5.14 Two simple examples of the two-segment robot manipulator: ϑ1 = −90◦ (left) and
ϑ2 = 0◦ (right)
complicated expression has been reduced. For small oscillations (ϑ2 1) we have
sin ϑ2 ≈ ϑ2 and the equation becomes
ϑ̈2 + (
g
r2
)ϑ2 = 0. (5.54)
This is the equation of the simple pendulum with angular frequency ω0 =

g
r2
and
oscillation period T = 2π

r2
g
.
Next assume ϑ2 = 0 so we have one rigid rod rotating around one end, which is
fixed at the coordinate frame origin (Fig.5.14 right). If we also “switch off” gravity
(g = 0), we obtain for the torque in the first joint
M1 = ϑ̈1(m1r2
1 + m2r2
2 + 4m2r2
1 + 4m2r1r2) =
= ϑ̈1[m1r2
1 + m2(2r1 + r2)2
] = J12α1,
(5.55)
where α1 = ϑ̈1 is the angular acceleration and J12 is the combined moment of inertia
of the two masses. Alternatively, one might take the torque in the first joint equal
to zero, include gravity and one gets a relatively simple pendulum with two point
masses on one massless rigid rod.
Let us mention that the above full equations for M1 and M2 (Eqs.(5.48) and
(5.52)), with minor adaptations of notation, are valid for a double pendulum with

5.4 Dynamics 67
S
RS−E
RE−M
E
M
Fig. 5.15 Schematic trajectory (not in scale) of the Earth (dashed curve) and the Moon (full curve)
in the reference frame of the Sun
friction. The torques of the actuators are in this case replaced by the torques of friction
in the joints.
An amusing exercise would be to compare the trajectories of the endpoints of the
two segments of our simple robot with the trajectories of the Earth and the Moon,
as seen from the reference frame of the Sun. Let us approximate the Earth and the
Moon as point particles (mE mM ) in coplanar circular orbits. As the gravitational
force acts only along the line joining the two particles, it cannot transmit torques,
so the angular accelerations are zero and the angular velocities are constant. The
orbital velocity of the Earth around the Sun (RS−E ≈ 150 · 106
km, T = 365 days,
vE ≈ 2.6 · 106
km/day) is much greater than the orbital velocity of the Moon around
the Earth (RE−M ≈ 0.38 · 106
km, T = 28 days, vM ≈ 0.08 · 106
km/day), so the
trajectory of the Moon as seen in the Sun’s reference frame would be approximately
a sine curve superimposed on the Earth’s circular orbit around the Sun (Fig.5.15).
With our two-segment robot one could have higher angular velocities of the second
segment resulting in different shapes of the trajectory of its endpoint (Ptolemy’s
epicycles for example).
Returning to our relatively complicated equations for the torques M1 and M2
(Eqs.(5.48) and (5.52)), due to actuators in the joints, we see that these equations
may be condensed into matrix form representing the robot dynamic model as
τ = B(q)q̈ + C(q, q̇)q̇ + g(q). (5.56)

In this equation the vector τ unites the torques of both actuators
τ =

M1
M2

. (5.57)
Vectors q, q̇ and q̈ belong to the segment trajectories, velocities and accelerations
respectively. For the two-segment robot we have
q =

ϑ1
ϑ2

, q̇ =

ϑ̇1
ϑ̇2

, q̈ =

ϑ̈1
ϑ̈2

.
The first term of the equation for τ is called the inertial term. In our case of the
planar, two-segment robot manipulator with r1 = r2 = l
2
and by simplifying the
notation with s1 = sin ϑ1, c12 = cos(ϑ1 + ϑ2) etc., we get
B(q) =
1
4
m1l2
+ 5
4
m2l2
+ m2l2
c2 1
4
m2l2
+ 1
2
m2l2
c2
1
4
m2l2
+ 1
2
m2l2
c2 1
4
m2l2

. (5.58)
The second term of this matrix equation is called the Coriolis term and includes
velocity and centrifugal effects. For the two-segment robot we have the following
matrix
C(q, q̇) =

−m2l2
s2ϑ̇2 −1
2
m2l2
s2ϑ̇2
1
2
m2l2
s2ϑ̇1 0

. (5.59)
The gravitational column g(q) has in our case the following form
g(q) =
1
2
m1glc1 + m2glc1 + 1
2
m2glc12
1
2
m2glc12

. (5.60)

Chapter 6
Parallel Robots
This chapter deals with the increasingly popular and high-performing robots that are
known as parallel robots. Standard mechanisms of industrial robots possess serial
kinematic chains in which links and joints alternate as shown in Fig.6.1 (left). These
are referred to as serial robots. Lately, we have seen a significant advancement of
parallel robots. They include closed kinematic chains, an example is shown in Fig.6.1
(right).
In industry, parallel robots have started to gain ground in the last two decades.
However, the initial developments date back to 1962 when Gough and Whitehall
developed a parallel robot for testing automobile tires. At about the same time, a
similar parallel robot was introduced by Stewart to design a flight simulator. The
parallel robot, in which a mobile platform is controlled by six actuated legs, is
therefore called the Stewart-Gough platform. The breakthrough of parallel robots
was also largely due to the robot developed by Clavel in the eighties. His mechanism
was patented in the USA in 1990 under the name of the Delta robot. The parallel
mechanisms in robotics had become a subject of systematic scientific research in the
early eighties. These activities intensified significantly in the nineties and culminated
with some key achievements in robot kinematics in general.
6.1 Characteristics of Parallel Robots
In serial robots, the number of degrees of freedom is identical to the total number
of degrees of freedom in joints. Thus, all joints must be actuated, and usually only
simple one degree of freedom translational and rotational joints are used. In parallel
robots, the number of degrees of freedom is lower than the total number of degrees
of freedom in joints so that many joints are passive. Passive joints can be more
complex; typical representatives are the universal joint and the spherical joint. The
universal joint consists of two perpendicular rotations while three perpendicular
69

70 6 Parallel Robots
Fig. 6.1 Serial kinematic chain (left) and closed kinematic chain (right)
T R U S
Fig. 6.2 Types of joints often used in parallel mechanisms
rotations compose the spherical joint as shown in Fig.6.2. Here, letters T, R, U, and
S are used to mark the translational joint, the rotational joint, the universal joint, and
the spherical joint, respectively.
In parallel robots, the last (top) link of the mechanisms is the so called platform
(Fig.6.3). The platform is the active link to which the end-effector is attached. It is
connected to the fixed base by a given number of (usually) serial mechanisms called
legs. The whole structure contains at least one closed kinematic chain (minimum
two legs). The displacements in the legs produce a displacement of the platform as
shown in Fig.6.3. The motions of the platform and the legs are connected by often
very complex trigonometric expressions (direct and inverse kinematics) depending
on the geometry of the mechanism, on the type of joints, the number of legs and on
their kinematic arrangements.
Unfortunately, unique and uniform denominations for parallel robots do not exist.
In this work, a parallel robot is denominated by the type of kinematic chains repre-

6.1 Characteristics of Parallel Robots 71
T
T
R
R
R
U
S
S
S
base
legs
platform
Fig. 6.3 Basic structure of a parallel robot
senting the legs. Thus, the robot in Fig.6.3 is denominated as UTR-SS-RRTS. When
legs of the same type are repeated, for example, in the TRR-TRR-TRR robot, the
denomination can be simplified as 3TRR.
Number of degrees of freedom
Each joint contributes to the mobility of the robot by introducing a given number
of degrees of freedom or, alternatively, by introducing a corresponding number of
constraints, which are defined as follows. Let λ denote the maximum number of
degrees of freedom of a freely moving body (in space λ = 6 and in plane λ = 3), and
let fi be the number of degrees of freedom of the i − th joint. The corresponding
number of constraints is
ci = λ − fi . (6.1)
In robotic practice where serial robots dominate, we usually consider joints as
elements that add degrees of freedom to the motion of the robot end-effector. In
parallel robots, on the contrary, it is more advantageous to consider the movement of
the platform (to which the end-effector is attached), taking into account the number
of constraints introduced by the joints. Thus, a universal joint U in a space where λ =
6 introduces fi = 2 degrees of freedom and ci = λ − fi = 6 − 2 = 4 constraints.
Or, for example, in a plane where λ = 3, a rotational joint R introduces fi = 1
degrees of freedom and ci = λ − fi = 3 − 1 = 2 constraints, while the same joint
in space introduces ci = λ − fi = 6 − 1 = 5 constraints. Note that rotational and
translational joints can operate both in a plane and in space, whereas spherical and
universal joints produce only spatial movements and cannot be used in planar robots.
The number of degrees of freedom of a parallel robot is less than the total number
of degrees of freedom contributed by the robot joints, unlike in a serial robot where
these two numbers are identical. Let N be the number of moving links of the robot
and n the number of joints. The joints are referred to as i = 1, 2, . . . , n. Each joint
possesses fi degrees of freedom and ci constraints. The N free moving links possess
Nλ degrees of freedom. When they are combined into a mechanism, their motion

is limited by the constraints introduced by joints, so that the number of degrees of
freedom of a robot mechanism is
F = Nλ −
n

i=1
ci . (6.2)
Here, by substituting ci with λ − fi we obtain the well known Grübler’s formula as
follows
F = λ(N − n) +
n

i=1
fi . (6.3)
We must not forget that the number of motors which control the motion of a robot is
equal to F.
Note that in serial robots the number of moving links and the number of joints
are identical (N = n), so that the first part of Grübler’s formula is always zero
(λ(N − n) = 0). This explains why the number of degrees of freedom in serial robots
is simply
F =
n

i=1
fi . (6.4)
A very practical form of Grübler’s formula to calculate the degrees of freedom of a
parallel robot can be obtained as follows. Suppose that a parallel mechanism includes
k = 1, 2, . . . , K legs, and that each of the legs possesses νk degrees of freedom and
consequently ξk = λ − νk constraints. When the platform is not connected to the
legs and can freely move in space, it contains λ degrees of freedom. The number of
degrees of freedom of a connected platform can thus be computed by subtracting the
sum of constraints introduced by the legs
F = λ −
K

k=1
ξk. (6.5)
Equations (6.3) and (6.5) are mathematically identical and can be transformed from
one to another by simple algebraic operations.
Now we can calculate the degrees of freedom for the robot shown in Fig.6.3. This
robot possesses N = 7 moving links and n = 9 joints. The total number of degrees
of freedom in joints is 16 (3 rotational joints, 2 translational joints, 1 universal and
3 spherical joints). Using the standard Grübler’s formula given in Eq.(6.3), we get
F = 6(7 − 9) + 16 = 4.
If we now use the modified form of Grübler’s formula we need to calculate the
constraints introduced by each leg. This is rather simple because we only need to
subtractthenumberofdegreesoffreedomofeachlegfromλ.Forthegivenrobot(legs

6.1 Characteristics of Parallel Robots 73
are counted from left to right) we have ξ1 = 2, ξ2 = 0, and ξ3 = 0. By introducing
these values in Eq.(6.5), as expected, we obtain
F = 6 − 2 = 4.
Advantages and disadvantages of parallel robots
The introduction of parallel robots in industry is motivated by the number of signif-
icant advantages that parallel robots have in comparison to serial robots. The most
evident are the following:
Load capacity, rigidity, and accuracy. The load carrying capacity of parallel robots
is considerably larger than that of serial robots. Parallel robots are also more rigid,
and their accuracy in positioning and orienting an end-effector is several times
better than with serial robots.
Excellent dynamic properties. The platform can achieve high velocities and accel-
erations. Furthermore, the resonant frequency of a parallel robot is orders of
magnitude higher.
Simple construction. Several passive joints in parallel robots enable less expensive
and simple mechanical construction. When building parallel robots standard bear-
ings, spindles, and other machine elements can be used.
The use of parallel robots is, nevertheless, limited. Because of the tangled legs,
parallel robots can have difficulties in avoiding obstacles in their workspace. Other
significant disadvantages are:
Small workspace. Parallel robots have considerably smaller workspaces than serial
robots of comparable size. Their workspace may be further reduced since during
motion of the platform the legs may interfere with each other.
Complex kinematics. The computation of kinematics of parallel robots is complex
and lengthy. In contrast to serial robots, where the difficulty arises when solving
the inverse kinematics problem, in parallel mechanisms the difficulty arises in
solving the direct kinematics problem.
Fatal kinematic singularities. Serial robots in kinematically singular poses lose
mobility. Parallel robots in singular poses gain degrees of freedom, which cannot
be controlled. This is a fatal situation because it cannot be resolved.
6.2 Kinematic Arrangements of Parallel Robots
We can create an immense number of kinematic arrangements of parallel robots.
In industrial practice, however, only few of these are used. The most popular and
general in the kinematic sense is the Stewart-Gough platform as shown in Fig.6.4.

T
U
S
Fig. 6.4 The Stewart-Gough platform
Stewart-Gough platform
A general Stewart-Gough platform is shown on the left side of Fig.6.4. According
to our denomination, the mechanism is of type 6UTS. The robot contains n = 18
joints, N = 13 moving links and the sum of fi , i = 1, 2, . . . n is 36. This gives the
expected result
F = 6(13 − 18) + 36 = 6
degreesoffreedom.Theplatformofthisrobotcanbespatiallypositionedandoriented
under the control of six motors, which are typically the six translations. By shortening
or expanding the legs (changing the lengths of the legs) the platform can be moved
into a desired pose (position and orientation). A special advantage of the Stewart-
Gough platform with the UTS legs is that loads acting on the platform are transferred
to each particular leg in the form of a longitudinal force in the direction of the leg and
there is no transverse loading on the legs. This peculiarity allows excellent dynamic
performances.
ThenumberofdegreesoffreedomofaUTSlegissixandthenumberofconstraints
is zero. If we consider Grübler’s formula (6.5) it is easy to verify that the number
of UTS legs does not affect the number of degrees of freedom of the robot and that
the mobility of the Stewart-Gough platform does not depend on the number of legs.
A robot with only one UTS leg, which is a serial robot, possesses six degrees of
freedom, the same as the fully parallel original six-legs Stewart-Gough robot.
The six-legged mechanism on the right side of Fig.6.4 schematically represents
the original Stewart-Gough platform which has a central-symmetrical star shape. In
this arrangement, two by two legs are clamped in one point in which two overlap-
ping coincident spherical (or universal) joints are placed. Therefore, the number of
independent spherical joints is six and the same is the number of universal joints.
The overlapping joints not only simplify the construction but also allow easier com-
putations of the robot kinematics and dynamics.

6.2 Kinematic Arrangements of Parallel Robots 75
R
U
U
S
S
T0
R0
U0
U0
Fig. 6.5 The Delta robot
Delta robot
Due to its specific motion characteristics and its numerous applications in industry,
the Delta robot found its place among robot manufacturers (see Fig.6.5). The kine-
matics of this robot is very sophisticated. The main objective of its creator was to
create a lightweight robot with extreme dynamic performances.
The fixed base of the robot is the upper hexagon while the lower hexagon repre-
sents the moving platform. The robot has three lateral legs. Only one is presented
in the figure, with one R joint, two S joints and two U joints; the other two legs
are symbolically drawn with a dotted line. There is also an independent middle leg
R0U0T0U0 which has no influence on the motion of the platform. There is a parallel-
ogram mechanism between the middle of the leg and the base, which consists of two
spherical joints S and two universal joints U. Each leg, therefore, has 3 links and 5
joints. Without considering the middle leg, the number of degrees of freedom of the
mechanism is
F = 6(10 − 15) + 33 = 3.
The pose of the platform is determined by only three variables. In the original version
of the Delta robot the three rotation angles R in lateral legs are controlled by motors.
Due to the parallelogram structure of the legs, the platform executes only translation
and is always parallel to the base.
The purpose of the middle leg is to transfer the rotation R0 across the platform
to the gripper at the end-point of the robot. It acts as a telescoping driveshaft for
rotating the gripper. This leg is a cardan joint with two universal joints U0 separated
by a translational joint T0. In all, the mechanism has four degrees of freedom: three
translational, enabling the spatial position of the gripper and one rotational, enabling

rotation of the gripper about an axis perpendicular to the platform. All actuators
of the Delta mechanism are attached to the base and do not move. Therefore the
mechanism is extremely lightweight and the platform can move with high velocities
and accelerations.
Planar parallel robots
The following examples are planar parallel robots which operate in a given plane
where λ = 3. The first example is given in Fig.6.6 left. The robot contains three
legs of the type RTR-RRR-RRR. As a result we have N = 7 and n = 9 and the total
number of degrees of freedom in joints is 9. According to Eq.(6.3), the number of
degrees of freedom of this robot is
F = 3(7 − 9) + 9 = 3.
The result is expected since all legs introduce zero constraints (6.5). Consequently,
the platform can achieve any desired pose inside the workspace. Note that in plane
two degrees of freedom are needed for the position (translations in x–y plane) and
one degree of freedom for the orientation (a rotation about z axis). To activate this
robot three motors are needed. To attach the motors, we can select any of the nine
joints. Usually we prefer the joints attached to the base so that the motors are not
moving and their weight does not influence the robot dynamics. In a specific case,
the translational joint can also be motorized using an electric spindle or a hydraulic
cylinder.
In Fig.6.6 right a similar planar parallel robot is presented, its structure is RTR-
RR-RR. Here, we can see that each of the two RR legs introduce one constraint.
According to Eq.(6.5), the number of degrees of freedom of this parallel robot is
F = 3 − 2 = 1.
The robot is controlled using one motor. The platform has limited mobility and can
only move along a curve in plane x–y. We can, for example, either position the
T
T
R
R
R
R
R
R
R
R
R
R
R
R
R
R
x
y
Fig. 6.6 Planar parallel robots

6.2 Kinematic Arrangements of Parallel Robots 77
platform along x axis without having control over y and the platform’s orientation
or, alternatively, orient the platform without having control over its position in x
and y.
Parallel humanoid shoulder
Parallel mechanisms are very common in nature, in the human body or in animals.
It is, therefore, no surprise that the models of parallel robots can be efficiently used
in simulating biomechanical properties of humans where muscles and ligaments
stretched over the joints form various parallel kinematic structures. For instance, the
shoulder complex can be represented by two basic compositions, the so-called inner
joint, which includes the motion of the clavicle and the scapula with respect to the
trunk,andtheso-calledouterjoint,whichisassociatedwiththeglenohumeraljoint.In
today’shumanoidrobotics,themotionoftheinnerjointistypicallyneglectedbecause
of its mechanical complexity. Nevertheless, its contribution to human motion, reach-
ability of the arm and dynamics is crucial.
A parallel shoulder mechanism representing the inner shoulder was proposed in
the literature. Its motion is shown in Fig.6.7. The proposed structure is TS-3UTS.
There is a central leg T0S0 with four degrees of freedom and two constraints. Around
the axis of the central leg three UTS lateral legs are attached possessing six degrees of
freedom each, their number of constraints is zero. According to Eq.(6.5), the number
of degrees of freedom of the robot is
F = 6 − 2 = 4.
The robot can produce a complete orientation of the platform (about three principal
orientation angles), and can expand or shrink similarly to the human shoulder. The
arm is attached to this platform through the glenohumeral joint. The inner shoulder
joints, as it is proposed, precisely mimic the motion of the arm, including shrugging
T
U
S
T0
S0
Fig. 6.7 Parallel robot mimicking the inner shoulder mechanism

and avoiding collisions with the body, and provides excellent static load capacity and
dynamic capabilities.
6.3 Modelling and Design of Parallel Robots
The majority of parallel robots which appear in industry or in research laborato-
ries possess symmetrical kinematic arrangements. From the point of view of their
construction, it is useful that they are composed of the same mechanical elements.
Symmetry also contributes to making their mathematical treatment simpler.
One common group of kinematic arrangements is represented by the previously
described shoulder robot. This group contains a central leg with ν1 degrees of freedom
around which there are symmetrically placed lateral legs, which are often of type
UTS possessing ν2, ν3, ..., νK = λ degrees of freedom (and zero constraints). The
central leg is therefore crucial to determine the kinematic properties of the whole
robot, as the number of degrees of freedom of the robot is F = ν1.
The second group of kinematic arrangements are represented by the Stewart-
Gough platform in which all the legs are identical and are usually of type UTS
so that ν1, ν2, ..., νK = λ. When ν1, ν2, ..., νK λ only a small number of such
robots are movable, most of their structures are with zero or negative degrees of
freedom. Robots with a negative number of degrees of freedom are referred to as
overconstrained.
Consider the second group of robots (Gough-Stewart-like kinematic structure)
with a single motor in each leg. Such a robot must have K = F legs, as a robot with
K F cannot be controlled. It is easy to verify that only the following robots can
exist in space (where λ = 6)
K = 1, ν1 = 1
K = 2, ν1 = ν2 = 4
K = 3, ν1 = ν2 = ν3 = 5
K = 6, ν1 = ν2 = ... = ν6 = 6
Robots in this group with four and five legs do not exist. In plane, where λ = 3, only
the following robots can exist
K = 1, ν1 = 1
K = 3, ν1 = ν2 = ν3 = 3
In the planar case, robots with two legs do not exist.
Kinematic parameters and coordinates of parallel robots
In Fig.6.8 the coordinate frame x–y–z is attached to the moving platform, while
x0–y0–z0 is fixed to the base. The position of the platform is given with respect to
the fixed coordinate frame by vector r; its components are rx,ry,rz. The orientation

6.3 Modelling and Design of Parallel Robots 79
x
y
z
x0
y0
z0
r
bk
ck
dk
ϑ
ϕ
ψ
Fig. 6.8 Kinematic parameters of a parallel robot
of the platform can be described by a chosen triplet of orientation angles ψ, ϑ, ϕ
occurring between both coordinate frames (see Chap. 4 for details).
Vector bk defines the attachment of leg k to the base expressed in frame x0–y0–z0,
while vector ck defines the attachment of the same leg to the platform in frame x–y–z.
The vectors
dk = r + Rck − bk, k = 1, 2, ..., K, (6.6)
describe the geometry of the robot legs expressed in coordinate frame x0–y0–z0.
Here, R = R(ψ, ϑ, ϕ) is the 3 × 3 rotation matrix which transforms the coordinate
frame x–y–z into x0–y0–z0. Equation (6.6) can also be formulated in a homogeneous
form as follows
dk = Hck, k = 1, 2, . . . , K, (6.7)
where the homogeneous transformation matrix is
H =

R r − bk
0 0 0 1

. (6.8)

We assume that the leg lengths are the joint coordinates of the robot
qk = dk, k = 1, 2, . . . , K, (6.9)
where · indicates vector norm. They are elements of the vector of joint coordinates
q = (q1, q2, ..., qK )T
.
The robot kinematic parameters are vectors bk, k = 1, 2, . . . , K expressed in frame
x0–y0–z0 and vectors ck expressed in frame x–y–z.
Once we have defined the internal coordinates, let’s look at what the robot’s
external coordinates are. In parallel robots they usually represent some characteristics
in the motion of the platform to which the end-effector is attached. In most cases, the
chosen external coordinates are the position and orientation of the platform, the so-
called Cartesian coordinates. In space where λ = 6 they include the three components
rx,ry,rz of the position vector in Fig.6.8, and the three orientation angles ψ, ϑ, ϕ,
so that the vector of external coordinates is defined as follows
p = (rx,ry,rz, ψ, ϑ, ϕ)T
.
Inverse and direct kinematics of parallel robots
From the control point of view, the relation between the external and internal coor-
dinates is of utmost importance. Their relationship is, similarly to serial robots,
determined by very complicated algebraic trigonometric equations.
The inverse kinematics problem of parallel robots requires determining the inter-
nal coordinates q, which are the leg lengths, from a given set of external coordinates
p, which represent the position and orientation of the platform. For a given set of
external coordinates p the internal coordinates can be obtained by simply solving
Eq.(6.7). Here, unlike in serial robots, it is important to recognize that the values of
the external coordinates uniquely define the leg lengths of the parallel robot and the
computation is straightforward.
The direct kinematics problem of parallel robots requires determining the external
coordinates p from a given set of joint coordinates q (Fig.6.9). This problem is
extremely complicated in mathematical terms and the computation procedures are
cumbersome. In general, it is not possible to express the external coordinates as
explicit functions of the internal coordinates, whereas with serial robots this is quite
straightforward. Usually, these are coupled trigonometric and quadratic equations
which can be solved in closed-form only in special cases. There exist no rules as how
to approach symbolic solutions. The following difficulties are common:
Nonexistence of a real solution. For some values of internal coordinates real solu-
tions for the external coordinates do not exist. Intervals of internal coordinates
when this can happen cannot be foreseen in advance.

1
1
2
2 3
3
4
4
base
legs
platform
Fig. 6.9 The direct kinematics problem consists of finding the pose of the platform corresponding
to the length of the legs. Leg end-points need to match corresponding points on the platform
(e.g., 1 − 1)
Multiple solutions. For a given set of internal coordinates, there exist multiple solu-
tions for the external coordinates. The number of solutions for a given combination
of leg lengths depends on the kinematic structure of the mechanism. The general
Stewart-Gough platform has forty possible solutions of the direct kinematics prob-
lem. For a selected combination of leg lengths there exist forty different poses of
the platform. In addition, sometimes two poses of the platform cannot be transi-
tioned between as the legs get entangled. In such cases, the platform could transit
from one pose into another only by dismantling the legs in the first pose and
reassembling them in the new pose.
Nonexistence of closed-form solutions. Generally for a given set of joint coordinates,
it is not possible to find an exact solution to the direct kinematics problem, even
if a real solution exists. In such cases we use numerical techniques which may
not necessarily converge and may not find all the solutions.
Design of parallel robots
The design of parallel robots depends on desired performance, flexibility, mobility,
and load capacity as well as the actual workspace.
In considering the workspaces for both parallel and serial robots, we are referring
to the reachable workspace and the dexterous workspace. One of the main drawbacks
of parallel robots is their small workspace. The main goal in workspace analysis is,
therefore, to determine if a desired trajectory lies inside the robot workspace. The size
of the workspace in parallel robots is limited by the ranges in the displacements of

the legs, displacements of passive joints, and, particularly, by interference between
the legs of the robot. Even with small movements, the legs can collide with each
other. The interlacing of legs is in practice a major obstacle in a robot’s motion and
its reachability. The determination and analysis of robot workspace is in general a
tedious process. In parallel robots it is usually even more complex, depending on the
number of degrees of freedom and the mechanism’s architecture.
The effect of load in serial robots is usually seen in terms of dynamics, which to a
large extent includes the inertia of the links. The contribution of an external force is
typically smaller and in many cases can be neglected. In parallel robots with a large
number of legs, where the links are very light and the motors typically attached to the
fixed base, the robot statics plays an important role. The computation of robot statics
is related to the well-known Jacobian matrix which represents the transformation
between the external and the internal coordinates. This goes beyond the scope of our
book, but considerable literature, articles, and textbooks are available to the interested
reader.
In practice, we can often see a Stewart-Gough platform that has a structure as
presented in Fig.6.10. The robot contains (instead of six legs of type UTS) six legs
of type STS. Kinematically, this architecture is quite unusual and redundant. The
robot has too many degrees of freedom. Each leg possesses 7 degrees of freedom
T
S
S
Fig. 6.10 A modification of the Stewart-Gough platform

which corresponds to −1 constraints. According to Grübler’s formula (6.5), the
number of degrees of freedom of the robot is
F = 6 − (−6) = 12.
It is important to note that six of the twelve degrees of freedom are manifested as
rotations of the legs around their own axes. These rotations have no influence on the
movement of the platform. Thus, the robot can still be motorized by only six motors
that change the length of the legs, affecting the translation T, while the rotations
around the leg axes can be left passive and can freely change. The advantages of
this construction are that S joints are easier to build than U joints (and therefore
cheaper), and that the passive rotations around the leg axes enable more flexibility
when connecting power and signal cables, as these are often arranged along the legs
from the base to the robot platform.

Chapter 7
Robot Sensors
The human sensory system encompasses sensors of vision and hearing, kinesthetic
sensors (movement, force, and touch), sensors of taste and smell. These sensors
deliver input signals to the brain which uses this sensory information to build its own
image of the environment and takes decisions for further actions. Similar require-
ments are valid also for robot mechanisms. However, because of the complexity of
human sensing, robot sensing is limited to fewer sensors.
The use of sensors is of crucial importance for efficient and accurate robot opera-
tion.Robotsensorscanbegenerallydividedinto:(1)proprioceptivesensorsassessing
the internal states of the robot mechanism (positions, velocities, and torques in the
robot joints); and (2) exteroceptive sensors delivering to the controller the informa-
tion about the robot environment (force, tactile, proximity and distance sensors, robot
vision).
7.1 Principles of Sensing
In general, sensors convert the measured physical variable into an electrical signal
which can be in a digital form assessed by the computer. In robotics we are predom-
inantly interested in the following variables: position, velocity, force, and torque. By
the use of special transducers these variables can be converted into electrical signals,
such as voltage, current, resistance, capacity, or inductivity. Based on the principle
of conversion the sensors can be divided as follows:
• Electrical sensors—the physical variable is directly transformed into an electrical
signal; such sensors are for example potentiometers or strain gauges;
• Electromagnetic sensors—use the magnetic field for the purposes of physical vari-
able conversion; an example is the tachometer;
85

86 7 Robot Sensors
• Optical sensors—use light when converting the signals; an example of such a
sensor is the optical encoder.
7.2 Sensors of Movement
Typical sensors of robot movements are potentiometers, optical encoders, and
tachometers. They all measure the robot movements inside the robot joint. Where in
the joint to place the movement sensor is important, as well as how to measure the
motion parameters.
7.2.1 Placing of Sensors
Let us first consider a sensor of angular displacement. It is our aim to measure
the angle in a robot joint which is actuated by a motor through a reducer with the
reduction ratio kr . Using a reducer we decrease the joint angular velocity by the
factor kr with respect to the angular velocity of the motor. At the same time the joint
torque is increased by the same factor. It is important whether the movement sensor
is placed before or after the reducer. The choice depends on the task requirements
and the sensor used. In an ideal case we mount the sensor before the reducer (on the
side of the motor), as shown in Fig.7.1. In this way we measure directly the rotations
of the motor. The sensor output must be then divided by the reduction ratio, to obtain
the joint angle.
Let us denote by ϑ the angular position of the joint, ϑm as the angular position of
the corresponding motor, and kr the reduction ratio of the reducer. When the sensor
motor
sensor
reducer
joint segment
Fig. 7.1 Mounting of the sensor of movement before the reducer

7.2 Sensors of Movement 87
motor
sensor
reducer
joint segment
Fig. 7.2 Mounting of the sensor of movement behind the reducer
is placed before the reducer, its output is equal to the angle ϑm. The variable which
we need for control purposes is the joint angle ϑ, which is determined by the ratio
ϑ =
ϑm
kr
. (7.1)
By differentiating the Eq.(7.1) with respect to ϑm we have
dϑ
dϑm
=
1
kr
thus dϑ =
1
kr
dϑm, (7.2)
which means that the sensor measurement error is reduced by the factor kr . The
advantageof theplacement of thesensor beforethereducer is ingettingmoreaccurate
information about the joint angular position.
Another sensor mounting possibility is shown in Fig.7.2. Here, the sensor is
mounted behind the reducer. In this way the movements of the joint are measured
directly. The quality of the control signal is decreased, as the sensor measurement
error (which is now not reduced) directly enters the joint control loop. As the range
of motion of the joint is by the factor kr smaller than that of the motor, sensors
with smaller range of motion can be used. Sometimes we cannot avoid mounting the
motion sensor into the joint axis. It is important, therefore, that we are aware of the
deficiency of such a placement.
7.2.2 Potentiometer
Figure7.3 presents a model of a rotary potentiometer and its components. The
potentiometer consists of two parts: (1) resistive winding and (2) movable wiper.

88 7 Robot Sensors
wiper
axis
Uout
Uin
digital
signal
A
B
C
A/D
+
-
Fig. 7.3 The model of a potentiometer
The potentiometer represents a contact measuring method, because the wiper slides
along the circular resistive winding.
Potentiometers are generally placed behind the reducer in such a way that the
potentiometer axis is coupled to the joint axis. Let us suppose that point B represents
the reference position of the potentiometer belonging to the joint. The resistance of
the potentiometer along the winding
AB equals R, while r represents the resistance
of the
C B part of the winding. The angle of the wiper with respect to the reference
position B is denoted by ϑ (in radians). When the resistance along the circular
winding of the potentiometer is uniform and the distance between the points A and
B is negligible, we have the following equation
r
R
=

C B

AB
=
ϑ
2π
. (7.3)
Let us suppose that the potentiometer is supplied by the voltage Uin. The output
voltage measured on the wiper is equal to
Uout
Uin
=
r
R
=
ϑ
2π
, (7.4)
or
Uout =
Uin
2π
ϑ. (7.5)
By measuring the output voltage Uout , the angular position ϑ is determined.
7.2.3 Optical Encoder
The contact measurement approach to the robot joint angle using potentiometers
has several deficiencies. The most important is the relatively short lifespan because
of high wear and tear. In addition, the most adequate placement is directly in the
joint axis (behind the reducer) and not on the motor axis (before the reducer). The

light transmitter
LED
5V
5V
R1
R2
input logic
lens
rotating disc
phototransistor
output logic
signal
signal
light receiver
NOT
+
+
−
−
Fig. 7.4 The model of optical encoder
most widely used movement sensors in robotics are therefore optical encoders which
provide contact-less measurement.
The optical encoder is based on the transformation of the joint movement into a
series of light pulses, which are further converted into electrical pulses. To generate
thelightpulses,alightsourceisneeded,usuallyalightemittingdiode.Theconversion
of light into electrical pulses is performed by a phototransistor or a photodiode
converting light into electrical current.
The model of an optical encoder assessing the joint angular position is presented
in Fig.7.4. It consists of a light source with lens, light detector, and a rotating disc
with slots, which is connected to either motor or joint axis. On the rotating disc there
is a track of slots and interspaces, which alternately either transfer or block the light
from the light emitting diode to the phototransistor. The logical output of the sensor
is high when the light goes through the slot and hits the phototransistor on the other
side of the rotating plate. When the path between the light emitting diode and the
phototransistor is blocked by the interspace between two slots, the logical output is
low.
The optical encoders are divided into absolute and incremental. In the further text
we shall learn about their most important properties.
7.2.3.1 Absolute Encoder
The absolute optical encoder is a device which measures the absolute angular position
of a joint. Its output is a digital signal. In a digital system each logical signal line
represents one bit of information. When connecting all these bits into a single logical
state variable, the number of all possible logical states determines the number of all
absolute angular positions that can be measured by the encoder.

90 7 Robot Sensors
Let us suppose that we wish to measure the angular rotation of 360◦
with the
resolution of 0.1◦
. The absolute encoder must distinguish between 3600 different
logical states, which means that we need at least 12 bits to assess the joint angles
with the required resolution. With 12 bits we can represent 4096 logical states. An
important design parameter of absolute encoders is therefore the number of logical
states, which depends on the task requirements and the placement of the encoder
(before or after the reducer). When the encoder is placed before a reducer with
the reduction ratio kr , the resolution of the angle measurement will be increased
by the factor kr . When the encoder is behind the reducer, the necessary resolution
of the encoder is directly determined by the required resolution of the joint angle
measurement. All logical states must be uniformly engraved into the rotating disc of
the encoder. An example of absolute encoder with sixteen logical states is shown in
Fig.7.5. The sixteen logical states can be represented by four bits. All sixteen logical
states are engraved into the surface of the rotating disc. The disc is in the radial
direction divided into four tracks representing the four bits. Each track is divided
into sixteen segments corresponding to the logical states. As the information about
the angular displacement is represented by four bits, we need four pairs of light
emitting diodes and phototransistors (one pair for each bit). With the rotation of the
disc, which is connected to either motor or joint axis, the output signal will change
according to the logical states defined by the order of segments (Grey code, where
two successive values differ in only one bit, is typically used in absolute encoders).
0000 0001
1001
1101
0101
0111
1111
1011
0011
0010
1010
1110
0110
0100
1100
1000
motor
sensor
reducer
joint
disc
trace
axis
segment
light receiver
light transmitter
slots
segment
Fig. 7.5 Model of absolute encoder

The absolute encoder does not determine only the angular position of the joint but
also the direction of rotation.
7.2.3.2 Incremental Encoder
In contrast to absolute encoders, the incremental encoders only supply information
about the changes in angular joint position. The advantages of incremental encoders,
compared to the absolute encoders, are their simplicity, smaller dimensions, and
(most importantly) low cost. This can be achieved by lowering the number of the
tracks on the rotating disc to only a single track. Instead of having as many tracks as
the number of the bits necessary for the representation of all required logical states,
we have now only one track with even graduation of the slots along the rim of the disc.
Figure7.6 shows a model of an incremental encoder. A single track only requires a
single pair of light emitting diode and phototransistor (optical pair). During rotation
of the encoded disc a series of electrical pulses is generated. The measurement of the
joint displacement is based on counting of these pulses. Their number is proportional
to the robot joint displacement. The incremental encoder shown in Fig.7.6 generates
eight pulses during each rotation. The resolution of this encoder is
Δϑ =
2π
8
=
π
4
. (7.6)
By increasing the number of the slots on the disc, the resolution of the encoder
is increased. By denoting the number of the slots as nc, a general equation for the
encoder resolution can be written
Δϑ =
2π
nc
. (7.7)
The encoder with one single track is only capable of assessing the change in the
joint angular position. It cannot provide information about the direction of rotation
or the absolute joint position. If we wish to apply the incremental encoders in robot
control, we must determine: (1) the home position representing the reference for the
measurement of the change in the joint position and (2) the direction of rotation.
The problem of the home position is solved by adding an additional reference slot
on the disc. This reference slot is displaced radially with respect to the slotted track
measuring the angular position. For detection of the home position, an additional
optical pair is needed. When searching for the reference slot, the robot is programmed
to move with low velocity, as long as the reference slot or the end position of the
joint range of motion is reached. In the latter case the robot moves in the opposite
direction towards the reference slot.
The problem of determining the direction of rotation is solved by another pair of
light emitting diode and phototransistor. This additional optical pair is tangentially
and radially displaced from the first optical pair as shown in Fig.7.6. When the disc

92 7 Robot Sensors
motor
sensor
reducer
joint
disk
light receiver
light transmitter
slots
sig A
sig A
sig B
sig B
home
segment
A
B
A
A
B
B
Fig. 7.6 Model of incremental encoder. The series of pulses for positive (above) and negative
(below) direction of rotation
is rotating, two signals are obtained, which are, because of the displacement of the
optical pairs, shifted in phase. This shift in phase occurs because each slot on the
disc first reaches the first optical pair and after a short delay also the second pair.
The optical components are usually placed in such a way that the phase shift of π/2
is obtained between the two signals. During the rotation in clockwise direction the
signal B is phase-lagged for π/2 behind the signal A. During counter clockwise
rotation the signal B is in phase-lead of π/2 with respect to the signal A (Fig.7.6).
The direction of the encoder rotation can be determined upon the basis of the phase
shifts between signals A and B. Another advantage of having two optical pairs is the
possibility of counting all the changes in both the A and B signals. The approach
known as quadrature decoding enables measurement resolution of four-times the
nominal encoder resolution.
7.2.4 Magnetic Encoder
Incontrasttoopticalencodersthemagneticencoderusesmagneticfieldformeasuring
position. This can be achieved by using a series of magnetic poles (2 or more) on the
sensor rotor to represent the encoder position to a magnetic sensor. The rotor turns

90◦
magnet
Hall sensor
Hall sensor
air gap
N
N
N
S
S
S
S
(a) (b)
Fig. 7.7 Model of magnetic encoder: a Hall sensor and alternating north and south poles and b
multiple Hall sensors with diametrically magnetized rotating magnet
with the shaft and contains alternating evenly spaced north and south poles around its
circumference. The magnetic sensor (typically magneto-resistive or Hall effect) reads
the magnetic pole positions. Hall sensors generate output voltage proportional to the
strength of an applied magnetic field. Magneto-resistive sensors detect changes in
resistance caused by a magnetic field. The principle of operation is shown in Fig.7.7a.
Hall sensors can be used for angle measurement also when placed near a dia-
metrically magnetized magnet that generates a sinusoidal waveform. The limitation
of this method is the ambiguity at angles 90◦
in both directions from the zero
crossing point. In order to extend the measurement range to 360◦
, refinement of the
method is required. The problem can be solved by using multiple Hall sensors, rather
than one, and placing them underneath a diametrically magnetized rotating magnet
to generate multiple sinusoidal waveforms. Figure7.7b shows four equally spaced
Hall sensors generating four sinusoidal signals, each phase-shifted by 90◦
from its
neighbor. Magnetic encoders are typically more robust than optical encoders.
7.2.5 Tachometer
The signal of the joint velocity can be obtained by numerical differentiation of the
position signal. Nevertheless, direct measurement of the joint velocity with the help
of a tachometer is often used in robotics. The reason is the noise introduced by
numerical differentiation, which can greatly affect the quality of the robot control.

94 7 Robot Sensors
Tachometers can be divided into: (1) direct current (DC) and (2) alternate current
(AC) tachometers. Inrobotics it is generallythesimpler DCtachometers that areused.
The working principle is based on a DC generator whose magnetic field is provided
by permanent magnets. As the magnetic field is constant, the tachometer output
voltage is proportional to the angular velocity of the rotor. Because commutators
are used in the DC tachometers, a slight ripple appears in the output voltage, which
cannot be entirely filtered out. This deficiency, together with other imperfections, is
avoided by using AC tachometers.
7.2.6 Inertial Measurement Unit
Potentiometers and optical encoders measure joint displacements in robot mecha-
nisms. When considering, for example, a robotic aerial vehicle, or a wheeled robot,
these sensors do not provide information about the orientation of the device in space.
Measuring of object (robot) orientation in space is typically based on the magneto-
inertial principle. This method combines a gyroscope (angular velocity sensor),
accelerometer (linear acceleration sensor), and magnetometer (measures orientation
relative to the Earth’s magnetic field and is not regarded as an inertial sensor).
The method will be illustrated with the example of a rigid pendulum equipped
with a two-axis accelerometer (measures accelerations along two perpendicular axes)
and a single-axis gyroscope (Fig.7.8). Both sensors give the measured quantities in
their own coordinate frames, which are attached to the center of the sensor and
have their axes parallel to the x and y axes of the coordinate frame attached to
the pendulum. Figure7.8a shows a stationary pendulum while Fig.7.8b shows a
swinging pendulum. We are interested in the orientation of the pendulum relative
to the reference coordinate frame x0–y0–z0. Since the pendulum is only swinging
around z axis, we are only actually interested in angle ϕ.
We first analyze stationary conditions. Since the angular velocity of a stationary
pendulum is equal to zero, the gyroscope’s output is also zero and the gyroscope
tells us nothing about the pendulum’s orientation. However, we can see that the
accelerometer still measures the gravitational acceleration. Since the accelerometer
is at an angle of ϕ relative to the gravitational field, two acceleration components
are measured: ax and ay. The vector sum of both components gives the gravitational
acceleration. Figure7.8a shows that the angle between vectors g and ay is equal to ϕ.
Since the scalar values of accelerations ax and ay are known, we can now determine
the pendulum angle
ϕ = arctan
ax
ay
. (7.8)
The accelerometer thus allows the pendulum’s angle to be measured in stationary
conditions. For this reason, accelerometers are frequently used as inclinometers.

axis
axis x0
x0
y0
y0
x
x
y
y
r
r
ω
ϕ
ϕ
ϕ
ϕ
ω
ω
ωz
gyroscope
gyroscope
accelerometer
accelerometer
g
g
ax
ax
ay
ay
at
ar
at+r
a
(a) (b)
Fig. 7.8 Example of using an inertial measurement system to measure the angle of a pendulum:
left figure—stationary pendulum, right figure—swinging pendulum
The conditions in a swinging pendulum are quite different. Since swinging is
an accelerated rotational movement, the accelerometer is affected not only by the
gravitational acceleration g, but also by centripetal acceleration
ar = ω × (ω × r) (7.9)
and tangential acceleration
at = ω̇ × r. (7.10)
The total acceleration acting on the accelerometer is thus
a = g + ar + at . (7.11)
The equation used to calculate angle in stationary conditions (7.8) is no longer valid,
therefore, the accelerometer cannot be used to calculate the angle of a swinging pen-
dulum. However, the output of the gyroscope, which measures the angular velocity
of the pendulum, is now also available. Since the angle of the pendulum can be
calculated as the temporal integral of angular velocity, the following relation can be
stated
ϕ = ϕ0 +

ωdt, (7.12)
where the initial orientation of the pendulum ϕ0 must be known.

96 7 Robot Sensors
The given example makes it clear that an accelerometer is suitable for orienta-
tion measurements in static or quasi-static conditions while a gyroscope is suitable
for orientation measurements in dynamic conditions. However, two weaknesses of
accelerometers and gyroscopes must be mentioned. An accelerometer cannot be used
to measure angles in a horizontal plane, as the output of the sensor is zero when its
axis is perpendicular to the direction of gravity.
For this purpose, we can use a magnetometer, which also allows measurement
of rotation around the gravity field vector (think of how a compass works). Further-
more, neither the gyroscope’s nor the accelerometer’s output is ideal. In addition to
the measured quantity, the output includes an offset and noise. Integrating the offset
causes a linear drift, so Eq.(7.12) does not give an accurate pendulum orientation
measurement. Due to the weaknesses of the individual sensors, it is common to com-
bine three perpendicular accelerometers, three perpendicular gyroscopes, and three
perpendicular magnetometers into a single system, referred to as a magneto-inertial
measurement unit (IMU). Combination of the best properties of an accelerometer,
gyroscope, and magnetometer can give an accurate and reliable measurement of
spatial orientation.
The angular velocity measured by the gyroscope is integrated, giving an estimate
of orientation. Measurements from the accelerometer and magnetometer are used to
directly calculate the sensor orientation with reference to the gravity and magnetic
field vectors. This is achieved through sensor fusion, which can be done by using the
Kalman filter.
7.3 Contact Sensors
The sensors considered so far provide information about robot pose and motion. They
enable closing of the position and velocity control loop. In some robot tasks contact
of the end-effector with the environment is required. Typical contact sensors used
in robotics are tactile sensors and force and torque sensors. Tactile sensors measure
parameters that define the contact between the sensor and an object.
Sensing consists in measurement of a point contact force and the spatial distribu-
tion of forces perpendicular to an area. By contrast, force and torque sensors measure
the total forces being applied to an object.
7.3.1 Tactile Sensor
Robots can collect information about the environment also through touch. In order
to increase robot manipulation capabilities, tactile sensors can be used in robotic
fingersasshowninFig.7.9a.Thesensorprovidesdataaboutcontactforcedistribution
between the finger and the manipulated object. To increase robot safety (e.g., when

7.3 Contact Sensors 97
tactile sensor
tactile sensor
(a) (b)
Fig. 7.9 Tactile sensor used in robot finger (left) and as robot skin (right)
sensing elements
base
outer layer
Fig. 7.10 A model of a tactile sensor
working with a human), tactile sensors can be used as an artificial robot skin that
enables the robot to sense contacts with objects in the environment (Fig.7.9b).
Tactile sensing is based on an array of touch sensors as shown in Fig.7.10. The
following sensing principles can be implemented in the array:
• deformation-based sensors—material surface deforms (changes length), when it
is subjected to external forces; deformation is converted to electrical signals with
strain gauges connected in a Wheatstone bridge;
• resistive sensors—electrical resistance changes with pressure of a material placed
between two electrodes;
• capacitive sensors—sensing element is a capacitor whose capacitance changes
with the applied force; force can produce either a change in the distance between
capacitor plates or its area;
• optical sensors—sensing is typically based on light intensity measurement; inten-
sity of light can be modulated by moving an obstruction or a reflective surface into
the light path; the intensity of the received light is a function of displacement and
hence of the applied force;
• piezoelectric sensors—materials, like quartz, have piezoelectric properties and
can thus be used for tactile sensing; piezoelectric transducers are not adequate for
static force transduction; this problem can be overcome by vibrating the sensor
and detecting the difference in the vibration frequency due to the applied force;

98 7 Robot Sensors
• magneticsensors—changesofmagneticfluxdensityormagneticcouplingbetween
circuits are the most widely used principles in magnetic tactile sensing;
• mechanical sensors—sensing elements are mechanical micro-switches with on
and off states.
7.3.2 Limit Switch and Bumper
Limit switches are often used to control robot mechanisms. They can sense a single
position of a moving part and are therefore suitable for ensuring that movement
doesn’t exceed a predefined limit. A bumper sensor, a special type of limit switch,
for instance, will tell the robot whether it is in contact with a physical object or not.
If the sensor is mounted on the front bumper of a mobile robot, the robot could use
this information to tell whether it has run into an obstacle, like a wall (Fig.7.11).
Robotic vacuum cleaners typically rely on bumper sensors for navigating inside the
home environment.
7.3.3 Force and Torque Sensor
In the simplest case the force measurement enables disconnection of the robot when
the contact force exceeds a predetermined safety limit. In a more sophisticated case
we use force sensors for control of the force between the robot end-effector and the
environment. It is therefore not difficult to realize that the force sensor is placed into
the robot wrist and is therefore often called the wrist sensor.
Strain gauges are usually used for the force measurements. The strain gauge is
attached to an elastic beam which is deformed under the stress caused by the applied
force. The strain gauge therefore behaves as a variable resistor whose resistance
changes proportionally to its deformation. The wrist sensor must not influence the
bumper
Fig. 7.11 Bumper sensors to be used on a mobile robot

a)
b)
c)
c)
c)
c)
d)
d)
d)
d)
x
y
z
fx
fy
fz
μx
μy
μz
w1
w2
w3
w4
w5 w6
w7
w8
Fig. 7.12 Model of the force and torque sensor: a rigid body which is in contact with the robot
end-effector, b rigid ring which is in contact with the robot environment, c elastic beams, and d
strain gauge
interactionoftherobotwiththeenvironment.Thismeansthatthewristsensormustbe
sufficiently rigid. The robot wrist sensors are usually designed as shown in Fig.7.12.
The structure of the sensor is based on three components: (a) rigid inner part which
is in contact with the robot end-effector; (b) rigid outer ring which is in contact with
the robot environment; and (c) elastic beams interconnecting the outer and the inner
ring. During contact of the robot with the environment, the beams are deformed by
the external forces which causes a change in the resistance of the strain gauges.
The vector of the forces and torques acting at the robot end-effector is in the
three-dimensional space represented by six elements, three forces and three torques.
The rectangular cross-section of a beam (shown in Fig.7.12) enables the mea-
surement of deformations in two directions. To measure the six elements of the force
and torque vector, at least three beams, which are not collinear, are necessary. Four
beams are used in the example in Fig.7.12. There are two strain gauges attached
to the perpendicular surfaces of each beam. Having eight strain gauges, there are
eight variable resistances, R1 to R8. As the consequence of the external forces and
torques, elastic deformations w1 to w8 occur resulting in changes in the resistances
ΔR1 to ΔR8. The small changes in the resistance are, by the use of the Wheat-
stone bridge, converted into voltage signals (Fig.7.13). To each of the eight variable

100 7 Robot Sensors
+Uin
Uouti
U1i U2i
Ri,1
Ri,1
Ri,2
Ri,3
Ri
Fig. 7.13 The Wheatstone bridge
resistors {R1 . . . R8}, three additional resistors are added. The three resistors are,
together with the strain gauge, connected into the measuring bridge. The bridge is
supplied with the Uin voltage, while the output voltage Uouti
is determined by the
difference U1i
− U2i
. The U1i
voltage is
U1i
=
Ri,2
Ri,1 + Ri,2
Uin, (7.13)
while the U2i
voltage is
U2i
=
Ri
Ri + Ri,3
Uin. (7.14)
The output voltage is equal to
Uouti
=

Ri,2
Ri,1 + Ri,2
−
Ri
Ri + Ri,3

Uin. (7.15)
By differentiating the Eq.(7.15) with respect to the variable Ri , the influence of the
change of the strain gauge resistance on the output voltage can be determined
ΔUouti
= −
Ri,3Uin
(Ri + Ri,3)2
ΔRi . (7.16)
Before application, the force sensor must be calibrated, which requires the deter-
mination of a 6 × 8 calibration matrix transforming the six output voltages into the
three forces

fx fy fz
T
and three torques

μx μy μz
T

fx fy fz μx μy μz
T
= K

Uout1
Uout2
Uout3
Uout4
Uout5
Uout6
Uout7
Uout8
T
,
(7.17)

where
K =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 0 K13 0 0 0 K17 0
K21 0 0 0 K25 0 0 0
0 K32 0 K34 0 K36 0 K38
0 0 0 K44 0 0 0 K48
0 K52 0 0 0 K56 0 0
K61 0 K63 0 K65 0 K67 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(7.18)
is the calibration matrix with constant values Ki j .
7.3.4 Joint Torque Sensor
Often it is required or preferable to measure joint torques instead of robot end-effector
forces. In such cases a joint torque sensor must be used. By measuring joint torques
the robot can respond to forces applied anywhere on its mechanism. If the robot
dynamic model is known, it is also possible to estimate end-effector forces.
As an example, consider Eq.(5.20). The inverse of this equation would give
f = J−T
(q)τ. (7.19)
It should be noted that the above equation would give exact end-effector forces only in
static conditions and if gravity force does not affect joint torques. In other conditions,
robot dynamic model (5.56) must be taken into account.
The principle of operation of torque sensor is typically similar to that of the
wrist sensor. However, its mechanical structure is designed to fit onto the joint axis.
x
y
y
z
fx
fy
fz
μx
μz
)
a
)
a b)
c)
c)
c)
c)
c)
c)
Fig. 7.14 Joint torque sensor structure measures torque μz: a frame, b elastic element, and c strain
gauge

102 7 Robot Sensors
Thus, the sensor is integrated between the actuator (and possibly gear) and the robot
segment. The sensor needs to guarantee high sensitivity to torsion, low sensitivity
to non-torsional components, and high stiffness in all axes of forces and moment.
Deformation of mechanical structure due to joint torque is measured by using strain
gauges. A schematic representation of joint torque sensor is shown in Fig.7.14.
7.4 Proximity and Ranging Sensors
Proximity and ranging sensors detect the presence of nearby objects without any
physical contact. Consecutively they enable distinguishing between obstacles of dif-
ferent shapes and sizes as well as more efficient obstacle avoidance than contact
sensors. Different methods can be used to detect obstacles from a distance. Methods
based on magnetic and capacitive principles typically enable detecting proximity
of an object but not its distance. When distance is relevant, active methods such as
ultrasonic rangefinder, laser rangefinder, and infrared proximity sensor as well as
passive methods based on cameras can be considered. All methods are characterized
by high reliability and long operational life as they operate without physical contact
between the sensor and the sensed object.
7.4.1 Ultrasonic Rangefinder
An ultrasonic rangefinder measures the distance to an object by using sound waves.
Distance is measured by sending out a sound wave at an ultrasonic frequency (higher
frequencies are better for short range and high precision needs) and listening for that
sound wave to bounce back (Fig.7.15a). The elapsed time between the sound wave
being generated and the sound wave bouncing back, is used to calculate the distance
between the sensor and the object (considering that the speed of sound in the air is
approximately 343m/s).
Understanding of the detection zone is important for successful object detection
and avoidance. The beam width of ultrasonic rangefinder is typically described as
a cone of a certain angle. This angle describes the arc at which the ultrasonic wave
emanates from the transducer. However, at a certain distance the rate of expansion
starts to decay as shown in Fig.7.15b. An extension of the measurement area of
an ultrasonic rangefinder can be achieved by using multiple sensor units facing at
different angles (Fig.7.15c). The problem of cross-talk needs to be considered in
such case.
Different other factors affect performance of an ultrasonic rangefinder. The size,
composition, shape, and orientation of objects must be considered. In the cases
presented in upper images in Fig.7.16, measurements are normally correct, while in
the scenarios presented in lower images in Fig.7.16 the ultrasonic rangefinder would
give false results.

7.4 Proximity and Ranging Sensors 103
ultrasonic rangefinder
object
detection zone
(a)
(b)
(c)
Fig. 7.15 Ultrasonic rangefinder: a principle of operation, b detection zone, and c combination of
multiple sensors
d
d
Fig. 7.16 Ultrasonic rangefinder distance measurement and limitations: correct measurements of
distance d (upper row) and false results (lower row)
7.4.2 Laser Rangefinder and Laser Scanner
A laser rangefinder uses a laser beam to determine the distance to an object. The most
common form of laser rangefinder operates on the time of flight principle. Distance
can be determined by measuring the time it takes for the laser pulse to return to the

104 7 Robot Sensors
motor
encoder
distance
object
ϑ
ϑ̇
laser pulse
reflected signal
rotating base
Fig. 7.17 Laser scanner
sensor and it requires precise time measuring. With the speed of light known and
an accurate measurement of the time taken, the distance can be calculated. Another
possibility is to compute light-wave phase shift by analyzing the incoming light and
comparing it to a reference signal. The most accurate method to measure changes in
distance rather than absolute distances is interferometry.
The laser rangefinder measures distance to one object at a time. Therefore, it
is a one dimensional sensor. The laser scanner uses a laser that sweeps across the
sensor’s field of view. As the name implies, the instrument principally consists of a
laser and a scanner. Distances are measured as with the laser rangefinder. The laser
scanner produces an array of points by sampling the environment at a high rate. This
is typically achieved by using rotating assemblies or rotating mirrors to sweep 360
degrees around the environment. The principle of laser scanner operation is shown
in Fig.7.17.
Sampled points represent object positions relative to the sensor. Generation of
array of points is presented in Fig.7.18. Distance dL is measured by using the laser
and rotation angle ϑL is typically measured by using an encoder on the rotating
assembly. Points are therefore defined in polar coordinates. They can be transformed
into Cartesian coordinates (xL , yL ) relative to the sensor with
xL = dL cos ϕL and yL = dL sin ϕL. (7.20)
Sampled points can be used to generate a map of the environment, for path planning,
and avoidance of obstacles. A three-dimensional (3-D) scanner enables scanning of

7.4 Proximity and Ranging Sensors 105
xL
yL
dL
ϑL
(a)
(b) (c)
Fig. 7.18 Laser scanner used to create a map of environment: a environment, b scanning, and c
map
a complete 3-D space and collecting a 3-D point cloud data by using another degree
of freedom at the rotating assembly. These 3-D laser-scanners are typically termed
LiDAR (Light Detection And Ranging) and are often used in autonomous vehicles
for scanning the environment.

Chapter 8
Robot Vision
The task of robot vision is to recognize the geometry of the robot workspace from a
digital image. It is our aim to find the relation between the coordinates of a point in
the two-dimensional (2D) image and the coordinates of the point in the real three-
dimensional (3D) robot environment.
8.1 System Configuration
The robot vision system is based on the use of one, two or more cameras. If several
cameras are used to observe the same object, information about the depth of the
object can be derived. In such case, we talk about 3D or stereo vision. Of course, the
3D view can also be achieved with a single camera if two images of the object are
available, captured from different poses. If only one image is available, the depth can
be estimated based on some previously known geometric properties of the object.
When analyzing the configuration of the robotic vision system, it is necessary
to distinguish between possible placements of the cameras. The cameras can be
placed in a fixed configuration, where they are rigidly mounted in the workcell,
or in a mobile configuration, where the camera is attached to a robot. In the first
configuration, the camera observes objects from a fixed position with respect to the
robot base coordinate frame. The field of view of the camera does not change during
the execution of the task, which means that basically the accuracy of the measurement
is constant. In some tasks, it is difficult to prevent the manipulator from reaching into
the field of view of the camera and thereby occluding the objects. Therefore, in such
case, it is necessary to put a camera on a robot (in a mobile configuration).
The camera can be attached before or after the robot wrist. In the first case, the
camera observes the situation from a favorable position and the manipulator gen-
erally does not occlude its field of view. In the second case, the camera is attached
to the robot end-effector and typically only observes the object that is being manip-
107

108 8 Robot Vision
p = (u,v) fc
xc
yc
zc
P = (xc,yc,zc)
Fig. 8.1 Perspective projection
ulated. In both cases, the field of view of the camera changes with movements of
the manipulator. When the manipulator approaches the object, the accuracy of the
measurement typically increases.
8.2 Forward Projection
The basic equations of optics determine the position of a point in the image plane
with respect to the corresponding point in 3D space (Fig.8.1). We will therefore
find the geometrical relation between the coordinates of the point P = (xc, yc, zc)
in space and the coordinates of the point p = (u, v) in the image.
As the aperture of the camera lenses, through which the light falls onto the image
plane, is small compared to the size of the objects manipulated by the robot, we can
replace the lenses in our mathematical model by a simple pinhole. In perspective
projection points from space are projected onto the image plane by lines intersecting
in a common point called the center of projection. When replacing a real camera
with a pinhole camera, the center of projection is located in the center of the lenses.
When studying robot geometry and kinematics, we attached a coordinate frame to
each rigid body (e.g., to robot segments or to objects manipulated by the robot). When
considering robot vision, the camera itself represents a rigid body and a coordinate
frame should be assigned to it. The pose of the camera will be from now on described
by a corresponding coordinate frame. The zc axis of the camera frame is directed
along the optical axis, while the origin of the frame is positioned at the center of
projection. We shall choose a right-handed frame where the xc axis is parallel to the
rows of the imaging sensor and the yc axis is parallel with its columns.

8.2 Forward Projection 109
Oc
fc
xc
yc
zc
yi
xi
p
oi
q P = (0,yc,zc)
Q = (xc,0,zc)
Q1
P1
Fig. 8.2 Equivalent image plane
The image plane is in the camera, which is placed behind the center of projection.
The distance fc between the image and the center of projection is called the focal
length. In the camera frame the focal length has a negative value, as the image plane
intercepts the negative zc axis. It is more convenient to use the equivalent image
plane placed at a positive zc value (Fig.8.2). The equivalent image plane and the
real image plane are symmetrical with respect to the origin of the camera frame. The
geometrical properties of the objects are equivalent in both planes and differ only in
the sign.
From now on we shall call the equivalent image plane simply the image plane.
Also the image plane can be considered as a rigid body to which a coordinate frame
should be attached. The origin of this frame is placed in the intersection of the optical
axis with the image plane. The xi and yi axes are parallel to the xc and yc axes of the
camera frame.
In this way the camera has two coordinate frames, the camera frame and the image
frame. Let the point P be expressed in the camera frame, while the point p represents
its projection onto the image plane. It is our aim to find the relations between the
coordinates of the point P and the coordinates of its image p.
Let us first assume that the point P is located in the yc–zc plane of the camera
frame. Its coordinates are
P =
⎡
⎣
0
yc
zc
⎤
⎦ . (8.1)
The projected point p is in this case located in the yi axis of the image plane
p =

0
yi

. (8.2)

110 8 Robot Vision
Because of similarity of the triangles P P1 Oc and poOc we can write
yc
yi
=
zc
fc
or
yi = fc
yc
zc
. (8.3)
Let us consider also the point Q laying in the xc–zc plane of the camera frame.
After the perspective projection of the point Q, its image q falls onto the xi axis of
the image frame. Because of similar triangles QQ1 Oc and qoOc we have
xc
xi
=
zc
fc
or
xi = fc
xc
zc
. (8.4)
In this way we obtained the relation between the coordinates (xc, yc, zc), of the
point P in the camera frame and the coordinates (xi , yi ), of the point p in the
image plane. Equations (8.3) and (8.4) represent the mathematical description of the
perspective projection from a 3D onto a 2D space. Both equations can be written in
the form of perspective matrix equation
s
⎡
⎣
xi
yi
1
⎤
⎦ =
⎡
⎣
fc 0 0 0
0 fc 0 0
0 0 1 0
⎤
⎦
⎡
⎢
⎢
⎣
xc
yc
zc
1
⎤
⎥
⎥
⎦ . (8.5)
In Eq.(8.5) s is a scaling factor, while (xi , yi ) are the coordinates of the projected
point in the image frame and (xc, yc, zc) are the coordinates of the original point in
the camera frame.
FromthematrixEq.(8.5)itisnotdifficulttorealizethatwecanuniquelydetermine
the coordinates (xi , yi ) and the scaling factor s when knowing (xc, yc, zc). On the
contrary, we cannot calculate the coordinates (xc, yc, zc) in the camera frame when
only the coordinates (xi , yi ) in the image frame are known, but not the scaling factor.
Equation (8.5) represents the forward projection in robot vision. The calculation of
(xc, yc, zc) from (xi , yi ) is called backward projection. When using a single camera
and without a priori information about the size of the objects in the robot environment,
a unique solution of the inverse problem cannot be found.
For ease of programming it is more convenient to use indices, marking the position
of a pixel (i.e., the smallest element of a digital image) in a 2D image instead of metric
units along the xi and yi axes of the image frame. We shall use two indices which
we shall call index coordinates of a pixel (Fig.8.3). These are the row index and the
column index. In the memory storing the digital image the row index runs from the

8.2 Forward Projection 111
u0
u
v0
v
xi
yi
q
p
Dx
Dy
Fig. 8.3 The image plane and the index coordinate frame
top of the image to the bottom while the column index starts at the left and stops at
the right edge of the image. We shall use the u axis for the column indices and the v
axis for the row indices. In this way the index coordinate frame u–v belongs to each
particular image. The upper left pixel is denoted either by (0, 0), or (1, 1). The index
coordinates have no measuring units.
In the further text we shall find the relation between the image coordinates (xi , yi )
and the index coordinates (u, v). Let us assume that the digital image was obtained as
a direct output from the image sensor (A/D conversion was performed at the output
of the image sensor). In this case each pixel corresponds to a particular element of
the image sensor. We shall assume that the area of the image sensor is rectangular.
The origin of the image frame is in the point (u0, v0) of the index frame. The size
of a pixel is represented by the pair (Dx , Dy). The relation between the image frame
xi –yi and the index frame u–v is described by the following two equations
xi
Dx
= u − u0
yi
Dy
= v − v0.
(8.6)
Equations (8.6) can be rewritten as
u = u0 +
xi
Dx
v = v0 +
yi
Dy
.
(8.7)

112 8 Robot Vision
In Eq.(8.7), xi
Dx
and yi
Dy
represent the number of digital conversions along the row and
column respectively. Equation(8.7) can be rewritten in the following matrix form
⎡
⎣
u
v
1
⎤
⎦ =
⎡
⎣
1
Dx
0 u0
0 1
Dy
v0
0 0 1
⎤
⎦
⎡
⎣
xi
yi
1
⎤
⎦ . (8.8)
Using the pinhole camera model, we can now combine Eq.(8.5), relating the image
coordinates to the camera coordinates, and Eq.(8.8), describing the relation between
the image and index coordinates
s
⎡
⎣
u
v
1
⎤
⎦ =
⎡
⎣
1
Dx
0 u0
0 1
Dy
v0
0 0 1
⎤
⎦
⎡
⎣
fc 0 0 0
0 fc 0 0
0 0 1 0
⎤
⎦
⎡
⎢
⎢
⎣
xc
yc
zc
1
⎤
⎥
⎥
⎦ =
=
⎡
⎢
⎣
fc
Dx
0 u0 0
0 fc
Dy
v0 0
0 0 1 0
⎤
⎥
⎦
⎡
⎢
⎢
⎣
xc
yc
zc
1
⎤
⎥
⎥
⎦ .
(8.9)
The above matrix can be written also in the following form
P =
⎡
⎣
fx 0 u0 0
0 fy v0 0
0 0 1 0
⎤
⎦ . (8.10)
The P matrix represents the perspective projection from the camera frame into the
corresponding index coordinate frame. The variables
fx =
fc
Dx
(8.11)
fy =
fc
Dy
are the focal lengths of the camera along the xc and yc axes. The parameters fx , fy,
u0, and v0 are called the intrinsic parameters of a camera.
In general the intrinsic parameters of the camera are not known. The specifications
of the camera and the lenses are not sufficiently accurate. The intrinsic parameters
of the camera are therefore obtained through the camera calibration process. When
knowing the intrinsic parameters of the camera we can uniquely calculate the index
coordinates (u, v) from the given coordinates (xc, yc, zc). The coordinates (xc, yc, zc)
cannot be determined from the known (u, v) coordinates without knowing the scaling
factor.

8.3 Backward Projection 113
8.3 Backward Projection
The digital image is represented by a matrix of pixels. As the index coordinates (u, v)
do not have measuring units, this means that characteristic features of the image are
described more qualitatively than quantitatively. If we wish to express the distances
in metric units, we must know the relation between the index coordinates (u, v)
and the coordinates (xr , yr , zr ) in the 3D reference frame. Without knowing the real
dimensions or the geometry of the scene it is impossible to recognize the features of
the image.
8.3.1 Single Camera
Let us assume that we have a robot vision system with a single camera. The system
has the image of the robot workspace as the input and is required to reproduce
geometrical measurements as its output. The necessary transformations between the
coordinate frames are evident from Fig.8.4.
Let us suppose that we are now in a position to recognize the point q in the image.
It is our aim to determine the coordinates of the real point Q from the coordinates of
its image q. This is the problem of backward projection. In order to solve the problem,
we must know how the coordinates of the point q are related to the coordinates of
the real point Q in the reference frame, which is the problem of forward projection.
Q
Oc
xc yc
zc
xr
yr
zr
u v
q
camera frame
image plane
reference frame
Fig. 8.4 The coordinate frames in a robot vision system

114 8 Robot Vision
Let us solve first the problem of forward projection. The point Q is given by the
coordinates (xr , yr , zr ) in the reference coordinate frame. We wish to determine the
coordinates of its image q = (u, v), expressed in the index frame. The frame xc–yc–
zc is attached to the camera. The matrix M represents the transformation from the
reference into the camera frame
⎡
⎢
⎢
⎣
xc
yc
zc
1
⎤
⎥
⎥
⎦ = M
⎡
⎢
⎢
⎣
xr
yr
zr
1
⎤
⎥
⎥
⎦ . (8.12)
By combining Eqs.(8.12) and (8.9), we obtain
s
⎡
⎣
u
v
1
⎤
⎦ = PM
⎡
⎢
⎢
⎣
xr
yr
zr
1
⎤
⎥
⎥
⎦ . (8.13)
The relation (8.13) describes the forward projection. The elements of the P matrix are
the intrinsic parameters of the camera, while the elements of the M matrix represent
its extrinsic parameters. The 3 × 4 matrix
H = PM (8.14)
is called the calibration matrix of the camera. It is used in the calibration process in
order to determine both the intrinsic and extrinsic parameters of the camera.
In the further text we shall consider backward projection. It is our aim to determine
the coordinates (xr , yr , zr ) of the real point Q from the known coordinates of the
image point (u, v) and the calibration matrix H. The scaling factor s is not known. In
(8.13) we have four unknowns s, xr , yr , and zr and only three equations for a single
point in space.
Let us try with three points A, B, and C (Fig.8.5). We know the distances between
these three points. Their coordinates in the reference frame are
(xrj
, yrj
, zrj
), j = 1, 2, 3 .
The coordinates of the corresponding image points are
(u j , vj ), j = 1, 2, 3 .
The forward projection can be written in the following form
sj
⎡
⎣
u j
vj
1
⎤
⎦ = H
⎡
⎢
⎢
⎣
xrj
yrj
zrj
1
⎤
⎥
⎥
⎦ . (8.15)

A
B
C
L23
L13
L12
v
u
xr
yr
zr
a
b
c
xc
yc
zc
Fig. 8.5 Example of projecting three points
In Eq.(8.15) we have 12 unknowns and 9 equations. To solve the problem we need
additional three equations. These equations can be obtained from the size of the
triangle represented by the points A, B, and C. We shall denote the triangle sides
AB, BC, and C A as the lengths L12, L23, and L31
L2
12 = (xr1
− xr2
)2
+ (yr1
− yr2
)2
+ (zr1
− zr2
)2
L2
23 = (xr2
− xr3
)2
+ (yr2
− yr3
)2
+ (zr2
− zr3
)2
L2
31 = (xr3
− xr1
)2
+ (yr3
− yr1
)2
+ (zr3
− zr1
)2
.
(8.16)
Now we have twelve equations for the twelve unknowns. Thus, the solution of the
inverse problem exists. It is inconvenient that the last three equations are nonlinear,
requiring a computer for numerical solving of the equations. The approach is called
model based backward projection.
8.3.2 Stereo Vision
Since the model of the observed object is usually not available or the object changes
with time, other solutions to the backward projection problem need to be found.
One possible solution is the use of stereo vision: sensing based on two cameras. The
principle is similar to human visual perception where the images seen by the left and
right eyes differ slightly due to parallax and the brain uses the differences between
images to determine the distance to the observed object.
For simplicity we will assume two parallel cameras that observe point Q as shown
in Fig.8.6. Point Q is projected onto the image plane of the left and right cameras.
The left camera’s image plane contains projection ql with coordinates xi,l and yi,l

116 8 Robot Vision
Q = (xQ,yQ,zQ)
xQ
yQ
zQ
xi,l
yi,l
z0
xi,r
yi,r
x0
y0
qr
ql
fc
fc
Ocr
Ocl
dc
left camera
right camera
Fig. 8.6 Stereo view of point Q using two parallel cameras
while the right camera’s image plane contains projection qr with coordinates xi,r
and yi,r . The axes of the vision system coordinate frame x0–y0–z0 have the same
directions as the left camera’s coordinate frame.
Figure8.7a shows the top view, while Fig.8.7b shows the side view of the situation
in Fig.8.6. These views will help us calculate the coordinates of point Q. From the
geometry in Fig.8.7a we can extract the following relations (distances xQ, yQ, and
zQ are with regard to the coordinate frame x0–y0–z0)
zQ
fc
=
xQ
xi,l
zQ
fc
=
xQ − dc
xi,r
,
(8.17)
where dc is the distance between the cameras. From the first equation in (8.17) we
express
xQ =
xi,l
fc
zQ (8.18)
and insert into the second equation to get
xi,l zQ
xi,r fc
−
zQ
fc
=
dc
xi,r
. (8.19)
We can then determine the distance zQ to point Q as
zQ =
fcdc
xi,l − xi,r
. (8.20)

Q = (xQ,yQ,zQ)
Q = (xQ,yQ,zQ)
xQ
yQ
zQ
xi,l
yi,l
z0
z0
xi,r
x0
y0
qr
ql
fc
fc
fc
Ocr
Ocl
Ocl
, Ocr
dc
left camera
right camera
(a)
(b)
Fig. 8.7 Projections of point Q on the planes of the left and right cameras. The upper figure a shows
a view of both cameras from above, while the lower figure b shows a side view of the cameras
The distance xQ can be determined from Eq.(8.18). To determine distance yQ we
refer to Fig.8.7b. From the geometry we can extract relation
zQ
fc
=
yQ
yi,l
, (8.21)
allowing us to calculate the remaining coordinate
yQ =
yi,l
fc
zQ. (8.22)
Use of two cameras enables computation of the position (and orientation) of an
object in space without an accurate model of the object.

118 8 Robot Vision
(a) (b) (c)
Fig. 8.8 a Model definition, b recognized object’s features, c located object
8.4 Image Processing
In contrast to most other sensory systems, visual systems provide very reach infor-
mation, which requires complex processing algorithms before it can be used for robot
control. The goal of image processing is to obtain numerical information from the
image, which provides a robust description of the object in the scene. An example
of the result of image processing is shown in Fig.8.8. An object is first identified in
the scene and then its pose is determined as marked with the coordinate frame.
Image processing is beyond the scope of this book and it will not be specifically
addressed here.
8.5 Object Pose from Image
Inordertocontroltherobotrelativetotheobjectofinterest,theobjectposeneedstobe
defined relative to the robot coordinate frame x–y–z. As shown in Fig.8.8, the pose
of the object is known in the image coordinate frame after the image processing.
In order to determine its pose in the robot frame, the transformation between the
image and the robot coordinate frame must be defined, which is the result of camera
calibration. Figure8.9 presents a simple approach for the calibration problem, where
the image plane is parallel to the horizontal plane. For simplicity, the image frame
xi –yi –zi is located at the same point as the index frame u–v (the zi axis was added
to the image frame to emphasize the rotation around the vertical axis).
8.5.1 Camera Calibration
Camera is mounted in a fixed position over the robot workspace. The calibration is
performed with the calibration pattern (checkerboard), and the calibration tip at the
robot end-effector. The calibration pattern can be augmented with a fiducial marker,
which appears in the image for use as a point of reference or a measure. The goal of
the calibration procedure is to find the transformation matrix Hi between the image

8.5 Object Pose from Image 119
x y
z
xi
xi
yi
yi
zi
x
x
y
y
z
iϕcp
Hi
iHcp
Hcp
calibration pattern
calibration tip
Fig. 8.9 Transformations used for camera calibration
and the robot coordinate frames xi –yi –zi and x–y–z. Based on relations in Fig.8.9
the following equality can be written
Hcp = Hi
i
Hcp, (8.23)
where Hcp and i
Hcp are the poses of the calibration pattern expressed in the robot
and the image coordinate frame, respectively.

120 8 Robot Vision
The pose of the calibration pattern i
Hcp expressed in the image coordinate frame
xi –yi –zi is the result of image processing
i
Hcp =
⎡
⎢
⎢
⎣
cos i
ϕcp − sin i
ϕcp 0 i
xcp
sin i
ϕcp cos i
ϕcp 0 i
ycp
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦ , (8.24)
where i
ϕcp and (i
xcp, i
ycp) are the orientation and position of the calibration pattern
relative to the image plane, respectively. Position is expressed in metric units as
i
xcp
i
ycp

= λ

ucp
vcp

, (8.25)
where (ucp, vcp) are the calibration pattern origin coordinates in pixels and λ is the
ratio between position expressed in metric units and pixels on the image (the ratio
can be obtained from the calibration pattern with the known size of black and white
fields). Matrix i
Hcp represents a rotation around the zi axis and translation along xi
and yi axes of the image coordinate frame.
The pose of the calibration pattern Hcp expressed in the robot coordinate frame
x–y–z can be determined with the calibration tip at the robot end-effector and the
calibration points marked on the calibration pattern. By placing the calibration tip
on the calibration point, recording the robot end-effector coordinates and repeating
the procedure for the three calibration points, a set of coordinates is obtained that
enables the definition of the calibration pattern pose relative to the robot coordinate
frame as
Hcp =
⎡
⎢
⎢
⎣
cos ϕcp − sin ϕcp 0 xcp
sin ϕcp cos ϕcp 0 ycp
0 0 1 zcp
0 0 0 1
⎤
⎥
⎥
⎦ , (8.26)
where ϕcp and (xcp, ycp, zcp) are the orientation and position of the calibration pattern
relative to the robot frame, respectively.
From Eqs.(8.23), (8.24) and (8.26) the transformation matrix between the image
and the robot coordinate frames can be obtained as
Hi = Hcp
i
H−1
cp . (8.27)
8.5.2 Object Pose
With the known Hi , the object pose Ho relative to the robot coordinate frame can be
determined as shown in Fig.8.10.

8.5 Object Pose from Image 121
x y
z
xi
xi
yi
yi
zi
x
x
y
y
z
iϕo
Hi
iHo Ho
Fig. 8.10 Transformations used for object pose computation
The pose of the object i
Ho expressed in the image coordinate frame xi –yi –zi is
the result of image processing
i
Ho =
⎡
⎢
⎢
⎣
cos i
ϕo − sin i
ϕo 0 i
xo
sin i
ϕo cos i
ϕo 0 i
yo
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦ , (8.28)

122 8 Robot Vision
where i
ϕo and (i
xo, i
yo) are the orientation and position of the object relative to the
image plane, respectively. Position is expressed in metric units as
i
xo
i
yo

= λ

uo
vo

, (8.29)
where (uo, vo) are the object origin coordinates in pixels.
Finally, Ho can be determined as
Ho = Hi
i
Ho. (8.30)

Chapter 9
Trajectory Planning
In previous chapters we studied mathematical models of robot mechanisms. First
of all we were interested in robot kinematics and dynamics. Before applying this
knowledge to robot control, we must become familiar with the planning of robot
motion. The aim of trajectory planning is to generate the reference inputs to the
robot control system, which will ensure that the robot end-effector will follow the
desired trajectory.
Robot motion is usually defined in the rectangular world coordinate frame placed
in the robot workspace most convenient for the robot task. In the simplest task
we only define the initial and the final point of the robot end-effector. The inverse
kinematic model is then used to calculate the joint variables corresponding to the
desired position of the robot end-effector.
9.1 Interpolation of the Trajectory Between Two Points
When moving between two points, the robot manipulator must be displaced from the
initial to the final point in a given time interval t f . Often we are not interested in the
precise trajectory between the two points. Nevertheless, we must determine the time
course of the motion for each joint variable and provide the calculated trajectory to
the control input.
The joint variable is either the angle ϑ for the rotational or the displacement d
for the translational joint. When considering the interpolation of the trajectory we
shall not distinguish between the rotational and translational joints, so that the joint
variable will be more generally denoted as q. With industrial manipulators moving
between two points we most often select the so called trapezoidal velocity profile.
The robot movement starts at t = 0 with constant acceleration, followed by the phase
of constant velocity and finished by the constant deceleration phase (Fig.9.1).
123

124 9 Trajectory Planning
q
q̇
q̈
qf
qm
qc
qi
q̇c
q̈c
−q̈c
tc tm
tf −tc tf
t
t
t
0
0
0
0
0
Fig. 9.1 The time dependence of the joint variables with trapezoidal velocity profile
The resulting trajectory of either the joint angle or displacement consists of the
central linear interval, which is started and concluded with a parabolic segment. The
initial and final velocities of the movement between the two points are zero. The
duration of the constant acceleration phase is equal to the interval with the constant
deceleration. In both phases the magnitude of the acceleration is q̈c. In this way we
deal with a symmetric trajectory, where
qm =
q f + qi
2
at the moment tm =
t f
2
. (9.1)
The trajectory q(t) must satisfy several constraints in order that the robot joint
will move from the initial point qi to the final point q f in the required time interval
t f . The velocity at the end of the initial parabolic phase must be equal to the constant
velocity in the linear phase. The velocity in the first phase is obtained from the
equation describing the constant acceleration motion
q̇ = q̈ct. (9.2)
At the end of the first phase we have
q̇c = q̈ctc. (9.3)

9.1 Interpolation of the Trajectory Between Two Points 125
The velocity in the second phase can be determined with the help of Fig.9.1
q̇c =
qm − qc
tm − tc
, (9.4)
where qc represents the value of the joint variable q at the end of the initial parabolic
phase (i.e., at the time tc). Until that time the motion with constant acceleration q̈c
takes place, so the velocity is determined by Eq.(9.2). The time dependence of the
joint position is obtained by integrating Eq.(9.2)
q =

q̇dt = q̈c

tdt = q̈c
t2
2
+ qi , (9.5)
where the initial joint position qi is taken as the integration constant. At the end of
the first phase we have
qc = qi +
1
2
q̈ct2
c . (9.6)
The velocity at the end of the first phase (9.3) is equal to the constant velocity in the
second phase (9.4)
q̈ctc =
qm − qc
tm − tc
. (9.7)
By inserting Eq.(9.6) into Eq.(9.7) and considering the expression (9.1), we obtain,
after rearrangement, the following quadratic equation
q̈ct2
c − q̈ct f tc + q f − qi = 0. (9.8)
The acceleration q̈c is determined by the selected actuator and the dynamic properties
of the robot mechanism. For chosen qi , q f , q̈c, and t f the time interval tc is
tc =
t f
2
−
1
2

t2
f q̈c − 4(q f − qi )
q̈c
. (9.9)
Togeneratethemovementbetweentheinitialqi andthefinalpositionq f thefollowing
polynomial must be generated in the first phase
q(t) = qi +
1
2
q̈ct2
0 ≤ t ≤ tc. (9.10)
In the second phase a linear trajectory must be generated starting in the point (tc, qc),
with the slope q̇c
(q − qc) = q̇c(t − tc). (9.11)
After rearrangement we obtain

q(t) = qi + q̈ctc(t −
tc
2
) tc t ≤ (t f − tc). (9.12)
In the last phase the parabolic trajectory must be generated similarly to the first phase,
only that now the extreme point is in (t f , q f ) and the curve is turned upside down
q(t) = q f −
1
2
q̈c(t − t f )2
(t f − tc) t ≤ t f . (9.13)
In this way we obtained analytically the time dependence of the angle or displacement
of the rotational or translational joint moving from point to point.
9.2 Interpolation by Use of via Points
In some robot tasks, movements of the end-effector more complex than point to point
motions, are necessary. In welding, for example, the curved surfaces of the objects
must be followed. Such trajectories can be obtained by defining, besides the initial
and the final point, also the so called via points through which the robot end-effector
must move.
In this chapter we shall analyze the problem, where we wish to interpolate the
trajectory through n via points {q1, . . . , qn}, which must be reached by the robot in
time intervals {t1, . . . , tn}. The interpolation will be performed with the help of trape-
zoidal velocity profiles. The trajectory will consist of a sequence of linear segments
describing the movements between two via points and parabolic segments represent-
ing the transitions through the via points. In order to avoid the discontinuity of the
first derivative at the moment tk, the trajectory q(t) must have a parabolic course in
the vicinity of qk. By doing so the second derivative in the point qk (acceleration)
remains discontinuous.
Theinterpolatedtrajectory,definedasasequenceoflinearfunctionswithparabolic
transitions through the via points (the transition time tk), is analytically described
by the following constraints
q(t) =

a1,k(t − tk) + a0,k tk + tk
2
≤ t tk+1 − tk+1
2
b2,k(t − tk)2
+ b1,k(t − tk) + b0,k tk − tk
2
≤ t tk + tk
2
.
(9.14)
The coefficients a0,k and a1,k determine the linear parts of the trajectory, where k
represents the index of the corresponding linear segment. The coefficients b0,k, b1,k
and b2,k belong to the parabolic transitions. The index k represents the consecutive
number of a parabolic segment.
First, the velocities in the linear segments will be calculated from the given posi-
tions and the corresponding time intervals, as shown in Fig.9.2. We assume that the
initial and final velocities are equal to zero. In this case we have

9.2 Interpolation by Use of via Points 127
q
t
q1 = qi
q2
q3
qn = qf
Δt2
Δt2,3
t1 = 0 t2 t3 tn = tf
Δt1
ek
Fig. 9.2 Trajectory interpolation through n via points—linear segments with parabolic transitions
are used
q̇k−1,k =
⎧
⎪
⎨
⎪
⎩
0 k = 1
qk −qk−1
tk −tk−1
k = 2, . . . , n
0 k = n + 1.
(9.15)
Further, we must determine the coefficients of the linear segments a0,k and a1,k. The
coefficient a0,k can be found from the linear function (9.14), by taking into account
the known position at the moment tk, when the robot segment approaches the point
qk
q(tk) = qk = a1,k(tk − tk) + a0,k = a0,k, (9.16)
therefore
t = tk ⇒ a0,k = qk k = 1, . . . , n − 1 . (9.17)
The coefficient a1,k can be determined from the time derivative of the linear function
(9.14)
q̇(t) = a1,k. (9.18)
By considering the given velocities in the time interval tk,k+1, we obtain
a1,k = q̇k,k+1 k = 1, . . . , n − 1. (9.19)

In this way the coefficients of the linear segments of the trajectory are determined and
we can proceed with the coefficients of the parabolic functions. We shall assume that
the transition time is predetermined as tk. If the transition time is not prescribed,
the absolute value of the acceleration |q̈k| in the via point must be first defined and
then the transition time is calculated from the accelerations and velocities before and
after the via point. In this case only the sign of the acceleration must be determined
by considering the sign of the velocity difference in the via point
q̈k = sign(q̇k,k+1 − q̇k−1,k)|q̈k|, (9.20)
where sign(·) means the sign of the expression in the brackets. Given the values of
the accelerations in the via points and the velocities before and after the via point, the
time of motion through the via point tk is calculated (deceleration and acceleration)
tk =
q̇k,k+1 − q̇k−1,k
q̈k
. (9.21)
We shall proceed by calculating the coefficients of the quadratic functions. The
required continuity of the velocity during the transition from the linear into the
parabolic trajectory segment at the instant (tk − tk
2
) and the required velocity con-
tinuity during the transition from the parabolic into the linear segment at (tk + tk
2
)
represents the starting point for the calculation of the coefficients b1,k and b2,k. First,
we calculate the time derivative of the quadratic function (9.14)
q̇(t) = 2b2,k(t − tk) + b1,k . (9.22)
Assuming that the velocity at the instant (tk − tk
2
), is q̇k−1,k, while at (tk + tk
2
), it
is q̇k,k+1, we can write
q̇k−1,k = 2b2,k tk −
tk
2
− tk + b1,k = −b2,ktk + b1,k t = tk −
tk
2
q̇k,k+1 = 2b2,k tk +
tk
2
− tk + b1,k = b2,ktk + b1,k t = tk +
tk
2
.
(9.23)
By adding Eq.(9.23), the coefficient b1,k can be determined
b1,k =
q̇k,k+1 + q̇k−1,k
2
k = 1, . . . , n (9.24)
and by subtracting Eq.(9.23), the coefficient b2,k is calculated
b2,k =
q̇k,k+1 − q̇k−1,k
2tk
=
q̈k
2
k = 1, . . . , n . (9.25)

By taking into account the continuity of the position at the instant (tk + tk
2
), the
coefficient b0,k of the quadratic polynomial can be calculated. At (tk + tk
2
), the
position q(t), calculated from the linear function
q tk +
tk
2
= a1,k tk +
tk
2
− tk + a0,k = q̇k,k+1
tk
2
+ qk (9.26)
equals the position q(t), calculated from the quadratic function
q tk +
tk
2
= b2,k tk +
tk
2
− tk
2
+ b1,k tk +
tk
2
− tk + b0,k
=
q̇k,k+1 − q̇k−1,k
2tk
tk
2
2
+
q̇k,k+1 + q̇k−1,k
2
·
tk
2
+ b0,k .
(9.27)
By equating (9.26) and (9.27) the coefficient b0,k is determined
b0,k = qk + (q̇k,k+1 − q̇k−1,k)
tk
8
. (9.28)
It can be verified that the calculated coefficient b0,k ensures also continuity of
positionattheinstant(tk − tk
2
).Suchachoiceofthecoefficientb0,k preventsthejoint
trajectory going through point qk. The robot only more or less approaches this point.
The distance of the calculated trajectory from the reference point depends mainly on
the decelerating and accelerating time interval tk, which is predetermined by the
required acceleration |q̈k|. The error ek of the calculated trajectory can be estimated
by comparing the desired position qk with the actual position q(t) at the instant tk,
which is obtained by inserting tk into the quadratic function (9.14)
ek = qk − q(tk) = qk − b0,k = −(q̇k,k+1 − q̇k−1,k)
tk
8
. (9.29)
It can be noticed that the error ek equals zero only when the velocities of the linear
segments before and after the via points are equal or when the time interval tk is
zero, meaning infinite acceleration (which in reality is not possible).
The described approach to the trajectory interpolation has a minor deficiency.
From Eq.(9.29) it can be observed that, instead of reaching the via point, the robot
goes around it. As the initial and final trajectory points are also considered as via
points, an error is introduced into the trajectory planning. At the starting point of the
trajectory, the actual and the desired position differ by the error e1 (Fig.9.3, the light
curve shows the trajectory without correction), arising from Eq.(9.29). The error
represents a step in the position signal, which is not desired in robotics. To avoid
this abrupt change in position, the first and the last trajectory point must be handled
separately from the via points.
The required velocities in the starting and the final points should be zero. The
velocity at the end of the time interval t1 must be equal to the velocity in the first

q
t
q1
q2
1
2 Δt2
Δt1,2
t1 = 0 t2
Δt1
1
2 Δt1
e2
e1
Fig.9.3 Trajectoryinterpolation—enlargedpresentationofthefirstsegmentofthetrajectoryshown
in Fig.9.2. The lighter curve represents the trajectory without correction, while the darker curve
shows the corrected trajectory
linear segment. First, we calculate the velocity in the linear part
q̇1,2 =
q2 − q1
t2 − t1 − 1
2
t1
. (9.30)
Equation (9.30) is similar to Eq.(9.15), only that now 1
2
t1 is subtracted in the
denominator, as in the short time interval (the beginning of the parabolic segment in
Fig.9.3) the position of the robot changes only to a very small extent. By doing so, a
higher velocity in the linear segment of the trajectory is obtained. At the end of the
acceleration interval t1 we have
q2 − q1
t2 − t1 − 1
2
t1
= q̈1t1 (9.31)
We must determine also the acceleration q̈1 at the starting point of the trajectory.
Assuming that its absolute value |q̈1| was predetermined, only the sign must be
adequately selected. The choice of the sign will be performed on the basis of the
positional difference. In principle the velocity difference should be taken into account

when determining the sign of acceleration, however the initial velocity is zero, and
the sign can therefore depend on the difference in positions.
q̈1 = sign(q2 − q1)|q̈1|. (9.32)
From Eq.(9.31), the time interval t1 is calculated
(q2 − q1) = q̈1t1(t2 − t1 −
1
2
t1). (9.33)
After rearrangement we obtain
−
1
2
q̈1t2
1 + q̈1(t2 − t1)t1 − (q2 − q1) = 0, (9.34)
so the time interval t1 is
t1 =
−q̈1(t2 − t1) ± q̈2
1 (t2 − t1)2 − 2q̈1(q2 − q1)
−q̈1
, (9.35)
and after simplifying Eq.(9.35)
t1 = (t2 − t1) −

(t2 − t1)2 −
2(q2 − q1)
q̈1
. (9.36)
In Eq.(9.36), the minus sign was selected before the square root, because the time
interval t1 must be shorter than (t2 − t1). From Eq.(9.30), the velocity in the linear
part of the trajectory can be calculated. As is evident from Fig.9.3 (the darker curve
represents the corrected trajectory), the introduced correction eliminates the error in
the initial position.
Similarly, as for the first segment, the correction must be calculated also for the
last segment between points qn−1 and qn. The velocity in the last linear segment is
q̇n−1,n =
qn − qn−1
tn − tn−1 − 1
2
tn
. (9.37)
In the denominator of Eq.(9.37) the value 1
2
tn was subtracted, as immediately
before the complete stop of the robot, its position changes only very little. At the
transition from the last linear segment into the last parabolic segment the velocities
are equal
qn − qn−1
tn − tn−1 − 1
2
tn
= q̈ntn. (9.38)

The acceleration (deceleration) of the last parabolic segment is determined on the
basis of the positional difference
q̈n = sign(qn−1 − qn)|q̈n|. (9.39)
By inserting the above equation into Eq.(9.38), we calculate, in a similar way as for
the first parabolic segment, also the duration of the last parabolic segment
tn = (tn − tn−1) −

(tn − tn−1)2 −
2(qn − qn−1)
q̈n
. (9.40)
From Eq.(9.37), the velocity of the last linear segment can be determined. By con-
sidering the corrections at the start and at the end of the trajectory, the time course
through the via points is calculated. In this way the entire trajectory was interpolated
at the n points.

Chapter 10
Robot Control
The problem of robot control can be explained as a computation of the forces or
torques which must be generated by the actuators in order to successfully accomplish
the robot’s task. The appropriate working conditions must be ensured both during
the transient period as well as in the stationary state. The robot task can be presented
either as the execution of the motions in a free space, where position control is
performed, or in contact with the environment, where control of the contact force is
required. First, we shall study the position control of a robot mechanism which is
not in contact with its environment. Then, in the further text we shall upgrade the
position control with the force control.
The problem of robot control is not unique. There exist various methods which
differ in their complexity and in the effectiveness of robot actions. The choice of the
control method depends on the robot task. An important difference is, for example,
between the task where the robot end-effector must accurately follow the prescribed
trajectory (e.g., laser welding) and another task where it is only required that the robot
end-effector reaches the desired final pose, while the details of the trajectory between
the initial and the final point are not important (e.g., palletizing). The mechanical
structure of the robot mechanism also influences the selection of the appropriate
control method. The control of a cartesian robot manipulator in general differs from
the control of an anthropomorphic robot.
Robot control usually takes place in the world coordinate frame, which is defined
by the user and is called also the coordinate frame of the robot task. Instead of world
coordinate frame we often use a shorter expression, namely external coordinates. We
are predominantly interested in the pose of the robot end-effector expressed in the
external coordinates and rarely in the joint positions, which are also called internal
coordinates. Nevertheless, we must be aware that in all cases we directly control
the internal coordinates (i.e., joint angles or displacements). The end-effector pose
is only controlled indirectly. It is determined by the kinematic model of the robot
mechanism and the given values of the internal coordinates.
133

134 10 Robot control
qr
xr x
u τ
q
q̃ Robot
mechanism
Control Actuators
Sensors
Inverse
kinematics
Trajectory
planning
+
−
Fig. 10.1 A general robot control system
Figure10.1 shows a general robot control system. The input to the control system
is the desired pose of the robot end-effector, which is obtained by using trajectory
interpolation methods, introduced in the previous chapter. The variable xr represents
the desired (i.e., the reference pose) of the robot end-effector. The x vector, describing
the actual pose of the robot end-effector, in general comprises six variables. Three
of them define the position of the robot end-point, while the other three determine
the orientation of the robot end-effector. Thus, we write x =

x y z ϕ ϑ ψ
T
.
The position of the robot end-effector is determined by the vector from the origin
of the world coordinate frame to the robot end-point. The orientation of the end-
effector can be presented in various ways. One of the possible descriptions is the so
called RPY notation, arising from aeronautics and shown in Fig.4.4. The orientation
is determined by the angle ϕ around the z axis (Roll), the angle ϑ around the y axis
(Pitch), and the angle ψ around the x axis (Yaw).
By the use of the inverse kinematics algorithm the internal coordinates qr , corre-
sponding to the desired end-effector pose, are calculated. The variable qr represents
the joint position (i.e., the angle ϑ for the rotational joint and the distance d for
the translational joint). The desired internal coordinates are compared to the actual
internal coordinates in the robot control system. On the basis of the positional error
q̃ = qr − q the control system output u is calculated. The output u is converted from
a digital into an analogue signal, amplified and delivered to the robot actuators. The
actuators ensure the forces or torques necessary for the required robot motion. The
robot motion is assessed by the sensors which were described in the chapter devoted
to robot sensors.
10.1 Control of the Robot in Internal Coordinates
The simplest robot control approach is based on controllers where the control loop is
closed separately for each particular degree of freedom. Such controllers are suitable
for control of independent second order systems with constant inertial and damp-
ing parameters. This approach is less suitable for robotic systems characterized by
nonlinear and time varying behavior.

10.1 Control of the Robot in Internal Coordinates 135
qr q̃
Kp
Kd
u
q
q̇
Robot
+ +
−
−
Fig. 10.2 PD position control with high damping
10.1.1 PD Control of Position
First, a simple proportional-derivative (PD) controller will be analyzed. The basic
control scheme is shown in Fig.10.2. The control is based on calculation of the
positional error and determination of control parameters, which enable reduction or
suppression of the error. The positional error is reduced for each joint separately,
which means that as many controllers are to be developed as there are degrees of
freedom. The reference positions qr are compared to the actual positions of the robot
joints q
q̃ = qr − q. (10.1)
The positional error q̃ is amplified by the proportional position gain Kp. As a robot
manipulator has several degrees of freedom, the error q̃ is expressed as a vector, while
Kp is a diagonal matrix of the gains of all joint controllers. The calculated control
input provokes robot motion in the direction of reduction of the positional error. As
the actuation of the robot motors is proportional to the error, it can happen that the
robot will overshoot instead of stopping in the desired position. Such overshoots are
not allowed in robotics, as they may result in collisions with objects in the robot
vicinity. To ensure safe and stable robot actions, a velocity closed loop is introduced
with a negative sign. The velocity closed loop brings damping into the system. It is
represented by the actual joint velocities q̇ multiplied by a diagonal matrix of velocity
gains Kd. The control law can be written in the following form
u = Kp(qr − q) − Kdq̇, (10.2)
where u represents the control inputs (i.e., the joint forces or torques), which must be
provided by the actuators. From Eq.(10.2) we can notice that at higher velocities of
robot motions, the velocity control loop reduces the joint actuation and, by damping
the system, ensures robot stability.

qr
q̇r
q̃
˙
q̃
Kp
Kd
u
q
q̇
Robot
+
+ +
−
−
Fig. 10.3 PD position control
The control method shown in Fig.10.2 provides high damping of the system in
the fastest part of the trajectory, which is usually not necessary. Such behavior of the
controller can be avoided by upgrading the PD controller with the reference velocity
signal. This signal is obtained as the numerical derivative of the desired position.
The velocity error is used as control input
˙
q̃ = q̇r − q̇. (10.3)
The control algorithm demonstrated in Fig.10.3 can be written as
u = Kp(qr − q) + Kd(q̇r − q̇). (10.4)
As the difference between the reference velocity q̇r and q̇ is used instead of the total
velocity q̇, the damping effect is reduced. For a positive difference the control loop
can even accelerate the robot motion.
The synthesis of the PD position controller consists of determining the matrices
Kp and Kd. For fast response, the Kp gains must be high. By proper choice of the
Kd gains, critical damping of the robot systems is obtained. The critical damping
ensures fast response without overshoot. Such controllers must be built for each joint
separately. The behavior of each controller is entirely independent of the controllers
belonging to the other joints of the robot mechanism.
10.1.2 PD Control of Position with Gravity Compensation
In the chapter on robot dynamics we found that the robot mechanism is under the
influence of inertial, Coriolis, centripetal, and gravitational forces (5.56). In general,
friction forces occurring in robot joints, must also be included in the robot dynamic
model. In a somewhat simplified model, only viscous friction, being proportional
to the joint velocity, will be taken into account (Fv is a diagonal matrix of the

joint friction coefficients). The enumerated forces must be overcome by the robot
actuators, which is evident from the following equation, similar to Eq.(5.56)
B(q)q̈ + C(q, q̇)q̇ + Fvq̇ + g(q) = τ. (10.5)
When developing the PD controller, we did not pay attention to the specific forces
influencing the robot mechanism. The robot controller calculated the required actu-
ation forces solely on the basis of the difference between the desired and the actual
joint positions. Such a controller cannot predict the force necessary to produce the
desired robot motion. As the force is calculated from the positional error, this means
that in general the error is never equal to zero. When knowing the dynamic robot
model, we can predict the forces which are necessary for the performance of a par-
ticular robot motion. These forces are then generated by the robot motors regardless
of the positional error signal.
In quasi-static conditions, when the robot is standing still or moving slowly, we
can assume zero accelerations q̈ ≈ 0 and velocities q̇ ≈ 0. The robot dynamic model
is simplified as follows
τ ≈ g(q). (10.6)
According to Eq.(10.6), the robot motors must above all compensate for the gravity
effect. The model of gravitational effects ĝ(q) (the circumflex denotes the robot
model), which is a good approximation of the actual gravitational forces g(q), can
be implemented in the control algorithm shown in Fig.10.4. The PD controller,
shown in Fig.10.2, was upgraded with an additional control loop, which calculates
the gravitational forces from the actual robot position and directly adds them to the
controller output. The control algorithm shown in Fig.10.4 can be written as follows
u = Kp(qr − q) − Kdq̇ + ĝ(q). (10.7)
By introducing gravity compensation, the burden of reducing the errors caused
by gravity is taken away from the PD controller. In this way the errors in trajectory
tracking are significantly reduced.
10.1.3 Control of the Robot Based on Inverse Dynamics
When studying the PD controller with gravity compensation, we investigated the
robot dynamic model in order to improve the efficiency of the control method. With
the control method based on inverse dynamics, this concept will be further upgraded.
From the equations describing the dynamic behavior of a two-segment robot manip-
ulator (5.56), we can clearly observe that the robot model is nonlinear. A linear
controller, such as the PD controller, is therefore not the best choice.

qr q̃
Kp
Kd
u
q
q̇
ĝ(q)
Robot
+ + +
−
−
Fig. 10.4 PD control with gravity compensation
We shall derive the new control scheme from the robot dynamic model described
by Eq.(10.5). Let us assume that the torques τ, generated by the motors, are equal
to the control outputs u. Equation (10.5) can be rewritten
B(q)q̈ + C(q, q̇)q̇ + Fvq̇ + g(q) = u. (10.8)
In the next step we will determine the direct robot dynamic model, which describes
robot motions under the influence of the given joint torques. First we express the
acceleration q̈ from Eq.(10.8)
q̈ = B−1
(q) (u − (C(q, q̇)q̇ + Fvq̇ + g(q))) . (10.9)
By integrating the acceleration, while taking into account the initial velocity value,
the velocity of robot motion is obtained. By integrating the velocity, while taking
into account the initial position, we calculate the actual positions in the robot joints.
The direct dynamic model of a robot mechanism is shown in Fig.10.5.
In order to simplify the dynamic equations, we shall define a new variable n(q, q̇),
comprising all dynamic components except the inertial component
n(q, q̇) = C(q, q̇)q̇ + Fvq̇ + g(q). (10.10)
The robot dynamic model can be described with the following shorter equation
B(q)q̈ + n(q, q̇) = τ. (10.11)
In the same way Eq.(10.9) can also be written in a shorter form
q̈ = B−1
(q) (u − n(q, q̇)) . (10.12)

x
g(q)
C(q,q̇)
Fv
B−1(q)
n(q,q̇)
Robot
q̈ q̇ q
u
+
+
+
−
Fig. 10.5 The direct dynamic model of a robot mechanism
Let us assume that the robot dynamic model is known. The inertial matrix B̂(q) is an
approximation of the real values B(q), while n̂(q, q̇) represents an approximation of
n(q, q̇), as follows
n̂(q, q̇) = Ĉ(q, q̇)q̇ + F̂vq̇ + ĝ(q). (10.13)
The controller output u is determined by the following equation
u = B̂(q)y + n̂(q, q̇), (10.14)
where the approximate inverse dynamic model of the robot was used. The system,
combining Eqs.(10.12) and (10.14), is shown in Fig.10.6.
Let us assume the equivalence B̂(q) = B(q) and n̂(q, q̇) = n(q, q̇). In Fig.10.6
we observe that the signals n̂(q, q̇) and n(q, q̇) subtract, as one is presented with a
positive and the other with a negative sign. In a similar way, the product of matrices
B̂(q) and B−1
(q) results in a unit matrix, which can be omitted. The simplified
system is shown in Fig.10.7. By implementing the inverse dynamics (10.14), the
control system is linearized, as there are only two integrators between the input y
and the output q. The system is not only linear, but is also decoupled (e.g. the first
element of the vector y only influences the first element of the position vector q).
From Fig.10.7 it is also not difficult to realize that the variabley has the characteristics
of acceleration, thus
y = q̈. (10.15)

x
B−1(q)
n(q,q̇)
Robot
q̈
q̇
q̇
q
q
u
B̂(q)
n̂(q,q̇)
y
+ +
−
Fig. 10.6 Linearization of the control system by implementing the inverse dynamic model
x
Robot
q̇
q̇
q
q
u y
y
B̂(q)
n̂(q,q̇)
+
Fig. 10.7 The linearized system
In an ideal case, it would suffice to determine the desired joint accelerations as
the second derivatives of the desired joint positions and the control system will track
the prescribed joint trajectories. As we never have a fully accurate dynamic model
of the robot, a difference will always occur between the desired and the actual joint
positions and will increase with time. The positional error is defined by
q̃ = qr − q, (10.16)
where qr represents the desired robot position. In a similar way also the velocity
error can be defined as the difference between the desired and the actual velocity
˙
q̃ = q̇r − q̇. (10.17)
The vector y, having the acceleration characteristics, can be now written as
y = q̈r + Kp(qr − q) + Kd(q̇r − q̇). (10.18)
It consists of the reference acceleration q̈r and two contributing signals which depend
on the errors of position and velocity. These two signals suppress the error arising
becauseoftheimperfectlymodeleddynamics.Thecompletecontrolschemeisshown
in Fig.10.8.

Robot
Robot
q̇
q̇
q
q
u
u
B̂(q)
n̂(q,q̇)
q̈r
q̈r
q̇r
q̇r
qr
qr
˙
q̃
q̃
Kp
Kd
y
y
Position control
Position control
Inverse dynamics
Inverse dynamics
+
+
+ +
−
−
Fig. 10.8 Control of the robot based on inverse dynamics
By considering Eq.(10.18) and the equality y = q̈, the differential equation
describing the robot dynamics can be written as
¨
q̃ + Kd
˙
q̃ + Kpq̃ = 0, (10.19)
where the acceleration error ¨
q̃ = q̈r − q̈ was introduced. The differential Eq.(10.19)
describes the time dependence of the control error as it approaches zero. The dynam-
ics of the response is determined by the gains Kp and Kd.
10.2 Control of the Robot in External Coordinates
All the control schemes studied up to now were based on control of the internal
coordinates (i.e., joint positions). The desired positions, velocities and accelerations
were determined by the robot joint variables. Usually we are more interested in the
motion of the robot end-effector than in the displacements of particular robot joints.
At the tip of the robot, different tools are attached to accomplish various robot tasks.
In the further text we shall focus on the robot control in the external coordinates.

10.2.1 Control Based on the Transposed Jacobian Matrix
The control method is based on the already known Eq.(5.18), connecting the forces
acting at the robot end-effector with the joint torques. The relation is defined by the
use of the transposed Jacobian matrix
τ = JT
(q)f, (10.20)
where the vector τ represents the joint torques and f is the force at the robot end-point.
It is our aim to control the pose of the robot end-effector, where its desired pose is
defined by the vector xr and the actual pose is given by the vector x. The vectors xr
and x in general comprise six variables, three determining the position of the robot
end-point and three for the orientation of the end-effector, thus x =

x y z ϕ ϑ ψ
T
.
Robots are usually not equipped with sensors assessing the pose of the end-effector;
robot sensors measure the joint variables. The pose of the robot end-effector must
therefore, be determined by using the equations of the direct kinematic model x =
k(q), introduced in the chapter on robot kinematics (5.4). The positional error of the
robot end-effector is calculated as
x̃ = xr − x = xr − k(q). (10.21)
The positional error must be reduced to zero. A simple proportional control system
with the gain matrix Kp is introduced
f = Kpx̃. (10.22)
When analyzing Eq.(10.22) more closely, we find that it reminds us of the equa-
tion describing the behavior of a spring (in external coordinates), where the force is
proportional to the spring elongation. This consideration helps us to explain the intro-
duced control principle. Let as imagine that there are six springs virtually attached
to the robot end-effector, one spring for each degree of freedom (three for position
and three for orientation). When the robot moves away from the desired pose, the
springs are elongated and pull the robot end-effector into the desired pose with the
force proportional to the positional error. The force f therefore pushes the robot end-
effector towards the desired pose. As the robot displacement can only be produced
by the motors in the joints, the variables controlling the motors must be calculated
from the force f. This calculation is performed by the help of the transposed Jacobian
matrix as shown in Eq.(10.20)
u = JT
(q)f. (10.23)
The vector u represents the desired joint torques. The control method based on the
transposed Jacobian matrix is shown in Fig.10.9.

10.2 Control of the Robot in External Coordinates 143
Robot
q
u
xr
x
x̃ Kp
f JT (q)
k(q)
+
−
Fig. 10.9 Control based on the transposed Jacobian matrix
10.2.2 Control Based on the Inverse Jacobian Matrix
The control method is based on the relation between the joint velocities and the
velocities of the robot end-point (5.10), which is given by the Jacobian matrix. In
Eq.(5.10) we emphasize the time derivatives of external coordinates x and internal
coordinates q
ẋ = J(q)q̇ ⇔
dx
dt
= J(q)
dq
dt
. (10.24)
As dt appears in the denominator on both sides of Eq.(10.24), it can be omitted.
In this way we obtain the relation between changes of the internal coordinates and
changes of the pose of the robot end-point
dx = J(q)dq. (10.25)
Equation (10.25) is valid only for small displacements.
As with the previously studied control method, based on the transposed Jacobian
matrix, we can also in this case first calculate the error of the pose of the robot end-
point by using Eq.(10.21). When the error in the pose is small, we can calculate the
positional error in the internal coordinates by the inverse relation (10.25)
q̃ = J−1
(q)x̃. (10.26)
In this way the control method is translated to the known method of robot control in
theinternalcoordinates.Inthesimplestexample,basedontheproportionalcontroller,
we can write
u = Kpq̃. (10.27)
The equation describes the behavior of a spring (in internal coordinates). The control
method, based on the inverse Jacobian matrix, is shown in Fig.10.10.

Robot
q
u
xr
x
x̃ q̃ Kp
J−1(q)
k(q)
+
−
Fig. 10.10 Control based on the inverse Jacobian matrix
10.2.3 PD Control of Position with Gravity Compensation
The PD control of position with gravity compensation was already studied in detail
for the internal coordinates. Now we shall derive the analogue control algorithm in
the external coordinates. The starting point will be Eq.(10.21), expressing the error
of the pose of the end-effector. The velocity of the robot end-point is calculated with
the help of the Jacobian matrix from the joint velocities
ẋ = J(q)q̇. (10.28)
The equation describing the PD controller in external coordinates is analogous to
that written in the internal coordinates (10.2)
f = Kpx̃ − Kdẋ. (10.29)
In Eq.(10.29), the pose error is multiplied by the matrix of the positional gains Kp,
while the velocity error is multiplied by the matrix Kd. The negative sign of the
velocity error introduces damping into the system. The joint torques are calculated
from the force f, acting at the tip of the robot, with the help of the transposed Jacobian
matrix(inasimilar wayas inEq.(10.23)) andbyaddingthecomponent compensating
gravity (as in Eq.(10.7)). The control algorithm is written as
u = JT
(q)f + ĝ(q). (10.30)
The complete control scheme is shown in Fig.10.11.
10.2.4 Control of the Robot Based on Inverse Dynamics
In the chapter on the control of robots in the internal coordinates, the following
controller based on inverse dynamics was introduced
u = B̂(q)y + n̂(q, q̇). (10.31)

Robot q
q̇
u
ẋ
xr
x
x̃ f
Kp
Kd
JT (q)
J(q)
k(q)
ĝ(q)
+ + +
−
−
Fig. 10.11 PD control with gravity compensation in external coordinates
We also learned that the vector y has the characteristics of acceleration
y = q̈, (10.32)
which was determined in such a way, that the robot tracked the desired trajectory
expressed in the internal coordinates. As it is our aim to develop a control method in
the external coordinates, the y signal must be adequately adapted. Equation (10.31),
linearizing the system, remains unchanged.
We shall again start from the equation relating the joint velocities to the robot
end-effector velocities
ẋ = J(q)q̇. (10.33)
By calculating the time derivative of Eq.(10.33), we obtain
ẍ = J(q)q̈ + J̇(q, q̇)q̇. (10.34)
The error of the pose of the robot end-effector is determined as the difference between
its desired and its actual pose
x̃ = xr − x = xr − k(q). (10.35)
In a similar way the velocity error of the robot end-effector is determined
˙
x̃ = ẋr − ẋ = ẋr − J(q)q̇. (10.36)
Theaccelerationerroristhedifferencebetweenthedesiredandtheactualacceleration
¨
x̃ = ẍr − ẍ. (10.37)
When developing the inverse dynamics based controller in the internal coordinates,
Eq.(10.19) was derived describing the dynamics of the control error in the form

¨
q̃ + Kd
˙
q̃ + Kpq̃ = 0. An analogous equation can be written for the error of the end-
effector pose. From this equation the acceleration ẍ of the robot end-effector can be
expressed
¨
x̃ + Kd
˙
x̃ + Kpx̃ = 0 ⇒ ẍ = ẍr + Kd
˙
x̃ + Kpx̃. (10.38)
From Eq.(10.34) we express q̈ taking into account the equality y = q̈
y = J−1
(q)

ẍ − J̇(q, q̇)q̇

. (10.39)
By replacing ẍ in Eq.(10.39) with expression (10.38), the control algorithm based
on inverse dynamics in the external coordinates is obtained
y = J−1
(q)

ẍr + Kd
˙
x̃ + Kpx̃ − J̇(q, q̇)q̇

. (10.40)
ẍr
ẍr
ẋr
ẋr
xr
xr
ẋ
ẋ
x
x
˙
x̃
x̃
Kp
Kd
J−1(q)
y
y
B̂(q)
u
u
q
q
q̇
q̇
n̂(q,q̇)
J̇(q,q̇)
J(q)
k(q)
Robot
Robot
Position control
Position control
Inverse dynamics
Inverse dynamics
Direct kinematics
Direct kinematics
+
+
+ +
−
−
−
Fig. 10.12 Robot control based on inverse dynamics in external coordinates

The control scheme encompassing the linearization of the system based on inverse
dynamics (10.31) and the closed loop control (10.40) is shown in Fig.10.12.
10.3 Control of the Contact Force
The control of position is sufficient when a robot manipulator follows a trajectory in
free space. When contact occurs between the robot end-effector and the environment,
position control is not an appropriate approach. Let us imagine a robot manipulator
cleaning a window with a sponge. As the sponge is very compliant, it is possible to
control the force between the robot and window by controlling the position between
the robot gripper and the window. If the sponge is sufficiently compliant and when
we know the position of the window accurately enough, the robot will appropriately
accomplish the task.
If the compliance of the robot tool or its environment is smaller, then it is not so
simple to execute the tasks which require contact between the robot and its environ-
ment. Let us now imagine a robot scraping paint from a glassy surface while using a
stiff tool. Any uncertainty in the position of the glassy surface or malfunction of the
robot control system will prevent satisfactory execution of the task; either the glass
will break, or the robot will wave uselessly in thin air.
In both robot tasks, i.e. cleaning a window or scraping a smooth surface, it is
more reasonable that instead of position of the glassy surface we determine the force
that the robot should exert on the environment. Most of the modern industrial robots
carry out relatively simple tasks, such as spot welding, spray painting, and various
point-to-point operations. Several robot applications, however, require control of
the contact force. A characteristic example is grinding or a similar robot machining
task. An important area of industrial robotics is also robot assembly, where several
component parts are to be assembled. In such robot tasks, sensing and controlling
the forces is of utmost importance.
Accurate operation of a robot manipulator in an uncertain, non-structured, and
changeable environment is required for efficient use of robots in an assembly task.
Here, several component parts must be brought together with high accuracy. Mea-
surement and control of the contact forces enable the required positional accuracy of
the robot manipulator to be reached. As relative measurements are used in robot force
control, the absolute errors in positioning of either the manipulator or the object are
not as critical as in robot position control. When dealing with stiff objects, already
small changes in position produce large contact forces. Measurement and control of
those forces can lead to significantly higher positional accuracy of robot movement.
When a robot is exerting force on the environment, we deal with two types of
robot tasks. In the first case we would like the robot end-effector to be brought into
a desired pose while the robot is in contact with the environment. This is the case of
robot assembly. A characteristic example is that of inserting a peg into a hole. The
robot movement must be of such nature that the contact force is reduced to zero or to
a minimal value allowed. In the second type of robot task, we require from the robot

end-effector to exert a predetermined force on the environment. This is the example
of robot grinding. Here, the robot movement depends on the difference between the
desired and the actual measured contact force.
The robot force control method will be based on control of the robot using inverse
dynamics. Because of the interaction of the robot with the environment, an additional
component, representing the contact force f, appears in the inverse dynamic model.
As the forces acting at the robot end-effector are transformed into the joint torques
by the use of the transposed Jacobian matrix (5.18), we can write the robot dynamic
model in the following form
B(q)q̈ + C(q, q̇)q̇ + Fvq̇ + g(q) = τ − JT
(q)f. (10.41)
On the right hand side of the Eq.(10.5) we added the component −JT
(q)f represent-
ing the force of interaction with the environment. It can be seen that the force f acts
through the transposed Jacobian matrix in a similar way as the joint torques (i.e., it
tries to produce robot motion). The model (10.41) can be rewritten in a shorter form
by introducing
n(q, q̇) = C(q, q̇)q̇ + Fq̇ + g(q), (10.42)
whichgivesusthefollowingdynamicmodelofarobotincontactwithitsenvironment
B(q)q̈ + n(q, q̇) = τ − JT
(q)f. (10.43)
10.3.1 Linearization of a Robot System Through Inverse
Dynamics
Let us denote the control output, representing the desired actuation torques in the
robot joints, by the vector u. Equation (10.43) can be written as follows
B(q)q̈ + n(q, q̇) + JT
(q)f = u. (10.44)
From Eq.(10.44) we express the direct dynamic model
q̈ = B−1
(q)

u − n(q, q̇) − JT
(q)f)

. (10.45)
Equation (10.45) describes the response of the robot system to the control input u.
By integrating the acceleration, while taking into account the initial velocity value,
the actual velocity of the robot motion is obtained. By integrating the velocity, while
taking into the account the initial position, we calculate the actual positions in the
robot joints. The described model is represented by the block Robot in Fig.10.13.

10.3 Control of the Contact Force 149
In a similar way as when developing the control method based on inverse dynam-
ics, we will linearize the system by including the inverse dynamic model into the
closed loop
u = B̂(q)y + n̂(q, q̇) + JT
(q)f, (10.46)
The use of circumflex denotes the estimated parameters of the robot system. The dif-
ference between Eqs.(10.46) and (10.14), representing the control based on inverse
dynamics in internal coordinates, is the component JT
(q)f, compensating the influ-
ence of external forces on the robot mechanism. The control scheme, combining
Eqs.(10.45) and (10.46), is shown in Fig.10.13. Assuming that the estimated param-
eters are equal to the actual robot parameters, it can be observed, that by introducing
the closed loop (10.46), the system is linearized because there are only two integrators
between the input y and the output q, as already demonstrated in Fig.10.7.
10.3.2 Force Control
After linearizing the control system, the input vector y must be determined. The
force control will be translated to control of the pose of the end-effector. This can
be, in a simplified way, explained with the following reasoning: if we wish the robot
to increase the force exerted on the environment, the robot end-effector must be
displaced in the direction of the action of the force. Now we can use the control
system which was developed to control the robot in the external coordinates (10.40).
The control scheme of the robot end-effector including the linearization, while taking
into account the contact force, is shown in Fig.10.14.
Up to this point we mainly summarized the knowledge of the pose control of the
robot end-effector as explained in the previous chapters. In the next step we will
determine the desired pose, velocity and acceleration of the robot end-effector, on
the basis of the force measured between the robot end-point and its environment.
Let us assume that we wish to control a constant desired force fr . With the force
wrist sensor, the contact force f is measured. The difference between the desired and
measured force represents the force error
f̃ = fr − f. (10.47)
The desired robot motion will be calculated based on the assumption that the force f̃
must displace a virtual object with inertia Bc and damping Fc. In our case the virtual
object is in fact the robot end-effector. For easier understanding, let us consider a
system with only one degree of freedom. When a force acts on such a system, an
accelerated movement will start. The movement will be determined by the force,
the mass of the object and the damping. The robot end-effector therefore behaves as
a system consisting of a mass and a damper, which are under the influence of the

x
J
T
(q)
J
T
(q)
f
f
B
−1
(q)
y
B̂(q)
n̂(q,
q̇)
u
q
q
q̇
q̇
q̈
n(q,
q̇)
Robot
Linearization
with
dynamic
model
Linearization
of
contact
force
+
+
+
+
−
Fig.
10.13
Linearization
of
the
control
system
by
implementing
the
inverse
dynamic
model
and
the
measured
contact
force

10.3 Control of the Contact Force 151
ẍr
ẋr
xr
ẋ
x
y
f
u q
q̇
Robot
Position control Inverse dynamics
Direct kinematics
Fig. 10.14 Robot control based on inverse dynamics in external coordinates including the contact
force
x
B−1
c
Fc
f̃ xc
xc
ẋc
ẍc
+
−
Fig. 10.15 Force control translated into control of the pose of robot end-effector
force f̃. For more degrees of freedom we can write the following differential equation
describing the movement of the object
f̃ = Bcẍc + Fcẋc. (10.48)
The matrices Bc and Fc determine the movement of the object under the influence of
the force f̃. From Eq.(10.48) the acceleration of the virtual object can be calculated
ẍc = B−1
c

f̃ − Fcẋc

. (10.49)
By integrating the Eq.(10.49), the velocities and the pose of the object are calculated,
as shown in Fig.10.15. In this way the reference pose xc, reference velocity ẋc, and
referenceaccelerationẍc aredeterminedfromtheforceerror.Thecalculatedvariables
are inputs to the control system, shown in Fig.10.14. In this way the force control
was translated into the already known robot control in external coordinates.
In order to also simultaneously control the pose of the robot end-effector, parallel
composition is included. Parallel composition assumes that the reference control
variables are obtained by summing the references for force control (xc, ẋc, ẍc) and
references for the pose control (xd, ẋd, ẍd). The parallel composition is defined by
equations

xc
ẋc
ẍc
xd
ẋd
ẍd
ẍr
ẋr
xr
ẋ
x
y
f
fd u q
q̇
Robot
Position
control
Inverse
dynamics
Force
controller
Parallel
composition
Direct
kinematics
Fig. 10.16 Direct force control in the external coordinates
xr = xd + xc
ẋr = ẋd + ẋc
ẍr = ẍd + ẍc
(10.50)
The control system incorporating the contact force control, parallel composition and
control of the robot based on inverse dynamics in external coordinates is shown in
Fig.10.16. The force control is obtained by selecting
xr = xc
ẋr = ẋc
ẍr = ẍc
(10.51)
The described control method enables the control of force. However, it does not
enable independent control of the pose of the robot end-effector as it is determined
by the error in the force signal.

Chapter 11
Robot Environment
This chapter will illustrate robot environments, exemplified by product assembly
processes where robots are a part of a production line or as completely independent
units. The example can be easily replicated also to other tasks, such as product
inspection and testing, welding, painting, pick and place operations etc.
As a matter of fact, robots represent an ideal solution for many industrial safety and
health problems, mainly because they are capable of performing hard and fatiguing
tasksinadangerousenvironment.Weldingandpaintingrobotsenablehumanworkers
to avoid toxic fumes and vapors. Robots also load power presses, which were frequent
causes of injuries to workers in the past. Robots work in foundries and radioactive
environments. With the increasing number of robots in industrial processes, there is,
however, an increased danger introduced by the robots themselves. Thus, considering
safety is of utmost importance when designing a robotic working cell.
11.1 Robot Safety
Industrial robots are strong devices which move quickly in their workspace. An
accident in most cases occurs only when a human worker enters the robot workspace.
A person steps into the robot vicinity either accidentally or even without knowing or
with the aim of robot reprogramming or maintenance. It is often difficult for a human
operator to judge what will be the robot’s next move. Particularly dangerous are the
unexpected robot motions, which are the consequence either of a robot failure or of a
programming error. Many governmental organizations and large companies, together
with robot producers, have developed safety standards. The approaches assuring safe
cooperation of human workers and industrial robots can be divided into three major
groups: (1) robot safety features, (2) robot workspace safeguards, and (3) personnel
training and supervision.
153

154 11 Robot environment
Today’s robots have safety features to a large extent already built-in for all three
modes of operation: normal work, programming and maintenance. Fault avoidance
features increase robot reliability and safety. Such a feature, for example, prevents
the robot from reaching into the press before it is open. The safety features built
into the robot control unit usually enable synchronization between the robot and
other machines in the robot environment. Checking the signals, indicating when a
device is ready to take an active part in the robot cell, must be part of safe robot
programming. The use of reliable sensors plays an important role when checking the
status of machines in the robot working area. Important safety features of any robot
system are also software and electric stops.
When programming or teaching a robot, the human operator must be in the robot
working area. In the programming phase the velocity of the robot motions must be
considerably lower than during normal work. The speed of the robot must be reduced
to such a value that the human operator can avoid unexpected robot motions. The
recommended maximal velocity of the robot, when there is a human worker inside
the workspace, is 0.25m/s.
The teach pendant unit can be a critical component in safe robot operation.
Programming errors during teaching of a robot often cause unexpected robot
motions. The design of a teach pendant unit can have a significant impact on safe
operation. The use of joystick control was found safer than the use of control push-
buttons. The size of emergency pushbuttons also has an important influence on the
human operator’s reaction times.
Special safety features facilitate safe robot maintenance. Such a feature is, for
example, the possibility of switching on the control system, while the robot arm is
not powered. Another feature enables passive manual motion of the robot segments,
while the robot actuators are switched off. Some robot features cause the robot to
stop as soon as possible, while some allow the control system to execute the current
command and stop afterwards.
Most robot accidents occur when persons intentionally or carelessly enter the
robot working area. The robot workspace safeguards prevent such entrance into the
robot cell. There are three major approaches to the robot workspace protection: (1)
barriers and fences, (2) presence sensing, and (3) warning signs, signals and lights.
Most commonly metal barriers or fences are used to prevent unauthorized workers
from entering the robot working area. The color of the fence plays an important
role, efficiently warning non-informed personnel. The fences are also an adequate
protection against various vehicles that are used for transporting materials in the
production hall. Safe opening of the gates, which enable entrance into the fenced-off
area, must also be provided. A human operator can only enter after switching-off the
robot system using a control panel outside the barriers. Well-designed safeguarding
barriers may also protect bystanders from objects flying out from the robot’s grasp.
Important safeguarding is provided by the devices detecting the presence of a
person in the robot working area. These can be pressure-sensitive floor mats, light
curtains, end-effector sensors, various ultrasound, capacitive, infrared or microwave
sensors inside the robot cell and computer vision. Instrumented floor mats or light
curtains can detect the entrance of a person into the robot working area. In such a

11.1 Robot Safety 155
case a warning signal is triggered and normal robot working can be stopped. The end-
effectorsensorsdetecttheunexpectedcollisionswithobjectsintherobotenvironment
and cause an emergency stop. Contactless sensors and computer vision detect various
intrusions into the robot working area.
Warning signs, signals and lights can to a large extent increase the safe operation
of robot cells. These warning signs alert the operators to the presence of a hazardous
situation. Instruction manuals and proper training are also important for effective
use of warning signs. Such signs are more effective with people who unintentionally
enter the robot working area, than with operators who are familiar with the operation
of the robot cell. Experienced operators often neglect the warnings and intentionally
enter the robot workspace without switching off the robot aiming to save some small
amount of time. Such moves are often causes of accidents. False alarms may also
reduce the effectiveness of warnings.
Selection of qualified workers, safety training and proper supervision are the pre-
requisites for safe working with robots. Especially critical moments are startup and
shutdown of a robot cell. Similarly, maintenance and programming of robots can
be dangerous. Some robot applications (e.g. welding) include specific dangerous
situations which must be well known to the workers. Those employed in the robot
environment must satisfy both physical and mental requirements for their job. The
selection of appropriate workers is an important first step. The second step, which is
equally important, is extensive safety training. Satisfactory safety is only achieved
with constant supervision of the employees. Additional training is an important com-
ponent of the application of industrial robots. In the training courses the workers must
be acquainted with the possible hazards and their severity. They must learn how to
identify and avoid hazardous situations. Common mistakes that are causes of acci-
dents should be explained in detail. Such training courses are usually prepared with
the help of robot manufacturers.
It is expected that future robots will not work behind safety guards with locked
doors or light barriers. Instead they will be working in close cooperation with humans
which leads to the fundamental concern of how to ensure a safe human-robot physical
interaction. The major progress is expected in the design of lightweight flexible robot
segments, compliant joints, novel actuators and advanced control algorithms.
The robot installation can be as an individual robot cell or as a part of a larger
industrial production line. Industrial robots are position controlled and often without
sensors for sensing their surroundings. For this reason the robots must be isolated
from human environment in case of improper activity of the robot or its peripheral
parts, to prevent human injuries or collisions with other equipment in the robot
working cell. The safety risk for each individual robot cell needs to be defined so
that appropriate precautions can be taken. Improper robot behavior can be the result
of robot system fault or human error, such as:
• unpredictable robot behavior because of a fault in the control system,
• cable connection fault because of robot movement,
• data transfer error producing unpredictable robot movement,
• robot tool fault, e.g. welding gun,

Switch
Robot
Robot
work
area
Gates
Fig. 11.1 Level 1: mechanical robot cell protection
• software errors,
• worn out robot mechanical components.
The potential dangers of system faults arising from these errors can be divided
into three categories:
• Risk of collision is the possibility that the moving robot or the tool attached to the
robot hits the operator.
• Danger of pinching is a situation where the robot, during the movement near the
objects in the robot cell (e.g. transport mechanisms), squeezes the operator.
• Other hazards that are specific to each robot application, such as the risk of elec-
trical shock, impact of welding arc, burns, toxic substances, radiation, excessive
sound levels.
For all these reasons the robot safety demands can be split in three levels.
Level 1 is the level of protection of the entire robot cell. It is usually achieved
with physical protection using a combination of mechanical fences, railings and gates
(Fig.11.1). Alongside physical protection also a human presence sensor (e.g. laser
curtains) can be installed.
Level 2 includes a level of protection while an operator is in the working area of
the robot. Normally, protection is performed by presence sensors. In contrast to the
previous level, which is based mainly on mechanical protection, level 2 is based on
the perception of the operators presence (Fig.11.2).

11.1 Robot Safety 157
Switch
Robot
Robot
work
area
Gates
Laser
Laser
curtain
curtain
Fig. 11.2 Level 2: opto-electrical robot cell protection
Fig. 11.3 Level 3: collaboration of human and the robot
Level 3 is the level of protection where people are in contact with the robot
referred to as collaborative robots. Security at this level is carried out by detecting
the presence of a human or obstacles nearby the robot or when the robot and the
human are in collaboration (Fig.11.3). In risk situations the robot system must slow
down or stop. These systems incorporate sensors for human tracking, various force
and torque sensors and contact or touch sensors. Collaborative robots are described
in more details in Chap.12.

11.2 Robot Peripherals in Assembly Processes
The robot systems installed in industry are usually a part of larger dedicated produc-
tion lines. The production lines are used for high-volume production of parts where
multiple processing operations are necessary. The production line is split in worksta-
tions where human workers, dedicated machines or robots perform necessary tasks.
Other peripherals can also be incorporated to increase the production line capabili-
ties. The properly selected peripherals also increase system reliability, flexibility and
efficiency.
11.2.1 Assembly Production Line Configurations
Assembly production lines in industry consist of conveyor belts, pallets traveling with
conveyor belts, vision systems, pneumatic cylinders, different sensors and robots or
manipulators. The pallets provide the mean to index, locate and track individual
manufactured parts traveling through the automation process. The robots provide
flexibility and can be integrated into any of the production line configurations. The
most usual assembly production line configurations with robot assistance are:
• In-line (direct, L-shaped, U-shaped, circular, rectangular),
• Rotary,
• Hybrid.
In Fig.11.4 an example of a circular in-line production line is presented. Line
workstations are served by humans, dedicated machines and robots. Parts for assem-
Finished
parts
part
pallet
pallet
pallets
Input
Input
input
output
for first
assembly
Human
Human
line
Production
robot
Collaborative
Non-collaborative robot
protected by a fence
worker in
worker in
workstation
workstation
pallet for second
Dedicated machine
assembly part
Fig. 11.4 Example of circular in-line assembly production line with human, machine and robot
workstations

11.2 Robot Peripherals in Assembly Processes 159
input
Robot
Robot
welding
Human
Human
worker in
worker in
workstation
workstation
workstation
output
Screw setting
Rotary table
object
manipulation
Conveyor belt
Fig. 11.5 Example of rotary table production line with human, machine and robot workstations
bly are manipulated by hand or by the robot and transferred among workstations
by pallets along a conveyor belt. The distance between the pallets is not necessarily
constant and their position is monitored by location sensors, usually capacitive or
inductive presence sensors. These sensors are necessary to signal the robots or a ded-
icated machine that the pallet is in the right position and the workstation operation
can be performed. The cycle time to transport the part from one workstation to the
next is usually constant, making the workstations synchronous. In certain cases the
production line developers integrate parts to buffer the pallets, making the production
line asynchronous. The need for a buffer arises in cases where some workstations
have variable cycle times; with a buffer the overall production line cycle time is not
affected.
Another very common assembly line configuration is a rotational or rotary table
(Fig.11.5), usually actuated by electrical motors. The speed and repeatability of
positioning are high. The rotary table is often called a dial table or an indexing
machine.Theadvantageoftherotaryconfigurationisthatrequireslessfloorspaceand
is often cheaper than other production line configurations. The rotary table is always
performing synchronous transfer of parts between workstations with a constant cycle
time.
As with the previous example, this configuration can also be served by humans,
robots or dedicated machines. The rotary table has a circular shape around which the
pallets or part-holders are traveling and transporting parts, in turn, into each manual

or automated workstation where production operations are performed. The rotary
table can be split in several workstations (minimum 2), making the rotation angle
of 90◦
. More common are rotary tables with more than 2 workstations, e.g. 4, 5, 6
workstations. The size of the rotary table is defined by the part size, equipment size
and number of workstations of the rotary production line. Closed loop controlled
turntables are also available.
Usually a combination of the above configurations is installed and is referred to as
hybrid production line configuration. Several factors declare the overall configuration
of the production line, such as:
• space needed for production line,
• cost of installation of the production line,
• production line cycle time.
11.3 Feeding Devices
The task of the feeding devices is to bring parts or assemblies to the robot or dedicated
machine in such a way that the part pose is known. Reliable operation of the feeding
devices is of utmost importance in the robot cells without robot vision. The position
of a part must be accurate, as the robot end-effector always moves along the same
trajectory and the part is expected to be always in the same place.
The requirements for the robot feeding devices are much more strict than in
manual assembly, unless the robot cell is equipped with a robot vision system. The
robot feeding devices must not deform the parts, must operate reliably, position the
parts accurately, work at sufficient speed, require minimal time of loading and contain
sufficient number of parts.
The feeding device should not cause any damage to the parts handled, as damaged
parts would afterwards be inserted by the robot into assemblies which cannot function
properly. The cost of such damaged assemblies is higher than the cost of a more
reliable feeding device. The feeding device must reliably handle all the parts whose
dimensions are within tolerance limits. It must also be fast enough to meet the
requirements of the whole production line cycle time and should never slow down
its operation. Further, the feeding device should require as little time as possible
for loading of the parts. It is more desirable to fill a large amount of parts into the
feeding device at once than inserting them manually one by one. The feeding devices
should contain as large number of parts as possible. This way the number of loadings
required per day is reduced.
The simplest feeding devices are pallets and fixtures; an every-day example is the
carton or plastic pallets used for eggs. The pallets store the parts, while determining
their position and sometimes also orientation. In an ideal situation the same pallet is
used for shipping the parts from the vendor and for later use in the consumer’s robot
cell. The pallets are either loaded automatically by a machine or manually. Fragile
parts, flexible objects or parts with odd shapes must be loaded manually. Loading of

11.3 Feeding Devices 161
Fixture
table
Fig. 11.6 Simultaneous loading of a fixture table
the pallets represents the weakest point of palletizing. Another disadvantage of pallets
is their rather large surface, taking up considerable area in the robot workspace.
The simplest way to bring parts into the robot cell is represented by a fixture table.
The human operator takes a part from a container, where the parts are unsorted, and
places it onto the fixture table inside the robot workspace (Fig.11.6). The fixture table
must contain special grooves which assure reliable positioning of a part into the robot
workspace. Such a fixture table is often used in welding where the component parts
must be also clamped onto the table before the robot welding takes place. The time
required for robot welding is considerably longer than loading and unloading which
can justify the use of a fixture table.
The pallets can be loaded in advance in some other place and afterwards brought
into the robot cell (Fig.11.7). This avoids a long waiting period for the robot while
the human operator is loading the pallets. The human worker must only bring the
pallet into the robot workspace and position it properly using special pins in the
working table. It is important that the pallet contains a sufficient number of the parts
to allow continuous robot operation. Exchanging the pallets in the robot workspace
represents a safety problem as the operator must switch off the robot or the robot
cell must be equipped with other safety solutions (e.g. rotary table or collaborative
robot).
A larger number of pallets can be placed on a rotary table (Fig.11.8). The rotary
table enables loading of the pallets on one side, while the robot activities take place on
the other side of the turntable. This way robot cell inactivity is considerably reduced
and the human operator is protected against the movements of the robot.
There are generally three types of pallets used: vacuum formed or, injection
molded plastic and metal pallets. Since the cost of vacuum formed pallets is low,

Fig. 11.7 Loading of the pallets in advance
Fig. 11.8 Rotary table with pallet stages
they are used both for packaging and shipping of the parts and for use in the robot
cell. Reference holes must be built into the all pallets to match pins in the worktable
to enable simple and fast positioning. As the vacuum formed pallets are inexpen-
sive, it is not difficult to understand that they are not the most accurate, reliable,
or durable. They are made of a thin sheet of plastic material which is heated and
vacuum formed over a mold. The inaccuracy of the pallet is the consequence of its
low rigidity. Injection molded plastic pallets are used when more accurate and more
durable pallets are required. The production of the mold is rather expensive, while
the cost of production of a single pallet is not high. We must keep in mind that most
vacuum and molded plastic pallets are flammable. Metal pallets are the only ones

11.3 Feeding Devices 163
Robot
In-line feeder Vibratory bowl
Transport direction
Fig. 11.9 Vibratory bowl feeder
which are non-flammable. They are produced by various machining approaches. The
metal pallets are the most reliable and durable, while their cost is higher than that of
plastic type. They are therefore only used inside the robot assembly process.
Part feeders represent another interesting family of feeding devices which are used
not only for storing parts, but also for positioning and even orienting them into the
pose appropriate for robot grasping. The most common are vibratory bowl feeders
(Fig.11.9). Here, the parts are disorderly loaded into the bowl. The vibration of the
bowl and the in-line feeder is produced by an electromagnet, and the proper vibration
is obtained by attaching the vibratory feeders to a large mass, usually a thick steel
table. The vibrations cause the parts to travel out of the bowl. Specially formed spiral
shaped fences force them into the required orientation. The same bowl feeder can
be used for different parts, however not at the same time. Another benefit is that the
bowl can hold a large number of parts while occupying only a small area in the robot
workspace. Bowl feeders are not appropriate for parts such as soft rubber objects
or springs. Another disadvantage is possible damage caused by the parts becoming
jammed in the bowl. The noise of vibratory feeders can also be disturbing.
A simple magazine feeder consists of a tube storing the parts and the sliding plate,
pneumatically or electrically actuated, which takes the parts one by one out of the
magazine (Fig.11.10). The magazine is loaded manually, so that the orientation of
the parts is known. Gravity pushes the parts into the sliding plate. The mechanism
of the sliding plate must be designed in such a way that it prevents jamming of the
parts, while only a single part is fed out from the feeder at a time. The sliding plate
must block all the parts except the bottom one.
Magazine feeders are excellent solutions for handling integrated circuits
(Fig.11.11). Integrated circuits are already shipped in tubes which can be used for
feeding purposes. The magazine feeder for integrated circuits usually consists of

Sliding
plate
Magazine feeder
Fig. 11.10 Magazine feeder
Robot
Magazine feeder
Fig. 11.11 Integrated circuit magazine feeder
several tubes. The tubes are aligned along a vibratory in-line feeder. The main dis-
advantage of magazine feeders is manual loading. They are also inappropriate for
handling large objects.
11.4 Conveyors
Conveyors are used for transport of parts, assemblies or pallets between the robot
cells. The simplest conveyor makes use of a plastic or metal chain which pushes the
pallets along a metal guide (Fig.11.12). An electrical motor drives the chain with
constant velocity. The driving force is represented by the friction between the chain
and the pallet. The pallet is stopped by special pins actuated by pneumatic cylinders.
The chain continues to slide against the bottom of the pallet. When another pallet
arrives, it is stopped by the first one. This way a queue of pallets is obtained in front
of the robot cell.
The turn of a conveyor is made by bending the metal guide. The advantages of
the sliding chain conveyor are low cost and simplicity in handling the pallets and
performing the turns. The disadvantage is that perpendicular intersections cannot be
made. Also, the turns must be made in wide arcs, which takes considerable floor

11.4 Conveyors 165
Pallet
Guide Chain
Fig. 11.12 Sliding chain conveyor (end view)
Pallet
Conveyor
Motor
Fig. 11.13 Belt conveyor
space in the production facility. The sliding chain conveyor is best suited when used
as a single loop feeding system.
With the belt-driven conveyor, the upper part of the belt is driving pallets or other
objects or material (Fig.11.13). A turn or intersection is made with the help of a
special device enabling lifting, transfer and rotation of pallets.
A conveyor can also consist of rollers which are actuated by a common driving
shaft (Fig.11.14). The driving shaft transmits torque through a drive belt to the roller
shaft. The advantage of the conveyor with rollers is in low collision forces occurring
between the pallets or objects handled by the conveyor. They are the consequence of
low friction between the rollers and the pallets. The turns are made by the use of lift
and transfer devices. The disadvantages of the conveyors with rollers are high cost
and low accelerations.
11.5 Robot Grippers and Tools
In the same way as robot manipulators are copies of the human arm, robot grippers
imitate the human hand. In most cases robot grippers are considerably simpler than
the human hand, encompassing wrist and fingers, altogether 22 degrees of freedom.
Industrial robot grippers differ to a large extent, so it is not difficult to understand
that their cost range from almost negligible to higher than the cost of a robot manip-

pallet
rollers
driving shaft
Fig. 11.14 Conveyor with rollers
Fig. 11.15 Robot gripper with two fingers
ulator. Although many various robot grippers are commercially available, it is often
necessary to develop a special gripper to meet the requirements of a specific robotic
task.
The most characteristic robot grippers are those with fingers. They can be divided
into grippers with two fingers (Fig.11.15) and multi-fingered grippers. Most multi-
fingered grippers have three fingers (Fig.11.16), to achieve a better grasping. In
industrial applications we usually encounter grippers with two fingers. The simplest
two-finger grippers are only controlled between the two states, open and closed. Two-
finger grippers, where the distance or force between the fingers can be controlled,
are also available. Multi-fingered grippers usually have three fingers, each having
three segments. Such a gripper has 9 degrees of freedom which is more than robot
manipulator. The cost of such grippers is high. In multi-fingered grippers the motors
are often not placed into the finger joints, as the fingers can become heavy or not
strong enough. Instead, the motors are all placed into the gripper palm, while tendons
connect them with pulleys in the finger joints. Apart from grippers with fingers,
in industrial robotics there are also vacuum, magnetic, perforation and adhesive
grippers. Different end-effector tools, used in spray painting, finishing or welding,
are not considered robot grippers.
Two-fingered grippers are used for grasping the parts in a robotic assembly pro-
cess. An example of such a gripper is shown in Fig.11.15. Different end-points can

11.5 Robot Grippers and Tools 167
Fig. 11.16 Robot gripper with three fingers
be attached to the fingers to adapt the robot grasp to the shape and surface of the part
or assembly to be grasped. With two-fingered robotic grippers pneumatic, hydraulic
or electrical motors are used. Hydraulic actuation enables higher grasping forces
and thus handling of heavier objects. Different structures of two-fingered grippers
are presented in Fig.11.17. Simple kinematic presentations enable the choice of an
appropriate gripper for the selected task. The gripper on the right side of Fig.11.17
enables parallel finger grasping.
In industrial processes, robot manipulators are often used for machine loading. In
such cases the robot is more efficient when using a twofold gripper. The robot can
simultaneously bring an unfinished part into the machine while taking a finished part
out of it. A twofold gripper is shown in Fig.11.18.
Specific grippers are used for grasping hot objects. Here, the actuators are placed
far from the fingers. When handling hot objects air cooling is applied, while often
the gripper is immersed into water as part of the manipulation cycle. Of utmost
importance is also the choice of appropriate material for the fingers.
When grasping lightweight and fragile objects, grippers with spring fingers can
be used. This way the maximal grasping force is constrained, while at the same time
it enables a simple way of opening and closing of the fingers. An example of a simple
gripper with two spring fingers is shown in Fig.11.19.
The shape of the object requires careful design of a two-fingered robot gripper.
A reliable grasp can be achieved either by form or force closure of the two fingers.
Also possible is the combination of the two grasp modes (Fig.11.20).
When executing a two-fingered robot grasp, the position of the fingers with respect
to the object is also important. The grasping force can be applied only on the external

Fig. 11.17 Kinematic presentations of two-fingered grippers
Fig. 11.18 Twofold robot gripper
surfaces or only on the internal surfaces of a work-piece. An intermediate grip is also
possible where the object is grasped on internal and external surfaces (Fig.11.21).
Among the robot grippers without fingers, vacuum grippers are by far the most
frequently used. Vacuum grippers or grippers with negative pressure are successfully
applied in cases, where the surface of the grasped object is flat or evenly curved,
smooth, dry and relatively clean. The advantages of these grippers are reliability, low
cost and small weight. Suction heads of various shapes are commercially available.
Often several suction heads are used together, being put into a pattern that suits the

Fig. 11.19 Gripper with spring fingers
Fig. 11.20 Form closure, force closure and combined grasp
Fig. 11.21 External, internal, and intermediate grip
shape of the object to be grasped. Figure11.22 shows the shape of two frequently
used suction heads. The head on the left is appropriate in cases when the surface is not
completely smooth. The soft material of the head adapts to the shape of the object.
The small nipples on the head presented on the right side of Fig.11.22 prevent damage
to surface of the object. Vacuum is produced either with Venturi or vacuum pumps.
The Venturi pump needs more power and produces only 70% vacuum. However, it
is often used in industrial processes because of its simplicity and low cost. Vacuum
pumps provide 90% vacuum and produce considerably less noise. In all grippers,
fast grasping and releasing of the objects is required. Releasing very lightweight and
sticky objects can be critical with vacuum grippers. In this case we release the objects
with the help of positive pressure as demonstrated in Fig.11.23.
Magneticgrippersareanotherexampleofgripperswithoutfingers:theseuseeither
permanent magnets or electromagnets. The electromagnets are used to a larger extent.

Fig. 11.22 Suction heads of vacuum grippers
Fig. 11.23 Grasping and releasing of an object with the help of negative and positive pressure
With permanent magnets the releasing of the object presents a difficulty. The problem
is solved by using a specially planned trajectory of the end-effector where the object is
retained by a fence in the robot workspace. In magnetic grippers several magnets are
used together, placed into various patterns corresponding to the shape of the object.
Already small air fissures between the magnet and the object considerably decrease
the magnetic force. The surfaces of the objects being grasped must be therefore even
and clean.
Perforation grippers are considered as special robot grippers. Here the objects are
simply pierced by the gripper. Usually these are used for handling material such as
textile or foam rubber. Such grippers can be used only in cases when perforation does
not cause damage to the object. Sheets of textile can be grasped by large brushes
made of stiff nylon hairs or simply of Velcro straps.
Adhesive grippers can be used when grasping very lightweight parts. Release of
the parts must be solved by special robot end-point trajectories where the part collides
with the fence in the robot workspace and is thus removed from the adhesive gripper.
Sufficient adhesive force is provided using adhesive tape which must move during
the operation.
Besides grippers the robot can have other tools attached to its end. The shape
and the function of the tool depends on the task of the robot cell. The most frequent
operation that robots perform is welding. For welding purposes several different
approaches can be used. Among them the most frequent tool attached to the robot

Robot
Welding gun
Fig. 11.24 Robot with welding gun attached to its end
Robot
Spot welding gun
Fig. 11.25 Robot with spot welding gun attached to its end
end is an arc welding gun or torch (Fig.11.24) to transmit welding current from a
cabletotheelectrode.Thetaskisperformedinmanydifferentareasofmanufacturing.
Besides arc welding also spot welding guns (Fig.11.25) can frequently be found in
manufacturing processes, mainly in the automotive industry.

Chapter 12
Collaborative Robots
In 1942 Isaac Asimov published the science fiction novel “I, Robot”, where the three
laws of robotics were introduced. First rule stated that “A robot may not injure a
human being or, through inaction, allow a human being to come to harm”.
Until now, industrial robots have always been fast and robust devices that work on
specific tasks designed for them. To stay in accordance with the aforementioned rule
they were performing behind fixed and interlocked guards and sensitive protective
equipment to prevent human intrusion into their workspace. With the introduction
of collaborative robots the cages are omitted as those robots are designed to work
with humans. They are built with different safety features to prevent collisions, but
if a collision occurs, the mechanism will move in the opposite direction or stop
completely to avoid causing injury.
The technical specification ISO/TS 15066:2016: Robots and robotic devices—
Collaborative robots supplements the requirements and guidance on collabora-
tive industrial robot operation provided in ISO 10218-1:2011 and ISO 10218-
2:2011 (ANSI/RIA R15.06:2012). It specifies safety requirements for collabo-
rative industrial robot systems and the work environment. Specifically, ISO/TS
15066:2016 provides comprehensive guidance for risk assessment in collaborative
robot applications.
12.1 Collaborative Industrial Robot System
A collaborative robot is a robot that can be used in a collaborative operation, where
a purposely designed robot system and a human operator work in direct cooperation
within a defined workspace. The term robot defines robot arm and robot control
and does not include the robot end-effector or part. With the term robot system we
describe robot, end-effector, and workpiece.
For the collaborative robot system we can define different workspaces (Fig.12.1):
173

174 12 Collaborative Robots
collaborative
workspace
operating
workspace
maximum
workspace
restricted
workspace
Fig. 12.1 Maximum workspace (limited by dotted line), restricted workspace (limited by dashed
line), operating workspace (grey areas), and collaborative workspace (dark grey area)
• maximum workspace: space which can be swept by the moving segments of the
robot as defined by the manufacturer plus the space which can be swept by the
end-effector and the workpiece;
• restricted workspace: portion of the maximum space restricted by limiting devices
that establish limits which will not be exceeded;
• operating workspace: portion of the restricted space that is actually used while
performing all motions commanded by the task program;
• collaborative workspace: portion of the operating space where the robot system
and a human can perform tasks concurrently during production operation.
The collaborative workspace must be designed in a way that the operator can perform
all intended tasks. The location of machinery and equipment should not introduce any
additionally safety hazards. In the collaborative workspace strict limitations about
the speed, space limits, and torque sensing are applied to guarantee operator safety.
Outside the collaborative workspace the robot can act as a traditional industrial robot
without any particular limitations excluding those that are task-related.
The term operator includes all personnel that are in contact with the robot sys-
tem, not only production operators. It includes maintenance, troubleshooting, setup,
cleaning, and production personnel.
The operational characteristics of collaborative robot systems are significantly
different from those of traditional industrial robot system presented in ISO 10218-
1:2011 and ISO 10218-2:2011. In collaborative robot operations, operators can work
in direct proximity to the robot system while the system is active, and physical con-
tact between an operator and the robot system can occur within the collaborative
workspace. As such, adequate protective measures must be introduced to collabo-

12.1 Collaborative Industrial Robot System 175
rative robot systems to ensure the operator’s safety at all times during collaborative
robot operation.
12.2 Collaborative Robot
The design of collaborative robots is moving away from heavy, stiff, and rigid indus-
trial robots towards lightweight devices with an active and/or passive compliance.
The use of lightweight high-strength metals or composite materials for robot links
contributes to small moving inertia which further affects the power consumption of
the motors. Serial manipulators can be equipped with high power/torque motors with
high transmission ratio gears in each joint or have motors positioned at the base while
the power is transferred via tendons. If the transmission ratio is small the system is
inherently back-drivable.
Use of intrinsically flexible actuators enables the design of biologically inspired
robots, as the actuators mimic the performances of human/animal muscles. The
actuators can have fixed mechanical impendence controlled via active control, such
as series elastic actuator (SEA), or the impedance can be adjusted by changing
parameters of a mechanical joint, as in variable stiffness actuator (VSA). SEA is a
combinationofmotor,gearbox,andaspring,wherethetwistofthespringismeasured
to control the force output, while that measurement of the twist of the spring is used
as a force sensor. VSA can be used to make the robot safer in the case of collision
as the joint stiffness and impact inertia are reduced. Conceptual designs of SEA and
VSA are presented in Fig.12.2.
Collaborativerobotsalsohavespecialgeometriesthatminimizethecontactenergy
transfer by maximizing the impact area. Robots have round shapes and integrated
features that reduce the risk of pinch points and the severity of an impact. Main
features of the collaborative robot are presented in Fig.12.3
To ensure a high level of safety, the robot system must include different sensors
for monitoring the state of the robot and its workspace as presented in Chap.7.
Robots can be equipped with joint torque sensors, force/torque sensors at the end-
effector, and different tactile sensors used as a soft skin or a hard shell for the robot.
All these sensors enable the robot to detect contact with the environment (operator)
or avoid collision by anticipating it and responding accordingly. Some robots use
(a) (b)
ml
ml rm
rm rg
rg
Ks Ka
τl
τl
Fig. 12.2 a Series elastic actuator (SEA), b variable stiffness actuator (VSA); rm and rg represent
motor and gearbox, Ks compliant element with fixed stiffness, Ka adjustable compliant element,
ml moving link’s mass, and τl joint torque resulting in link movement

compliant actuator
with elastic element
tactile skin
joint torque
sensor
lightweight design
with round edges
“no pinch”
design
wrist force and
torque sensor
Fig. 12.3 Design features of a collaborative robot
redundant encoders in every joint to substitute for expensive joint torques; force
can be derived from the known motor current and joint position. Robot systems can
include other safety rated sensors, e.g., safety cameras, laser scanners, laser curtains,
safety mats and other electro-sensitive protective equipment, to detect the presence
of the operator in the robot surroundings. This information can be then used for a
proper robot response to prevent clamping, compression, or crushing of the operator.
The incorporated sensors can be used for safe control of the robot. The main
paradigm is how to handle physical contact between the mechanism and the sur-
roundings. One of the most popular control schemes is impedance control, that is
based on the dynamic robot model (5.56). The dynamic model is used to assess the
necessary joint torques for proper robot movement. If the measured joint torques
deviate from the assessed one, then the difference is detected as a collision. When
a collision has been detected, the proper response strategy should be activated to
prevent potential danger to the operator. The robot can ignore the contact and fol-
low the reference trajectory, or the robot can be stopped. Other possibilities include
switching from position control to zero-gravity torque control (very high compliancy
of the robot), switch to torque control with the use of signals from joint torques to
minimize link and motor inertia (even “lighter” robot), or to use external measured
torques and switch to admittance control, where robot and collided object act as two
magnets facing with the same poles together.
The objective of collaborative robots is to combine the best of robots and of
human operator: the robot’s precision, power, and endurance coupled with the human
operator’s excellent capability for solving imprecise problems. As the robot and
the operators are collaborating in the same workspace, contact between robots and
humans is allowed. If an incidental contact does occur, then that contact should not

12.2 Collaborative Robot 177
result in pain or injury. As such, collaborative robots can be used alongside operators
and enhance the productivity of the workers. Robots are lightweight and have a small
footprint so can be easily moved around workshop, thus increasing their versatility.
Programming of collaborative robots is simple, mostly done by hand guiding, so the
use of the robot is very flexible; the robot can be operational at a new workstation in
a very short time.
12.3 Collaborative Operation
Collaborative operation is not defined with the use of the robot alone but is con-
ditioned by the task, what the robot system is doing, and the space in which the
task is being performed. Four main techniques (one or combination of more) can be
included into collaborative operation:
• safety-rated monitored stop;
• hand guiding;
• speed and separation monitoring;
• power and force limiting.
With all four techniques the robot performs in automatic mode. The main details of
all four methods are presented in Table12.1. More detailed descriptions are available
further below.
Table 12.1 Types of collaborative operations
Speed Torques Operator controls Technique
Safety-rated
monitored stop
Zero while
operator is in
collaborative
workspace
Gravity and load
compensation
only
None while
operator is in
collaborative
workspace
No motion in the
presence of the
operator
Hand guiding Safety-rated
monitored speed
As by direct
operator input
Emergency stop,
enabling device,
motion input
Motion only by
direct operator
input
Speed and
separation
monitoring
Safety-rated
monitored speed
As required to
maintain min.
separation
distance and to
execute the
application
None while
operator is in
collaborative
workspace
Prevented contact
between the robot
system and the
operator
Power and force
limiting
Max. determined
speed to limit
impact forces
Max. determined
torque to limit
static forces
As required by
application
Robot cannot
impart excessive
force (by design
or control)

12.3.1 Safety-Rated Monitored Stop
In this method the robot system must be equipped with safety-rated devices which
detect the presence of the operator inside the collaborative workspace (e.g., light
curtains or laser scanners). The operator is permitted to interact with the robot system
in the collaborative workspace only when the robot’s safety-rated monitored stop
function is active and the robot motion is stopped before the operator enters the
shared workspace. During collaborative task the robot is in standstill with the motors
powered. Robot system motion can resume only when the operator has exited the
collaborative workspace. If there is no operator in the collaborative workspace, the
robot may operate as classical industrial robot, e.g., non-collaboratively.
The operations of the safety-rated monitored stop are presented in Table12.2.
When the operator is outside the collaborative workspace the robot can perform
without any limitations. But in the case that the robot is present in the workspace
at the same time as the operator, the robot’s safety-rated monitored stop should be
active. Otherwise the robot must engage category 0 protective stop (uncontrolled
stop of the robot by immediately removing power to the actuators) in case of fault
(IEC 60204-1).
This method can be applied to applications of manual loading or unloading of
end-effector, work-in-progress inspections, and applications where only one moves
in collaborative workspace, (e.g., robot or operator). Safety-rated monitored stops
can also be integrated with other collaborative techniques.
12.3.2 Hand Guiding
For hand guiding the robot must be equipped with a special guiding device located
at or near the robot end-effector that serves for transmitting motion commands to
the robot system. The device must incorporate an emergency stop and an enabling
device unless the robot system meets inherently safe design measures or safety-
limiting functions. The location of the guiding device should enable the operator to
Table 12.2 Robot actions for safety-rated monitored stop
Operator’s proximity to collaborative workspace
Outside Inside
Robot’s proximity to
collaborative
workspace
Outside Continue Continue
Inside and moving Continue Protective stop
Inside, safety-rated
monitored stop
Continue Continue

12.3 Collaborative Operation 179
Robot in
collaborative
workspace
Operator in
collaborative
workspace
Operator in
collaborative
workspace
Continue
Continue
Continue in
automatic mode
Safety-rated
monitored
stop
Safety-rated
monitored
stop
Protective
stop
Hand
guiding
enabled
Hand
guiding
disabled
Task
YES
YES
YES
YES
YES
YES
YES
NO
NO
NO
NO
NO
NO
Collaborative
operation
Fig. 12.4 The operating sequence for hand guiding
directly observe the robot motion and prevent any hazardous situations (e.g., operator
is standing under heavy load). The control of the robot and end-effector should be
intuitively understandable and controllable.
The robot system is ready for hand guiding when it enters the collaborative
workspace and issues a safety-rated monitored stop. At this point the operator
can enter the collaborative workspace and take control of the robot system with
the hand guiding device. If the operator enters the collaborative workspace before
the system is ready for hand guiding, a protective stop must be issued. After the
safety-monitored stop is cleared the operator can perform the hand guiding task.
When the operator releases the guiding device the safety-rated monitored stop is
issued. Non-collaborative operation resumes when the operator leaves the collabo-
rative workspace. The operating sequence for hand guiding is presented in Fig.12.4.
This collaboration technique is suitable for implementation within applications
where the robot system acts as a power amplifier, in highly variable applications,

where robot system is used as a tool, and in applications where coordination of
manual and partially automated steps is needed. Hand guiding collaboration can be
successfully implemented into limited or small-batch productions.
12.3.3 Speed and Separation Monitoring
In this method the operator and robot system may move concurrently in the collabora-
tive workspace. During joint operations, the minimum protective separation distance
between the operator and robot system is maintained at all time. Protective separation
distance is the shortest permissible distance between any moving hazardous part of
the robot system and operator in the collaborative workspace.
The protective separation distance Sp at time t0 can be described by (12.1):
Sp(t0) = Sh + Sr + Ss + C + Zd + Zr , (12.1)
where Sh is the contribution to the protective separation distance attributed to the
operator’s change in location. The formula takes into account the braking distance Sr ,
which is the distance due to the robot’s reaction time, and Ss describing the distance
due to the robot system’s stopping distance. C presents the intrusion distance, which
is the distance that a part of the body can intrude into the sensing field before it is
detected. The protective separation distance Sp also includes the position uncertainty
oftheoperator Zd ,resultingfromthesensingmeasurementtolerance,andtheposition
uncertainty of the robot system Zr , resulting from the accuracy of the robot position
measurement system. The maximum permissible speeds and the minimum protective
separation distances in an application can be either variable or constant. The various
contributions to the protective separation distance are illustrated in Fig.12.5.
The robot must be equipped with a safety-rated monitored speed function and a
safety-rated monitored stop. The robot system includes also additional safety-rated
peripheral for human monitoring (e.g., safety-rated camera systems). The robot sys-
tem can maintain minimum protective separation distance by speed reduction, which
could be followed by safety-rated monitored stop, or execution of an alternate path
which does not violate the protective separation distance, as presented in Fig.12.6. If
the actual separation distance between the robot system and the operator falls below
the protective separation distance, the robot system should initiate a protective stop
and initiate safety-related functions connected to the robot system (e.g., turn off any
hazardous tools). When the operator moves away from the robot, the actual separa-
tion distance meets and exceeds the protective separation distance; at this point the
robot can resume motion automatically.
Speed and separation monitoring is useful in applications where robot system’s
and operator’s tasks run simultaneously.

speed
separation
distance
application
motion
reaction
time Tr
stopping time Ts after robot stop
time
0 t0 t0 +Tr t0 +Tr +Ts
stop
issued
stopping
begins
stopping
ends
vr
vh
S(t)
Sp(t0)
C +Zd +Zr
Sh +Sr +Ss
LEGEND
robot speed vr
operator speed vh
separation distance S(t)
protective separation
distance Sp(t0)
intrusion distance C
and uncertainty Zd +Zr
Fig. 12.5 Graphical representation of the contributions to the protective separation distance
between an operator and a robot
medium
speed
slow
speed
safety-rated
monitored
stop
protective
stop
Fig. 12.6 Safety-rated levels for maintaining minimum protective separation distance
12.3.4 Power and Force Limiting
The method of power and force limiting allows physical contact between the robot
system and the operator, that can occur either intentionally or unintentionally. The
method demands that robots be specifically designed by means of low inertia,
suitable geometry (rounded edges and corners, smooth and compliant surfaces),
materials (padding, cushioning, deformable components), and control functions. The
former includes active safety design methods, such as limiting forces and torques,

(a) (b)
Fc
vc
Fig. 12.7 a Quasi-static and b transient contact
limiting velocities of moving parts, limiting momentum by limiting moving masses,
and limiting mechanical power or energy as a function of masses and velocities. The
design of the robot can also include use of safety-rated soft axis, space limiting func-
tions, and safety-rated monitored stop functions. Some robots also include sensing
to anticipate or detect contact.
The contact between the collaborative robot and operator’s body parts could be:
• intended as part of the application sequence;
• incidental due to not following the working procedure, but without technical fail-
ure;
• a failure mode that leads to contact situations.
There are two possible types of contact between moving part of the robot system
and areas on the operator’s body. The quasi-static contact (Fig.12.7a) includes a
clamping or crushing situation in which the operator’s body part is trapped between
a moving part of the robot system and another fixed or moving part of the work
cell. In this situation, the pressure or force Fc of the robot system is applied for an
extended period of time until the conditions are alleviated. The transient contact (i.e.,
dynamic impact, Fig.12.7b) describes the contact between the moving part of the
robot system and the operator’s body part without clamping or trapping of that part.
The actual contact is shorter than the aforementioned quasi-static contact (50ms),
and depends on the inertia of the robot, the inertia of the operator’s body part, and
the relative speed vc of the two.
The robot system must be adequately designed to reduce risk to an operator by not
exceeding the applicable threshold limit values of force and pressure for quasi-static
and transient contact. The limits can apply to forces, torques, velocities, momentum,
mechanical power, joint ranges of motion, or space ranges. Threshold limit value for
the relevant contact event on the exposed body region are determined for a worst-case
scenario for both contact types.
The limit values presented in ISO/TS 15066:2016 are based on a conservative
estimate and scientific research on pain sensations. Some informative values for

Table 12.3 Biomechanical limits for quasi-static contact
Body area Maximum permissible
pressure pQS/N/cm2
Maximum permissible
force FQS/N
Seventh neck muscle 210 150
Shoulder joint 160 210
Sternum 120 140
Abdomen 140 110
Pelvis 210 180
Humerus 220 150
Forearm 180 160
Palm 260 140
Forefinger pad 300 140
Forefinger end joint 280 140
Back of the hand 200 140
Thigh 250 220
Kneecap 220 220
Shin 220 130
Calf 210 130
maximum permissible pressure and maximum permissible force between the robot’s
part and operator’s body region in quasi-static contact are presented in Table12.3.
Pressure and force values for transient contact (pT , FT ) can be at least two times the
values for quasi-static contact (pQS, FQS).
pT = 2 · pQS (12.2)
FT = 2 · FQS . (12.3)
Contact with face, skull, or forehead is not permissible and needs to be prevented.
For proper robot system reactions, both pressure and force limits must be taken
into consideration, depending on the situation. In case of clamping of operator’s body
part (e.g., operator’s hand), the resulting force can be well below the limit threshold
so the pressure limit will be the limiting factor. On the other hand, if the contact
is between two fairly large and soft areas (e.g., padded robot part and operator’s
abdomen), the resulting pressure will be below the limit threshold and the limiting
factor will then be the force limit.
In case of contact, the robot system must react in a way that the effect of the
identified contact remains below the identified threshold limit values, as presented
in Fig.12.8. In case of clamping or pinning a body part between a robot segment
and some other object, the robot must limit the speed to comply with the protective
limits. The robot should also have an integrated option for the operator to manually
extricate the affected body area.

time
force or
pressure
FT , pT
maximum actual
transient value
FS, pS
maximum actual
quasi-static value
0.5 s
transient limit for relevant body region
quasi-static limit for relevant body region
unacceptable region for force or pressure
acceptable region for force or pressure
Fig. 12.8 Graphical representation of the acceptable and unacceptable forces or pressures in case
of quasi-static or transient contact
The power and force limiting method can be used in collaborative application
where the presence of the operator is frequently needed, in time-dependant operations
(where delay due to safety-rated stops is unwanted but physical contact between the
robot system and the operator can occur), and applications with small parts and high
variability of assembly.
12.4 Collaborative Robot Grippers
The design and control of a collaborative robot enables the robot to be safe while
working together with the operator. But the robot itself is just a part of the robot
system. Grippers represent an important part of the robot system as they are used
for object manipulation in the direct vicinity of the operator. As such, grippers must
attain high level of safety.
The grippers are usually rigidly attached to the already-safe robot with built-in
speed and force limitations. The shape and materials of the gripper must coincide
with the safety design preventing exceeded pressure limits on the contact area of the
operator’s body. In addition, the grippers at the tip of the robot should create as little
inertia as possible to minimally interfere with robot’s safety features.
The design of the grippers should prevent the operator from getting their fingers
stuck in the gripper or in the connecting cables. The grippers must have implemented
a safe mode under an emergency stop, which function depends on the application.
If there is a gripped part, the operator usually wants the part to stay safely gripped.
Whenteachingandclosingthegripper,theoperatorwantsthegrippertostopapplying
the force.

12.4 Collaborative Robot Grippers 185
capacitive
sensor
force
sensor
tactile
sensor
camera
user
interface
Fig. 12.9 Conceptual design of a gripper for collaborative gripping
When the gripper is interacting with the part, the operator wants a good solid grip.
The grip also has to be secure under an emergency stop or power loss as a dropped
part could represent a danger for an operator, robot, or environment. If the robot is
moving fast, the dropped part could become a projectile.
Grippers can be equipped with different sensors to increase the operator’s safety
(Fig.12.9). Capacitive sensors are used for early operator detection and thus preven-
tion of unwanted contacts. Camera systems can detect the robot’s surroundings and
aids in object search. Tactile sensors are used to differentiate between workpiece and
operator. To set adequate gripping force, different force sensors can be integrated.
The gripper design can also include different user interfaces, such as LCD screen,
signal lights, and control buttons.
Grippers used in the collaborative robot systems should be easy to install and
program. The future design of the grippers is tending away from user programming
towards grippers that will be capable of automatic adaptation depending of the parts
and applications.
12.5 Applications of Collaborative Robotic System
The document ISO 10218-2:2011 provides the division of collaborative applications
into five categories presented in Fig.12.10.
Hand-over window application (see Fig.12.10a) covers loading/unloading, test-
ing, benching, cleaning, and service tasks. The robot is positioned behind fixed or

(a) (b) (c)
(d) (e)
Fig. 12.10 Conceptual applications of collaborative robots: a hand-over window, b interface win-
dow, c collaborative workspace, d inspection, and e hand-guided robot (ISO 10218-2:2011)
sensitive guards around the workspace where the application is performed in auto-
matic mode without limitations. Interaction with the operator is performed through a
window. In the vicinity of the window the robot reduces its speed. The window also
acts as the limit for the robot workspace.
The interface window (Fig.12.10b) acts as a barrier for the robot system. On the
robot side the robot can perform autonomous automatic operations. The robot system
is also guarded by fixed or sensitive guards around the workspace. The robot stops
at the interface window and can be then manually moved outside the interface. For
guided movement the robot must be equipped with hand guiding device. This method
is used for automatic stacking, guided assembly, guided filling, testing, benching,
and cleaning.
Applications including simple assembling and handling can take advantages of the
collaborative workspace (Fig.12.10c). Inside the common workspace the robot can
perform automatic operations. When the operator enters the collaborative workspace,
the robot reduces speed and/or stops. In this type of application, additional person-
detection systems using one or more sensors are needed.
Applications including inspection and parameters tuning (e.g., welding appli-
cation, see Fig.12.10d) require guarded workspace and person-detection systems.

12.5 Applications of Collaborative Robotic System 187
When the operator enters the shared workspace, the robot continues operation with
reduced speed. The application needs to have additional measures to prevent misuse.
Hand-guided robots (Fig.12.10e) are used for hand-guided applications (e.g.,
assembling or painting). The robot is equipped with hand-guiding device. The oper-
ator guides the robot by hand along a path in a task-specific workspace with reduced
speed. The area of collaborative workspace is mainly dependent on the hazards of
the required application.

Chapter 13
Mobile Robots
A mobile robot is a device that is capable of locomotion. It has the ability to move
around its environment using wheels, tracks, legs, or a combination of them. It
may also fly, swim, crawl, or roll. Mobile robots are used for various applications
in factories (automated guided vehicles), homes (floor cleaning devices), hospitals
(transportation of food and medications), in agriculture (fruit and vegetable picking,
fertilization, planting), for military as well as search and rescue operations. They
address the demand for flexible material handling, the desire for robots to be able to
operate on large structures, and the need for rapid reconfiguration of work areas.
Though mobile robots move in different ways, the focus in this chapter will be on
devices that use wheels for locomotion (walking robots are presented in Chap. 14).
In industrial applications automated guided vehicles (AGVs) are of special interest
to move materials around a manufacturing facility or a warehouse. Tuggers typically
pull carts (Fig.13.1a), unit loaders use a flat platform to transport a unit load stacked
on the platform (Fig.13.1b), and mobile forklifts are used to automatically pickup
and drop loads off from various heights (Fig.13.1c). AGVs typically follow markers
or wires in the floor, or use vision, magnets, or lasers for moving around the facility.
This organized movement is called navigation; a process or activity to plan and direct
a robot along a route or path to move safely from one location to another without
getting lost or colliding with other objects.
Navigationistypicallyacomplextaskconsistingoflocalization,pathplanningand
motion control. Localization denotes robot’s ability to establish its own position and
orientation within the global coordinate frame. Autonomous path planning represents
determination of a collision-free path for a robot between start and goal positions
between obstacles cluttered in a workspace. This also includes interactions between
mobile robots and humans and between groups of mobile robots. Motion control
must guarantee execution of movement along the planned path with simultaneous
obstacle avoidance.
In collaborative settings humans and robots share a workspace resulting in a need
for improved human-robot communication and for robot awareness of people around
it. The robot must typically keep a safe distance from people. However, devices like
189

190 13 Mobile Robots
(a) (b) (c)
Fig. 13.1 Automated guided vehicles: a Tugger, b unit loader, and c mobile forklift
(a) (b) (c) (d) (e)
Fig. 13.2 Wheel designs: a Standard fixed wheel, b standard steered wheel, c castor wheel,
d spherical wheel, and e Swedish wheel
personal care robots, require close proximity between the human and the robot and
these machines are examples of advanced human-robot interactive systems.
13.1 Mobile Robot Kinematics
With its simple mechanical design, the wheel is the most popular locomotion mech-
anism in mobile robotics. Wheels provide traction and three wheels guarantee stable
robot balance. Wheels can be designed in different forms as shown in Fig.13.2.
The fixed wheel, the standard steered wheel and the castor wheel have a primary
axis of rotation and are directional. Movement in different direction is not possible
without first steering the wheel around the vertical axis. The spherical wheel is
omnidirectional as it enables movement in all directions without steering first. The
Swedish wheel tries to achieve omnidirectional behavior with passive rollers attached
around the circumference of the wheel. Thus, the wheel can move along different
trajectories, as well as forwards and backwards.
Selection of wheel type, number of wheels, as well as their attachment to the robot
chassis significantly affect mobile robot kinematics. Examples of kinematic designs
are shown in Fig.13.3. They range from two-wheel to four-wheel configurations.
The two platforms in the righthand column are omnidirectional.

13.1 Mobile Robot Kinematics 191
(a) (b) (c) (d)
(e) (f) (g) (h)
Fig. 13.3 Mobile robot configuration examples: a two-wheel differential drive, b differential drive
with castor wheel, c three synchronously motorized and steered wheels, d three omnidirectional
wheels in triangle, e four wheels with car-like steering, f two differential traction wheels and two
omnidirectional wheels, g four motorized and steered castor wheels, and h four omnidirectional
wheels in rectangular configuration
For the purpose of analysis, a mobile robot will be represented as a rigid body on
wheels that can move only in a horizontal plane. With these assumptions the pose
of the robot can be defined with three coordinates, two representing position in the
horizontal plane and one describing orientation around the vertical axis. Relations
are presented in Fig.13.4 for a simple differential drive mechanism. Axes xG and yG
define the global coordinate frame. The robot local coordinate frame is defined with
axes xm and ym. The xm axis points in the robot forward direction.
Robot position and orientation are defined with the following vector
x =
⎡
⎣
x
y
ϕ
⎤
⎦ , (13.1)
where x and y coordinates define robot position relative to the global coordinate
frame and angle ϕ determines its orientation (rotation around vertical axis). Robot
orientation can be described also in the form of a rotation matrix
R =
⎡
⎣
cos ϕ − sin ϕ 0
sin ϕ cos ϕ 0
0 0 1
⎤
⎦ . (13.2)

x
y
xG
yG
xm
ym
l
2r
vl
vr
v
ω
ϕ
Fig. 13.4 Position and orientation of a mobile robot—differential drive robot example
Homogenous transformation matrix describing the pose of the mobile robot is then
T =
⎡
⎢
⎢
⎣
cos ϕ − sin ϕ 0 x
sin ϕ cos ϕ 0 y
0 0 1 0
0 0 0 1
⎤
⎥
⎥
⎦ . (13.3)
The differential drive robot presented in Fig.13.4 has a simple mechanical struc-
ture. Its movement is based on two separately driven wheels attached on either side
of the robot body. The robot changes its direction by varying the relative speed of
rotation of its wheels. Thus, it does not require an additional steering motion. If
wheels are driven in the same direction and with equal speed, the robot will follow a
straight line. If wheels are turned with equal speed in opposite directions, the robot
will rotate about the middle point between the wheels. In general, the center of robot
rotation may lay anywhere on the line through wheel axes and will depend on each
wheel speed of rotation and its direction.
With its simple kinematics it is an ideal model for studying robot movement. By
representing robot width (distance between tire contact points with the ground) with
l and wheel radius with r the robot motion can be analyzed. The wheels rotate with

xG
yG
xm
ym
x
y
l
vl
vr
v
ω
ϕ
R
L
D
ICR
Fig. 13.5 Differential drive robot kinematics
angular rates ωr (right wheel) and ωl (left wheel), resulting in wheel speeds vr and
vl of the right and left wheel, respectively
vr = ωrr,
vl = ωlr.
(13.4)
The two wheel rotations result in the robot translational speed along robot xm axis
and angular rate around its vertical axis. With reference to Fig.13.5 the angular rate
can be defined as
ω =
vl
D − l
2
=
vr
D + l
2
, (13.5)
where D is the distance between the middle point on the robot (in this case the origin
of the frame xm–ym) and the point that defines the instantaneous center of rotation
(ICR). The ICR is the point in the horizontal plane around which the robot rotates
at a specific instant of time. From equality in (13.5) the following relation can be
derived
ω =
vr − vl
l
=
r
l
(ωr − ωl). (13.6)
Translational speed along the xm axis can then be determined as

v = ωD =
vr + vl
2
=
r
2
(ωr + ωl). (13.7)
Equations (13.6) and (13.7) define relations between wheels’ angular rates and
mobile robot velocity. However, from the control perspective it is the more relevant
inverse relation that defines wheels’ angular rates from the desired robot velocity.
By combining (13.6) and (13.7) the following relations are obtained
ωr =
2v + ωl
2r
,
ωl =
2v − ωl
2r
.
(13.8)
Robot velocity determined as a pair [v, ω] is defined relative to the local coordinate
frame of the mobile robot xm–ym. Robot velocity in the global coordinate frame
xG–yG defined as time derivative of robot pose vector x (13.1) can be computed by
rotating the locally expressed velocity using the rotation matrix R (13.2) as
⎡
⎣
cos ϕ − sin ϕ 0
sin ϕ cos ϕ 0
0 0 1
⎤
⎦
⎡
⎣
v
0
0
⎤
⎦ =
⎡
⎣
v cos ϕ
v sin ϕ
0
⎤
⎦ ,
⎡
⎣
cos ϕ − sin ϕ 0
sin ϕ cos ϕ 0
0 0 1
⎤
⎦
⎡
⎣
0
0
ω
⎤
⎦ =
⎡
⎣
0
0
ω
⎤
⎦ .
(13.9)
By combining translational and rotation parts of the above equations and omitting
elements that are zero, the mobile robot velocity in the global coordinate frame can
be written as
ẋ =
⎡
⎣
ẋ
ẏ
ϕ̇
⎤
⎦ =
⎡
⎣
v cos ϕ
v sin ϕ
ω
⎤
⎦ . (13.10)
From Eq. (13.10) it is clear that relevant quantities for describing mobile robot
movement are translational velocity along robot xm axis v, rotational velocity around
vertical axis ω, and robot orientation with respect to the global coordinate frame ϕ.
With this in mind we may further simplify the differential drive robot into a unicycle
model (as shown in Fig.13.6). Now the above-mentioned three quantities describe
the movement of the unicycle represented as a single wheel with marked forward
direction in the middle of the differential drive robot in Fig.13.6. The unicycle can
be easily transformed back to the differential drive robot based on Eq. (13.8).
The attractive property of the unicycle model is its simplicity. Therefore, it will be
used throughout this chapter for analysis. However, the model can be in general con-
verted back to any other kinematically more complex mobile robot. As an example,
we review a mobile platform based on the car steering principle shown in Fig.13.7.
The car steering geometry solves the problem of wheels on the inside and outside
of a turn needing to trace circles of different radii. Therefore, steering angles of left
and right front wheels are different. In the unicycle model the orientation of the
unicycle is defined with angle ϕ, the same as the orientation of the differential drive
robot.

xG
yG
xm
ym
x
y
vl
vr
v
ω
ϕ
Fig. 13.6 Unicycle model of a differential drive mobile robot
In the car-like problem the orientation of the mobile robot is defined by angle ϕ.
The unicycle model is positioned in the middle of the front wheels and its orientation
is defined such to achieve the same instantaneous center of rotation as defined by the
orientation of the car’s left and right wheels. The unicycle is now the third front wheel
and the ICR is positioned at the intersection point of all the three lines perpendicular to
the front wheels. Angle ψ is now defined as the deviation of the unicycle orientation
from the robot xm axis (as shown in Fig.13.7). By computing angle ψ the relation
between the car-like robot and the unicycle will be established.
By following the same principle as in (13.7), translational velocity of the unicycle
can be defined as
v = Dω, (13.11)
where D is the distance between the unicycle and the ICR. Distance D can then be
computed as
D =
v
ω
. (13.12)
Path curvature for the unicycle Ku can be defined as the inverse of the instantaneous
radius of rotation as
Ku =
1
D
=
ω
v
. (13.13)

h
xG
yG
xm
ym
x
y
l
vl
vr
v
ω
ϕ
D ψ
ψ
R
ICR
Fig. 13.7 Unicycle model of the car-like steering mobile robot
By considering car kinematics, the following relation can be written from Fig.13.7.
h = D sin ψ, (13.14)
where angle ψ is also the angle between lines D and R (the distance between ICR
and the middle point between the rear wheels of the vehicle) and h is the distance
between the center of the unicycle and the middle point between the rear wheels of
the robot. Distance D can then be computed as
D =
h
sin ψ
(13.15)
and the curvature for the car Kc is then defined as
Kc =
1
D
=
sin ψ
h
. (13.16)
With equal Kc and Ku the following relation can be obtained
Kc = Ku ⇒ sin ψ =
ωl
v
. (13.17)

Finally, the angle ψ equals
ψ = arcsin
ωl
v
. (13.18)
Angle ψ is the desired steering angle for the car and it can be computed from the
known speed v, angular rate ω, and width of the car l.
With the defined relation between the unicycle and a mobile robot with other
kinematics the analysis can be based on a simple unicycle model and generalized to
the other robot.
13.2 Navigation
Mobile robots often operate in unknown and unstructured environments and need to
self-localize, plan a path to a goal, build and interpret the map of the environment,
and then control their motion through that environment.
13.2.1 Localization
An important difference between a manipulator and a mobile robot is in position
estimation. A manipulator has a fixed base and by measuring robot joint positions
and knowing its kinematic model it is possible to determine the pose of its end-
effector. A mobile robot can move as one unit through the environment and there
is no direct way for measuring its position and orientation. A general solution is to
estimate the robot position and orientation through integration of motion (velocity)
over time.
However, more accurate and often also more complex approaches are typically
required. If the map of the environment is known in advance mobile robot paths can
be preplanned. This is specifically useful when the environment is relatively static
and robust operation is required, such as in industrial applications. More complex
approaches are based on dynamic path planning based on sensor information and
recognition of features in the environment. The robot first determines its own position
and plans its movement through traversable areas. When the workspace or the tasks
change frequently it is typically better to plan dynamically. Often a trade-off is
required between preplanning and dynamic generation of plans. In order to simplify
the task, markers may be placed in the environment. These markers can be easily
recognized by sensors on the robot and provide accurate localization.
Automated guided vehicles in industrial environments make use of various navi-
gation/guidance technologies: magnetic tape, wire, magnetic spot, laser, and natural.
Localization and path planning are often based on electrified wires embedded in
the floor using inductive guidance. A guide path sensor is mounted on the vehicle.
The wire can be replaced by magnetic tape or a painted line (Fig.13.8a). In the latter

(a) (b)
(c) (d)
wire or magnetic tape magnetic spots
reflective marker
spinning laser
laser scanning
Fig. 13.8 Sensor abstraction disk from the suit of sensors on board the robot
case the robot uses a camera to determine its relative position to the floor line. Paths
are fixed and continuous. Unique markers may be placed along the line to indicate
specific positions. Instead of placing lines and markers on the floor, markers (two-
dimensional patterns) can also be put on the ceiling to be identified by an onboard
camera. Magnetic spot guidance uses path marked with magnetic pucks (Fig.13.8b).
Paths are open and changeable.
Floor-based localization techniques are often replaced by laser-based methods.
Laser triangulation methods, in which a spinning laser senses range and azimuth
to wall-mounted reflectors, provide accurate localization information without the
need to follow specific lines on the floor. Laser guidance technology uses multiple,
fixed reference points (reflective strips) located within the operating area that can
be detected by a laser head mounted on the vehicle (Fig.13.8c). As the facility is
mapped in advance, paths can be easily changed and expanded.
Natural navigation is based on information of the existing environment scanned
by laser scanners, with the aid of a few fixed reference points (Fig.13.8d). Area is
mapped in advance. Natural navigation is flexible and expendable. It is suitable for
environments that change frequently but not significantly. In confined spaces the
robot may follow the wall through the environment range-basing from the wall.

13.2 Navigation 199
Radio-based indoor positioning systems are also being introduced that enable
robot localization in a similar manner as the outdoor global positioning system.
Localization is based on triangulation with fixed beacons mounted in the facility and
the sensor mounted on the robot. Distances are computed by measuring travel time
of radio waves from the beacon to the sensor.
13.2.1.1 Odometry
A simple and commonly-used approach for robot localization is to rely on odometry,
which uses information from motion sensors (typically wheel encoders) to estimate
change in position over time. These position changes are accumulated using integra-
tion principles providing the robot position relative to a starting location. The method
is sensitive to errors due to integration of velocity measurements over time to give
position estimates.
Analysis of robot motion starts with the understanding of the contribution of
each wheel to the velocity of the robot. For the specific case of a differential drive
robot these relations are defined in (13.6) and (13.7). Wheel speed may be directly
measured using a tachometer. If such a sensor is not available, the speed can be
estimated through numerical differentiation of the position obtained from encoders.
In such case speeds for the right and left wheel can be computed as
vr = 2πr
nr(t) − nr(t − Δt)
NΔt
,
vl = 2πr
nl(t) − nl(t − Δt)
NΔt
,
(13.19)
where r is the wheel radius, N is the encoder resolution in terms of counts per revo-
lution, nr and nl are encoder counts of the right and left wheel at time t, respectively,
and nr(t − Δt) and nl(t − Δt) are the same quantities at the previous sampling time.
Robot position and orientation can then be estimated with numerical integration
of Eq. (13.10) and consideration of (13.6) and (13.7) as
x(t) = x(t − Δt) + v cos ϕΔt = x(t − Δt) +
vr + vl
2
cos ϕΔt,
y(t) = y(t − Δt) + v sin ϕΔt = x(t − Δt) +
vr + vl
2
sin ϕΔt,
ϕ(t) = ϕ(t − Δt) + ωΔt = ϕ(t − Δt) +
vr − vl
l
Δt.
(13.20)
Different factors reduce the effectiveness of odometry-based methods for robot
position estimation. A very important factor is wheel slippage that significantly
reduces precision of position estimation. Performance may be improved by using
models of the errors and of the vehicle. Floor spots or magnets may be used to cor-
rect for odometry errors that accumulate between these points. Odometry can also

be augmented by sensor-based measurements from lasers, cameras, radiofrequency
identification systems, and beacons.
13.2.1.2 Simultaneous Localization and Mapping
More advanced systems make use of algorithms that accomplish the navigation sub-
tasks (localization, path planning) simultaneously. The approach that is concerned
with the problem of building a map of an unknown environment by a mobile robot
while at the same time navigating the environment using the map is called simultane-
ous localization and mapping (SLAM). By observing the same features in multiple
views using sensors that move with the vehicle, the SLAM algorithm accumulates
and combines together the sensor information. By combining the robot position esti-
mation with the gathered information, a local map can be constructed by stitching
together available data. Over time, the complete environment can be mapped and the
maps can be used to plan the robot paths.
SLAM consists of multiple parts, such as landmark extraction, data association,
state estimation, state update and landmark update. There are many ways to solve
each of the smaller parts, but they are beyond the scope of this book.
13.2.1.3 Sensor Abstraction Disk
When the mobile robot is moving through the environment it must also observe its
surroundings. Sensors on-board the robot look for obstacles or unexpected objects
in the path of the vehicle and the robot may be able to plan a way around them
before returning to the pre-planned route. A typical suite of sensors includes infrared
proximity sensors, ultrasonic distance sensors, laser scanners, vision, tactile sensing,
and global positioning sensors. Sensors are strategically placed onboard the robot
and around its circumference. Each sensor provides different information in terms
of quantity, quality, range, and resolution. However, typically information from all
sensors is combined to provide an accurate image of the robot environment. Without
dealing specifically with analysis of individual sensors and integration of sensory
information it is possible to assume that distance and direction to all obstacles from
the robot’s perspective can be obtained from the sensor suite. The sensor abstraction
disk presented in Fig.13.9 is an example of sensory integration providing information
about obstacles within the radius of the disk around the robot.
From the known position do and orientation ϕo of the obstacle and the known
pose of the robot [x, y, ϕ]T
, it is possible to determine the obstacle position (xo, yo)
in the global coordinate frame as
xo = x + do cos(ϕ + ϕo),
yo = y + do sin(ϕ + ϕo).
(13.21)

13.2 Navigation 201
xG
yG
xm
ym
xo
yo
x
y
ϕ
ϕo
do
Fig. 13.9 Sensor abstraction disk from the suit of sensors on board the robot
The following analysis will be based on the assumptions of a unicycle robot model
and the information about objects obtained from the sensor abstraction disk.
13.2.2 Path Planning
Path planning enables autonomous mobile robots to track an optimal collision free
path from the starting position to the goal without colliding with obstacles in the
workspace area. An ideal path planner must be able to handle uncertainties in the
sensed world model, to minimize the impact of objects to the robot, and to find the
optimum path in minimum time especially if the path is to be negotiated regularly.
In general, the path planning should result in the path with the lowest possible cost,
it should be fast and robust as well as generic with respect to different maps.
Different algorithms are available for (real-time) path planning. A simple method
consists of combining straight-line segments connected with vertices. Another stan-
dard search method for finding the optimal path is the A* algorithm with its modi-
fications. The algorithm finds a directed path between multiple points, called nodes.
The robot environment represented with a map can be decomposed into free and
occupied spaces. Then A* search can be performed to find a piecewise linear path
through the free nodes.
An artificial potential field algorithm can be used for obstacle avoidance. The
algorithm uses repulsive potential fields around the obstacles to force away the robot
subjected to this potential and use an attractive potential field around the goal to
attract the robot to go to the goal. Repulsive and attractive fields modify the robot’s
path. The algorithm enables real-time operations of a mobile robot in a complex
environment.

xG
yG
xm
ym
xmr
ymr
x
y
ϕ
ϕr
Fig. 13.10 Unicycle orientation control; grey unicycle represents actual robot and white unicycle
represents desired orientation
13.2.3 Path Control
In order to complete the task, the mobile robot needs to move from its initial location
to the desired final position and orientation. A control system is required to control
the vehicle along its path.
13.2.3.1 Control of Orientation
Based on the unicycle model presented in Fig.13.10 control of orientation will first
be considered. A similar approach would be valid for mobile robots that can change
orientationwithoutchangingtheirposition(adifferentialdriverobotissuchavehicle,
but the car is not).
The control goal is to minimize the orientation error
ϕ̃ = ϕr − ϕ, (13.22)
where ϕr is the desired orientation and ϕ is the actual orientation. We assume that
the control is based on proportional-integral-derivative (PID) control approach
PID(ϕ̃) = Kpϕ̃ + Ki

ϕ̃dt + Kd
˙
ϕ̃ (13.23)
or one of its subversions, such as proportional-derivative controller. Then the desired
angular velocity of the mobile robot can be computed as

13.2 Navigation 203
xG
yG
xm
ym
xr
yr
xmr
ymr
x
y
S
ϕr
ϕ
Fig. 13.11 Unicycle position and orientation control; grey unicycle represents actual robot and
white unicycle represents goal location
ω = Kpϕ̃ + Ki

ϕ̃dt + Kd
˙
ϕ̃. (13.24)
It should be noted that angles are periodic functions and if we assume configuration
ϕr = 0 ∧ ϕ = 2π ⇒ ϕ̃ = −2π, (13.25)
the robot will spin once before it will reach the final orientation. This is usually not
desirable robot behavior. Therefore, orientation error must be limited such to require
at maximum π radians rotation in either direction
ϕ̃ ∈ [−π, π]. (13.26)
A simple solution is to use a four-quadrant arctan function as
ϕ̃ = arctan(sin ϕ̃, cos ϕ̃) ∈ [−π, π]. (13.27)
With the combination of (13.27) and (13.24) the robot will reach the desired orien-
tation without rotating more than half circle in positive or negative direction.
13.2.3.2 Control of Position and Orientation
The mobile robot typically moves from its initial location to its final (goal) location
which requires change of position and orientation. Since the robot needs to move to
its goal location we will refer to this task as go-to-goal. Figure13.11 represents such

conditions. Coordinate frame xm–ym defines the robot current pose and frame xmr
–ymr
defines the goal pose. Line segment S represents the shortest path for completing the
task.
The desired robot orientation for completing the task can be defined as the angle
between line segment S and the horizontal axis of the global coordinate frame. With
the known desired position (xr, yr) and robot current position (x, y), angle ϕr can
be computed at every time instant during robot motion as
ϕr = arctan
yr − y
xr − x
. (13.28)
By assuming that the robot is moving at constant forward speed v0, robot movement
in the global coordinate frame can be described with the following set of equations
ẋ = v0 cos ϕ,
ẏ = v0 sin ϕ,
ϕ̇ = ω = PID(ϕ̃).
(13.29)
With this approach the control goal is to maintain constant speed v0 and track the
desired angle ϕr computed from (13.28). If we assume a differential drive robot,
wheel angular rates can then be computed from (13.8) as
ωr =
2v0 + ωl
2r
,
ωl =
2v0 − ωl
2r
.
(13.30)
When moving with constant velocity v0 the robot would overshoot its goal loca-
tion. Therefore, it is reasonable to define robot forward speed based on the distance
to the goal
G = (xr − x)2 + (yr − y)2. (13.31)
With a proportional controller, the desired speed can be defined as
vG = KvG, (13.32)
where Kv is the velocity gain. Equations (13.29) can then be rewritten as
ẋ = vG cos ϕ,
ẏ = vG sin ϕ,
ϕ̇ = ω = PID(ϕ̃)
(13.33)

13.2 Navigation 205
xG
yG
xm
ym
xmr
ymr
xr
yr
x
y
ϕr
ϕ
Obstacle
Fig. 13.12 Unicycle position and orientation control with obstacle avoidance; grey unicycle repre-
sents actual robot and white unicycle represents goal location; gray circle is the obstacle and dashed
circular line is safe zone around the obstacle
and in (13.30) v0 must be replaced by vG. With this approach the robot will decelerate
when approaching the goal location. Since desired speed increases with the distance
to the goal, a maximum limit can be set on vG ∈ [0, vGmax
].
13.2.3.3 Obstacle Avoidance
Figure13.12 shows conditions with an obstacle in the robot’s path to the goal position.
The robot cannot proceed directly to the target location without first avoiding the
obstacle. Based on the concept of the sensor abstraction disk we assume that the robot
is capable of detecting and locating the obstacle from a safe distance and using this
information, can plan avoidance activities. The obstacle in Fig.13.12 is represented
by a gray circle and the dashed circular line around the obstacle represents a safe
zone around the obstacle. The robot would not be allowed to enter the dashed circle.
With this in mind, we now have two control objectives. The first is go-to-goal
and the second is avoid-obstacle. A more detailed representation of the two control
objectives is shown in Fig.13.13, where do indicates distance from the robot to the
obstacle, ug is the control variable associated with the go-to-goal objective and uo is
the control variable associated with avoid-obstacle objective. In order to successfully
complete the task, the ug needs to point to the goal while the uo needs to point away
from the obstacle. The actual control variable u is the result of blending ug and uo.
The go-to-goal control part can be defined based on the distance to the goal
position as
ugx
ugy
= Kg
xr − x
yr − y
. (13.34)

xG
yG
xr
yr
xo
yo
do u
uo
uo
ug
x
y
ϕr
Obstacle
Fig. 13.13 Unicycle obstacle avoidance; grey unicycle represents actual robot, white unicycle
represents goal location and grey circle is obstacle
Similarly, the avoid-obstacle control variable can be defined based on the distance
to the obstacle
uox
uoy
= Ko
x − xo
y − yo
. (13.35)
It should be noted that ug points to the goal and uo points away from the obstacle
as seen by the definition of distances in the above two equations. Blending of the
two control variables must be made based on the distance to the obstacle, which is
defined as
do = (xo − x)2 + (yo − y)2. (13.36)
When the robot is far away from the obstacle, it only needs to proceed directly to
the goal. However, in the vicinity of the obstacle the primary task becomes obstacle
avoidance. Consecutively, blending can be implemented as
ux
uy
= λ(do)
ugx
ugy
+ (1 − λ(do))
uox
uoy
, λ(do) ∈ [0, 1]. (13.37)
Parameter λ can, for example, be defined as an exponential function based on distance
to the obstacle λ = 1 − e−κdo
and parameter κ defines convergence rate of the
function toward 1. As seen from Fig.13.13 control variable u defines desired robot
velocities in the global coordinate frame
ẋ
ẏ
=
ux
uy
. (13.38)

13.2 Navigation 207
xG
yG
xm
ym
xmr
ymr
xr
yr
x
y
x̃m
ỹm
ϕr
ϕ
Fig. 13.14 Unicycle path following control; grey unicycle represents actual robot and white uni-
cycle represents virtual vehicle on the path
Desired robot orientation can then be computed as
ϕr = arctan
uy
ux
, (13.39)
resulting in angular rate
ϕ̇ = ω = PID(ϕ̃). (13.40)
The forward robot speed can be computed as
v = ẋ2 + ẏ2 = v2 cos2 ϕ + v2 sin2
ϕ = u2
x + u2
y. (13.41)
Again, by assuming a differential drive robot, wheel angular rates can be computed
from (13.8).
13.2.3.4 Path Following
Often the robot cannot just take the shortest path to the goal and it must follow
a predefined path. In this case the control goal is to stay on the path. The task
can be simplified by considering a virtual vehicle that moves along the path with
a predefined speed. Then the control goal becomes tracking the virtual vehicle as
shown in Fig.13.14.

The tracking error can be defined as
x̃ = xr − x, (13.42)
where xr and x represent position and orientation of the virtual vehicle and the mobile
robot, respectively. All quantities are expressed in the global coordinate frame and
can be transformed into the robot coordinate frame as
x̃m
=
⎡
⎣
x̃m
ỹm
ϕ̃m
⎤
⎦ = RT
x̃, (13.43)
where R is defined as in (13.2). The robot forward speed can be computed from the
tracking error along xm axis as
v = Kxx̃m
, (13.44)
where Kx is the controller proportional gain. The angular rate must take into account
the angle tracking error ϕ̃m
= ϕ̃ as well as distance to the path ỹm
. Namely, when the
robot is away from the path it must steer toward the path. Thus, the control algorithm
becomes
ω = Kyỹm
+ Kϕϕ̃m
, (13.45)
where Ky and Kϕ are controller proportional gains. Since velocity of the virtual
vehicle is known (angular rate can be computed as the change of tangential direction
along the path when the virtual vehicle moves forward), it can be taken into account
as a feedforward control term. If vr is the forward speed of the virtual vehicle and
ωr its angular rate, Eqs. (13.44) and (13.45) can be rewritten with the feedforward
term as
v = vr cos ϕ̃ + Kxx̃m
(13.46)
and
ω = ωr + Kyỹm
+ Kϕϕ̃m
. (13.47)

Chapter 14
Humanoid Robotics
Even before modern robotics began to develop, philosophers, engineers, and artists
were interested in machines similar to humans. The first known example of a
humanoid mechanism, which design has been preserved and can still be rebuilt today,
is a mechanical knight created by Leonardo da Vinci and presented to the Milanese
ruler Ludovico Sforza around 1495. The mechanism had a kinematic structure sim-
ilar to present humanoid robots and it could move by a system of wires and pulleys.
More recently writers like Karel Čapek and Isaac Asimov thought of robots that
have a form similar to humans. There are several reasons why humanoid robots are
thought to be interesting:
• Human environments are built for humans, therefore a general-purpose robot
designed for human environments, e.g., homes, factories, hospitals, schools, etc.,
should have a form similar to humans to successfully operate in such environments.
• It is more natural for humans to interact and communicate with robots that look
and behave in like humans.
• A humanoid robot can serve as an experimental tool to test the theories about
human behavior created by computational neuroscientists, interested in how the
human brain operates.
It can be said that modern humanoid robotics started with a series of humanoid robots
created at the University of Waseda in Tokyo, Japan. The first of these robots was
WABOT-1 created in 1973.
Despite recent progress in related areas such a soft robotics and artificial intelli-
gence, humanoid robots that can operate in human-populated environments, where
they collaborate and communicate with people in a natural way, are still only a dis-
tant dream. Currently, humanoid robots are at the stage where they can execute a
variety of tasks. Tasks that are for example used in humanoid robot competitions,
e.g. DARPA Robotics Challenge, include:
1. Drive: drive a utility vehicle down a lane blocked with barriers.
2. Egress: get out of the vehicle and locomote to a specified area.
3. Door: open a door and travel through a doorway.
4. Valve: turn a valve actuated by a hand-wheel.
209

210 14 Humanoid Robotics
5. Wall: use a tool (drill or saw) to cut through a concrete panel.
6. Surprise task, which was not known until the day of competition: remove a
magnetic plug from one socket, insert it in a different socket.
7. Rubble: cross a debris field or negotiate irregular terrain.
8. Stairs: climb the stairs.
Modern humanoid robots can already execute such tasks autonomously, providing
the approximate state of the environment is known in advance. However, it is still
difficult for modern humanoid robots to perform such tasks without some prior infor-
mation about the environmental conditions that can be exploited by a programmer
to prepare the humanoid robot for the execution of multiple tasks. Integration and
continuous sequencing of multiple robot actions remains a problem and some degree
of teleoperation is still needed when performing longer task sequences.
While most of the standard robotics methodologies regarding robot kinematics,
dynamics, control, trajectory planning, and sensing are relevant also when develop-
ing humanoid robots, humanoid robotics needs to deal with several specific issues.
The foremost is the problem of biped locomotion and balance. Unlike other robots,
humanoid robots must walk and keep balance during their operation. In the afore-
mentioned robotics challenge, locomotion turned out to be one of the biggest issues.
The basic indicator that describes the balance of a humanoid robot is the concept
of zero-moment point, usually abbreviated as ZMP. The concept of ZMP was intro-
duced by Miomir Vukobratović in 1968. It is still the most widely used approach for
generating dynamically stable walking movements in which the supporting foot or
feet keep contact with the ground surface at all times. This is important to prevent
the robot from falling. The basic concepts related to ZMP are described in Sect.14.1.
Another specific issue that arises when programming humanoid robots is the very
high number of degrees of freedom they require compared to standard industrial
robots. While typical industrial robots only have 6 and seldom 7 degree of freedom,
humanoid robots often have more than 30 degree of freedom. For example, one of
the best known humanoid robots Honda Asimo has 34 degree of freedom: 3 in the
head, 7 in each arm (3 in the shoulder, 1 in the elbow, and 3 in the wrist), 1 in the
waist, 6 in each leg, and 2 in each hand. Such a large number of degrees of freedom
makes classical robot programming with teach pendants and textual programming
languagesimpractical.Insteadwecanexploitthesimilaritybetweenhumanoidrobots
and humans. Because of this similarity, humanoid robots can perform tasks in a sim-
ilar way as humans do. This fact gives rise to an idea that instead of programming a
humanoid robot, a human teacher can show to the robot how to execute the desired
task. The robot can then attempt to replicate the human execution. This way of robot
programming is called programming by demonstration or imitation learning. Its suc-
cessful application requires that a robot transfers the demonstrated motion to its own
kinematic and dynamic structure. Furthermore, since natural environments are rarely
static but often change, the robot cannot simply replicate the observed movements.
Instead, the observed movements should be adapted to the current environmental
conditions. These topics are discussed in Sect.14.2.

14.1 Biped Locomotion 211
14.1 Biped Locomotion
Biped locomotion is an important topic in humanoid robotics. Here we focus on
walking, which is distinguished from other forms of biped locomotion such as run-
ning by the constraint that at least one foot must always be in contact with the ground.
As explained in the introduction, most of the modern humanoid robots exploit the
zero-moment point principle to generate stable walking patterns.
14.1.1 Zero-Moment Point
Throughout this section, we assume that the floor is flat and orthogonal to gravity. We
start by analyzing the distribution of a vertical component of ground reaction forces
(i.e., the component orthogonal to the ground, as shown in Fig.14.1). The zero-
moment point is defined as the point where the resultant of these forces intersects
with the ground. We first focus on the motion in the sagittal plane (i.e., the plane that
divides the body into the left and right part). As shown in Fig.14.1, a component of
the ground reaction force orthogonal to the ground must be positive at all contact
points, otherwise the foot would lose contact with the ground as it is not rigidly
attached to it. The zero-moment point px according to the above definition can be
calculated as follows
px =
x f
xb
x fz(x)dx
fn
, (14.1)
fn =
x f
xb
fz(x)dx, (14.2)
where fz(x) is the vertical component of the ground reaction force at contact point
x and fn the net vertical ground reaction force. The reason why px is called zero-
moment point becomes clear if the moment at px is calculated:
τ(px ) = −
x f
xb
(x − px ) fz(x)dx
fn
= −
⎛
⎜
⎜
⎝
x f
xb
x fz(x)dx
fn
− px
x f
xb
fz(x)dx
fn
⎞
⎟
⎟
⎠
= −(px − px ) = 0. (14.3)
Here we integrated the moment τ = −(x − px ) fz across the whole sole area, i.e.
xb ≤ x ≤ x f . Thus the net moment at the zero-moment point px is equal to zero. The
zero-moment point is usually abbreviated as ZMP. It is the point on the ground surface
where the net angular momentum is equal to zero. If it exists, ZMP is constrained to
lie within the support polygon.

xb xf ground surface
fz(xb) fz(xi) fz(xf )
fn
px
··· ···
Fig. 14.1 Ground reaction forces fz(xi ) at different contact points xi . The zero-moment point px
and the net ground reaction force orthogonal to the support surface fn are calculated according to
Eqs. (14.1) and (14.2), respectively
full contact
with one foot
full contact
with both feet
support
polygon
support
polygon
ZMP
ZMP
vertical ground
reaction forces
vertical ground
reaction forces
Fig. 14.2 Support polygon (the area enclosed by a gray line) is defined as the convex hull of all
points in contact with the ground. Left: support polygon corresponds to the sole area when only
one foot is in full contact with the ground. Right: support polygon corresponds to the convex hull
of the corners of both feet when both feet are in full contact with the ground
For general humanoid robot walking in 3-D, lateral motion should also be consid-
ered. As shown in Fig.14.2, we must distinguish between two cases: either only one
foot is in full contact with the ground or both feet are in full contact with the ground.
The ground is assumed to be flat at height pz. The derivation of ZMP is based on the
relationship between the moment about point p
p
p = (px , py, pz) of the vertical ground
reaction force [0, 0, fz(ξ
ξ
ξ)]T
at all points ξ
ξ
ξ = (ξx , ξy, pz) on the contact surface. The
moment is given by
τ
τ
τ(p
p
p) = (ξ
ξ
ξ − p
p
p) ×
⎡
⎣
0
0
fz(ξ
ξ
ξ)
⎤
⎦ =
⎡
⎣
(ξy − py) fz(ξ
ξ
ξ)
−(ξx − px ) fz(ξ
ξ
ξ)
0
⎤
⎦ . (14.4)

To obtain the moment about point p
p
p = (px , py, pz) due to the orthogonal ground
reaction forces [0, 0, fz(ξ
ξ
ξ)]T
arising at all points of contact ξ
ξ
ξ between the sole and
the ground, we need to integrate across all points of contact
τ
τ
τn(p
p
p) =

S
⎡
⎣
ξx − px
ξy − py
0
⎤
⎦ ×
⎡
⎣
0
0
fz(ξ
ξ
ξ)
⎤
⎦ dS =
⎡
⎣
S(ξy − py) fz(ξ
ξ
ξ)dS
− S(ξx − px ) fz(ξ
ξ
ξ)dS
0
⎤
⎦ , (14.5)
where S denotes the area of contact. Similarly as in 2D case, the point on the ground
where the moment of the normal of the ground reaction force becomes zero (i.e., the
zero-moment point τ
τ
τn(p
p
p) = 0), is given by
p
p
p =
⎡
⎣
px
py
pz
⎤
⎦ =

S
ξx fz(ξ
ξ
ξ)dS
fn
,

S
ξy fz(ξ
ξ
ξ)dS
fn
, pz
T
, (14.6)
where
fn =

S
fz(ξ
ξ
ξ)dS (14.7)
is the sum of ground reaction forces orthogonal to the ground at all contacts between
the sole and the ground.
On a real humanoid robot, ZMP (if it exists), is guaranteed to lie within the support
polygon because if the contact between the sole and the ground surface exists, the
component of the ground reaction force orthogonal to the ground must be positive.
Otherwise the contact between the sole and the ground surface would be lost as the
robot is not fixed to the ground and therefore cannot generate negative vertical ground
reaction forces. The humanoid robot can control its posture with its feet only if the
ZMP exists inside the support polygon. Otherwise the robot loses contact with the
ground and cannot control the posture with its feet any more.
14.1.2 Generation of Walking Patterns
In biped walking, the robot’s feet alternate between two phases:
• stance phase in which the foot’s location should not change,
• swing phase in which the foot moves.
Figure14.3 shows these two distinct phases in the gait cycle: when both feet are in
contact with the ground, the robot is in double support phase. The feet do not move
in this phase. Once one of the feet starts moving, the robot transitions from double
to single support phase, in which one of the two feet moves. The single support
phase is followed by another double support phase once the foot in the swing phase
establishes a contact with the ground.

COM
COM
Fig. 14.3 Single and double support phase. In the double support phase, both feet are in contact
with the ground and the robot’s weight is supported by both legs. In the single support phase, one
foot is in motion, whereas the other foot supporting the robot is in contact with the ground
In ZMP-based walking, one or both feet of the robot are always in contact with
the ground. Thus ZMP exists and the robot can keep balance by making sure that
the support polygon contains the ZMP. However, the robot cannot directly control
the ZMP as defined in Eqs. (14.1) and (14.6). We therefore introduce the concept
of center of mass (COM). ZMP can be controlled by exploiting its relationship with
COM.
Center of mass (COM) is defined as the average position of all body parts of a
humanoid robot, weighted with the mass of body parts. For a robot with D rigid
links, COM can be calculated as:
c
c
c =
D
i=1 mic
c
ci
M
, M =
D

i=1
mi , (14.8)
where mi is the mass of i-th link and c
c
ci its position, which can be calculated by
direct kinematics provided the center of mass of each link is known in the link’s
local coordinates. With some approximations, the relationship between ZMP and
COM can be specified as follows
px = cx −
(cz − pz)c̈x
c̈z + g
, (14.9)
py = cy −
(cz − pz)c̈y
c̈z + g
, (14.10)

where pz denotes the height of the ground floor, g is the gravity constant, and c
c
c =
(cx , cy, cz) and p
p
p = (px , py, pz) are the coordinates of COM and ZMP, respectively.
Note that if the robot is at rest, i.e. c̈x = c̈y = 0, then ZMP and the projection of COM
coincide as px = cx and py = cy. Note also that if the ground is flat and orthogonal
to gravity, as we assumed in Sect.14.1.1, pz is a constant.
In general we distinguish between static and dynamic walking. Static walking is
defined as any stable walking motion where the projection of COM always stays
inside the support polygon. This means that if the robot completely stops moving at
any moment during walking, it does not fall down because for the robot at rest, the
projection of COM onto the ground surface is equal to the ZMP (see Eqs. (14.9) and
(14.10)). In static walking the motion must generally be slow so that the projection
of COM is close to the ZMP. This kind of walking typically requires large feet and
strong ankle joints to generate sufficient forces at the ankles. As the robot’s motion
becomes faster, ZMP and the projection of COM become more different and stability
cannot be ensured by controlling the projection of COM only.
More effective walking behaviors are generated by dynamic walking patterns,
where the projection of COM is not equal to ZMP and can fall outside of the support
polygon during some period of motion. A ZMP-based dynamic walking pattern is
shown in Fig.14.4. Such patterns are planned so that the ZMP remains within the
boundary of the support polygon in all phases of walking. This can be accomplished
as follows:
• Specify the Cartesian motion of the robot’s feet. Here the robot’s step length and
timing of foot motion is prescribed.
• Specify the reference ZMP trajectory so that ZMP remains within the support
polygon at all times.
• Determinethehumanoidrobot’supperbodymotioninordertorealizethereference
ZMP motion. This can be accomplished using Eqs. (14.9) and (14.10).
• The humanoid robot’s leg motion is finally calculated from the body and feet
motion using inverse kinematics.
The motion of COM is not fully specified by Eqs. (14.9) and (14.10) as there are
only two equations and three unknown parameters. To fully specify the motion of
COM and consequently the motion of the humanoid robot’s upper body, an additional
constraint must be imposed. There are several possible approaches. The simplest
among them is to set the height of COM to a constant value (i.e., cz = const, c̈z = 0).
Withthisassumption,themotionofCOMisfullyspecifiedbyEqs.(14.9)and(14.10).
A more adaptable and active motion can be achieved if cz is allowed to vary.
Note that the above approach determines the motion of COM without considering
the legs. However, since most of the mass is usually concentrated in the upper body
of a humanoid robot and since it is not necessary to follow the prescribed ZMP
trajectory exactly, the above approach is sufficient to generate dynamically stable
walking patterns.
If an accurate model of the robot is available, biped walking can be realized by
simply following a predetermined walking pattern. Due to noise and model inaccura-
cies, in practice such an approach usually does not result in a stable walking behavior

LF
RF
TT
YLAB
DS
SS
DS SS
DS SS
ZMP
Fig. 14.4 An example ZMP-based walking pattern in sagittal plane. The robot starts with both feet
placed roughly parallel on the ground and then generates three steps, starting with the left foot. The
shaded areas show the extent of support polygon during single support phase (dark shaded area)
and double support phase (light shaded area). The ZMP trajectory (dotted) is planned in such a way
that it remains within the support polygon during the whole duration of walking. The trajectories
of both feet (left: dashed dotted, right: dashed) are also shown
without supplementing the precomputed walking pattern with a stabilizer that mod-
ifies the pattern according to the sensory input provided by gyros, accelerometers,
force sensors, cameras, etc.
It should be pointed out that ZMP is not the only principle that can be used to
generate stable walking patterns. It is possible to generate a walking pattern where a
robot is unstable for some period of motion. Such walking patterns must be planned
so that the robot can recover from instabilities before falling to the ground.
14.2 Imitation Learning
To fully exploit their potential, humanoid robots should be able to perform a variety
of tasks in unstructured environments (e.g., people’s homes, hospitals, shops, offices,
and even outdoor environments). The aforementioned robotics challenge was geared

14.2 Imitation Learning 217
towards humanoid robots at disaster sites. Unlike many industrial environments,
whererobotsarewidelyusedtoday,suchenvironmentscannotbepreparedinadvance
to ease the operation of a humanoid robot. The programming of humanoid robots is
further complicated by the large number of degrees of freedom involved in humanoid
robot motion. Hence classic robot programming techniques based on teach pendants,
carefully prepared off-line simulation systems, and programming languages are not
sufficient for humanoids. Instead, it is necessary to equip humanoid robots with
learning and adaptation capabilities. This way they can be programmed more easily
and even autonomously acquire additional knowledge.
Learning of humanoid robot behaviors is a difficult problem because the space
of all humanoid robot motions that needs to be explored is very large and increases
exponentially with the number of degrees of freedom. A solution to this problem is
to focus learning on those parts of the robot motion space that are actually relevant
for the desired task. This can be achieved by imitation learning, also referred to as
programming by demonstration. With this approach, a human teacher demonstrates
to a robot how to perform the desired task. For it to work, a robot must be able
to extract the important information from human demonstration and replicate the
essential parts of task execution. While in most cases it is not necessary to exactly
replicate the demonstrated movements to successfully execute the desired task, it
is advantageous if the robot can mimic the demonstrated movement as much as
possible. Since the body of a humanoid robot is similar to a human body, imitation
learning is often a good approach to focus learning on the relevant parts of humanoid
robot motion space.
14.2.1 Observation of Human Motion and Its Transfer
to Humanoid Robot Motion
There are many possible measurement systems and technologies that can be used to
observe and measure human movements. They include
• optical motion capture systems,
• ensembles of inertial measurement units (IMU),
• computer vision methods for the estimation of human motion,
• passive exoskeletons,
• hand guiding.
In the following we explain the major advantages and drawbacks of these systems.
14.2.1.1 Optical Tracking Devices for Human Motion Capture
Optical trackers are based on a set of markers attached to a human body. Markers
can either be passive or active. Passive markers are made of retroreflective materi-
als, which reflect light in the direction from where it came. In systems with passive

markers, cameras are equipped with a band of infrared light emitting diodes (LEDs).
The emitted light bounces off the marker back in the direction of the camera, making
the marker much brighter than any other point in the image. This property makes
retroreflective markers easy to detect in camera images. Using triangulation, a 3D
marker location can be calculated if the marker is detected in at least two simulta-
neously acquired camera images. The predicted motion of visible markers is used to
match the visible markers extracted at two successive measurement times.
Unlike passive markers that reflect light, active markers are equipped with LEDs
and thus emit their own light. Consequently they must be powered. Optical trackers
with active markers usually illuminate only one marker at a time for a very short
time. Thus the system always knows which marker is currently visible, thereby
providing the identity of the marker. For this reason, optical tracking systems with
active markers can cope with temporary occlusions more effectively than systems
with passive markers because an occluded active marker can be identified once it
becomes visible again. This is not the case with passive markers. On the other hand,
since active markers require power, they need to be connected to a power source with
cables. This makes them more cumbersome to use than passive markers that require
no cables.
To measure human motion, both passive and active markers must be attached to
the human body segments at appropriate locations. Usually at least three markers
are attached to each body segment, otherwise the location of rigid body segments
cannot be estimated. Various special motion capture suits were designed in the past
to ease the attachment of markers to the relevant body segments.
Optical tracking systems with active or passive markers provide 3-D locations of
markers attached to the human body that are currently in view. The 3-D position and
orientation of a body segment can be estimated if at least three markers attached to
the segment are visible. In order to reproduce the observed motion with a robot, this
information needs to be related to the robot motion. To a certain degree of accuracy,
human motion can be modelled as an articulated motion of rigid body parts. If a
humanoid robot kinematics is close enough to the human body kinematics, we can
embed it into a human body as shown in Fig.14.5. Such an embedding can later be
used to estimate the joint angles from the orientations of successive body segments.
Let us assume that the orientation of two successive body segments is given by
orientation matrices R1 and R2 and that the joint linking the two segments consists
of three successive joint axes j1, j2 and j3 with rotation angles denoted by ϕ, θ, and
ψ, respectively. We further assume that two consecutive joint axes are orthogonal and
that all three axes intersect in a common point. In such an arrangement, the three joint
angles correspond to the Euler angles introduced in Chap. 4. There are altogether 12
different joint axis combinations that cover every possible arrangement of axes in
joints with three degrees of freedom. In Fig.14.5, torso, neck, shoulder, wrist, and
ankle joints can be described by an appropriate combination of Euler angles. The
relationship between these values is given by
R1 = R(j1, ϕ)R(j2, θ)R(j3, ψ)R2 = R(ϕ, θ, ψ)R2. (14.11)

y
z
Fig. 14.5 Kinematic structure of a humanoid robot. In the upright position with extended arms and
legs, all joint axes are parallel to one of the three main axes of the body (forward/backward: x axis,
left/right: y axis, up/down: z axis)
The joint angles φ, θ, and ψ can then be calculated by solving equation
R(ϕ, θ, ψ) = R1RT
2 . (14.12)
This equation depends on the choice of joint axes j1, j2, and j3. The observed motion
can be replicated by a robot once all relevant joint angles from the embedded model
have been estimated.
Optical tracking systems can also accurately estimate the absolute position and
orientationofthehumanbodyinaworldcoordinatesystem.Astherootofahumanoid
robot’s kinematics is typically assumed to be at the local coordinate frame attached
to the torso, the estimated position and orientation of the torso corresponds to the
absolute position and orientation of the human body in world coordinates.
14.2.1.2 Inertial Measurement Units (IMUs)
Inertial measurement units (IMUs) contain different sensors including accelerom-
eters to measure 3D linear acceleration and gyroscopes for measuring the rate of
change of 3D orientation (i.e., angular velocity). IMUs also often include magne-
tometers to provide redundant measurements to improve accuracy and reduce the

drift. From these data, the position and orientation of an IMU can be estimated as
explained in Sect. 7.2.6.
In the context of transferring human motion to humanoid robot motion, IMU data
can be used to estimate the position and orientation of each body segment which
has an IMU attached. Just like with marker-based trackers, the joint angles can be
estimated from orientations of successive body segments using Eq. (14.12).
Unlike optical tracking systems, IMUs do not suffer from occlusions as no external
cameras are needed to measure the IMU motion. On the other hand, IMUs are not as
accurate as optical tracking systems as they involve integration of linear acceleration
and angular velocity. The integration can also cause drift, especially when estimating
the absolute body position and orientation in space. Drift can be reduced by develop-
ing appropriate filters that exploit redundancy existing in the measurements obtained
from accelerometers, gyros, and magnetometers.
14.2.1.3 Passive Exoskeletons and Hand Guiding
A crucial issue that all of the above systems must deal with is that they measure human
motion without considering the differences between the human and robot kinematics
and dynamics. Such measurements must often be adapted to the robot constraints,
otherwise the robot cannot execute the demonstrated movements. Alternatively, a
nonlinear optimization problem can be formulated to adapt the demonstrated motion
to the capabilities of a target robot.
The problem of transferring human motion to robot motion can be avoided by
applyingdifferentmeasurementsystems.Onepossibilityistodesignaspecialpassive
device, which is worn like exoskeleton with the degrees of freedom that correspond
to the robot degrees of freedom. The passive exoskeleton must be designed in such
a way that it does not restrict motion for most movements. It has no motors, but it
should be equipped with goniometers to measure the joint angles. The joint angles
measured by the exoskeleton can be used to directly control the robot if the kinematics
of the target robot corresponds to the kinematic of the exoskeleton. One drawback
of passive exoskeletons is that like clothes, they must be built to the specific size of
a human demonstrator.
As explained in Sect. 12.3.2, some robots can be physically guided through
the desired movements (see also Fig.14.6). During hand guiding the movement is
recorded by the robot’s own joint angle sensors and is thus by default kinematically
feasible. This approach is effective if the robot is compliant and can compensate for
gravity, so that a human demonstrator can easily move it in the desired direction.
The main drawback of hand guiding is that the demonstration of the desired
motion is less natural for a human demonstrator than for example when marker-
based tracking systems are used. Thus with such systems it is sometimes not as easy
to demonstrate complex movements. For example, hand guiding is not effective to
demonstrate complex dancing movements. On the other hand, dancing can be easily
demonstrated by a human directly and measured with an optical tracker, IMUs, or a
passive exoskeleton.

Fig. 14.6 Demonstration of peg-in-hole task by kinesthetic teaching. Human demonstrator guides
the anthropomorphic arm through the task execution with its own hands
14.2.2 Dynamic Movement Primitives
In Sect.14.2.1 we discussed how to measure human demonstrations and how to
transform the measured movements into the robot joint angle trajectories. In some
cases it is also necessary to adapt the measured motion to the kinematic and dynamic
capabilities of the target robot. Typically, we end up with a measurement sequence
{yd(tj ), tj }T
j=1, (14.13)
where yd(tj ) ∈ RD
are the measured joint angles at time tj , D is the number of
degrees of freedom, and T is the number of measurements. This sequence defines
the reference trajectory. However, for effective control we need to generate motor
commands with the servo rate of the target robot. The robot’s servo rate is often higher
than the capture rate of the measurement system. Thus from the measurement data
(14.13) we need to generate a continuous reference trajectory in order to generate
motor commands to control the robot at the appropriate rate.
In this section we introduce Dynamic Movement Primitives (DMPs), which pro-
vide a comprehensive framework for the effective imitation learning and control of
robot movements. DMPs are based on a set of nonlinear differential equations with
well-defined attractor dynamics. For a single robot degree of freedom, here denoted
by y and taken to be one of the D recorded joint angles, the following system of
linear differential equations with constant coefficients is analyzed to derive a DMP
τż = αz(βz(g − y) − z), (14.14)
τ ẏ = z. (14.15)

Note that auxiliary variable z is just a scaled velocity of the control variable y. Con-
stants αz and βz have an interpretation in terms of spring stiffness and damping. For
the appropriately selected constants αz, βz, τ 0, these equations form a globally
stable linear dynamic system with g as a unique point attractor. We often refer to g
as the goal of the movement. This means that for any start configuration y(0) = y0,
variable y reaches the goal configuration g after a certain amount of time, just as a
stretched spring, upon release, will return to its rest position. τ is referred to as the
time constant. It affects the speed of convergence to the attractor point g.
14.2.3 Convergence Properties of Linear
Dynamic Systems
Let us analyze why the above system is useful. We start by writing down a gen-
eral solution of the non-homogenous linear differential equation system (14.14) and
(14.15). It is well known that the general solution of such a system can be written as
a sum of the particular and homogeneous solution

z(t)
y(t)

=

zp(t)
yp(t)]

+

zh(t)
yh(t)

. (14.16)
Here [zp(t), yp(t)]T
denotes any function that solves the linear system (14.14)–
(14.15), while [zh(t), yh(t)]T
is the general solution of the homogeneous part of Eqs.
(14.14)–(14.15), i.e.,

ż
ẏ

=
1
τ

−αz(βz y + z)
z

= A

z
y

, A =
1
τ

−αz −αzβz
1 0

.
It is easy to check that constant function [zp(t), yp(t)]T
= [0, g]T
solves the equation
system (14.14) and (14.15). Additionally, it is well known that the general solution of
homogeneous system (14.17) is given by [zh(t), yh(t)]T
= exp (At) c, where c ∈ R2
is an arbitrary constant. Thus, the general solution of Eqs. (14.14) and (14.15) can
be written as
z(t)
y(t)

=

0
g

+ exp (At) c. (14.17)
Constantc shouldbecalculatedfromtheinitialconditions,[z(0), y(0)]T
= [z0, y0]T
.
The eigenvalues of A are given by λ1,2 =

−αz ±

α2
z − 4αzβz

/(2τ). Solution
(14.17) converges to [0, g]T
if the real part of eigenvalues λ1,2 is smaller than 0, which
is true for any αz, βz, τ 0. The system is critically damped, which means that y
converges to g without oscillating and faster than for any other choice of A, if A has
two equal negative eigenvalues. This happens at αz = 4βz where λ1,2 = −αz/(2τ).

14.2.4 Dynamic Movement Primitives
for Point-to-Point Movements
Differential equation system (14.14)–(14.15) ensures that y converges to g from any
starting point y0. It can therefore be used to realize simple point-to-point movements.
To increase a rather limited set of trajectories that can be generated by (14.14) and
(14.15) and thus enable the generation of general point-to-point movements, we can
add a nonlinear component to Eq. (14.14). This nonlinear function is often referred
to as forcing term. A standard choice is to add a linear combination of radial basis
functions Ψi
f (x) =
N
i=1 wi Ψi (x)
N
i=1 Ψi (x)
x(g − y0), (14.18)
Ψi (x) = exp

−hi (x − ci )2

, (14.19)
where ci are the centers of radial basis functions distributed along the phase of the
trajectory and hi 0. The term g − y0, y0 = y(t1), is used to scale the trajectory
if the initial and / or final configuration change. As long as the beginning and the
end of movement are kept constant, this scaling factor has no effect and can be
omitted. Phase variable x is used in forcing term (14.18) instead of time to make
the dependency of the resulting control policy on time more implicit. Its dynamics
is defined by
τẋ = −αx x, (14.20)
with the initial value x(0) = 1. A solution to (14.20) is given by
x(t) = exp (−αx t/τ) . (14.21)
The appealing property of using the phase variable x instead of explicit time is that by
appropriately modifying Eq. (14.20), the evolution of time can be stopped to account
for perturbations during motion. There is no need to manage the internal clock of the
system. We obtain the following system of nonlinear differential equations
τż = αz(βz(g − y) − z) + f (x), (14.22)
τ ẏ = z. (14.23)
The phase variable x and consequently f (x) tend to 0 as time increases. Hence
the influence of nonlinear term f (x) decreases with time. Consequently, through the
integration of system (14.22)–(14.23) the system variables [z, y]T
are guaranteed to
converge to [0, g]T
, just like the linear system (14.14)–(14.15). The control policy
specified by variable y and its first- and second-order derivatives defines what we
call a dynamic movement primitive (DMP). For a system with many degrees of

freedom, each degree of freedom is represented by its own differential equation
system (14.22)–(14.23), whereas the phase x is common across all the degrees of
freedom. This can be done because phase Eq. (14.20) does not include variables y
and z.
It is usually sufficient to determine the parameters ci and hi of Eq. (14.19) by
setting a predefined distribution pattern and increasing the number of base functions
N until the desired reconstruction accuracy can be achieved. For example, for a given
N we can define
ci = exp

−αx
i − 1
N − 1

, i = 1, . . . , N, (14.24)
hi =
2
(ci+1 − ci )2
, i = 1, . . . , N − 1, hN = hN−1. (14.25)
Note that c1 = 1 = x(0) and cN = exp (−αx ) = x(tT ).
In the equations above, αx , αz, and βz are set to constant values. The values must
be chosen in such a way that the convergence of the underlying dynamic system
is ensured as explained in Sect.14.2.3. This is the case if we set αx = 2, βz = 3,
αz = 4βz = 12.
DMPs were designed to provide a representation that enables accurate encoding
of the desired point-to-point movements and at the same time permits modulation of
different properties of the encoded trajectory. In this context, the shape parameters
wi are determined so that the robot can accurately follow the desired trajectory by
integrating the equation system (14.20), (14.22), and (14.23). The other parameters
are used for modulation and to account for disturbances.
For a movement with two degrees of freedom, Fig.14.7 shows a graphical plot
of attractor fields generated by the dynamic movement primitive. The attractor field
changes with the evolution of phase x. As long as the robot follows the demon-
strated trajectory, the attractor field directs the robot to move along the demonstrated
trajectory. However, if the robot is perturbed and deviates from the demonstrated
trajectory, the attractor fields generated along the phase x directs the robot so that it
reaches the desired final configuration (goal), albeit along a modified trajectory.
A trajectory can be reproduced from a fully specified DMP by integrating Eqs.
(14.22), (14.23), and (14.20) using Euler integration method:
zk+1 = zk +
1
τ
(αz(βz(g − yk) − zk) + f (xk)) Δt, (14.26)
yk+1 = yk +
1
τ
zi Δt, (14.27)
xk+1 = xk −
1
τ
αx xkΔt, (14.28)
where Δt 0 is the integration constant usually set to the robot’s servo rate. The
initial parameters for integration must be set to the current state of the robot, which

y1
y1 y1
y1 y1
y1
y
2
y
2
y
2
y
2
y
2
y
2 x = 1.00 x = 0.56 x = 0.42
x = 0.32 x = 0.24 x = 0.14
0.0
0.0 0.0
0.0 0.0
0.0
0.5
0.5 0.5
0.5 0.5
0.5
1.0
1.0 1.0
1.0 1.0
1.0
1.5
1.5 1.5
1.5 1.5
1.5
2.0
2.0 2.0
2.0 2.0
2.0
2.5
2.5 2.5
2.5 2.5
2.5
3.0
3.0 3.0
3.0 3.0
3.0
−3.5
−3.5 −3.5
−3.5 −3.5
−3.5
−3.0
−3.0 −3.0
−3.0 −3.0
−3.0
−2.5
−2.5 −2.5
−2.5 −2.5
−2.5
−2.0
−2.0 −2.0
−2.0 −2.0
−2.0
−1.5
−1.5 −1.5
−1.5 −1.5
−1.5
−1.0
−1.0 −1.0
−1.0 −1.0
−1.0
−0.5
−0.5 −0.5
−0.5 −0.5
−0.5
Fig. 14.7 Plots of attractor fields generated by a DMP specifying the motion of a robot with two
degrees of freedom y1, y2 as it is integrated along the phase x. The arrows in each plot show ż1, ż2 at
different values of y1, y2 at the given phase x, assuming that only y1 and y2 have changed compared
to the unperturbed trajectory. The circles show the desired configurations y1, y2 at the given phase x
at the beginning of motion is assumed to be at the given initial position and with
zero velocity. This results in the following initialization formulas: y0 = y0, z0 = 0,
x = 1.
14.2.5 Estimation of DMP Parameters
from a Single Demonstration
To estimate the DMP representing the measurement sequence (14.13), we first com-
pute the derivatives ẏj and ÿj by numerical differentiation. For any of the D degrees
of freedom y, we obtain the following measurement sequence
{yd(tj ), ẏd(tj ), ÿd(tj )}T
j=1, (14.29)
where yd(tj ), ẏd(tj ), ÿd(tj ) ∈ R are the measured positions, velocities, and accel-
erations on the training trajectory and T is the number of sampling points. Using
the DMP movement representation, the trajectory of any smooth movement can
be approximated by estimating parameters wi of Eq. (14.18). For this purpose we

rewrite the system of two first-order linear Eqs. (14.22) and (14.23) as one second-
order equation. This is done by replacing z with τ ẏ in Eq. (14.22). We obtain
τ2
ÿ + αzτ ẏ − αzβz(g − y) = f (x), (14.30)
with f defined as in Eq. (14.18). Note that time constant τ must be the same for all
degrees of freedom. A possible choice is τ = tT − t1, where tT − t1 is the duration
of the training movement. On the other hand, the attractor point g varies across the
degrees of freedom. It can be extracted directly from the data: g = yd(tT ). Writing
Fd(tj ) = τ2
ÿd(tj ) + αzτ ẏd(tj ) − αzβz(g − yd(tj )), (14.31)
f =
⎡
⎣
Fd(t1)
. . .
Fd(tT )
⎤
⎦ , w =
⎡
⎣
w1
. . .
wN
⎤
⎦ ,
we obtain the following system of linear equations
Xw = f, (14.32)
which must be solved to estimate the weights of a DMP encoding the desired motion.
The system matrix X is given by
X = (g − y0)
⎡
⎢
⎣
Ψ1(x1)
N
i=1 Ψi (x1)
x1 . . . ΨN (x1)
N
i=1 Ψi (x1)
x1
. . . . . . . . .
Ψ1(xT )
N
i=1 Ψi (xT )
xT . . . ΨN (xT )
N
i=1 Ψi (xT )
xT
⎤
⎥
⎦ . (14.33)
The phase sampling points xj are obtained by inserting measurement times tj into
Eq. (14.21). The parameters w can be calculated by solving the above system of
linear equations in a least-squares sense. An example DMP estimation is shown in
Fig.14.8. The calculated DMP ensures that the robot reaches the attractor point g
at time tT . Since DMPs have been designed to represent point-to-point movements,
the demonstrated movement must come to a full stop at the end of the demonstration
if the robot is to stay at the attractor point after tT . If any other type of motion is
approximated by a DMP, the robot will overshoot the attractor point and return back
to it after the dynamics of the second-order linear system of differential equations
starts dominating the motion. At least theoretically, the velocity does not need to be
zero at the beginning of movement, but it is difficult to imagine a real programming
by demonstration system in which such a trajectory would be acquired.

−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−2
−4
0
0.2
0.4
0.6
0.8
1.0
−40
−20
20
30
40
2
4
6
8
0
0
0
0
0
0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.0
1.0
1.2
1.2
1.2
1.2
time / s
time / s
time / s
time / s
y(t)
ẏ(t)
ÿ(t)
ψ
i
(x(t))
Fig. 14.8 Time evolution of an example dynamic movement primitive: control variable y and its
derivatives, phase x, and radial basis functions ψi are all shown with solid lines. Dashed lines show
the demonstrated values of y, ẏ and ÿ
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0 0
−1
−2
−3
−4
0
0 0.5
0.5 1.0
1.0 1.5
1.5 2.0
time / s
time / s
y(t)
y(t)
Fig. 14.9 DMP modulations. The dashed trajectories show the original DMP without applying any
modulation. Left: Time modulation. Solid trajectories show DMPs with changed τ. Right: Goal
modulation. Solid trajectories show DMPs with changed goal g. Circles show the goal position
14.2.6 Modulation of DMPs
An important advantage of DMPs is that they enable easy modulation of the learnt
movement. Figure14.9 left shows that by changing parameter τ, the movement can
be sped up or slowed down. The same figure also shows that by changing the goal
parameter g, the final configuration on the trajectory can be changed so that the robot
moves to a new goal. The term y0 − g in the forcing term (14.18) ensures that the
movement is appropriately scaled as the goal or initial configuration changes.

−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0
−2
2
4
6
0
0 0.5
0.5 1.0
1.0 1.5
1.5
time / s
time / s
y(t)
ẏ(t)
Fig. 14.10 DMP modulation with joint limit avoidance at y = −2.8. The solid trajectories show
the DMP trajectory and its velocity obtained by integrating (14.34) instead of (14.23), while the
dashed trajectories show the original DMP and its velocity without applying any modulation
More complex modulations involve changing the underlying differential Eqs.
(14.22), (14.23), and/or (14.20). For example, Eq. (14.23) can be changed to
τy = z −
ρ
(yL − y)3
(14.34)
to implement the avoidance of a lower joint limit. This happens because once y
starts approaching yL , the denominator in Eq. (14.34) becomes small and there is a
significant difference between integrating Eq. (14.23) or (14.34). Figure14.10 right
shows that the second term in Eq. (14.34) acts as a repulsive force, preventing y
from approaching yL too closely. On the other hand, the denominator in Eq. (14.34)
remains large as long as the joint angle y is far away from the joint limit yL . Thus
in this case there is little difference between integrating Eq. (14.23) or (14.34) and
the DMP generated trajectory follows the demonstrated movement. Note that it is
not necessary to learn new parameters wi , goal g, or time constant τ because of
modulation. They can remain as they were initially learnt. Only Eq. (14.23) must be
changed to (14.34) to ensure joint limit avoidance during on-line control.
The appealing property of applying the phase variable instead of time is that we
can easily modulate the time evolution of phase, e.g., by speeding up or slowing
down a movement as appropriate by means of coupling terms. Instead of integrating
Eqs. (14.20) and (14.23) at time of execution, the modified Eqs. (14.20) and (14.36)
could be integrated
τẋ = −
αx x
1 + αpx (y − ỹ)2
, (14.35)
τ ẏ = z + αpy(y − ỹ), (14.36)
where y and ỹ respectively denote the desired and actual robot joint angle position,
respectively. If the robot cannot follow the desired motion, αpx (y − ỹ)2
becomes
large, which in turn makes the phase change ẋ small. Thus the phase evolution
is stopped until the robot catches up with the desired configuration y. This will

−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
−2
0
0
0
2
4
6
−30
−20
−10
10
20
30
0.2
0.4
0.6
0.8
1.0
0
0
0
0
0.5
0.5
0.5
0.5
1.0
1.0
1.0
1.0
1.5
1.5
1.5
1.5
2.0
2.0
2.0
2.0
2.5
2.5
2.5
2.5
time / s
time / s
time / s
time / s
y(t)
ẏ(t)
ÿ(t)
phase(x)
Fig. 14.11 The effect of phase stopping caused by blocking the evolution of joint position ỹ in
the time interval [0.6, 1.4] (grey area). The dashed trajectories show the original DMP, velocity,
acceleration and phase evolution, while the solid trajectories show their counterparts from the
perturbed motion with phase stopping enabled. Note that outside of the time interval [0.6, 1.4]
where the joint motion is blocked, the robot accurately follows the desired motion
eventually happen due to the added term in Eq. (14.36). On the other hand, if the
robot follows the desired movement precisely, then ỹ − y ≈ 0 and Eqs. (14.35) and
(14.36) are no different from Eqs. (14.20) and (14.23), respectively. Thus in this case
the DMP-generated movement is not altered. Figure14.11 illustrates the effect of
phase stopping when the robot’s motion is temporarily blocked.
In summary, DMPs provide an effective representation for learning humanoid
robot trajectories and to control humanoid robots. They are based on autonomous,
nonlinear differential equations that are guaranteed to create smooth kinematic con-
trol policies. An important property of DMPs is that they can be learnt from a single
demonstration of the desired task. They have several advantages compared to other
motor representations including
• they possess free parameters that are easy to learn in order to reproduce any desired
movement,
• they are not explicitly dependent on time and allow for time modulation,
• they are robust against perturbations,
• they are easy to modulate by adapting various parameters and equations.
Due to their flexibility and robustness, DMPs are considered a method of choice
when learning robot trajectories from single demonstrations.

Chapter 15
Accuracy and Repeatability
of Industrial Manipulators
In this chapter we shall briefly consider performance criteria and the methods for
testing of industrial robot manipulators as described in the ISO 9283 standard. Before
addressing accuracy and repeatability of industrial manipulators we will summarize
basic information about robot manipulators.
The basic robot data typically includes a schematic drawing of the robot mechan-
ical structure:
• cartesian robot (Fig.15.1 left),
• cylindrical robot (Fig.15.1 right),
• polar (spherical) robot (Fig.15.2 left),
• anthropomorphic robot (Fig.15.2 right),
• SCARA robot (Fig.15.3).
In all drawings the degrees of freedom of the robot mechanism must be marked.
Thedrawingmustincludealsothebasecoordinateframeandthemechanicalinterface
frame which are determined by the manufacturer.
Of special importance is the diagram showing the boundaries of the workspace
(Fig.15.4). The maximal reach of the robot arm must be clearly shown in at least
two planes. The range of motion for each robot axis (degree of freedom) must be
indicated. The manufacturer must specify also the center of the workspace cw, where
most of the robot activities take place.
The robot data must be accompanied by the characteristic loading parameters,
such as mass (kg), torque (Nm), moment of inertia (kgm2
), and thrust (N). The
maximal velocity must be given at a constant rate, when there is no acceleration or
deceleration. The maximal velocities for particular robot axes must be given with
the load applied to the end-effector. The resolution of each axis movement (mm
or ◦
), description of the control system and the programming methods must also be
presented.
The three most relevant robot coordinate frames (right-handed) are shown in
Fig. 15.5. First is the world coordinate frame x0–y0–z0. The origin of the frame
is defined by the user. The z0 axis is parallel to the gravity vector, however in the
opposite direction. Second is the base coordinate frame x1–y1–z1, whose origin is
defined by the manufacturer. Its axes are aligned with the base segment of the robot.
231

232 15 Accuracy and Repeatability of Industrial Manipulators
x1
x1
y1
y1
z1 z1
xm
xm
ym
ym
zm
zm
1 1
2
2
3
3
4
4
Fig. 15.1 Mechanical structures of the cartesian robot (left) and the cylindrical robot (right)
x1
x1
y1
y1
z1
z1
xm
xm
ym
ym
zm
zm
1
1
2
2
3
3
4
4
Fig. 15.2 Mechanical structure of the polar robot (left) and the anthropomorphic robot (right)
The positive z1 axis is pointing perpendicularly away from the base mounting surface.
The x1 axis passes through the projection of the center of the robot workspace cw.
The frame xm–ym–zm is called the mechanical interface coordinate frame. Its origin
is placed in the center of the mechanical interface (robot palm) connecting the robot
arm with the gripper. The positive zm axis points away from the mechanical interface
toward the end-effector. The xm axis is located in the plane defined by the interface,
which is perpendicular to the zm axis.
The positive directions of robot motions, specified as the translational and rota-
tional displacements are shown in Fig.15.6.

15 Accuracy and Repeatability of Industrial Manipulators 233
x1
y1
z1
xm
ym
zm
1
2
3
4
Fig. 15.3 Mechanical structure of the SCARA robot
The ISO 9283 standard deals with criteria and methods for testing of industrial
robot manipulators. This is the most important standard as it facilitates the dialogue
between manufacturers and users of the robot systems. It defines the way by which
particular performance characteristics of a robot manipulator should be tested. The
tests can be performed during the robot acceptance phase or in various periods of
robot usage in order to check the accuracy and repeatability of the robot motions.
The robot characteristics, which significantly affect the performance of a robot task,
are the following:
• pose accuracy and repeatability (pose is defined as position, and orientation of a
particular robot segment, usually end-effector),
• distance accuracy and repeatability,
• pose stabilization time,
• pose overshoot,
• drift of the pose accuracy and repeatability.
These performance parameters are important in the point-to-point robot tasks.
Similar parameters are defined for cases when the robot end-effector moves along
a continuous path. These parameters will not be considered in this book and can be
found in the original documents.
When testing the accuracy and repeatability of a robot mechanism, two terms
are important, namely the cluster and the cluster barycenter. The cluster is defined
as a set of attained end-effector poses, corresponding to the same command pose.
The barycenter is a point whose coordinates are the mean values of the x, y and
z coordinates of all the points in the cluster. The measured position and orien-
tation data must be expressed in a coordinate frame parallel to the base frame.

x1
y1
z1
cw
cw
Fig. 15.4 Robot workspace
The measurement point should lie as close as possible to the origin of the mechani-
cal interface frame. Contact-less optical measuring methods are recommended. The
measuring instrumentation must be adequately calibrated. The robot accuracy and
repeatability tests must be performed with maximal load at the end-effector and
maximal velocity between the specified points.
The standard defines the poses which should be tested. The measurements must be
performed at five points, located in a plane which is placed diagonally inside a cube
(Fig.15.7). Also specified is the pose of the cube in the robot workspace. It should
be located in that portion of the workspace where most of the robot activities are
anticipated. The cube must have maximal allowable volume in the robot workspace
and its edges should be parallel to the base coordinate frame. The point P1 is located
in the intersection of the diagonals in the center of the cube. The points P2 – P5 are
located at a distance from the corners of the cube equal to 10% ± 2% of the length
of the diagonal L. The standard also determines the minimum number of cycles to
be performed when testing each characteristic parameter:

x0
y0
z0
x1
y1
z1
xm
ym zm
Fig. 15.5 The coordinate frames of the robot manipulator
+x
+y
+z
+a
+b
+c
Fig. 15.6 Positive directions of translational and rotational displacements
• pose accuracy and repeatability: 30 cycles,
• distance accuracy and repeatability: 30 cycles,
• pose stabilization time: 3 cycles,
• pose overshoot: 3 cycles,
• drift of pose accuracy and repeatability: continuous cycling during 8 h.
When testing the accuracy and repeatability of the end-effector poses we must
distinguish between the so-called command pose and the attained pose (Fig.15.8).
The command pose is the desired pose, specified through robot programming or
manual input of the desired coordinates using a teach pendant. The attained pose
is the actually achieved pose of the robot end-effector in response to the command

P1
P2
P3
P4
P5
x
y
z
base frame
L = diagonal length
(0.1±0.02)L (0.1±0.02)L
Fig. 15.7 The cube with the points to be tested
c
x
y
y
z xj
yj
z
j
xc
z
c
mechanical interface
j-th attained pose
command pose
Fig. 15.8 The command pose and the attained end-effector pose
pose. The pose accuracy evaluates the deviations, which occur between the command
and the attained pose. The pose repeatability estimates the fluctuations in the attained
poses for a series of repeated visits to the same command pose. The pose accuracy and
repeatability are, therefore, very similar to the accuracy and repeatability of repetitive
shooting at a target. The reasons for the deviations are: errors caused by the control
algorithm, coordinate transformation errors, differences between the dimensions of
the robot mechanical structure and the robot control model, mechanical faults, such
as hysteresis or friction, and external influences such as temperature.

x
y
z
ΔLx
ΔLy
Δ
L
z
Δ
L
Oc
G r
cluster of attained positions
command position
Fig. 15.9 The position accuracy and repeatability
The pose accuracy is defined as the deviation between the command pose and
the mean value of the attained poses when the end-effector was approaching the
command pose from the same direction. The position and orientation accuracy are
treated separately. The position accuracy is determined by the distance between the
command pose and the barycenter of the cluster of attained poses (Fig.15.9). The
position accuracy L = [Lx Ly Lz]T
is expressed by the following equation
L =

(x̄ − xc)2 + (ȳ − yc)2 + (z̄ − zc)2, (15.1)
where (x̄, ȳ, z̄) are the coordinates of the cluster barycenter, obtained by averaging
the 30 measurement points, assessed when repeating the motions into the same
command pose Oc with the coordinates (xc, yc, zc).
The orientation accuracy is the difference between the commanded angular orien-
tation and the average of the attained angular orientations. It is expressed separately
for each axis of the base coordinate frame. The orientation accuracy around the z
axis has the following form
Lc = C̄ − Cc, (15.2)
where C̄ is the mean value of the orientation angles around the z axis, obtained in 30
measurements when trying to reach the same command angle Cc. Similar equations
are written for the orientation accuracy around the x and y axes.
The standard exactly defines also the course of the measurements. The robot starts
from point P1 and moves into points P5, P4, P3, P2, P1. Each point is always reached
from the same direction

0 cycle P1
1st
cycle P5 → P4 → P3 → P2 → P1
2nd
cycle P5 → P4 → P3 → P2 → P1
.
.
.
30th
cycle P5 → P4 → P3 → P2 → P1
For each point the position accuracy L and the orientation accuracies La, Lb
and Lc are calculated.
For the same series of measurements also the pose repeatability is to be deter-
mined. The pose repeatability expresses the closeness of the positions and orien-
tations of the 30 attained poses when repeating the robot motions into the same
command pose. The position repeatability (Fig.15.9) is determined by the radius of
the sphere r whose center is the cluster barycenter. The radius is defined as
r = D̄ + 3SD. (15.3)
The calculation of the radius r according Eq. (15.3) is further explained by the
following equations
D̄ =
1
n
n

j=1
Dj
Dj =

(xj − x̄)2 + (yj − ȳ)2 + (z j − z̄)2 (15.4)
SD =
n
j=1(Dj − D̄)2
n − 1
.
In the above equations we again select n = 30, while (xj , yj , z j ) are the coordinates
of the j-th attained position.
The orientation repeatability for the angle around the z axis is presented in
Fig.15.10. The orientation repeatability expresses how dispersed are the 30 attained
angles around their average for the same command angle. It is described by the
threefold standard deviations. For the angle around the z axis we have
rc = ±3Sc = ±3
n
j=1(Cj − C̄)2
n − 1
. (15.5)
In Eq. (15.5) Cj represents the angle measured at the j-th attained pose. The course
of the measurements is the same as in testing of the accuracy. The radius r and the
angular deviations ra, rb and rc are calculated for each pose separately.
The distance accuracy and repeatability are tested in a similar way. The distance
accuracy quantifies the deviations which occur in the distance between two command
positions and two sets of the mean attained positions. The distance repeatability

¯
C
C
c
Δ
L
c
3
S
c
base frame
mean value of attained angles
command angle
Fig. 15.10 The orientation accuracy and repeatability
determines thefluctuations indistances for aseries of repeatedrobot motions between
two selected points. The distance accuracy is defined as the deviation between the
command distance and the mean of the attained distances (Fig.15.11). Assuming
that Pc1 and Pc2 are the commanded pair of positions and P1 j and P2 j are the j-th
pair from the 30 pairs of the attained positions, the distance accuracy B is defined
as
B = Dc − D̄. (15.6)
where
Dc =

Pc1
− Pc2

=

(xc1
− xc2
)2 + (yc1
− yc2
)2 + (zc1
− zc2
)2
D̄ =
1
n
n

j=1
Dj
Dj =

P1j
− P2j

=

(x1j
− x2j
)2 + (y1j
− y2j
)2 + (z1j
− z2j
)2.
In the above equations describing the distance accuracy Pc1
= (xc1
, yc1
, zc1
) and
Pc2
= (xc2
, yc2
, zc2
) represent the pair of desired positions while P1j
= (x1j
, y1j
, z1j
)
and P2j
= (x2j
, y2j
, z2j
) are the pair of attained positions. The distance accuracy test
is performed at maximal loading of the robot end-effector, which must be displaced
30 times between points P2 and P4 of the measuring cube. The distance repeatability
RB is defined as
RB = ±3
n
j=1(Dj − D̄)2
n − 1
. (15.7)
Let us consider another four characteristic parameters which should be tested
in industrial robots moving from point to point. The first is the pose stabilization

Pc1
Pc2
P1 j P2 j
mean attained distance
command distance
Fig. 15.11 Distance accuracy
time. The stabilization time is the time interval between the instant when the robot
gives the “attained pose” signal and the instant when either oscillatory or damped
motion of the robot end-effector falls within a limit specified by the manufacturer.
The definition of the pose stabilization time is evident from Fig.15.12. The test is
performed at maximal loading and velocity. All five measuring points are visited in
the following order P1 → P2 → P3 → P4 → P5. For each pose the mean value of
three cycles is calculated.
A similar parameter is the pose overshoot, also shown in Fig.15.12. The pose
overshoot is the maximum deviation between the approaching end-effector trajectory
and the attained pose after the robot has given the “attained pose” signal. In Fig.15.12
a negative overshoot is shown in the first example and a positive overshoot in the
second example. The instant t = 0 is the time when the “attained pose” signal was
delivered. The measuring conditions are the same as when testing the stabilization
time.
The last two parameters to be tested in the industrial robot manipulator moving
from point to point are drift of the pose accuracy and the drift of the pose repeatability.
The drift of the position accuracy LDR is defined as
LDR = |Lt=0 − Lt=T | , (15.8)
where Lt=0 and Lt=T are the position accuracy values at time t = 0 and time
t=T, respectively. The drift of the orientation accuracy LDRC is equal to
LDRC =

Lc,t=0 − Lc,t=T

, (15.9)

t = 0
attained pose
overshoot
(1)
overshoot
(2)
case (1)
case (2)
deviation
stabilization
time (1)
stabilization
time (2)
specified
limit
time
Fig. 15.12 Pose stabilization time and overshoot
whereLc,t=0 andLc,t=T aretheorientationaccuracyvalues at time t = 0 andtime
t=T, respectively. The drift of the position repeatability is defined by the following
equation
rDR = rt=0 − rt=T , (15.10)
where rt=0 and rt=T are the position repeatability values at time t = 0 and time t=T,
respectively. The drift of the orientation repeatability is for the rotation around the z
axis defined as
rDRC = rc,t=0 − rc,t=T , (15.11)
where rc,t=0 and rc,t=T are the orientation repeatability values at time t = 0 and
time t=T, respectively. The measurements are performed at maximal robot loading
and velocity. The robot is cyclically displaced between points P4 and P2. The cyclic
motions last for eight hours. Measurements are only taken in point P4.

Appendix A
Derivation of the Acceleration in Circular
Motion
Let us first recall the definitions of position vector, velocity and acceleration of a
particle. In a given reference frame (i.e., coordinate system) the position of a particle
is given by a vector extending from the coordinate frame origin to the instantaneous
position of the particle. This vector could be a function of time, thus specifying the
particle trajectory
r(t) = (x(t), y(t), z(t)). (A.1)
The velocity of the particle is defined as the change of position per unit time
v = lim
t→0
r
t
=
dr
dt
. (A.2)
The acceleration is defined as the change of velocity per unit time,
a = lim
t→0
v
t
=
dv
dt
. (A.3)
We note that this is a vector equation, so the change in velocity refers to both a change
in the magnitude of velocity, as well as to a change of velocity direction.
Circular motion is described by a rotating vector r(t) of fixed length, |r| =
constant. The position vector is thus determined by the radius of the circle r, and
by the angle θ(t) of r with respect to the x-axis (Fig. A.1).
Let us now introduce a set of three orthogonal unit vectors: er in the direction of
r, et in the direction of the tangent to the circle and ez in the direction of the z axis.
The relation between the three unit vectors is given by et = ez × er .
We define the angular velocity vector as being perpendicular to the plane of the
circular trajectory with magnitude equal to the time derivative of the angle θ
ω = θ̇ez. (A.4)
Let us proceed to calculate the velocity
M. Mihelj et al., Robotics, https://doi.org/10.1007/978-3-319-72911-4
243

244 Appendix A: Derivation of the Acceleration in Circular Motion
Fig. A.1 Parameters and
variables in circular motion
v =
dr
dt
. (A.5)
The direction of velocity is given by the tangent to the circle: et = ez × er . The
magnitude of velocity is given by the length of the infinitesimal circular arc ds = rdθ
divided by the infinitesimal time dt, that the particle requires to traverse this path
ds
dt
= r
dθ
dt
= rθ̇. (A.6)
Including the tangential direction of velocity gives
v = rθ̇et = θ̇ez × rer = ω × r. (A.7)
In order to obtain the acceleration, we calculate the time derivative of velocity
a =
dv
dt
=
d
dt
(ω × r). (A.8)
We differentiate the vector product as one would differentiate a normal product of
two functions
a =
dω
dt
× r + ω ×
dr
dt
. (A.9)
Defining the angular acceleration α as the time derivative of angular velocity α =
dω
dt
, we see that the first term corresponds to the tangential acceleration
at = α × r. (A.10)
In the second term we insert the expression for velocity

Appendix A: Derivation of the Acceleration in Circular Motion 245
dr
dt
= v = ω × r, (A.11)
and we get a double vector product ω × (ω × r). Using the identity a × (b × c) =
b(a · c) − c(a · b) from vector algebra and noting that ω and r are orthogonal, we
obtain for the second term in the equation for acceleration
ω ×
dr
dt
= ω × (ω × r) = ω(ω · r) − r(ω · ω) = −ω2
r, (A.12)
which is the radial (or centripetal) component of acceleration. So we finally have
a = at + ar = α × r − ω2
r. (A.13)
Further Reading
1. Bajd T, Mihelj M, Munih M (2013) Introduction to Robotics, Springer
2. Craig JJ (2005) Introduction to Robotics—Mechanics and Control, Pearson
Prentice Hall
3. Kajita S, Hirukawa H, Harada K, Yokoi K (2014) Introduction to Humanoid
Robotics, Springer
4. Klančar G, Zdešar A, Blažič S, Škrjanc I (2017) Wheeled Mobile Robotics -
From Fundamentals Towards Autonomous Systems, Elsevier
5. Lenarčič J, Bajd T, Stanišić MM (2013) Robot Mechanisms, Springer
6. Merlet J-P (2006) Parallel Robots (Second Edition), Springer
7. Mihelj M, Podobnik J (2012) Haptics for Virtual Reality and Teleoperation,
Springer
8. Mihelj M, Novak D, Beguš S (2014) Virtual Reality Technology and Applica-
tions, Springer
9. Natale C (2003) Interaction Control of Robot Manipulators, Springer
10. Nof SY (1999) Handbook of Industrial Robotics, John Wiley Sons
11. Paul RP (1981) Robot Manipulators: Mathematics, Programming, and Control,
The MIT Press
12. Sciavico L, Siciliano B (2002) Modeling and Control of Robot Manipulators,
Springer
13. Spong MW, Hutchinson S, Vidyasagar M (2006) Robot Modeling and Control,
John Wiley Sons
14. Tsai LW (1999) Robot Analysis: The Mechanics of Serial and Parallel Manip-
ulators, John Wiley Sons
15. Xie M (2003) Fundamentals of Robotics—Linking Perception to Action, World
Scientific

Index
A
Acceleration
angular, 62, 244
centripetal, 62, 245
radial, 62, 245
tangential, 62, 244
Accelerometer, 94
Accuracy
distance, 239
drift, 233, 240
orientation, 237
pose, 233, 235
position, 237
Actuator
flexible, 175
series elastic, 175
variable stiffness, 175
Anthropomorphic robot, 6, 231
Application
hand-guided, 187
hand-over window, 185
Assembly, 147
Attractor
point, 222, 226
Automated guided vehicle, 189
B
Backward projection, 110, 113, 114
model based, 115
Balance, 210
Bumper, 98
C
Camera, 185
calibration, 114, 118
calibration pattern, 118
calibration tip, 118
extrinsic parameters, 114
intrinsic parameters, 112
Cartesian robot, 231
Center of Mass (COM), 214, 215
Closed kinematic chain, 69
Cobot, 9
Collaborative
application, 185
gripper, 184
operation, 173, 177
robot, 157, 173, 175, 184
workspace, 174, 186
Collision, 175
Compliance, 175
Constraints, 71, 77
Contact
quasi-static, 182
transient, 182
Contact sensor, 96
Control, 133
admittance, 176
force, 133, 147, 149, 151
gravity compensation, 136, 137, 144
impedance, 176
inverse dynamics, 137, 139, 144, 148,
152
inverse Jacobian, 143
orientation, 202
parallel composition, 151
position, 133
position and orientation, 203
proportional-derivative, 135, 136, 144
torque, 176
transposed Jacobian, 142
Conveyor, 164
M. Mihelj et al., Robotics, https://doi.org/10.1007/978-3-319-72911-4
247

248 Index
belt-driven, 165
chain, 164
metal chain, 164
roller, 165
Conveyor belt, 158
Cooperation, 173
Coordinate frame
base, 231
camera, 109
global, 189, 194, 200, 204, 206
image, 109
index, 111
local, 191
mechanical interface, 232
reference, 23, 31
robot, 208
world, 133, 231
Coordinates
external, 49, 80, 133, 141, 145, 151, 152
image, 111
index, 110
internal, 49, 133
joint, 80
Coriolis, 68
Cycle time, 159
Cylinder
pneumatic, 158
Cylindrical robot, 231
D
Danger, 153
Degrees of freedom, 5, 6, 210, 217
Delta robot, 69, 75
Differential drive, 192
Displacement, 16
Distance
braking, 180
intrusion, 180
protective separation, 180
stopping, 180
Dynamic Movement Primitives (DMP), 221,
223
estimation, 225
modulation, 227
Dynamic system
linear, 222
E
Electrical gripper, 167
Encoder
magnetic, 92
optical, 89
End-effector, 133, 142, 149, 160
Euler angles, 39, 218
Exoskeleton, 2, 3
passive, 220
F
Fault avoidance, 154
Feeder, 160
magazine, 163
part, 163
vibratory, 163
vibratory bowl, 163
vibratory in-line, 164
Feeding device, 160
Finger
spring, 167
Fixture, 160
Force
contact, 148, 149, 152
ground reaction, 211
limiting, 177, 181
magnetic, 170
permissible, 183
Force and torque sensor, 98
Force sensor, 98
Forcing term, 223
Forward projection, 108, 114
G
Grasp
by force, 167
by form, 167
reliable, 167
Gravity, 68
Gripper, 39, 41, 46, 165
electrical, 167
hydraulic, 167
magnetic, 169
multi-fingered, 166
pneumatic, 167
two-fingered, 166
vacuum, 168
Guiding device, 178
Gun
spot welding, 171
welding, 171
Gyroscope, 94
H
Hand guiding, 177, 178, 220
Haptic robot, 2

Index 249
Human motion capture, 217
Humanoid robot, 3, 209, 210
Hydraulic gripper, 167
I
Image coordinates, 111
Image plane, 109
Image processing, 118
Imitation learning, 210, 216, 217
Inclinometer, 94
Index coordinates, 110
Inertia, 68
Inertial measurement unit, 94, 96, 219
Instantaneous center of rotation, 193
Interface window, 186
J
Joint torque sensor, 101
K
Kinematic pair, 27
Kinematics, 49
direct, 49, 51, 73, 80, 214
inverse, 49, 51, 73, 80
L
Laser rangefinder, 103
Laser scanner, 104
Leg, 70
LiDAR, 105
Limit switch, 98
Localization, 189, 197
inductive guidance, 197
laser, 197
magnetic spot, 197
magnetic tape, 197
natural, 197
odometry, 199
radio-based, 199
wire, 197
Locomotion
biped, 210, 211
double support phase, 213
single support phase, 213
stance phase, 213
swing phase, 213
walking pattern, 213
M
Machine
indexing, 159
Magnet
electromagnet, 169
permanent, 170
Magnetic encoder, 92
Magnetic gripper, 169
Magnetometer, 94
Matrix
homogenous transformation, 11, 13, 16,
20, 27
Jacobian, 51, 55, 142–144, 148
rotation, 13, 39, 42
Mobile robot, 1, 189, 191
mobile forklift, 189
tugger, 189
unit loader, 189
Model
dynamic, 136, 138, 148, 176
geometrical, 19, 24
kinematic, 142
Monitoring speed and separation, 177
N
Nanorobot, 1
Navigation, 189, 197
O
Obstacle avoidance, 205
Odometry, 199
Omnidirectional, 190
Operation
collaborative, 173
Operator, 174
Optical encoder, 89
absolute, 89
incremental, 91
Orientation, 6, 16, 39
Overshoot, 240
pose, 233
P
Pallet, 158–160
injection molded, 161
metal, 161
vacuum formed, 161
Parallel robot, 69, 73, 76, 80
Part-holder, 159
Path
control, 202
Path following, 207
Path planning, 189, 201

250 Index
Perspective matrix equation, 110
Perspective projection, 108, 110
Phase
evolution, 228
variable, 223, 228
Platform, 70
Pneumatic cylinder, 158
Pneumatic gripper, 167
Point
final, 126, 129
initial, 126, 129
via, 126, 128, 129
Point-to-point, 123, 223, 226
Pose, 6, 16
initial, 23, 29, 31, 35
Position, 6, 16
Potentiometer, 87
Pressure
permissible, 183
Production line, 158
asynchronous, 159
hybrid, 158
in-line, 158
rotary, 158
synchronous, 159
Programming by demonstration, 210, 217
Projection
backward, 110, 113–115
forward, 108, 114
perspective, 108, 110
Proximity sensor, 102
Pump
vacuum, 169
Venturi, 169
Q
Quadrocopter, 2
Quaternion, 39, 44, 45
R
Ranging sensor, 102
Reducer, 86
Repeatability
distance, 239
drift, 233, 240
orientation, 238
pose, 233, 235, 238
position, 238
Robo-ethics, 3
Robot
anthropomorphic, 6, 231
cartesian, 231
collaborative, 9, 157, 173, 175, 184
cylindrical, 231
Delta, 69, 75
end-effector, 160
environment, 153
haptic, 2
humanoid, 3, 209, 210
industrial, 173, 174, 178
mobile, 1, 189, 191
parallel, 1, 69, 73, 76, 80
rehabilitation, 2
SCARA, 6, 22, 35, 231
segment, 27
soft, 9
spherical, 231
system, 173
vision system, 160
Robot arm, 4
Robot cell, 6, 8, 153
Robot dynamics, 60, 67
Robot gripper, 4
Robot hand, 9
Robotics, 1
humanoid, 209
Robot joint, 1, 4
prismatic, 4
revolute, 4
rotational, 4
translational, 4
Robot manipulator, 1, 4, 6, 7
Robot segment, 1, 4
Robot vehicle, 1
Robot vision, 107
Robot wrist, 4
Rotation, 6, 12, 16
Rotation matrix, 13, 39, 42
RPY angles, 39, 42, 134
S
Safety, 153–155
Safety hazard, 174
SCARA robot, 6, 231
Sensation
pain, 182
Sensor, 2, 4, 85
bumper, 98
capacitive, 185
contact, 96
electric, 85
electromagnetic, 85
exteroceptive, 85

Index 251
force, 98, 185
force and torque, 98, 175
inclinomater, 94
inertial measurement unit, 94
joint torque, 101, 175
laser rangefinder, 103
laser scanner, 104
LiDAR, 105
limit switch, 98
magnetic encoder, 92
optical, 86
optical encoder, 89
potentiometer, 87
proprioceptive, 85
proximity, 102
ranging, 102
safety rated, 176
tachometer, 93
tactile, 96, 175, 185
ultrasonic rangefinder, 102
Sensor abstraction disk, 200
Simultaneous Localization and Mapping
(SLAM), 200
Spherical robot, 231
Spring finger, 167
Stabilization time, 233, 240
Standards, 153
Statics, 54
Stereo vision, 115
Stewart-Gough platform, 69, 74, 78, 82
Stop
emergency, 178, 184
protective, 178–180
safety-rated monitored, 177–180, 182
Support polygon, 211, 214
T
Table
dial, 159
rotary, 161
Tachometer, 93
Tactile sensor, 96
capacitive, 97
deformation-based, 97
magnetic, 98
mechanical, 98
optical, 97
piezoelectric, 97
Tele-manipulator, 2, 3, 9
Torch
welding, 171
Tracking
optical, 217
Trajectory, 123, 132
interpolation, 126, 129
planning, 123
Translation, 5, 12, 16
Trapezoidal velocity profile, 123, 126
U
Ultrasonic rangefinder, 102
Unicycle model, 194
V
Variable
rotational, 29
translational, 29
Virtual environment, 2
Vision
robot, 107
Vision system, 158
W
Walking
dynamic, 215
static, 215
Wheel, 190
castor, 190
fixed, 190
spherical, 190
standard steered, 190
Sweedish, 190
Working area, 58
Workspace, 6, 55, 153, 231
collaborative, 174, 186
dexterous, 59
maximum, 174
operating, 174
reachable, 59
restricted, 174
Wrist sensor, 98
Z
Zero-Moment-Point (ZMP), 210, 211, 214

Robotics and automation an introduction-.pdf

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