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Wen Yu
PID Control with
Intelligent Compensation
for Exoskeleton Robots

PID CONTROL
WITH INTELLIGENT
COMPENSATION
FOR EXOSKELETON
ROBOTS

PID CONTROL WITH
INTELLIGENT
COMPENSATION FOR
EXOSKELETON ROBOTS
WEN YU
CINVESTAV-IPN (National Polytechnic Institute),
Mexico City, Mexico

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To my daughters
Huijia and Lisa

CONTENTS
About the Author xi
Preface xiii
Introduction xv
1. Preliminaries 1
1.1. Exoskeleton robots 1
1.2. Control of exoskeleton robots 3
1.3. Neural network and fuzzy systems 4
1.4. PD and PID control 5
1.4.1. PID parameters tuning 5
1.4.2. PID control in task space 6
1.4.3. PID control with velocity observer 7
1.5. PD and PID control with compensations 7
1.6. Robot admittance control 9
1.7. Trajectory generation of exoskeleton robots 10
2. Stable PID Control and Systematic Tuning of PID Gains 13
2.1. Stable PD and PID control for exoskeleton robots 13
2.1.1. Stable PD control 14
2.1.2. Stable PID control 17
2.2. PID parameters tuning in closed-loop 22
2.2.1. Linearization of the closed-loop system 25
2.2.2. PD/PID tuning 26
2.2.3. Reﬁne PID gains 28
2.2.4. Stability conditions for PID gains 28
2.3. Application to an exoskeleton 29
2.4. Conclusions 33
3. PID Control in Task Space 35
3.1. Linear PID control in task space 35
3.2. Linear PID control with velocity observers 44
3.3. Experimental results 48
3.4. Conclusions 53
4. PD Control with Neural Compensation 55
4.1. PD control with high gain observer 55
4.1.1. Singular perturbation method 56
4.1.2. Lyapunov method 63
4.2. PD control with neural compensator 65
4.2.1. PD control with single layer neural compensation 65
4.2.2. PD control with a multilayer feedforward neural compensator 66
vii

viii Contents
4.3. PD control with velocity estimation and neural compensator 71
4.4. Simulation 75
4.5. Conclusions 80
5. PID Control with Neural Compensation 81
5.1. Stable neural PID control 81
5.2. Neural PID control with unmeasurable velocities 91
5.3. Neural PID tracking control 96
5.4. Experimental results of the neural PID 101
5.5. Conclusions 106
6. PD Control with Fuzzy Compensation 109
6.1. PD control with fuzzy compensation 109
6.2. Membership functions learning and stability analysis 114
6.3. Experimental comparisons 120
6.4. Conclusion 124
7. PD Control with Sliding Mode Compensation 125
7.1. PD control with parallel neural networks and sliding mode 125
7.2. PD control with serial neural networks and sliding mode 129
7.3. Simulation 133
8. PID Admittance Control in Task Space 139
8.1. Human–robot cooperation via admittance control 139
8.2. PID admittance control in task space 141
8.3. PID admittance control in task space with neural compensation 145
8.4. Admittance PD control with Jacobian approximation 149
8.5. Admittance control with adaptive compensations 154
8.6.1. Pan and tilt robot 156
8.6.2. 4-DoF robot 156
9. PID Admittance Control in Joint Space 159
9.1. PD admittance control 159
9.2. PD admittance control with adaptive compensations 164
9.3. PD admittance control with sliding mode compensations 167
9.4. PID admittance control 168
9.5.1. Pan and tilt robot 170
9.5.2. 4-DoF exoskeleton 170
9.6. Conclusion 174

Contents ix
10. Robot Trajectory Generation in Joint Space 175
10.1. Codebook and key-points generation 175
10.2. Joint space trajectory generation with a modiﬁed hidden Markov model 179
10.3. Experiments of learning trajectory 185
10.3.1. Two-link planar elbow manipulator 186
10.3.2. 4-DoF upper limb exoskeleton 189
A. Design of Upper Limb Exoskeletons 195
A.1. Heavy duty exoskeleton robot 195
A.2. Portable exoskeleton robot 200
Bibliography 205
Index 213

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ABOUT THE AUTHOR
Wen Yu received the BS degree from Tsinghua University, Beijing, China, in 1990 and
the MS and PhD degrees, both in Electrical Engineering, from Northeastern Univer-
sity, Shenyang, China, in 1992 and 1995, respectively. From 1995 to 1996, he served as a
lecturer in the Department of Automatic Control at Northeastern University, Shenyang,
China. Since 1996, he has been with the Centro de Investigación y de Estudios Avanza-
dos, Instituto Politécnico Nacional (CINVESTAV-IPN), Mexico City, Mexico, where
he is currently a Professor with the Departamento de Control Automatico. From 2002
to 2003, he held research positions with the Instituto Mexicano del Petroleo. He was
a Senior Visiting Research Fellow with Queen’s University Belfast, Belfast, UK, from
2006 to 2007, and a Visiting Associate Professor with the University of California, Santa
Cruz, from 2009 to 2010. He also holds a visiting professorship at Northeastern Univer-
sity in China since 2006. Dr. Wen Yu serves as an associate editor of IEEE Transactions
on Cybernetics, Neurocomputing, and for the Journal of Intelligent and Fuzzy Systems.
He is a member of the Mexican Academy of Sciences.
xi

PREFACE
Proportional-integral-derivative (PID) control is widely used in robot control systems.
In the absence of robot knowledge, a PID controller may be the best controller, because
it is model-free, and its parameters can be adjusted easily and separately. The PID con-
troller has big advantages over the other controllers: simple and clear physical meaning.
The three gains of the PID controller need to be tuned to guarantee good performances,
which include rise-time, overshoot, settling time, and steady-state error. The integrator
in the PID controller reduces the bandwidth of the closed-loop system. In order to re-
move the steady-state error caused by uncertainties and noise, the integrator gain has to
be increased. This leads to worse transient performance, and even destroys the stability.
Therefore many robot manipulators use pure proportional-derivative (PD) control or
PD control with a small integral gain.
Model-based compensation for PD control is an alternative method for introducing
an integral component to PD control, i.e., PID control. However, it needs the structure
information of the robot. The controller becomes complex, and many good properties
of the linear PD and PID control do not exist. Intelligent compensations for PD and
PID control do not need a mathematical model; they are model-free compensators.
There are three different approaches to combine PID control with intelligent tech-
niques: 1) The intelligent controllers are formed into the PID structure. By proper
updating laws, the parameters of PID controllers are changed such that the closed-loop
systems are stable. They are not real industrial PID controllers; 2) Intelligent techniques,
such as fuzzy logic, neural networks, and the expert system, are used to tune the pa-
rameters of PID controllers; 3) The industrial linear PID controller adds an intelligent
compensator. The main obstacle of this PID control with intelligent compensation is
theoretically difficult in the performance analysis. Even for linear PID, it is not easy
to prove asymptotic stability. Without a theoretical guarantee for this PID control, in-
dustrial applications cannot be carried out safely. This book is intended for the third
one.
Throughout the last three decades, many designs of exoskeletons for human power
amplification have been developed and evaluated. In 2000 the author took his sabbatical
year in Professor Jacob Rosen’s bionics lab, University of California at Santa Cruz.
The author started his research on the control problems of the exoskeleton robots.
From 2012 to 2016, the author was supported by the CONACyT’s project “Intelligent
Human–Robot Interaction and Robotic Rehabilitation Exoskeleton,” and started to
study stable PID control for exoskeleton robots. After five years of work, the results
on closed-loop tuning of PID parameters, stability analysis of PD and PID control
with intelligent compensations, PID control in task space, PID admittance control, and
xiii

xiv Preface
trajectory generation of the exoskeleton robot, are developed. These results have been
published in a variety of journals and conferences. The author wishes to put together
all these results within this book.
Much of the material in this book was presented in the author’s two courses in
CINVESTA-IPN: Medical Robotics and the Introduction to Robotics. Part of the
book was written while the author was visiting the University of California at Santa
Cruz. This book is organized as a textbook for the course on the control of robots. It
could be used for self-learning. The level of competence expected for the reader is that
they have covered courses in robotics, nonlinear systems analysis, neural networks, and
some elements of the optimization theory.
Many people have contributed to shape and substance of this book. The author
would like to thank Professor Jacob Rosen, Dr. Ji Ma, Dr. Levi Miller, his former
PhD, and master students Debbie Crystal Hernández, Carlos Parga Ingenieur, Edgar
Oswaldo García, José Adolfo Perrusquía, and Javier Garrido Melendez. In addition,
he would like to thank Dr. Alberto Soria López and Dr. Rubén Alejandro Garrido
for helping to develop the exoskeleton robots. The author want to thank the financial
supports of CONACyT, Mexico, with the projects 167428, 50480Y, and the University
of California – CONACyT 2012. Last, but not least, the author is thankful for the time
and dedication of his wife, Xiaoou. Without her, this book would not be possible.
Wen Yu
Mexico, May of 2017

INTRODUCTION
Proportional-integral-derivative (PID) control is widely used in industrial robot manip-
ulators. The integrator in the PID controller reduces the bandwidth of the closed-loop
system, leads to worse transient performance, and even destroys the stability. Many
robot manipulators use proportional-derivative (PD) control with gravity and friction
compensations, and the gravity and friction models are needed. Intelligent control
has dramatically changed the face of industrial control engineering. The intelligent
compensation for PD/PID control has developed rapidly recently. Because of the inter-
disciplinary nature of the subject, there are only a few books consisting of the general
know-how in designing, implementing, and operating PD/PID control with intelligent
compensations.
This book introduces how to combine the traditional PD/PID control techniques
with the intelligent control, and presents several recent methods to design neural, fuzzy,
and sliding mode compensators, even with a high-gain velocity observer.
We also extend the neural PD control into neural PID control to reduce the inte-
gration gain. We give explicit conditions on how to select the linear PID gains by the
proofs of semiglobal asymptotic stability and local asymptotic stability with a velocity
observer. These conditions are applied in both task space and joint space. The desired
trajectory is generated with a modified hidden Markov model. Several experimental
studies on exoskeleton robots with these intelligent PID controllers are addressed.
The book is suitable for advanced undergraduate students and graduate engineering
students. In addition, practicing engineers will find it appropriate for self-study.
xv

CHAPTER 1
Preliminaries
Abstract
In this chapter, some background information about how to combine the traditional PD/PID control
techniques with the intelligent methods and how to apply them to the upper limb exoskeleton robots,
are proposed.
Keywords
Exoskeleton robot, Proportional-integral-derivative, Human–machine integration, Intelligent compen-
sation
1.1. EXOSKELETON ROBOTS
Throughout the last three decades, several designs of exoskeletons for human power
amplification have been developed and evaluated, such as Honda Exoskeleton Legs [46],
Berkeley Lower Extremity Exoskeleton [69], Hybrid Assistive Limb [47], and the MIT
Exoskeleton [33]. The first exoskeleton generation was developed based on the mission
profile of the US Department of Defense that defined the exoskeleton as a powered suit
to augment the lifting and carrying capabilities of soldiers. It was originally named the
“man-amplifier” [92].
The exoskeleton robot is worn by the human operator as an orthotic device. A wear-
able robot is a metachromatic system whose joints and links correspond to those of the
human body. The same system operated in different modes can be used for three fun-
damental applications [112]: human-amplifier assisted device sharing a portion of the
external load with the operator, haptic device, and automatic physiotherapy.
Application fields of the exoskeleton robot include tele-manipulation, man-
amplification, neuro-motor control and rehabilitation [49], and assisting with impaired
human [51]. The first generation prototype, known as Hardiman [92], was the first
attempt to mechanically design a man-amplifying exoskeleton using a hydraulically
powered articulating frame worn by an operator. The second generation of exoskele-
tons utilized the direct contact forces (measured by force sensors) between the human
and the machine as the main command signals to the exoskeleton. The operator was
in full physical contact with the exoskeleton throughout its manipulation [69]. The
third generation of exoskeletons is defined by at higher levels of the human physiologi-
cal (neurological) system hierarchy; one can overcome the electrochemical–mechanical
delay, usually referred to as the electromechanical delay (EMD) [37].
PID Control with Intelligent Compensation for Exoskeleton Robots
DOI: 10.1016/B978-0-12-813380-4.00001-3
Copyright © 2018 Elsevier Inc.
All rights reserved. 1

2 PID Control with Intelligent Compensation for Exoskeleton Robots
The common feature in both the first and second generation of exoskeletons is that
the operator must apply an action, either kinematic or dynamic in order to trigger the
exoskeleton response. Obviously, this sequence of events constitutes a source of delay
in both systems. The operator will not be able to act and respond quickly, for example,
to catch a falling object. The third generation of exoskeletons is defined by at higher
levels of the human physiological (neurological) system hierarchy; one can overcome
the EMD delay.
Two sets of databases are collected prior to the design of the exoskeleton arm: the
kinematics and the dynamics of the human arm during activities of daily living. They are
studied in part to define the engineering specifications for the exoskeleton arm design.
In addition, the neural activities in well-structured single and multijoint arm activities,
under various loading conditions, are needed to develop muscle models [110], which
enable development of the neural-control approach of the exoskeleton. Kinematic data
of the upper limb are acquired with a motion capture system while performing daily
activities from subjects, but the kinematics and dynamics of the exoskeleton robot are
still very complex.
For many physical tasks, human performance is limited by muscle strength. Simi-
larly, muscle weakness is the primary cause of disability for persons with a variety of
neuromuscular diseases including stroke, spinal cord injury, muscular dystrophies, and
other neuro-degenerative disorders. Opposite this limitation in muscular strength, hu-
mans possess specialized and complex algorithms for control of movement, involving
both higher and lower neural centers. These algorithms enable humans to perform very
complicated tasks such as locomotion and arm movement, while at the same time avoid-
ing object collisions. In contrast, robotic manipulators can be designed to perform tasks
requiring large forces or moments, depending on their structure and on the power of
their actuators. However, the control algorithms that govern their dynamics lack the
flexibility to perform in a wide range of conditions while preserving the same qual-
ity of performance as humans. It seems therefore that combining these two entities,
the human and the robot, into one integrated system under the control of the human,
may lead to a solution which will benefit from the advantages of each subsystem. The
mechanical power of the machine, integrated with the inherent human control system,
could allow efficient performance of tasks requiring higher forces than the human could
otherwise produce.
At the heart of this human–machine integration lie two fundamental scientific
and technological issues: (i) the exoskeleton (orthotic device) mechanism itself and its
biomechanical integration with the human body, and (ii) the human machine interface
(HMI). These two key issues will determine the quality of the integration between the
human and the exoskeleton in terms of how natural it will be for the operator to control
the exoskeleton device as a biological extension of his/her body.

Preliminaries 3
Figure 1.1 Control of exoskeleton robots.
1.2. CONTROL OF EXOSKELETON ROBOTS
The control structure of many human guide robots, such as robot exoskeleton, hu-
manoid robot, and surgical robot, have three levels: joint angle tracking, trajectory
planning, and path planning; see Fig. 1.1. The path planning is in task space. It generates
a path from a stating-point to an ending-point with respect to some restrictions. The
trajectory planning can be in task space or joint space. It gives desired trajectories of the
end-effector (task space), or desired joint angles (joint space), which are in the path gen-
erated by the path planning. The joint angle tracking usually uses PD/PID controller to
force the joints follow the desired angles generated by the trajectory planning.
The trajectory planning is to generate desired joint angles that satisfy human re-
quirements [151]. This is the object of transferring human skill to the robot through
demonstrations. We also call it as programming by demonstration (PbD) or learning
from demonstration (LfD) [8]. The robot trajectory generation can be broadly divided
into two trends: (1) Symbolic level. The human skill is decomposed into a sequence of
action-perception units, then a statistical model is used to deal with the demonstrations
[53,11]; (2) Trajectories level. A nonlinear mapping is used to model the sensor/motor
information. The trajectories level method is robust to the environment changes [59].
The symbolic level method is suitable to model human complex actions.
In order to control an exoskeleton robot, the interface between the human and
the mechanical system is needed. This interface, particularly in the field of hepatics,
maps human force into a motion. The input of an admittance is force and the output
is velocity or position. In other words, an admittance device would sense the input
force and “admit” a certain amount of motion. Path tracking accuracy and contact
forces are two contradiction objectives in stiffness control [149] and force control [28].
Improvement of the position tracking accuracy might give rise to larger contact forces.
The force/position control [115] and impedance control [56] used inverse dynamics

such that the task space motion is globally linearized and decoupled, and asymptotically
stable.
The heart of this human–machine integration has two fundamental scientific and
technological issues: the exoskeleton mechanism itself and its biomechanical integra-
tion with the human body [110]. There are three fundamental applications: device for
teleoperation [118], human-amplifier [33], and physical therapy modality as part of the
rehabilitation [111]. The exoskeleton robots can be divided into upper limbs [110] and
lower limbs [69].
1.3. NEURAL NETWORK AND FUZZY SYSTEMS
Both neural networks and fuzzy logic are universal estimators, they can approximate any
nonlinear function to any prescribed accuracy, provided that sufficient hidden neurons
and fuzzy rules are available. Resent results show that the fusion procedure of these
two different technologies seems to be very effective for nonlinear systems identifica-
tion [23].
The stability problem of fuzzy neural identification is very important in applica-
tions. It is well known that normal identification algorithms (e.g., gradient descent and
least square) are stable in ideal conditions. In the presence of unmodeled dynamics,
they might become unstable. The lack of robustness of the parameter identification was
demonstrated in [147] and became a hot issue in the 1980s, when some robust modifi-
cation techniques were suggested [60]. The learning procedure of fuzzy neural networks
can be regarded as a type of parameter identification. Gradient descent and backprop-
agation algorithms are stable if fuzzy neural models can match nonlinear plants exactly.
However, some robust modifications must be applied to assure stability with respect to
uncertainties. The projection operator is an effective tool to guarantee that fuzzy mod-
eling bounded [143]. Another general approach is to use robust adaptive techniques [60]
in fuzzy neural modeling, for example, [144] applied a switch σ-modification to prevent
parameters drift.
Fuzzy neural identification uses input–output data and model structure. It can be
regarded as black-box approximation. All uncertainties can be considered as parts of the
black-box, i.e., unmodeled dynamics are within the black-box model, not as structured
uncertainties. Therefore the robustifying techniques usually employed are not neces-
sary. In [143] the authors suggested a stable and optimal learning rate without robust
modification, and a genetic search algorithm was proposed to find the optimal rate.
However, the algorithm is complex, and difficult to realize. By using passivity theory,
we successfully proved that for continuous-time recurrent neural networks, gradient
descent algorithms without robust modification were stable and robust to any bounded
uncertainties [159], and for continuous-time identification they were also robustly sta-
ble [155].

Preliminaries 5
Gradient descent and backpropagation are always used to adjust the parameters of
membership functions (fuzzy sets) and the weights of defuzzification (neural networks)
for fuzzy neural networks. Slow convergence and local minimum are main drawbacks
of these algorithms [119]. Some modifications were derived in recently published liter-
atures. [23] suggested a robust backpropagation law to resist the noise effect and reject
errors drift during the approximation. [145] used B-spline membership functions to
minimize a robust object function; their algorithm can improve convergence speed. In
[150], RBF neural networks were applied to fuzzy systems, a novel approach of deter-
mining structure and parameters of fuzzy neural systems was proposed.
1.4. PD AND PID CONTROL
It is well known that most of the industrial manipulators are equipped with the simplest
proportional-derivative (PD) or proportional-integral-derivative (PID) control. Various
modified PD control schemes and successful experimental tests of these schemes have
been published [131,107]. With the absence of robot knowledge, the PID controller
may be the best controller for industrial robot manipulators [65], because it is model-free
and its parameters can be adjusted easily and separately [6,5]. The PID controller has
big advantages over the other controllers: simple and clear physical meaning. The three
gains of the PID controller need to be tuned to guarantee good performances, which
include rise-time, overshoot, settling time, and steady-state error.
From a control viewpoint, the regulation error caused by gravitational torques can be
removed by introducing an integral component to the PD control. There is widespread
use of PID control in an industrial manipulator, although it may reduce bandwidth of
the closed-loop system. In order to remove steady-state error caused by uncertainties
and noise, the integrator gain has to be increased. This leads to worse transient perfor-
mance and even destroys the stability. Therefore many robot manipulators use the pure
proportional-derivative (PD) control or PD control with a small integral gain [85].
Bounded stability can be guaranteed with a positive PD gains a controller for system
regulation [131]. The robotic system performance that utilizes a PD controller is limited
unless gravity compensation is applied, which requires a model of the system’s dynamic
[134,125,71,109]. Nonlinear PD controllers can also achieve asymptotic stability, such
as PD control with time-varying gains [113], nonlinear modification [101], and sliding
mode compensation [108].
1.4.1 PID parameters tuning
Since a PID controller is in linear form, the main study on PID tuning focused on linear
systems [102]. The tuning process for PID gains can be classified into five categories:

1. Model-based analytical tuning. According to the analytical relations between the
model and the control objective, the PID gain are calculated from the algebraic
equations [21,24,58].
2. Heuristic methods. These methods combine several techniques, such as practical
experience [161,20], manual tuning [5], and artificial intelligence [129,85,66].
3. Frequency domain methods. Frequency characteristics are easily obtained for linear
systems. When the controlled process are almost linear systems, the PID controller
can be tuned in the frequency domain [123].
4. Optimization methods. PID control can be transformed into a special optimal con-
trol form. PID tuning becomes an off-line numerical optimization problem [81].
5. Adaptive methods. Based on adaptive control and parameter on-line identification,
PID gains tuning can be realized as an automated online tuning process [143].
The above tuning methods cannot be applied to robot control directly, because the
robot dynamic is nonlinear. PID tuning for robot control can be grouped as:
1. Intelligent methods. The intelligent techniques, such as fuzzy logic [129], neural
networks [85], and the genetic method [66], are used to tune PID gains but the
final controllers are no longer industrial linear PID.
2. Impedance control. The inverse dynamics are applied to transfer the robot into a
linear system. Then some mechanical impedance ideas are applied to tune PID gains
[56]. In [21], discrete-time approximation of inverse dynamics was calculated such
that PID parameters could be adjusted.
3. Lyapunov approach. The Lyapunov approach was used to adjust the PID controller
such that it follows linearization control [22].
The above methods need exact models of the robot. The physical meanings of the
PID gains lost, because these PID controllers did not use the properties of robots. There
are the following difficulties to design a systematic tuning method for robot PID control:
• The control torque of each joint affects the other joints, and these influences are
strong nonlinear.
• There are too many gains to be tuned simultaneously and heuristically for a robot via
the Ziegler–Nichols [161] or Cohen–Coon [20] method. A six degree-of-freedom
robot manipulator has 18 gains to be tuned. When one gain is tuned, it requires to
tune the other 17 gains in turn because of dynamics coupling in the robot.
• The nonlinear methods, such as stability analysis, can obtain the upper and lower
bounds of PD gains. However, the desired performances are not guaranteed.
1.4.2 PID control in task space
Task space (or Cartesian space) is defined by the position and orientation of the end
effector of a robot. Joint space is defined by a vector whose components are the transla-
tional and angular displacements of each joint of a robotic link. The common linear PID
does not include any component of the robot dynamics into its control law whenever it

Preliminaries 7
is used in joint space or task-space. In order to analyze the stability of PID control in the
task-space, the simplest approach is to modify the linear PID into a nonstandard one, for
example, the integral term was modified into a linear combination of the velocity error
and the position error [128], the position error was filter by a scalar potential function
[27], or the input was saturated in [40].
Joint space and task space regulations are two common control schemes for robot
manipulators. Since the final control goal is normally in task-space, it is more natural to
design a robot controller in task-space. Task space PD control is the simplest scheme to
control robot manipulators. It can be classified into “transpose Jacobian” and “inverse Ja-
cobian” control. In the regulation case, similar with joint space [131], any positive gains
of PD controllers in the task space guarantee stability (bounded) [25]. However, asymp-
totic stability is not guaranteed when manipulators dynamics contain the gravitational
torques vector, friction, and the other uncertainties, unless model-based compensation
is applied [71]. Some nonlinear PD controllers can also achieve asymptotic stability, such
as PD control nonlinear gains [79] and sliding mode compensation [88].
1.4.3 PID control with velocity observer
There exists one weakness in PD control: PD control requires the measurements of joint
positions and joint velocities. It is necessary to implement position and velocity sensors
at each joint. The joint positions measurements can be obtained by means of encoders,
which give very accurate measurements. The joint velocities are usually measured by
velocity tachometers, which are expensive and often contaminated by noise [78].
One possible solution is to implement a velocity observer. Many papers have been
published devoted to the theory and implementation of velocity observers for manipu-
lators. Two kinds of observers may be used: the model-based observer and model-free
observer. The model-based observer assumes that the dynamics of the robot is com-
pletely known or partially known. For example, a sliding model observer was proposed
in [17] if the inertia matrix of the robotic’s dynamic is known; for similar conditions, an
adaptive observer was proposed in [18], and a passivity method was developed in [14].
The model-free observer means that no exact knowledge of robot dynamics is required.
The most popular used observers are the high-gain observers, which can estimate the
derivative of the output [100].
1.5. PD AND PID CONTROL WITH COMPENSATIONS
Due to the existence of friction, gravity forces, and unmodeled dynamics, the PD-
control cannot guarantee that the steady state error becomes zero [134,85]. Two kinds
of compensation can be used. The global asymptotic stability PD control was realized
by pulsing gravity compensation in [132]. If the parameter in the gravitational torque
vector are unknown, the adaptive version of PD control with gravity compensation was

introduced in [134]. A compensator could be used to make the tracking error zero, if we
knew the friction, gravity forces, and unmodeled dynamics [135]. In practice, this is not
always possible, for example, the modeling error and friction coefficients are regarded
to be unknown. From the proofs of stability of PD control [4,131], the gravity forces
should be compensated to guaranteed asymptotic stability.
Model-based compensation with PD control is an effective method for PID control
[131], such as adaptive gravity compensation [134], Lyapunov-based compensation [35],
desired gravity compensation [71], and PD+ with position measurement [109]. They
all need structure information of the robot gravity. Some nonlinear PD controllers can
also achieve asymptotic stability, for example, PD control with time-varying gains [113],
PD control with nonlinear gains [101], and PD control with sliding mode compensa-
tion [108]. These controllers are complex and many good properties of the linear PID
control do not exist.
Intelligent compensation does not need a mathematical model; it is a model-free
compensator. It can be classified into a fuzzy compensator [58], fuzzy PID [36], neu-
ral compensator [85], and fuzzy-neural compensator [24]. There are two different
approaches to combine PID control with the intelligent control, such as neural con-
trol. The first one is neural networks that are formed into PID structure [29,122,138].
By proper updating laws, the parameters of PID controllers are changed such that the
closed-loop systems are stable. They are not real industrial PID controllers, because the
PID gains (weights of the neural networks) are time-varying. The second method is that
intelligent techniques are used to tune the parameters of PID controllers, such as fuzzy
tuning [94], neural tuning [55,153], and expert tuning [70]. The controllers are still
industrial linear PID; however, the stability of the closed-loop system is not guaranteed.
By proper weight tuning algorithms, which are similar with robust adaptive control
methods, the derivative of the Lyapunov function is negative, as long as the filtered
tracking error is outside of the ball with radius B
Kv
. Here, B is the upper bound of
all unknown uncertainties and Kv is the derivative gain in PD control. These neural
PD controllers are uniformly ultimate boundedness (UUB), and tracking errors go to
smaller with increasing the gain Kv. The cost of large Kv is that the transient perfor-
mance becomes slow. Only when Kv → ∞, the tracking error converges to zero [48].
When the friction and gravity forces are unknown, neural networks, fuzzy systems,
and sliding mode techniques can compensate for them [86]. The neural networks are
black-box models, which use input/output data to train their weights. Fuzzy systems
are based on fuzzy rules, which are constructed from prior knowledge [22]. Sometimes
fuzzy systems are regarded as gray-box models. A neuro-adaptive controller by using a
neural networks plus a servo-feedback control was proposed in [77]. A hybrid neural
control for robot tracking was discussed in [13], where static neural networks are used
to learn mass matrix, centrifugal, and Coriolis forces. Because they used the theory of
function approximation, the PD control with neural networks (PD+NN) is sensitive

Preliminaries 9
to the training data and local minima. Due to neural modeling error, PD+NN cannot
assure that the regulation errors are asymptotically stable.
Sliding mode control (SMC) is obtained by means of injecting a nonlinear dis-
continuous term. This discontinuous term is the one that enables the system to reject
disturbances and also some classes of mismatches between the actual system and the
model used for design [93]. These standard SMCs are robust with respect to internal
and external perturbations, but they are restricted to the case in which the output rela-
tive degree is one. Besides, the high frequency switching that produces the sliding mode
may cause a chattering effect. The tracking error of SMC converges to zero if its gain is
bigger than the upper bound of the unknown nonlinear function. Boundary layer SMC
can assure no chattering happens when the tracking error is less than ε, but when the
tracking error converges to ε, it is not asymptotically stable [125]. A new generation of
SMC using the second-order sliding-mode has been recently developed by [104] and
[84]. This higher order SMC preserves the features of the first-order SMC and improves
it in eliminating the chattering and fast convergence.
Normal combinations of PD control with neural networks (PD+NN) and sliding
mode (PD+SMC) are to apply these three controllers at the same time [54], while NN
compensates the control error, and the SMC reduces the remaining error of neural PD
such that the final tracking error is asymptotically stable [85]. The chattering is still big,
because PD+SMC and PD+NN work parallel.
1.6. ROBOT ADMITTANCE CONTROL
Wearable robots such as exoskeletons combine the human and the robot into one inte-
grated system. Cooperative control is a rapidly emerging field in robotics, in the sense
to provide an interaction between the human and the manipulator [32]. The main idea
of cooperative control is to combine human skills and robot properties on a specific task
[39,15], e.g., comanipulation, haptic [3,69], learning from demonstrations [41], etc. Ad-
mittance control is one of the most common implementations for compliance control
of robotic manipulators [34]. In mechanical systems, particularly in the field of hepatics,
an admittance is a dynamic mapping from force to motion. The input of an admittance
is force and the output is velocity or position.
The admittance controller has forces/torque input, and “admits” a certain amount
of motions [156]. The relation between forces/torque and motion is imposed by a mass-
spring-damper system. These parameters give the ability of the manipulator following
the movement imposed by the human operator [32].
Two generations of human–machine integration are defined as: (1) it is a man-
amplifying exoskeleton using a hydraulically powered articulating frame worn by an
operator that would greatly increase the strength of a human operator [92]. This purely
positional control strategy may lead to poor responsiveness and instability; (2) the op-

erator is in full physical contact with the exoskeleton utilizing the direct contact forces
(measured by force sensors) between the human and the machine [69]. The most ef-
fective human–machine integration for the second one is the impedance/admittance
control strategy, which is devised aimed at limiting both internal and contact forces [56].
Path tracking accuracy and contact forces are two contradiction objectives in stiff-
ness control [149] and force control [28]. Improvement of the position tracking accuracy
might give rise to larger contact forces. The force/position control [115] and impedance
control [56] used an inverse dynamic such that the task space motion is globally lin-
earized and decoupled, and asymptotically stable. In [30], two three-axis force sensors
are used for admittance control of the upper and lower arm segments. It uses the top
half of the Jacobian to compute the forces and torques of the human arm. However,
all above impedance/admittance need robot models. A pure positional control strategy
may lead to the build up of large forces (both external and internal). The impedance
control strategy is devised aimed at limiting both internal and contact forces [56].
The normal admittance control used inverse dynamics such that the task space
motion is globally linearized and decoupled, and asymptotically stable [57,76]. This
controller has an inner position loop that enhances robustness against modeling error
[67]. The inner position loop can be any position controller. The problem with this con-
troller is that is impossible to design a model-based admittance control when a complete
dynamic model of the robot is unknown [154]. The modeling error prevents accu-
rate realization of the admittance dynamics; this is known as the “accuracy/robustness
dilemma in impedance control” [67]. The dynamic compensation uses the gravitational
torques, which is one of the principal factors of control accuracy. The inertia and Cori-
olis matrix have small values for small velocities that can be ignored, but the gravitational
torque vector cannot. For that reason, it is very important to compensate them, such as
in friction and disturbances [157].
There are many works on tracking control using adaptive admittance algorithms
[91], adaptive control [24,99], neural networks [157], fuzzy control [61,80], robust
control [93], and other algorithms [67]. In [124] the dynamics are estimated of the
manipulators, which is different with the model based method [96].
In admittance control field, PID control is used to establish a cooperative application
in task space between an exoskeleton and the operator [157,158]. Another practical
application of PID admittance control is to decouple the system [38].
1.7. TRAJECTORY GENERATION OF EXOSKELETON ROBOTS
Although the trajectory planning in joint space can avoid the calculation of the inverse
kinematics, the demonstrations in joint space are time-dependent. Fig. 1.2 shows the
trajectories in joint space and task space, when a two-link planar robot draws a broken
line. Since the trajectories in task space only give space relation, the three lines overlap

Preliminaries 11
Figure 1.2 A two-link planar robot draws a broken line: (1) left is in joint space, and (2) right is in task
space.
in task space. However, in joint space they are completely different because of different
drawing speeds. In this sense, training in task space is easier than in joint space [140,
82,143,43]. After the training in task space, the inverse kinematics need to be solved,
which requires complete knowledge of the robot.
There are few works in joint space [10,111]. The dynamic time warping (DTW) is
an effective tool to deal with the time-dependent problem. The computation time for
one-dimensional signals, such as time series, is in polynomial. The extension of DTW
for more than two-dimensions, like robots, becomes NP-complete. The accuracy of the
high dimension approximation is also very low [121].
Statistical learning techniques deal with the high variability inherent in the demon-
strations. They are not sensitive to disturbances. For instance, the spline smoothing
technique can deal with the uncertainty in several motion demonstrations [139]. The
mean and variance of the collected variables are applied in [103] to generate a model.
[62] realizes online imitation by encoding two different motor loops.
The Hidden Markov Model (HMM) generates a sequence, which is called the
Markov chain [116,152]. It can encode the motion of a robot, and find the highest
probability state path by the Viterbi algorithm [141]. HMMs use finite Gaussian mix-
ture models as their hidden state distributions. The Gaussian mixture model can encode
a set of trajectories [8]. The Gaussian mixture regression can retrieve a smooth trajec-
tory from several demonstrations [11]. There are many successful applications on robot

trajectory generation via the HMM [152,83]. The HMM offers many advantages over
other statistical models for human behavior modeling, such as better compression, vari-
ant structures, training incrementally, etc. One weakness of HMM is that the trajectory
generation can only use the current state for the emission and the transition probabilities.
HMM does not map well to many time-dependent domains, such as joint space [97].
To train HMM, it is necessary to map continuous trajectories into discrete values,
named codebook. It is impossible to use all sampled data to train HMM. The key-
points include necessary information for HMM. The normal method of selecting the
key-points uses the shape of the trajectory. It can be position evaluation [152] or posi-
tion/velocity evaluation [140]. Linde–Buzo–Gray (LBG) is the most popular method.
The above methods do not work well in joint space, because the trajectories in joint
space are time-dependent, while these methods use the shape information [42]. Lloyd’s
algorithm partitions data into well shaped and uniformly sized convex cells [90]. It
repeatedly finds the centroid of each set in the partition using Voronoi diagrams.

CHAPTER 2
Stable PID Control and Systematic
Tuning of PID Gains
Abstract
Although great progress has been made in a century-long effort to design and implement robotic
exoskeletons, many design challenges continue to limit the performance of the system. One of the
limiting factors is the lack of simple and effective control systems for the exoskeleton [56,137]. The
position error caused by gravitational torques can be reduced by introducing an integral component
to the PD control. In order to assure asymptotic stability, several components were previously added
to the classic linear PID controllers, for example, fourth-order filter [101], nonlinear derivative term [4],
nonlinear integral term (saturated function) [71], input saturation, and nonlinear observer [2].
Linear PID is the simplest and the most popular industrial controller, since tuning its internal parame-
ters does not require a model of the plant and can be performed experimentally. Lyapunov function
was previously used for the tuning procedure of a linear PID [72]. However, the inertia matrix and the
gravitational torque vector of the system have to be clearly defined [73,63]. If the robot dynamic can
be rewritten into a decoupled linear system with bounded nonlinear system, the stability of linear PID
can be proven [117]. The asymptotic stability was not achieved.
In this chapter, the semiglobal asymptotic stability is proven along with a new approach for tuning the
parameters of the PID controller. We apply this method on an upper limb exoskeleton. Experimental
results show that this new PID tuning method is simple, systematic, and effective for robot control.
Keywords
Semiglobal asymptotic stability, PID gains, Systematic tuning
2.1. STABLE PD AND PID CONTROL FOR EXOSKELETON ROBOTS
The dynamics of a serial n-link exoskeleton robot can be written as [131]
M

q
··
q + C

q,
·
q
·
q + g

q

+ f
·
q

= τ (2.1)
where q ∈ n denotes the joint positions,
·
q ∈ n denotes the joint velocities, M(q) ∈ n×n
is the inertia matrix, C(q,
·
q) ∈ n×n is the centripetal and Coriolis matrix, f ∈ Rn is the
frictional terms (Coulomb friction), τ ∈ n is the input control vector, and g(q) ∈ n is
the gravity vector, which satisfies
g

q

= ∂
∂q U

q

U =
n
i=1 mighi
(2.2)
DOI: 10.1016/B978-0-12-813380-4.00002-5

where hi = yi, yi is in the vector oi =

xi,yi,zi
T
, oi is given by the first three elements
of the fourth column of the homogeneous transformation matrix.
The robot (2.1) has the following structural properties, which will be used in the
stability analysis of the PID control.
Property 2.1. The inertia matrix is symmetric and positive definite, i.e.,
m1 x2
≤ xT
Mx ≤ m2 x2
where ∀x ∈ Rn, m1 and m2 are known positive scalar constant, and . denotes the
Euclidean vector norm.
Property 2.2. The centripetal and Coriolis matrix is skew-symmetric, i.e.,
xT
·
M(q) − 2C(q,
·
q) x = 0 (2.3)
C

q,q̇

q̇ ≤ kc q̇
2
, kc 0 (2.4)
Ṁ

q

= C

q,q̇

+ C

q,q̇
T
(2.5)
where Ck,ij(q) =

∂Bij
∂qk
+ ∂Bik
∂qj
−
∂Bjk
∂qi

, kc = 1
2 max
q∈Rn
n

k=1
Ck(q) , and C0(q) is a bounded ma-
trix.
Property 2.3. The gravitational torques vector g

q

in (2.2) is Lipschitz:
g(x) − g

y

≤ kg x − y (2.6)
2.1.1 Stable PD control
The classic industrial PD law for the robot (2.1) is
τ = −Kp(q − qd
) − Kd(
·
q −
·
q
d
) = PD1 (2.7)
where Kp and Kd are positive definite, symmetric, and constant matrices, which corre-
spond to proportional and derivative coefficients, qd ∈ n is the desired joint position,
·
q
d
∈ n is the desired joint velocity.
In regulation case, the desired position is constant,
·
q
d
= 0. We use the Lyapunov
function candidate as
VPD =
1
2
q̇T
Mq̇ +
1
2
q̃T
Kpq̃ (2.8)
where q̃ = q − qd.
By the property q̇T
·
M(q) − 2C(q,
·
q) q̇ = 0 and the matrix inequality [159]
XT
Y + YT
X ≤ XT
X + YT
−1
Y,

Stable PID Control and Systematic Tuning of PID Gains 15
which is valid for any X, Y ∈ Rn×m and any 0 = T ∈ Rn×n, the modeling error
q̇T (G + F) can be estimated as
q̇T
(G + F) ≤ q̇T
K1q̇ + (G + F)T
K−1
1 (G + F)
where K1 is any positive definite matrix, the derivative of (2.8) is
·
VPD = −q̇T
Kdq̇ + q̇T
(G + F) ≤ −q̇T
(Kd − K1)q̇ + d̄ (2.9)
where

g + f
T
K−1
1

g + f

≤ d̄, d̄ can be regarded as upper bound of G + F.
• If G and F are zero, the PD control (2.7) can assure
·
VPD ≤ −q̇T
(Kd − K1)q̇
When we choose Kd K1,
·
VPD ≤ 0, the closed-loop system is stable.
• If G and F are not zero, when we choose Kd K1, the regulation error q̃ is bounded
(stable), and q̇
Kd−K1
converges to d̄ [54]
• If G and F are known, the PD control is modified as
τ = −Kp(q − qd
) − Kd(
·
q −
·
q
d
) + F + G
then
·
VPD ≤ −q̇T
(Kd − K1)q̇
When we choose Kd K1,
·
VPD ≤ 0, and the closed-loop system is stable.
In tracking the case, the purpose is to make the joint motors follow the desired
positions. The desired joint positions are generated by the path planning or trajectory
planning, and are sent to the joint motors [72]. The tracking control can be divided
into several regulations by the path planning.
We can also use the following auxiliary error method. We define x1 = q as the link
position vector, x2 = q̇ as the link velocity vector. The tracking error is
x1 = x1 − xd
1
x2 = ẋ1 − ẋd
1
d
dt x1 = x2
where xd
1 is link position desired and xd
2 is link velocity desired. The auxiliary error is
defined as
r = x2 + x1

where = T 0. The PD control (2.7) becomes
τ = −Kvr (2.10)
When the dynamics of the robot (2.1) are known, the PD control with the known
model compensation is
τ = −Kvr − f + (G + F) (2.11)
where f = M
·
x1 −
··
q
d
+ C x1 −
·
q
d
.
We use the Lyapunov function candidate as
V =
1
2
rT
Mr
From Property 2.2, rT
·
M − 2C r = 0,
·
V = −rT
Kvr (2.12)
Then r → 0 when the dynamics of the robot are known.
• If F and G in (2.1) are unknown, similar with (2.9), the PD control (2.10) has
·
V ≤ −rT
Kvr + d̄
when Kd Kv, the regulation error r is bounded (stable) and converges to d̄.
• If G and F are known, or the dynamic of the robot (2.1) is known, the PD control
(2.11) makes the closed-loop system stable.
• If G and F are zero, the PD control (2.10) can assure
·
V ≤ 0, the closed-loop system
is stable.
In general, the regulation error of the PD control law (2.7) is bounded in a ball with
radius d̄.
The stability property is not enough for robot control. The steady-state error caused
by gravity and friction may be big. The derived gain Kd has to be increased to decrease
them. In this way the closed-loop system becomes slow. This big settling time does not
allow us to increase Kd as we want.
We use the stability property of PD control (2.7) to stabilize the open loop unstable
robot (2.1). For any Kd K1 (K1 0), the following closed-loop system is stable:
M

q
··
q + C

q,
·
q
·
q + g

q

+ f
·
q

= PD1 (2.13)

when the robot dynamics contain the gravitational torques vector g

q

. Gravity com-
pensation is a popular method to modify the PD control (2.7). The new PD control
is
τ = PD1 + ĝ

q

(2.14)
where g

q

= ĝ

q

+ g̃

q

, ĝ

q

, and g̃

q

are the gravity estimation and estimation error
and in this case (2.9) become
·
VPD ≤ −q̇T
(Kd − K1)q̇ + d̄1 (2.15)
where d̄1 is the upper bound of

g̃ + f

,

g̃ + f
T
K−1
1

g̃ + f

≤ d̄1. Normally, d̄1 d̄,
because g̃ is the estimation error of the gravity. The closed-loop system (2.13) becomes
M

q
··
q + C

q,
·
q
·
q + g

q

+ f
·
q

= PD1 + ĝ

q

Similar stability results as before can be obtained and only the upper bound of the
gravity becomes smaller.
2.1.2 Stable PID control
The position control objective is to evaluate the torque applied to the joints so that the
robot joint displacements tend asymptotically to constant desired joint positions. Given
a desired constant position qd ∈ Rn, semiglobal asymptotic stability of robot control is to
design the input torque τ in (2.1) to cause regulation error
q̃ = qd
− q
q̃ → 0 and
·
q̃ → 0 when initial conditions are in arbitrary large domain of attraction.
Classic linear PID law is
τ = Kpq̃ + Ki
t
0
q̃(τ)dτ + Kd
·
q̃
where Kp, Ki, and Kd are proportional, integral, and derivative gains of the PID con-
troller, respectively.
The common linear PID control does not include any component of the robot
dynamics into its control law. In order to assure asymptotic stability of PID control,
the simplest approach is to modify the linear PID into a nonlinear one. This chapter
will work on the linear PID, which is the most popular industrial controller. In [117]
the robot dynamic was rewritten in a decoupled linear system and bounded nonlin-
ear system; this linear PID control could not guarantee asymptotic stability. Sufficient
conditions of the linear PID in [72] were given via Lyapunov analysis. However, these

conditions are not explicit; the PID gains could not be decided with these conditions
directly, and a complex tuning procedure was needed [73].
Because q̇d = 0,
·
q̃ = −q̇, the PID control law can be expressed via the following
equations:
τ = Kpq̃ − Kdq̇ + ξ
ξ̇ = Kiq̃, ξ (0) = ξ0
(2.16)
In matrix form, it is
d
dt
⎡
⎢
⎣
ξ
q̃
·
q̃
⎤
⎥
⎦ =
⎡
⎢
⎣
Kiq̃
−q̇
q̈d + M−1

Cq̇ + g − Kpq̃ + Kdq̇ − ξ

⎤
⎥
⎦ (2.17)
The closed-loop system of the robot (2.13) is
M

q

q̈ + C

q,q̇

q̇ + g

q

= Kpq̃ − Kdq̇ + ξ, ξ̇ = Kiq̃
The equilibrium is ξ,q̃,
·
q̃ = [ξ∗
,0,0]. Since at equilibrium point q = qd the equi-
librium is

g

qd

,0,0

. In order to move the equilibrium to origin, we define ξ̃ =
ξ − g

qd

. The closed-loop equation becomes
M

q

q̈ + C

q,q̇

q̇ + g

q

= Kpq̃ − Kdq̇ + ξ̃ + g

qd

·
ξ̃ = Kiq̃
(2.18)
Theorem 2.1. Consider the robot dynamic (2.1) controlled by the linear PID controller
(2.16). The closed-loop system (2.18) is semiglobally asymptotically stable at the equilibrium
x = ξ − g

qd

,q̃,
·
q̃
T
= 0, provided that control gains satisfy
λm

Kp

≥ 3
2 kg
λM (Ki) ≤ β
λm

Kp

λM (M)
λm (Kd) ≥ β + λM (M)
(2.19)
where β =

λm(M)λm

Kp

3 , kg satisfies (2.6), λm (A) is the minimum eigenvalue of A, and λM (A)
is the maximum eigenvalue of A.

Proof. We construct a Lyapunov function as
V = 1
2 q̇T Mq̇ + 1
2 q̃T Kpq̃ + U

q

− ku + q̃T g

qd

+ α
2 ξ̃T K−1
i ξ̃ + q̃T ξ̃ + 3
2 g

qd
T
K−1
p g

qd

− αq̃T Mq̇ + α
2 q̃T Kdq̃
(2.20)
where ku = minq

U

q

, U

q

is defined in (2.2), ku is added such that V (0) = 0. α is
a design positive constant. We first prove V is a Lyapunov function, V ≥ 0. The term
1
2 q̃T Kpq̃ is separated into three parts, and V =
4
i=1 Vi:
V1 = 1
6 q̃T Kpq̃ + q̃T g

qd

+ 3
2 g

qd
T
K−1
p g

qd

V2 = 1
6 q̃T Kpq̃ + q̃T ξ̃ + α
2 ξ̃T K−1
i ξ̃
V3 = 1
6 q̃T Kpq̃ − αq̃T Mq̇ + 1
2 q̇T Mq̇
V4 = U

q

− ku + α
2 q̃T Kdq̃ ≥ 0
It is easy to find
V1 =
1
2

q̃
g

qd

T
1
3 Kp I
I 3K−1
p

q̃
g

qd

≥ 0
When α ≥ 3
λm

K−1
i

λm

Kp
,
V2 ≥ 1
2
1
6 λm

Kp

q̃
2
− q̃ ξ̃ +
αλm

K−1
i

2 ξ̃
2
= 1
2

1
3 λm

Kp

q̃ −

3
λm

Kp
ξ̃
2
≥ 0
Because
yT
Ax ≤ y Ax ≤ y Ax ≤ |λM (A)| y x
when α ≤

1
3 λm(M)λm

Kp

λM (M)
,
V3 ≥ 1
2 (λm (M) q̇
2
− 2αλM (M) q̃ q̇ + 1
3 λm

Kp

q̃
2
)
= 1
2
√
λm (M) q̇ −

1
3 λm

Kp

q̃
2
≥ 0
Obviously, if

1
3
λm

K−1
i

λ
3
2
m

Kp

λ
1
2
m (M) ≥ λM (M) (2.21)

there exists

1
3 λm (M)λm

Kp

λM (M)
≥ α ≥
3
λm

K−1
i

λm

Kp
(2.22)
This means if Kp is sufficiently large or Ki is sufficiently small, (2.21) is established,
and V

q̇,q̃,ξ̃

is globally positive definite. Now we compute its derivative. Taking the
derivative of V, we get
V̇ = q̇T Mq̈ + 1
2 q̇T
·
Mq̇ +
·
q̃
T
Kpq̃ + g

q
T
q̇ +
·
q̃
T
g

qd

+
·
αξ̃T K−1
i ξ̃ +
·
q̃
T
ξ̃ + q̃T
·
ξ̃ − α
·
q̃
T
Mq̇ + q̃T
·
Mq̇ + q̃T Mq̈ − αq̃T Kdq̇
(2.23)
Using (2.3), d
dt U

q

= q̇T g

q

, d
dt g

qd

= 0, and d
dt

q̃T g

qd

=
·
q̃
T
g

qd

, the first three
terms of (2.23) become
−q̇T
g

q

− q̇T
Kdq̇ + q̇T
ξ̃ + q̇T
g

qd

Because
·
q̃
T
g

qd

= −q̇T g

qd

and
·
ξ̃ = Kiq̃, the first seven terms of (2.23) are
−q̇T
Kdq̇ + αq̃T
ξ̃ + q̃T
Kiq̃ (2.24)
Now we discuss the last term of (2.23). From (2.5), we have
q̃T
·
Mq̇ = q̃T
Cq̇ + q̃T
CT
q̇
From (2.1),
q̃T
Mq̈ = −q̃T
Cq̇ − q̃T
g

q

+ q̃T
Kpq̃ − q̃T
Kdq̇ + q̃T
ξ̃ + q̃T
g

qd

For the regulation case,
·
q̃
T
Mq̇ = −q̇T Mq̇, using (2.4) and (2.6) the last two terms of
(2.23) are
−α{q̃T Kpq̃ − q̇T Mq̇ + q̃T CT q̇ + q̃T

g

qd

− g

q

+ q̃T ξ̃}
≤ αq̇T Mq̇ − αq̃T Kpq̃ + αkc q̃ q̇
2
+ αkg q̃
2
− αq̃T ξ̃
(2.25)
From (2.24) and (2.25),
V̇ ≤ −q̇T

Kd − αM − αkc q̃

q̇ − q̃T

αKp − Ki − αkg

q̃
≤ −

λm (Kd) − αλM (M) − αkc q̃

q̇
2
−

αλm

Kp

− λM (Ki) − αkg

q̃
2

If
q̃ ≤
λM (M)
αkc
(2.26)
and
λm (Kd) ≥ (1 + α)λM (M)
λm

Kp

≥ 1
α
λM (Ki) + kg
(2.27)
then V̇ ≤ 0, q̃ decreases. From (2.22), if
λm (Kd) ≥ λM (M) +

1
3 λm (M)

λm

Kp

λm

Kp

≥ 1
3 λm

K−1
i

λm

Kp

λM (Ki) + kg
(2.28)
then (2.27) is established. Using (2.21) and λm

K−1
i

= 1
λM (Ki)
, (2.28) is (2.19).
V̇ is negative semidefinite. Define a ball of radius σ 0 centered at the origin of
the state space, which satisfies these conditions:
=

q̃ : q̃ ≤
λM (M)
αkc
= σ

V̇ is negative semidefinite on the ball . There exists a ball of radius σ 0 centered
at the origin of the state space on which V̇ ≤ 0. The origin of the closed-loop equation
(2.18) is a stable equilibrium. Since the closed-loop equation is autonomous, we use
LaSalle’s theorem. Define as
=

x(t) =

q̃,q̇,ξ̃

∈ R3n : V̇ = 0
=

ξ̃ ∈ Rn : q̃ = 0 ∈ Rn,q̇ = 0 ∈ Rn
From (2.23), V̇ = 0 if and only if q̃ = q̇ = 0. For a solution x(t) to belong to for all
t ≥ 0, it is necessary and sufficient that q̃ = q̇ = 0 for all t ≥ 0. Therefore it must also hold
that q̈ = 0 for all t ≥ 0. We conclude that from the closed-loop system (2.18), if x(t) ∈
for all t ≥ 0, then
g

q

= g

qd

= ξ̃ + g

qd

·
ξ̃ = 0
implies that ξ̃ = 0 for all t ≥ 0. So x(t) =

q̃,q̇,ξ̃

= 0 ∈ R3n is the only initial condition
in for which x(t) ∈ for all t ≥ 0.
Finally, we conclude from all this that the origin of the closed-loop system (2.18)
is locally asymptotically stable. Because 1
α
≤ λm

K−1
i

λm

Kp

, the upper bound for q̃

can be
q̃ ≤
λM (M)
kc
λM (Ki)λm

Kp

It establishes the semiglobal stability of our controller, in the sense that the domain
of attraction can be arbitrarily enlarged with a suitable choice of the gains. Namely,
increasing Kp the basin of attraction will grow.
Remark 2.1. From above stability analysis, we see the three gain matrices of the linear PID
control (2.16) can be chosen directly from the conditions (2.19). The most important contribution
of our method is the tuning procedure of the PID parameters can be calculated directly. It is
more simple than the tuning procedures in [2,4,71,73,72,101,117]. No modeling information
is needed. This linear PID control is exactly the same as the industrial robot controllers, and is
semiglobally asymptotically stable. λM (M) can be estimated as [72]
λM (M) ≤ β, β ≥ n max
i,j
!
!mij
!
!
2.2. PID PARAMETERS TUNING IN CLOSED-LOOP
We use the following three important properties of robot PID control to derive a sys-
tematic tuning method:
1. In the regulation case, a robot can be stabilized by any PD controller providing their
gains are positive big enough.
2. The closed-loop behaviors of robot PID control are similar with linear systems.
3. The control torque of each joint is independent of the robot dynamics.
The turning steps are shown in Fig. 2.1. Here, PD1 is a PD controller, PID2 is the
PID controller after parameters tuning, PID3 is the PID controller after the parameters
refining, and PID4 is the PID controller after stability criterion. The final PID controller
is τ.
Since the robot dynamic is not stable in an open loop, it is impossible to send step
commands to all joints of the robot to tune PID gains. We use the following closed-loop
tuning method.
The PD control (2.14) cannot guarantee zero of the steady-state error. The integra-
tor is the most effective tool to eliminate steady-state error. PD control (2.7) becomes
PID control as
τ = Kpq̃ + Ki
t
0
q̃(τ)dτ + Kd
dq̃
dt
= PID2 (2.29)
where Kp, Ki, and Kd are proportional, integral, and derivative gains of the PID con-
troller, respectively. The integrator gain Ki has to be increased when the steady-state

Figure 2.1 PID tuning scheme.
error is big. This causes big overshoot, long settling time, and is less robust. PD1 is
defined in (2.7) as
PD1 = −Kp(q − qd
) − Kd(
·
q −
·
q
d
)
From (2.15), we know that any PD controller can guarantee stability (bounded) of
any robot manipulator in the regulation case. The stability property assures us to find
another PID controller for the closed-loop system as in Fig. 2.2A. The total control
system is
τ = −Kp1(q − qd1) − Kd1(
·
q −
·
q
d1
) + ĝ

q

qd1 = Kp2(q − qd) + Kd2(
·
q −
·
q
d
)Kpq̃ + Ki
t
0(q − qd)dτ
(2.30)
However, the controller (2.30) loses the PID form. Another tuning method in the
closed-loop is to change PD1 directly; see Fig. 2.2B. Although the closed-loop system
is different when PD1 is changed, the stability is always guaranteed with (2.15) with the
change of PD1. The introduction of an integrator may destroy the stability, and in the
next section we will show how to assure stability with PID control. We use Fig. 2.2B
to adjust PID gains such that we can tune the PID controllers one by one.
Since PD/PID control is linear, the change of PD1 is the same as adding another
controller PID2 to PD1. From (2.9), we know that the closed-loop system with PD1 is

Figure 2.2 PID tuning in closed-loop.
stable. When we apply a PID2 control to the closed-loop system (2.13), it is
M

q
··
q + C

q,
·
q
·
q + g̃

q

+ f
·
q

− PD1 = PID2 (2.31)
A gravity compensation to the closed-loop system (2.13) is
M

q
··
q + C

q,
·
q
·
q + g

q

+ f
·
q

− PD1 = ĝ

q

(2.32)
When we apply the PID control and the gravity compensation to the closed-loop system
(2.31), it is
M

q
··
q + C

q,
·
q
·
q + g

q

+ f
·
q

− PD1 = PID2 + ĝ

q

(2.33)
The total control torque to the robot is
τ = PID2 + PD1 + ĝ

q

(2.34)
From (2.31) to (2.34), we see that the control torque to the robot manipulator is
linearly independent of the robot dynamic (2.1). In the general case, if we tune PID
controllers m times, they can be expressed as
M

q

q̈ + C

q,q̇

q̇ + g

q

+ f
·
q

=
m
#
j=1
PIDj + ĝ

q

(2.35)

where
m
#
j=1
PIDj =
m
#
j=1
Kp,jq̃ +
m
#
j=1
Ki,j
t
0
q̃(τ)dτ +
m
#
j=1
Kd,j
dq̃
dt
PD1 is a special PID with Ki = 0. This property allows us to start a PID control with
small gains such that the closed-loop system is stable. Then any other tuning rule can
be applied to obtain new PID gains one by one. The final PID gains are the summation
of all these controllers (gains).
2.2.1 Linearization of the closed-loop system
Although the robot dynamics are strong nonlinear, the behaviors of the closed-loop
system with PD/PID control are similar with the transient responses of a linear system.
On the other hand, after PID control, each joint of the robot can be characterized as a
single input–single output (SISO) system.
Several methods can be used to linearize the robot models. If the velocity and gravity
are neglected, the terms C

q,q̇

q̇ and g

q

in the nonlinear dynamics (2.1) are zero. The
resulting system is a linear model [45]:
M

q

q̈ = u (2.36)
Obviously, it is an oversimplified model. Since the velocity dependent term C

q,q̇

q̇
representing Coriolis-centrifugal forces, it is negligible for small joint velocities. A rate
linearization scheme can be used as [44]
Aq̈ + Bq = u (2.37)
where A = M

q

|q=q0 , B =
∂g

q

∂q |q=q0 , q0 is an operating point. Many experiments show
that even at low speeds, C

q,q̇

is not zero [127].
The velocity and gravity are main control issues of robots, and they are dominant
components of the dynamics. When the robot model is completely known, Taylor series
expansion can be applied [89]. At the operating point q0 the nonlinear model (2.1) can
be approximated by
Aq̈ + Dq̇ + Bq = τ (2.38)
where A = M

q

|q=q0 , B =
∂

g

q

+C

q,q̇

∂q |q=q0 , D =
∂C

q,q̇

∂q̇ |q=q0 .
We use this identification-based linearization method. For each joint, the typical
linear model is a first-order system with transportation delay as
Gp =
Km
1 + Tms
e−tms
(2.39)

The response is characterized by three parameters: the plant gain Km, the delay time tm,
and the time constant Tm. These are found by drawing a tangent to the step response at
its point of inflection and noting its intersections with the time axis and the steady state
value.
Sometimes the first-order (2.39) model cannot describe the complete nonlinear dy-
namic of the robot. A reasonable linear model of the robot is a Taylor series model as in
(2.38). The model can be written in the frequency domain:
qi (s)
τi (s)
=
Km
T2
ms2 + 2ξmTms + 1
e−tms
(2.40)
or
qi (s)
τi (s)
=
Km
(1 + Tm1s)(1 + Tm2s)
e−tms
The responses of this second-order model are similar with mechanical motions. If there
exists a big overshoot a negative zero is added in (2.40)
qi (s)
τi (s)
=
Km (1 + Tm3s)
(1 + Tm1s)(1 + Tm2s)
e−tms
(2.41)
The normal input signals for the PID tuning are step and repeat inputs.
2.2.2 PD/PID tuning
Because the robot can be approximated by a linear system, some tuning rules for linear
systems can be applied for the closed-loop system tuning. We first give PD tuning rules.
When each joint can be approximated by a first-order system,
Gp =
Km
1 + Tms
e−tms
Here, Km, Tm, and tm are obtained from Fig. 2.3.
The linear PID law in time domain (2.29) can be transformed into the frequency
domain:
τ (s) = Kc 1 +
1
Tis
+ Tds E (s) = Gc (s)E (s) = PID2
Similar with Ziegler–Nichols [161] and Cohen–Coon [20] tuning methods, we use
a heuristic method to tune a PID controller. For PD control, our tuning law is shown
in Table 2.1. Here, we use the similar tuning formulas with Ziegler–Nichols. By several
experiments, we found a = 1 is good.

Figure 2.3 Step response of a linear system.
Table 2.1 Ziegler–Nichols and Cohen–Coon methods for PI/PD
Kc Ti Td
Ziegler–Nichols tuning a Tm
Kmτm
0.5τm
Cohen–Coon method Tm
Kmτm

4
3 + τm
4Tm

4Tmτm
11Tm+2τm
Our Method Tm2
Km
Tm1
Table 2.2 PD tuning for the second-order model
Kc Ti Td
Method 1 [58] 5Tm1ξm
KmTm3
Tm1+0.1ξm
0.8Tm1ξm
Method 2 [24] Tm2
Km
Tm1
Our Method Tm2
Km
Tm1
Table 2.3 PID tuning for the ﬁrst-order model
Kc Ti Td
Ziegler–Nichols tuning a Tm
Kmτm
, 2τm 0.5τm
Cohen–Coon method Tm
Kmτm
τm
4Tm
τm(32Tm+6τm)
13Tm+8τm
4Tmτm
11Tm+2τm
Our Method Tm2
Km
Tm2 Tm1
If each joint is approximated by a second-order system,
qi (s)
ui (s)
=
Km
T2
ms2 + 2ξmTms + 1
The PD gains are tuned similarly with [58] and [24]; see Table 2.2.
When PD control cannot provide good performances, PID control should be used.
The PID gains for the first-order model is decided by Table 2.3.

Table 2.4 PID tuning for the second-order model
Kc Ti Td
Method 1 5Tm1ξm
KmTm3
2Tm1ξm
Tm1+0.1ξm
0.8Tm1ξm
Method 2 Tm2
Km
Tm2 Tm1
Our Method 20ξmTm
Km
15ξmTm
T2
m
10
Table 2.5 Reﬁned PID (PID3)
Rise Overshoot Settling Steady Error Stability
Kp ↑ Decrease Increase Small Increase Decrease Degrade
Ki ↑ Small Decrease Increase Increase Large Decrease Degrade
Kd ↑ Small Decrease Decrease Increase Minor Decrease Improve
Here, Kc = Kp is a proportional gain, Ti = Kc
Ki
is a time constant, and Td = Kd
Kc
is a
derivative time constant.
The PID gains for the second-order model is decided by Table 2.4.
2.2.3 Reﬁne PID gains
From (2.35), we see the PID gains are linear independent, and we can modify them
directly. The refinement of PID2 is the same as adding a new PID controller, PID3, into
(2.34). We use Table 2.5 to refine PID gains.
After we refine the process, the robot control is
τ = PD1 + PID2 + PID3 + ĝ

q

(2.42)
In order to decrease the steady-state error, we should increase Ki. In order to get less
settling time, we should decrease Kd. In order to get less overshoot, we should decrease
Kp. However, the above tuning process does not guarantee stability of the closed-loop
system. In the next subsection, we will give the bounds of the PID gains to assure PID
control is stable.
2.2.4 Stability conditions for PID gains
For the robot dynamic controlled by the PID controller (2.35), the closed-loop system
(2.18) is semiglobally asymptotically stable at the equilibrium x = ξ − g

qd

,q̃,
·
q̃
T
= 0,
provided that control gains satisfy (2.19). The three gain matrices of the linear PID
control (2.35) can be chosen directly from the conditions (2.19). From (2.35), we know

Table 2.6 Comparison of the PID tuning method
PID tuning method system model stability systematic adaptive
Our Method linear/nonlinear no yes yes no
Ziegler–Nichols [161]
Cohen–Coon [20]
linear no no yes no
Model based [24,58] linear/nonlinear yes no no yes
Lyapunov method [21,22] nonlinear no yes no no
Frequency domain linear no no yes no
Optimization method [81] linear yes no no no
Adaptive tuning [143] linear yes no no yes
Intelligent method [129,85] linear/nonlinear no no no yes
Impedance method [56,21] robot yes no no no
that the PID control with gravity compensation (2.42) is
τ =
3
#
j=1
PIDj + ĝ

q

= PIDf + ĝ

q

Now we apply the condition (2.19) to PIDf . If the gains of PIDf are not in the bound
of (2.19), we add a new PID controller, PID4, such that the gains of PIDf + PID4 are
in the bounds of (2.19). The final control torque to the robot is
τ = PD1 + PID2 + PID3 + PID4 + ĝ

q

Table 2.6 gives a comparative analysis of the proposed method with other PID tuning
methods.
Here, we compare our method with the other eight PID tuning algorithms in six
properties. It can be seen that apart from adaptive ability, our method is better than the
others for robot control.
2.3. APPLICATION TO AN EXOSKELETON
The upper limb exoskeleton robot in UC-Santa Cruz is shown in Fig. 2.4. It has 7-DoF
(degrees-of-freedom) as in Fig. 2.5. The computer control platform of this exoskeleton
robot is a PC104 with an Intel Pentium4@2.4 GHz processor and 512 Mb RAM. The
motors for the first four joints are mounted in the base such that the large mass of the
motors can be removed. Torque transmission from the motors to the joints is achieved
using a cable system. The other three small motors are mounted in link five.
The real-time control program operated in Windows XP with Matlab 7.1, Windows
Real-Time Target and C++. All of the controllers employed a sampling frequency of

Figure 2.4 The UCSC 7-DoF exoskeleton robot.
Figure 2.5 Model of the 7-DoF exoskeleton robot.
Table 2.7 Parameters of the exoskeleton
Joint Mass (kg) Center (m) Length (m)
1 3.4 0.3 0.7
2 1.7 0.05 0.1
3 0.7 0.1 0.2
4 1.2 0.02 0.05
5 1.8 0.02 0.05
6 0.2 0.04 0.1
7 0.5 0.02 0.05
1 kHz. The properties of the exoskeleton with respect to base frame are shown in
Table 2.7.
We first use the following PD1 to stabilize the robot:
Kp1 = diag[150,150,100,150,100,100,100]
Kd1 = diag[330,330,300,320,320,300,300]
(2.43)

The joint velocities are estimated by the standard filters:
$̇
q(s) =
bs
s + a
q(s) =
18s
s + 30
q(s)
Here, the main weight of the exoskeleton is in the first four joints. The potential energy
is
U = m1gl1s1 + m2g

l1s1 + c2l2s1

+ m3ga3s1s2
+ m4g[a4c4 (c1s3 + c2c3s1) + a4s4 (c1c3 − c2s1s3) + a3s1s2]
The gravity compensation in (2.18) is calculated by
ĝ =
∂
∂q
U

q

Then we use the step responses of linear systems to approximate the closed-loop
responses of the robot via PD1. The step responses of the closed-loop systems of the
robot and the linear systems are shown in Fig. 2.6, where in the dash lines are the step
responses of the following second-order linear systems:
G1 = 0.93
60s2+9s+1
, G2 = 1
20s2+3s+1
G3 = 0.9
5.5s2+4s+1
, G4 = 0.85
30s2+8s+1
(2.44)
In order to tuning the gains of the PID (2.29), this PID is rewritten as
PIDt = Kc q̃ +
1
Ti
t
0
q̃(τ)dτ + Td
·
q̃
where Kc = Kp is proportional gain, Ti = Kc
Ki
is integral time constant, and Td = Kd
Kc
is
derivative time constant. We use the following tuning rule:
Kc =
20ξmTm
Km
, Ti = 15ξmTm, Td =
T2
m
10
(2.45)
to tune the PID parameters. This rule is similar with [58] and [24], in their case Kc =
5Tm1ξm
KmTm3
, Ti = 2Tm1ξm, Td = Tm1+0.1ξm
0.8Tm1ξm
. It is different with the other two famous rules,
Ziegler–Nichols and Cohen–Coon methods, where Kc = a Tm
Kmτm
, Ti = 2τm, Td = 0.5τm
or Kc = Tm
Kmτm

4
3 + τm
4Tm

, Ti = τm(32Tm+6τm)
13Tm+8τm
, Td = 4Tmτm
11Tm+2τm
. Because their rules are suitable
for the process control, our rule is for mechanical systems. By the rule (2.45) the PID2

Figure 2.6 The step responses of the closed-loop systems.
are
Kp2 = diag[90,30,40,90,10,150,10]
Ki2 = diag[1,2,20,1.5,3,1,2]
Kd2 = diag[500,410,350,400,50,30,400]
(2.46)
The control torque becomes u = PID1 + ĝ

q

+ PID2. The control results of Joint 1 are
shown in Fig. 2.7.
After this refine turning, PID3 is
Kp3 = diag[5,4,5,6,3,4,2]
Ki3 = diag[32,28,21,25,21,21,22]
Kd3 = diag[5,4,4,10,6,10,10]
(2.47)
The final control is
PIDf = PD1 + ĝ

q

+ PID2 + PID3 (2.48)

Figure 2.7 PID tuning process of Joint 1.
Figure 2.8 PID control for Joint 3 and Joint 7.
The stability condition (2.19) gives a sufficient condition for the minimal values of
proportional and derivative gains and maximal values of integral gains. We find that the
final control (2.48) satisfies the conditions (2.19). The final control results for Joint 3
and Joint 7 are shown in Fig. 2.8.
2.4. CONCLUSIONS
In this chapter the linear PID control for a class of exoskeleton robots is addressed.
The conditions of the semiglobal asymptotic stability is very simple, and the linear PID
control is exactly the same as the industrial controller. The systematic tuning method
for PID control is proposed. This method can be applied to any robot manipulator. By
using several properties of robot manipulators, the tuning process becomes simple and
is easily applied in real applications. Some concepts for PID tuning are novel, such as
step responses for the closed-loop systems under any PD control, and the joint torque is
separated into several independent PID. We finally apply this method on an upper limb
exoskeleton. Real experiment results give validation of our PID tuning method.

CHAPTER 3
PID Control in Task Space
Abstract
Task space (or Cartesian space) is defined by the position and orientation of the end-effector of a robot.
Joint space is defined by a vector whose components are the translational and angular displacements
of each joint of a robotic link. In this chapter, linear PID in the task-space is proposed. The sufficient
conditions for asymptotic stability are simple and explicit. The linear PID gains can be selected with
these conditions directly. When the measurement of velocities it is not available, a velocity observer
(position filter) is applied. The local asymptotic stability of the linear PID control with an observer is
proven. The analysis provides explicit conditions for choosing the linear PID gains and the parameters
of the velocity observer. We use a 4-DoF (degree-of-freedom) upper limb exoskeleton to verify our
PID tuning conditions. The experimental results show that the proposed methodology provides an
analytical tool for the robot controller design in the task space.
Keywords
Task space, Linear PID, Semiglobally asymptotic stability
3.1. LINEAR PID CONTROL IN TASK SPACE
The dynamics of the robot are derived from Euler–Lagrange equation as
M

q

q̈ + C

q,q̇

q̇ + g

q

= u (3.1)
where q ∈ Rn represents the joint positions. M

q

= KT + KR is the inertia matrix,
C

q,q̇

=

ckj

represents centrifugal force, ckj =
n
i=1 cijkq̇i, k,j = 1···n, cijk is Christoffel
symbols [131], and g

q

is the vector of gravity torques.
When the robot’s end-effector contacts the environment, or a desired path for the
end-effector is specified in task space such as visual space or Cartesian space, a task space
coordinate system defined with reference to the environment is convenient for the study
of contact motion.
We consider a nonredundant robot. The dimension of the task space is equal to the
dimension of the joint space. Let x ∈ Rn be a task-space vector defined by
x = h

q

, ẋ = Jq̇
where h(·) ∈ Rn → Rn is the forward kinematics of the robot, which is a nonlinear trans-
formation describing the relation between the joint and task space; x in the task-space
is assumed that the robot manipulator is operating in a finite work space such that the
Jacobian matrix J is of full rank.
DOI: 10.1016/B978-0-12-813380-4.00003-7

Since ẍ = Jq̈ + J̇q̇ the relations between the dynamic models of the task space and
the joint space are [75]
Mxẍ + Cxẋ + gx = ux (3.2)
where
Mx = J−T MJ−1
Cx = J−T

C − MJ−1J̇

J−1
gx = ḡx = J−T g
ux = J−T u
(3.3)
Mx, Cx, and gx depend on q and q̇.
q and q̇ can be computed from inverse kinematic and q̇ = J−1ẋ. So Mx, Cx, and gx
can be regarded as a function of x and ẋ. The PID attendance control will not use Mx
and Cx, only the following properties will be used to prove stability.
P3.1. The inertia matrix M (x) is symmetric positive definite, and
0 λm {Mx (x)} ≤ Mx ≤ λM {Mx (x)} ≤ β, β 0 (3.4)
where λM {M} and λm {M} are the maximum and minimum eigenvalues of the matrix A.
P3.2. For the centrifugal and Coriolis matrix C

q,q̇

, there exists a number kc 0
such that
Cx (x,ẋ)ẋ ≤ kc ẋ2
, kc 0 (3.5)
and Ṁx − 2Cx are skew symmetric, i.e.,
xT

Ṁx (x) − 2Cx (x,ẋ)

x = 0 (3.6)
also
Ṁx (x) = Cx (x,ẋ) + Cx (x,ẋ)T
(3.7)
P3.3. The gravitational torques vector g

q

and gx (x) is Lipschitz:
gx (x) − gx

y

≤ kg x − y (3.8)
The proof of the above properties is similar with the joint space case [85].
We design a linear stable PID control in task space to regulate the exoskeleton to the
desired position. We define the regulation error as
x̃ = xd − x
where xd is the desired position and orientation of the end-effector.

PID Control in Task Space 37
The objective of position control in task space is x̃ → 0 and
·
x̃ → 0 when initial
conditions are in arbitrary large domain of attraction. A linear PID control in task space
law is
ux = Kpx̃ + Ki
t
0
x̃(τ)dτ + Kd
·
x̃ (3.9)
where Kp, Ki, and Kd are proportional, integral, and derivative gains. x = h

q

, ẋ = Jq̇.
By (3.3) the final control torque applied on each joint is
u = JT
ux (3.10)
Here, the Jacobian matrix J is known. We do not discuss the case of an uncertain
Jacobian matrix [27].
Remark 3.1. Compared with the other task-space PID control, (3.9) has exactly the same
form as the classical linear PID control of robot manipulators. In order to prove the stability, in
[128] the PID is modified as
ux = Kpx̃ + Ki
t
0
y(τ)dτ + Kd
·
x̃ (3.11)
where the integral term is changed as y = q̇ + αx̃. In [27] the linear PID control is modified as
ux = Kps(x̃) + Ki
t
0
y(τ)dτ + Kd
·
x̃ (3.12)
where y(τ) = q̇ + αs(x̃), the position error x̃ is filter by a scalar potential function s(·). In [40],
s(·) is a saturation function.
We only discuss the regulation case, i.e., ẋd = 0,
·
x̃ = −ẋ. The PID control law can
be expressed via the following equations:
ux = Kpx̃ − Kdẋ + f + ξ
ξ̇ = Kix̃, ξ (0) = ξ0
(3.13)
We require that the linear control (3.13) is decoupled, i.e., Kp,Ki, and Kd are positive
definite diagonal matrices. The closed-loop system of the robot (3.2) is
Mxẍ + Cxẋ + gx (x) = Kpx̃ − Kdẋ + ξ
ξ̇ = Kix̃

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Returning homeward, I saw the palace of Cardinal Spada, where
is a most magnificent hall painted by Daniel de Volterra and Giulio
Piacentino, who made the fret in the little Court; but the rare
perspectives are of Bolognesi. Near this is the Mont Pieta, instituted
as a bank for the poor, who, if the sum be not great, may have
money upon pawns. To this joins St. Martino, to which belongs a
Schola, or Corporation, that do many works of charity. Hence we
came through Campo di Fiori, or herb-market, in the midst of which
is a fountain casting out water of a dolphin, in copper; and in this
piazza is common execution done.
19th February, 1645. I went, this afternoon, to visit my Lord John
Somerset, brother to the Marquis of Worcester, who had his
apartment in Palazzo della Cancellaria, belonging to Cardinal
Francesco Barberini, as Vice-chancellor of the Church of Rome, and
Protector of the English. The building is of the famous architect,
Bramante, of incrusted marble, with four ranks of noble lights; the
principal entrance is of Fontana's design, and all marble; the portico
within sustained by massy columns; on the second peristyle above,
the chambers are rarely painted by Salviati and Vasari; and so ample
is this palace, that six princes with their families have been received
in it at one time, without incommoding each other.
20th February, 1645. I went, as was my usual custom, and spent
an afternoon in Piazza Navona, as well as to see what antiquities I
could purchase among the people who hold market there for
medals, pictures, and such curiosities, as to hear the mountebanks
prate and distribute their medicines. This was formerly the Circus, or
Agonales, dedicated to sports and pastimes, and is now the greatest
market of the city, having three most noble fountains, and the
stately palaces of the Pamfilii, St. Giacomo de Spagnoli belonging to
that nation, to which add two convents for friars and nuns, all
Spanish. In this Church was erected a most stately catafalco, or
capellar ardente, for the death of the Queen of Spain; the church
was hung with black, and here I heard a Spanish sermon, or funeral
oration, and observed the statues, devices, and impresses hung

about the walls, the church and pyramid stuck with thousands of
lights and tapers, which made a glorious show. The statue of St.
James is by Sansovino; there are also some good pictures of Caracci.
The facciáta, too, is fair. Returning home, I passed by the stumps of
old Pasquin, at the corner of a street, called Strada Pontificia; here
they still paste up their drolling lampoons and scurrilous papers. This
had formerly been one of the best statues for workmanship and art
in all the city, as the remaining bust does still show.
21st February, 1645. I walked in the morning up the hill toward
the Capuchins, where was then Cardinal Unufrio (brother to the late
Pope Urban VIII.) of the same order. He built them a pretty church,
full of rare pictures, and there lies the body of St. Felix, that they say
still does miracles. The piece at the great altar is by Lanfranc. It is a
lofty edifice, with a beautiful avenue of trees, and in a good air. After
dinner, passing along the Strada del Corso, I observed the column of
Antoninus, passing under Arco Portugallo, which is but a relic,
heretofore erected in honor of Domitian, called now Portugallo, from
a Cardinal living near it. A little further on the right hand stands the
column in a small piazza, heretofore set up in honor of M. Aurelius
Antoninus, comprehending in a basso-relievo of white marble his
hostile acts against the Parthians, Armenians, Germans, etc; but it is
now somewhat decayed. On the summit has been placed the image
of St. Paul, of gilded copper. The pillar is said to be 161 feet high,
ascended by 207 steps, receiving light by fifty-six apertures, without
defacing the sculpture.
At a little distance, are the relics of the Emperor's palace, the
heads of whose pillars show them to have been Corinthian.
Turning a little down, we came to another piazza, in which stands
a sumptuous vase of porphyry, and a fair fountain; but the grace of
this market, and indeed the admiration of the whole world, is the
Pantheon, now called S. Maria della Rotonda, formerly sacred to all
the Gods, and still remaining the most entire antiquity of the city. It
was built by Marcus Agrippa, as testifies the architrave of the
portico, sustained by thirteen pillars of Theban marble, six feet thick,

and fifty-three in height, of one entire stone. In this porch is an old
inscription.
Entering the church, we admire the fabric, wholly covered with
one cupola, seemingly suspended in the air, and receiving light by a
hole in the middle only. The structure is near as high as broad, viz,
144 feet, not counting the thickness of the walls, which is twenty-
two more to the top, all of white marble; and, till Urban VIII.
converted part of the metal into ordnance of war against the Duke of
Parma, and part to make the high altar in St. Peter's, it was all over
covered with Corinthian brass, ascending by forty degrees within the
roof, or convex, of the cupola, richly carved in octagons in the stone.
There are niches in the walls, in which stood heretofore the statues
of Jupiter and the other Gods and Goddesses; for here was that
Venus which had hung in her ear the other Union28
that Cleopatra
was about to dissolve and drink up, as she had done its fellow.
There are several of these niches, one above another for the
celestial, terrestrial, and subterranean deities; but the place is now
converted into a church dedicated to the Blessed Virgin and all the
Saints. The pavement is excellent, and the vast folding-gates, of
Corinthian brass. In a word, it is of all the Roman antiquities the
most worthy of notice. There lie interred in this Temple the famous
Raphael di Urbino, Perino del Vaga, F. Zuccharo, and other painters.
Returning home, we pass by Cardinal Cajetan's Palace, a noble
piece of architecture of Vincenzo Ammanatti, which is the grace of
the whole Corso.
22d February, 1645. I went to Trinitá del Monte, a monastery of
French, a noble church built by Louis XI. and Charles VIII., the
chapels well painted, especially that by Zaccara da Volterra, and the
cloister with the miracles of their St. Francis de Paulo, and the heads
of the French Kings. In the pergolo above, the walls are wrought
with excellent perspective, especially the St. John; there are the
Babylonish dials, invented by Kircher, the Jesuit. This convent, so
eminently situated on Mons Pincius, has the entire prospect of

Campus Martius, and has a fair garden which joins to the Palazzo di
Medici.
23d February, 1645. I went to hear a sermon at St. Giacomo degli
Incurabili, a fair church built by F. da Volterra, of good architecture,
and so is the hospital, where only desperate patients are brought. I
passed the evening at St. Maria del Popolo, heretofore Nero's
sepulchre, where his ashes lay many years in a marble chest. To this
church joins the monastery of St. Augustine, which has pretty
gardens on Mons Pincius, and in the church is the miraculous shrine
of the Madonna which Pope Paul III. brought barefooted to the
place, supplicating for a victory over the Turks in 1464. In a chapel
of the Ghisi, are some rare paintings of Raphael, and noble
sculptures. Those two in the choir are by Sansovino, and in the
Chapel de Cerasii, a piece of Caravaggio. Here lie buried many great
scholars and artists, of which I took notice of this inscription:
Hospes, disce novum mortis genus; improba
felis,
Dum trahitur, digitum mordet, et intereo.
Opposite to the facciátæ of the church is a superb obelisk full of
hieroglyphics, the same that Sennesertus, King of Egypt, dedicated
to the Sun; brought to Rome by Augustus, erected in the Circus
Maximus, and since placed here by Pope Sextus V. It is eighty-eight
feet high, of one entire stone, and placed with great art and engines
by the famous Domenico Fontana.
Hence, turning on the right out of the Porto del Popolo, we came
to Justinian's gardens, near the Muro Torto, so prominently built as
threatening every moment to fall, yet standing so for these thousand
years. Under this is the burying place for the common prostitutes,
where they are put into the ground, sans ceremonie.
24th February, 1645. We walked to St. Roche's and Martine's,
near the brink of the Tiber, a large hospital for both sexes. Hence, to
the Mausoleum Augusti, between the Tiber and the Via Flaminia,

now much ruined, which had formerly contended for its sumptuous
architecture. It was intended as a cemetery for the Roman
Emperors, had twelve ports, and was covered with a cupola of white
marble, environed with stately trees and innumerable statues, all of
it now converted into a garden. We passed the afternoon at the
Sapienza, a very stately building full of good marbles, especially the
portico, of admirable architecture. These are properly the University
Schools, where lectures are read on Law, Medicine, and Anatomy,
and students perform their exercises.
Hence, we walked to the church of St. Andrea della Valle, near
the former Theater of Pompey, and the famous Piccolomini, but
given to this church and the Order, who are Theatins. The Barberini
have in this place a chapel, of curious incrusted marbles of several
sorts, and rare paintings. Under it is a place where St. Sebastian is
said to have been beaten with rods before he was shot with darts.
The cupola is painted by Lanfranc, an inestimable work, and the
whole fabric and monastery adjoining are admirable.
25th February, 1645. I was invited by a Dominican Friar, whom
we usually heard preach to a number of Jews, to be godfather to a
converted Turk and Jew. The ceremony was performed in the Church
of Santa Maria sopra la Minerva, near the Capitol. They were clad in
white; then exorcised at their entering the church with abundance of
ceremonies, and, when led into the choir, were baptized by a Bishop,
in pontificalibus. The Turk lived afterward in Rome, sold hot waters
and would bring us presents when he met us, kneeling and kissing
the hems of our cloaks; but the Jew was believed to be a
counterfeit. This church, situated on a spacious rising, was formerly
consecrated to Minerva. It was well built and richly adorned, and the
body of St. Catherine di Sienna lies buried here. The paintings of the
chapel are by Marcello Venuti; the Madonna over the altar is by
Giovanni di Fiesole, called the Angelic Painter, who was of the Order
of these Monks. There are many charities dealt publicly here,
especially at the procession on the Annunciation, where I saw his
Holiness, with all the Cardinals, Prelates, etc., in pontificalibus;

dowries being given to 300 poor girls all clad in white. The Pope had
his tiara on his head, and was carried on men's shoulders in an open
armchair, blessing the people as he passed. The statue of Christ, at
the Columna, is esteemed one of the masterpieces of M. Angelo:
innumerable are the paintings by the best artists, and the organ is
accounted one of the sweetest in Rome. Cardinal Bembo is interred
here. We returned by St. Mark's, a stately church, with an excellent
pavement, and a fine piece by Perugino, of the Two Martyrs.
Adjoining to this is a noble palace built by the famous Bramante.
26th February, 1645. Ascending the hill, we came to the Forum
Trajanum, where his column stands yet entire, wrought with
admirable basso-relievo recording the Dacian war, the figures at the
upper part appearing of the same proportion with those below. It is
ascended by 192 steps, enlightened with 44 apertures, or windows,
artificially disposed; in height from the pedestal 140 feet.
It had once the ashes of Trajan and his statue, where now stands
St. Peter's of gilt brass, erected by Pope Sextus V. The sculpture of
this stupendous pillar is thought to be the work of Apollodorus; but
what is very observable is, the descent to the plinth of the pedestal,
showing how this ancient city lies now buried in her ruins; this
monument being at first set up on a rising ground. After dinner, we
took the air in Cardinal Bentivoglio's delicious gardens, now but
newly deceased. He had a fair palace built by several good masters
on part of the ruins of Constantine's Baths; well adorned with
columns and paintings, especially those of Guido Reni.
27th February, 1645. In the morning Mr. Henshaw and myself
walked to the Trophies of Marius, erected in honor of his victory over
the Cimbrians, but these now taken out of their niches are placed on
the balusters of the Capitol, so that their ancient station is now a
ruin. Keeping on our way, we came to St. Croce of Jerusalem, built
by Constantine over the demolition of the Temple of Venus and
Cupid, which he threw down; and it was here, they report, he
deposited the wood of the true Cross, found by his mother, Helena;
in honor whereof this church was built, and in memory of his victory

over Maxentius when that holy sign appeared to him. The edifice
without is Gothic, but very glorious within, especially the roof, and
one tribuna (gallery) well painted. Here is a chapel dedicated to St.
Helena, the floor whereof is of earth brought from Jerusalem; the
walls are of fair mosaic, in which they suffer no women to enter,
save once a year. Under the high altar of the Church is buried St.
Anastasius, in Lydian marble, and Benedict VII.; and they show a
number of relics, exposed at our request; with a phial of our blessed
Savior's blood; two thorns of his crown; three chips of the real cross;
one of the nails, wanting a point; St. Thomas's doubting finger; and
a fragment of the title (put on the cross), being part of a thin board;
some of Judas's pieces of silver; and many more, if one had faith to
believe it. To this venerable church joins a Monastery, the gardens
taking up the space of an ancient amphitheatre.
Hence, we passed beyond the walls out at the Port of St.
Laurence, to that Saint's church, and where his ashes are enshrined.
This was also built by the same great Constantine, famous for the
Coronation of Pietro Altissiodorensis, Emperor of Constantinople, by
Honorius II. It is said the corpse of St. Stephen, the proto martyr,
was deposited here by that of St. Sebastian, which it had no sooner
touched, but Sebastian gave it place of its own accord. The Church
has no less than seven privileged altars, and excellent pictures.
About the walls are painted this martyr's sufferings; and, when they
built them, the bones of divers saints were translated to other
churches. The front is Gothic. In our return, we saw a small ruin of
an aqueduct built by Quintus Marcius, the prætor; and so passed
through that incomparable straight street leading to Santa Maria
Maggiore, to our lodging, sufficiently tired.
We were taken up next morning in seeing the impertinences of
the Carnival, when all the world are as mad at Rome as at other
places; but the most remarkable were the three races of the Barbary
horses, that run in the Strada del Corso without riders, only having
spurs so placed on their backs, and hanging down by their sides, as
by their motion to stimulate them: then of mares, then of asses, of

buffalos, naked men, old and young, and boys, and abundance of
idle ridiculous pastime. One thing is remarkable, their acting
comedies on a stage placed on a cart, or plaustrum, where the
scene, or tiring place, is made of boughs in a rural manner, which
they drive from street to street with a yoke or two of oxen, after the
ancient guise. The streets swarm with prostitutes, buffoons, and all
manner of rabble.
1st March, 1645. At the Greek Church, we saw the Eastern
ceremonies performed by a Bishop, etc., in that tongue. Here the
unfortunate Duke and Duchess of Bouillon received their ashes, it
being the first day of Lent. There was now as much trudging up and
down of devotees, as the day before of licentious people; all saints
alike to appearance.
The gardens of Justinian, which we next visited, are very full of
statues and antiquities, especially urns; among which is that of
Minutius Felix; a terminus that formerly stood in the Appian way, and
a huge colossé of the Emperor Justinian. There is a delicate aviary
on the hill; the whole gardens furnished with rare collections, fresh,
shady, and adorned with noble fountains. Continuing our walk a mile
farther, we came to Pons Milvius, now Mela, where Constantine
overthrew Maxentius, and saw the miraculous sign of the cross, In
hoc signo vinces. It was a sweet morning, and the bushes were full
of nightingales. Hence, to Aqua Claudia again, an aqueduct finished
by that Emperor at the expense of eight millions. In the afternoon,
to Farnese's gardens, near the Campo Vaccino; and upon the
Palatine Mount to survey the ruins of Juno's Temple, in the Piscina, a
piazza so-called near the famous bridge built by Antoninus Pius, and
re-edified by Pope Sextus IV.
The rest of this week, we went to the Vatican, to hear the
sermons, at St. Peter's, of the most famous preachers, who
discourse on the same subjects and text yearly, full of Italian
eloquence and action. On our Lady day, 25th March, we saw the
Pope and Cardinals ride in pomp to the Minerva, the great guns of
the Castle of St. Angelo being fired, when he gives portions to 500

zitelle (young women), who kiss his feet in procession, some
destined to marry, some to be nuns;—the scholars of the college
celebrating the blessed Virgin with their compositions. The next day,
his Holiness was busied in blessing golden roses, to be sent to
several great Princes; the Procurator of the Carmelites preaching on
our Savior's feeding the multitude with five loaves, the ceremony
ends. The sacrament being this day exposed, and the relics of the
Holy Cross, the concourse about the streets is extraordinary. On
Palm-Sunday, there was a great procession, after a papal mass.
11th April, 1645. St. Veronica's handkerchief (with the impression
of our Savior's face) was exposed, and the next day the spear, with a
world of ceremony. On Holy Thursday, the Pope said mass, and
afterward carried the Host in procession about the chapel, with an
infinity of tapers. This finished, his Holiness was carried in his open
chair on men's shoulders to the place where, reading the Bull In
Cœnâ Domini, he both curses and blesses all in a breath; then the
guns are again fired. Hence, he went to the Ducal hall of the
Vatican, where he washed the feet of twelve poor men, with almost
the same ceremony as it is done at Whitehall; they have clothes, a
dinner, and alms, which he gives with his own hands, and serves at
their table; they have also gold and silver medals, but their garments
are of white woolen long robes, as we paint the Apostles. The same
ceremonies are done by the Conservators and other officers of state
at St. John di Lateran; and now the table on which they say our
blessed Lord celebrated his last supper is set out, and the heads of
the Apostles. In every famous church they are busy in dressing up
their pageantries to represent the Holy Sepulchre, of which we went
to visit divers.
On Good Friday, we went again to St. Peter's, where the
handkerchief, lance, and cross were all exposed, and worshiped
together. All the confession seats were filled with devout people, and
at night was a procession of several who most lamentably whipped
themselves till the blood stained their clothes, for some had shirts,
others upon the bare back, having visors and masks on their faces;

at every three or four steps dashing the knotted and raveled whip
cord over their shoulders, as hard as they could lay it on; while some
of the religious orders and fraternities sung in a dismal tone, the
lights and crosses going before, making all together a horrible and
indeed heathenish pomp.
The next day, there was much ceremony at St. John di Laterano,
so as the whole week was spent in running from church to church,
all the town in busy devotion, great silence, and unimaginable
superstition.
Easter day, I was awakened by the guns from St. Angelo: we
went to St. Peter's, where the Pope himself celebrated mass, showed
the relics before-named, and gave a public Benediction.
Monday, we went to hear music in the Chiesa Nova; and, though
there were abundance of ceremonies at the other great churches,
and great exposure of relics, yet being wearied with sights of this
nature, and the season of the year, summer, at Rome being very
dangerous, by reason of the heat minding us of returning northward,
we spent the rest of our time in visiting such places as we had not
yet sufficiently seen. Only I do not forget the Pope's benediction of
the Gonfalone, or Standard, and giving the hallowed palms; and, on
May Day, the great procession of the University and the muleteers at
St. Anthony's, and their setting up a foolish May pole in the Capitol,
very ridiculous. We therefore now took coach a little out of town, to
visit the famous Roma Soterránea, being much like what we had
seen at St. Sebastians. Here, in a cornfield, guided by two torches,
we crept on our bellies into a little hole, about twenty paces, which
delivered us into a large entry that led us into several streets, or
alleys, a good depth in the bowels of the earth, a strange and fearful
passage for divers miles, as Bosio has measured and described them
in his book. We ever and anon came into pretty square rooms, that
seemed to be chapels with altars, and some adorned with very
ordinary ancient painting. Many skeletons and bodies are placed on
the sides one above the other in degrees like shelves, whereof some
are shut up with a coarse flat stone, having engraven on them Pro

Christo, or a cross and palms, which are supposed to have been
martyrs. Here, in all likelihood, were the meetings of the Primitive
Christians during the persecutions, as Pliny the Younger describes
them. As I was prying about, I found a glass phial, filled (as was
conjectured) with dried blood, and two lachrymatories. Many of the
bodies, or rather bones (for there appeared nothing else) lay so
entire, as if placed by the art of the chirurgeon, but being only
touched fell all to dust. Thus, after wandering two or three miles in
this subterranean meander, we returned almost blind when we came
into the daylight, and even choked by the smoke of the torches. It is
said that a French bishop and his retinue adventuring too far into
these dens, their lights going out, were never heard of more.
We were entertained at night with an English play at the Jesuits',
where we before had dined; and the next day at Prince Galicano's,
who himself composed the music to a magnificent opera, where
were present Cardinal Pamphilio, the Pope's nephew, the Governors
of Rome, the cardinals, the ambassadors, ladies, and a number of
nobility and strangers. There had been in the morning a joust and
tournament of several young gentlemen on a formal defy, to which
we had been invited; the prizes being distributed by the ladies, after
the knight-errantry way. The lancers and swordsmen running at tilt
against the barriers, with a great deal of clatter, but without any
bloodshed, giving much diversion to the spectators, and was new to
us travelers.
The next day Mr. Henshaw and I spent the morning in attending
the entrance and cavalcade of Cardinal Medici, the ambassador from
the Grand Duke of Florence, by the Via Flaminia. After dinner, we
went again to the Villa Borghese, about a mile without the city; the
garden is rather a park, or a Paradise, contrived and planted with
walks and shades of myrtles, cypress, and other trees, and groves,
with abundance of fountains, statues, and bass-relievos, and several
pretty murmuring rivulets. Here they had hung large nets to catch
woodcocks. There was also a vivary, where, among other exotic
fowls, was an ostrich; besides a most capacious aviary; and, in

another inclosed part, a herd of deer. Before the palace (which might
become the court of a great prince) stands a noble fountain, of
white marble, enriched with statues. The outer walls of the house
are encrusted with excellent antique bass-relievos, of the same
marble, incornished with festoons and niches set with statues from
the foundation to the roof. A stately portico joins the palace, full of
statues and columns of marble, urns, and other curiosities of
sculpture. In the first hall were the Twelve Cæsars, of antique
marble, and the whole apartments furnished with pictures of the
most celebrated masters, and two rare tables of porphyry, of great
value. But of this already: for I often visited this delicious place.
This night were glorious fire-works at the palace of Cardinal
Medici before the gate, and lights of several colors all about the
windows through the city, which they contrive by setting the candles
in little paper lanterns dyed with various colors, placing hundreds of
them from story to story; which renders a gallant show.
4th May, 1645. Having seen the entry of the ambassador of
Lucca, I went to the Vatican, where, by favor of our Cardinal
Protector, Fran. Barberini, I was admitted into the Consistory, heard
the ambassador make his oration in Latin to the Pope, sitting on an
elevated state, or throne, and changing two pontifical mitres; after
which, I was presented to kiss his toe, that is, his embroidered
slipper, two Cardinals holding up his vest and surplice; and then,
being sufficiently blessed with his thumb and two fingers for that
day I returned home to dinner.
We went again to see the medals of Signor Gotefredi, which are
absolutely the best collection in Rome.
Passing the Ludovisia Villa, where the petrified human figure lies,
found on the snowy Alps; I measured the hydra, and found it not a
foot long; the three necks and fifteen heads seem to be but patched
up with several pieces of serpents' skins.

5th May, 1645. We took coach, and went fifteen miles out of the
city to Frascati, formerly Tusculum, a villa of Cardinal Aldobrandini,
built for a country house; but surpassing, in my opinion, the most
delicious places I ever beheld for its situation, elegance, plentiful
water, groves, ascents, and prospects. Just behind the palace (which
is of excellent architecture) in the centre of the inclosure, rises a
high hill, or mountain, all over clad with tall wood, and so formed by
nature, as if it had been cut out by art, from the summit whereof
falls a cascade, seeming rather a great river than a stream
precipitating into a large theatre of water, representing an exact and
perfect rainbow, when the sun shines out. Under this, is made an
artificial grot, wherein are curious rocks, hydraulic organs, and all
sorts of singing birds, moving and chirping by force of the water,
with several other pageants and surprising inventions. In the centre
of one of these rooms, rises a copper ball that continually dances
about three feet above the pavement, by virtue of a wind conveyed
secretly to a hole beneath it; with many other devices to wet the
unwary spectators, so that one can hardly step without wetting to
the skin. In one of these theaters of water, is an Atlas spouting up
the stream to a very great height; and another monster makes a
terrible roaring with a horn; but, above all, the representation of a
storm is most natural, with such fury of rain, wind, and thunder, as
one would imagine oneself in some extreme tempest. The garden
has excellent walks and shady groves, abundance of rare fruit,
oranges, lemons, etc., and the goodly prospect of Rome, above all
description, so as I do not wonder that Cicero and others have
celebrated this place with such encomiums. The Palace is indeed
built more like a cabinet than anything composed of stone and
mortar; it has in the middle a hall furnished with excellent marbles
and rare pictures, especially those of Gioseppino d'Arpino; the
movables are princely and rich. This was the last piece of
architecture finished by Giacomo della Porta, who built it for Pietro
Cardinal Aldobrandini, in the time of Clement VIII.29
We went hence to another house and garden not far distant, on
the side of a hill called Mondragone, finished by Cardinal Scipio

Borghese, an ample and kingly edifice. It has a very long gallery,
and at the end a theatre for pastimes, spacious courts, rare grots,
vineyards, olive-grounds, groves and solitudes. The air is so fresh
and sweet, as few parts of Italy exceed it; nor is it inferior to any
palace in the city itself for statues, pictures, and furniture; but, it
growing late, we could not take such particular notice of these
things as they deserved.

TIVOLI
6th May, 1645. We rested ourselves; and next day, in a
coach, took our last farewell of visiting the circumjacent
places, going to Tivoli, or the old Tiburtum. At about six miles from Rome, we
pass the Teverone, a bridge built by Mammea, the mother of Severus, and so
by divers ancient sepulchres, among others that of Valerius Volusi; and near it
past the stinking sulphurous river over the Ponte Lucano, where we found a
heap, or turret, full of inscriptions, now called the Tomb of Plautius. Arrived at
Tivoli, we went first to see the palace d'Este, erected on a plain, but where
was formerly an hill. The palace is very ample and stately. In the garden, on
the right hand, are sixteen vast conchas of marble, jetting out waters; in the
midst of these stands a Janus quadrifrons, that cast forth four girandolas,
called from the resemblance (to a particular exhibition in fireworks so named)
the Fountana di Spéccho (looking-glass). Near this is a place for tilting. Before
the ascent of the palace is the famous fountain of Leda, and not far from that,
four sweet and delicious gardens. Descending thence are two pyramids of
water, and in a grove of trees near it the fountains of Tethys, Esculapius,
Arethusa, Pandora, Pomona, and Flora; then the prancing Pegasus, Bacchus,
the Grot of Venus, the two Colosses of Melicerta and Sibylla Tiburtina, all of
exquisite marble, copper, and other suitable adornments. The Cupids pouring
out water are especially most rare, and the urns on which are placed the ten
nymphs. The grots are richly paved with pietra-commessa, shells, coral, etc.
Toward Roma Triumphans, leads a long and spacious walk, full of
fountains, under which is historized the whole Ovidian Metamorphosis, in
rarely sculptured mezzo relievo. At the end of this, next the wall, is the city of
Rome as it was in its beauty, of small models, representing that city, with its
amphitheatres; naumachi, thermæ, temples, arches, aqueducts, streets, and
other magnificences, with a little stream running through it for the Tiber,
gushing out of an urn next to the statue of the river. In another garden, is a
noble aviary, the birds artificial, and singing till an owl appears, on which they
suddenly change their notes. Near this is the fountain of dragons, casting out
large streams of water with great noise. In another grotto, called Grotto di
Natura, is an hydraulic organ; and below this are divers stews and fish ponds,
in one of which is the statue of Neptune in his chariot on a seahorse, in
another a Triton; and lastly, a garden of simples. There are besides in the
palace many rare statues and pictures, bedsteads richly inlaid, and sundry
other precious movables: the whole is said to have cost the best part of a
million.

ROME
Having gratified our curiosity with these artificial miracles, and dined, we
went to see the so famous natural precipice and cascade of the river Anio,
rushing down from the mountains of Tivoli with that fury that, what with the
mist it perpetually casts up by the breaking of the water against the rocks,
and what with the sun shining on it and forming a natural Iris, and the
prodigious depth of the gulf below, it is enough to astonish one that looks on
it. Upon the summit of this rock stands the ruins and some pillars and
cornices of the Temple of Sibylla Tyburtina, or Albunea, a round fabric, still
discovering some of its pristine beauty. Here was a great deal of gunpowder
drying in the sun, and a little beneath, mills belonging to the Pope.
And now we returned to Rome. By the way, we were
showed, at some distance, the city Præneste, and the
Hadrian villa, now only a heap of ruins; and so came late to our lodging.
We now determined to desist from visiting any more curiosities, except
what should happen to come in our way, when my companion, Mr. Henshaw,
or myself should go to take the air: only I may not omit that one afternoon,
diverting ourselves in the Piazza Navona, a mountebank there to allure
curious strangers, taking off a ring from his finger, which seemed set with a
dull, dark stone a little swelling out, like what we call (though untruly) a
toadstone, and wetting his finger a little in his mouth, and then touching it, it
emitted a luculent flame as bright and large as a small wax candle; then,
blowing it out, repeated this several times. I have much regretted that I did
not purchase the receipt of him for making that composition at what price
soever; for though there is a process in Jo. Baptista Porta and others how to
do it, yet on several trials they none of them have succeeded.
Among other observations I made in Rome are these: as to coins and
medals, ten asses make the Roman denarius, five the quinarius, ten denarii
an aureus; which accompt runs almost exactly with what is now in use of
quatrini, baiocs, julios, and scudi, each exceeding the other in the proportion
of ten. The sestertius was a small silver coin, marked H. S. or rather LLs
, valued
two pounds and a half of silver, viz, 250 denarii, about twenty-five golden
ducati. The stamp of the Roman denarius varied, having sometimes a Janus
bifrons, the head of Roma armed, or with a chariot and two horses, which
were called bigi; if with four, quadrigi: if with a Victoria, so named. The mark
of the denarius was distinguished | thus, or X; the quinarius of half value,
had, on one side, the head of Rome and V; the reverse, Castor and Pollux on
horseback, inscribed Roma, etc.

I observed that in the Greek church they made the sign of the cross from
the right hand to the left; contrary to the Latins and the schismatic Greeks;
gave the benediction with the first, second, and little finger stretched out,
retaining the third bent down, expressing a distance of the third Person of the
Holy Trinity from the first two.
LORD WILLIAM RUSSELL TAKING LEAVE OF HIS CHILDREN, 1683
Photogravure after a painting by Bridges
For sculptors and architects, we found Bernini and Algardi were in the
greatest esteem; Fiamingo, as a statuary; who made the Andrea in St. Peter's,
and is said to have died mad because it was placed in an ill light. Among the
painters, Antonio de la Cornea, who has such an address of counterfeiting the
hands of the ancient masters so well as to make his copies pass for originals;
Pietro de Cortone, Monsieur Poussin, a Frenchman, and innumerable more.
Fioravanti, for armor, plate, dead life, tapestry, etc. The chief masters of

music, after Marc Antonio, the best treble, is Cavalier Lauretto, an eunuch;
the next Cardinal Bichi's eunuch, Bianchi, tenor, and Nicholai, bass. The Jews
in Rome wore red hats, till the Cardinal of Lyons, being short-sighted, lately
saluted one of them, thinking him to be a Cardinal as he passed by his coach;
on which an order was made, that they should use only the yellow color.
There was now at Rome one Mrs. Ward, an English devotée, who much
solicited for an order of Jesuitesses.
At executions I saw one, a gentleman, hanged in his cloak and hat for
murder. They struck the malefactor with a club that first stunned him, and
then cut his throat. At Naples they use a frame, like ours at Halifax.30
It is reported that Rome has been once no less than fifty miles in compass,
now not thirteen, containing in it 3,000 churches and chapels, monasteries,
etc. It is divided into fourteen regions or wards; has seven mountains, and as
many campi or valleys; in these are fair parks, or gardens, called villas, being
only places of recess and pleasure, at some distance from the streets, yet
within the walls.
The bills of exchange I took up from my first entering Italy till I went from
Rome, amounting to but 616 ducati di banco, though I purchased many
books, pictures, and curiosities.
18th May, 1645. I intended to have seen Loretto, but, being disappointed
of moneys long expected, I was forced to return by the same way I came,
desiring, if possible, to be at Venice by the Ascension, and therefore I diverted
to take Leghorn in the way, as well to furnish me with credit by a merchant
there, as to take order for transporting such collections as I had made at
Rome. When on my way, turning about to behold this once and yet glorious
city, from an eminence, I did not, without some regret, give it my last
farewell.
Having taken leave of our friends at Rome, where I had sojourned now
about seven months, autumn, winter, and spring, I took coach, in company
with two courteous Italian gentlemen. In the afternoon, we arrived at a
house, or rather castle, belonging to the Duke of Parma, called Caprarola,
situate on the brow of a hill, that overlooks a little town, or rather a natural
and stupendous rock; witness those vast caves serving now for cellarage,
where we were entertained with most generous wine of several sorts, being
just under the foundation. The palace was built by the famous architect,
Vignola, at the cost of Cardinal Alex. Farnese, in form of an octagon, the court
in the middle being exactly round, so as rather to resemble a fort, or castle;

SIENNA
yet the chambers within are all of them square, which makes the walls
exceedingly thick. One of these rooms is so artificially contrived, that from the
two opposite angles may be heard the least whisper; they say any perfect
square does it. Most of the paintings are by Zuccari. It has a stately entry, on
which spouts an artificial fountain within the porch. The hall, chapel, and a
great number of lodging chambers are remarkable; but most of all the
pictures and witty inventions of Hannibal Caracci; the Dead Christ is
incomparable. Behind are the gardens full of statues and noble fountains,
especially that of the Shepherds. After dinner, we took horse, and lay that
night at Monte Rossi, twenty miles from Rome.
19th May, 1645. We dined at Viterbo, and lay at St. Laurenzo. Next day, at
Radicofani, and slept at Turnera.
21st May, 1645. We dined at Sienna, where we could
not pass admiring the great church built entirely both
within and without with white and black marble in polished squares, by
Macarino, showing so beautiful after a shower has fallen. The floor within is of
various colored marbles, representing the story of both Testaments, admirably
wrought. Here lies Pius II. The bibliotéca is painted by P. Perrugino and
Raphael. The life of Æneas Sylvius is in FRESCO; in the middle are the Three
Graces, in antique marble, very curious, and the front of this building, though
Gothic, is yet very fine. Among other things, they show St. Catharine's
disciplining cell, the door whereof is half cut out into chips by the pilgrims and
devotees, being of deal wood.
Setting out hence for Pisa, we went again to see the Duomo in which the
Emperor Henry VII. lies buried, poisoned by a monk in the Eucharist. The
bending tower was built by Busqueto Delichio, a Grecian architect, and is a
stupendous piece of art. In the gallery of curiosities is a fair mummy; the tail
of a sea-horse; coral growing on a man's skull; a chariot automaton; two
pieces of rock crystal, in one of which is a drop of water, in the other three or
four small worms; two embalmed children; divers petrifactions, etc. The
garden of simples is well furnished, and has in it the deadly yew, or taxus, of
the ancients; which Dr. Belluccio, the superintendent, affirms that his
workmen cannot endure to clip for above the space of half an hour at a time,
from the pain of the head which surprises them.
We went hence from Leghorn, by coach, where I took up ninety crowns for
the rest of my journey, with letters of credit for Venice, after I had sufficiently
complained of my defeat of correspondence at Rome.

PISTORIA
The next day, I came to Lucca, a small but pretty territory and state of
itself. The city is neat and well fortified, with noble and pleasant walks of trees
on the works, where the gentry and ladies used to take the air. It is situate on
an ample plain by the river Serchio, yet the country about it is hilly. The
Senate-house is magnificent. The church of St. Michael is a noble piece, as is
also St. Fredian, more remarkable to us for the corpse of St. Richard, an
English king,31
who died here on his pilgrimage toward Rome. This epitaph is
on his tomb:
Hic rex Richardus requiescit, sceptifer, almus;
Rex Fuit Anglorum; regnum tenet iste Polorum.
Regnum demisit; pro Christo cuncta reliquit.
Ergo, Richardum nobis debit Anglia sanctum.
Hic genitor Sanctæ Wulburgæ Virginis almæ
Est Vrillebaldi sancti simul et Vinebaldi,
Suffragium quorum nobis det regna Polorum.
Next this, we visited St. Croce, an excellent structure all of marble both
without and within, and so adorned as may vie with many of the fairest even
in Rome: witness the huge cross, valued at £15,000, above all venerable for
that sacred volto which (as tradition goes) was miraculously put on the image
of Christ, and made by Nicodemus, while the artist, finishing the rest of the
body, was meditating what face to set on it. The inhabitants are exceedingly
civil to strangers, above all places in Italy, and they speak the purest Italian. It
is also cheap living, which causes travelers to set up their rest here more than
in Florence, though a more celebrated city; besides, the ladies here are very
conversable, and the religious women not at all reserved; of these we bought
gloves and embroidered stomachers, generally worn by gentlemen in these
countries. The circuit of this state is but two easy days' journey, and lies
mixed with the Duke of Tuscany's but having Spain for a protector (though
the least bigoted of all Roman Catholics), and being one of the fortified cities
in Italy, it remains in peace. The whole country abounds in excellent olives,
etc.
Going hence for Florence, we dined at Pistoria, where,
besides one church, there was little observable: only in the
highway we crossed a rivulet of salt water, though many miles from the sea.
The country is extremely pleasant, full of gardens, and the roads straight as a
line for the best part of that whole day, the hedges planted with trees at equal
distances, watered with clear and plentiful streams.

FLORENCE
Rising early the next morning we arrived at Peggio Imperiale, being a
palace of the Great Duke, not far from the city, having omitted it in my
passage to Rome. The ascent to the house is by a stately gallery as it were of
tall and overgrown cypress trees for near half a mile. At the entrance of these
ranges, are placed statues of the Tiber and Arno, of marble; those also of
Virgil, Ovid, Petrarch, and Dante. The building is sumptuous, and curiously
furnished within with cabinets of pietra-commessa in tables, pavements, etc.,
which is a magnificence, or work, particularly affected at Florence. The
pictures are, Adam and Eve by Albert Durer, very excellent; as is that piece of
carving in wood by the same hand standing in a cupboard. Here is painted the
whole Austrian line; the Duke's mother, sister to the Emperor, the foundress of
this palace, than which there is none in Italy that I had seen more
magnificently adorned, or furnished.
We could not omit in our passage to re-visit the same,
and other curiosities which we had neglected on our first
being at Florence. We went, therefore, to see the famous piece of Andrea del
Sarto, in the Annunciata. The story is, that the painter in a time of dearth
borrowed a sack of corn of the religious of that convent, and repayment being
demanded, he wrought it out in this picture, which represents Joseph sitting
on a sack of corn, and reading to the Blessed Virgin; a piece infinitely valued.
There fell down in the cloister an old man's face painted on the wall in fresco,
greatly esteemed, and broke into crumbs; the Duke sent his best painters to
make another instead of it, but none of them would presume to touch a pencil
where Andrea had wrought, like another Apelles; but one of them was so
industrious and patient, that, picking up the fragments, he laid and fastened
them so artificially together, that the injury it had received was hardly
discernible. Andrea del Sarto lies buried in the same place. Here is also that
picture of Bartolomeo, who having spent his utmost skill in the face of the
angel Gabriel, and being troubled that he could not exceed it in the Virgin, he
began the body and to finish the clothes, and so left it, minding in the
morning to work on the face; but, when he came, no sooner had he drawn
away the cloth that was hung before it to preserve it from the dust, than an
admirable and ravishing face was found ready painted; at which miracle all
the city came in to worship. It is now kept in the Chapel of the Salutation, a
place so enriched by devotees, that none in Italy, save Loretto, is said to
exceed it. This picture is always covered with three shutters, one of which is
of massy silver; methinks it is very brown, the forehead and cheeks whiter, as
if it had been scraped. They report that those who have the honor of seeing it
never lose their sight—happy then we! Belonging to this church is a world of

plate, some whole statues of it, and lamps innumerable, besides the costly
vows hung up, some of gold, and a cabinet of precious stones.
Visiting the Duke's repository again, we told at least forty ranks of porphyry
and other statues, and twenty-eight whole figures, many rare paintings and
relievos, two square columns with trophies. In one of the galleries, twenty-
four figures, and fifty antique heads; a Bacchus of M. Angelo, and one of
Bandinelli; a head of Bernini, and a most lovely Cupid, of Parian marble; at
the further end, two admirable women sitting, and a man fighting with a
centaur; three figures in little of Andrea; a huge candlestick of amber; a table
of Titian's painting, and another representing God the Father sitting in the air
on the Four Evangelists; animals; divers smaller pieces of Raphael; a piece of
pure virgin gold, as big as an egg. In the third chamber of rarities is the
square cabinet, valued at 80,000 crowns, showing on every front, a variety of
curious work; one of birds and flowers, of pietra-commessa; one, a descent
from the cross, of M. Angelo; on the third, our Blessed Savior and the
Apostles, of amber; and, on the fourth, a crucifix of the same. Between the
pictures, two naked Venuses, by Titian; Adam and Eve, by Durer; and several
pieces of Portdenone, and del Frate. There is a globe of six feet diameter. In
the Armory, were an entire elk, a crocodile, and among the harness, several
targets and antique horse-arms, as that of Charles V.; two set with turquoises,
and other precious stones; a horse's tail, of a wonderful length. Then, passing
the Old Palace, which has a very great hall for feasts and comedies, the roof
rarely painted, and the side walls with six very large pictures representing
battles, the work of Gio. Vassari. Here is a magazine full of plate; a harness of
emeralds; the furnitures of an altar four feet high, and six in length, of massy
gold; in the middle is placed the statue of Cosmo II., the bass-relievo is of
precious stones, his breeches covered with diamonds; the moldings of this
statue, and other ornaments, festoons, etc., are garnished with jewels and
great pearls, dedicated to St. Charles, with this inscription, in rubies:
Cosimus Secundus Dei gratiâ Magnus Dux Etruriæ ex voto.
There is also a King on horseback, of massy gold, two feet high, and an
infinity of such like rarities. Looking at the Justice, in copper, set up on a
column by Cosmo, in 1555, after the victory over Sienna, we were told that
the Duke, asking a gentleman how he liked the piece, he answered, that he
liked it very well, but that it stood too high for poor men to come at it.
Prince Leopold has, in this city, a very excellent collection of paintings,
especially a St. Catherine of P. Veronese; a Venus of marble, veiled from the
middle to the feet, esteemed to be of that Greek workman who made the

Venus at the Medici's Palace in Rome, altogether as good, and better
preserved, an inestimable statue, not long since found about Bologna.
Signor Gaddi is a lettered person, and has divers rarities, statues, and
pictures of the best masters, and one bust of marble as much esteemed as
the most antique in Italy, and many curious manuscripts; his best paintings
are, a Virgin of del Sarto, mentioned by Vassari, a St. John, by Raphael, and
an Ecce Homo, by Titian.
The hall of the Academy de la Crusca is hung about with impresses and
devices painted, all of them relating to corn sifted from the bran; the seats
are made like breadbaskets and other rustic instruments used about wheat,
and the cushions of satin, like sacks.
We took our farewell of St. Laurence, more particularly noticing that piece
of the Resurrection, which consists of a prodigious number of naked figures,
the work of Pontormo. On the left hand is the Martyrdom of St. Laurence, by
Bronzino, rarely painted indeed. In a chapel is the tomb of Pietro di Medici,
and his brother John, of copper, excellently designed, standing on two lions'
feet, which end in foliage, the work of M. Angelo. Over against this, are
sepulchres of all the ducal family. The altar has a statue of the Virgin giving
suck, and two Apostles. Paulus Jovius has the honor to be buried in the
cloister. Behind the choir is the superb chapel of Ferdinand I., consisting of
eight faces, four plain, four a little hollowed; in the other are to be the
sepulchres, and a niche of paragon, for the statue of the prince now living, all
of copper gilt; above, is a large table of porphyry, for an inscription for the
Duke, in letters of jasper. The whole chapel, walls, pavement, and roof, are
full of precious stones united with the moldings, which are also of gilded
copper, and so are the bases and capitals of the columns. The tabernacle,
with the whole altar, is inlaid with cornelians, lazuli, serpentine, agates,
onyxes, etc. On the other side are six very large columns of rock crystal, eight
figures of precious stones of several colors, inlaid in natural figures, not
inferior to the best paintings, among which are many pearls, diamonds,
amethysts, topazes, sumptuous and sparkling beyond description. The
windows without side are of white marble. The library is the architecture of
Raphael; before the port is a square vestibule of excellent art, of all the
orders, without confusion; the ascent to it from the library is excellent. We
numbered eighty-eight shelves, all MSS. and bound in red, chained; in all
about 3,500 volumes, as they told us.
The Arsenal has sufficient to arm 70,000 men, accurately preserved and
kept, with divers lusty pieces of ordnance, whereof one is for a ball of 300

BOLOGNA
pounds weight, and another for 160, which weighs 72,500 pounds.
When I was at Florence, the celebrated masters were: for pietra-commessa
(a kind of mosaic, or inlaying, of various colored marble, and other more
precious stones), Dominico Benetti and Mazotti; the best statuary, Vincentio
Brochi. This statuary makes those small figures in plaster and pasteboard,
which so resemble copper that, till one handles them, they cannot be
distinguished, he has so rare an art of bronzing them; I bought four of him.
The best painter, Pietro Beretino di Cortona.
This Duke has a daily tribute for every courtezan, or prostitute, allowed to
practice that infamous trade in his dominions, and so has his Holiness the
Pope, but not so much in value.
Taking leave of our two jolly companions, Signor
Giovanni and his fellow, we took horses for Bologna; and,
by the way, alighted at a villa of the Grand Duke's, called Pratolino. The house
is a square of four pavilions, with a fair platform about it, balustred with
stone, situate in a large meadow, ascending like an amphitheater, having at
the bottom a huge rock, with water running in a small channel, like a cascade;
on the other side, are the gardens. The whole place seems consecrated to
pleasure and summer retirement. The inside of the palace may compare with
any in Italy for furniture of tapestry, beds, etc., and the gardens are delicious,
and full of fountains. In the grove sits Pan feeding his flock, the water making
a melodious sound through his pipe; and a Hercules, whose club yields a
shower of water, which, falling into a great shell, has a naked woman riding
on the backs of dolphins. In another grotto is Vulcan and his family, the walls
richly composed of corals, shells, copper, and marble figures, with the hunting
of several beasts, moving by the force of water. Here, having been well
washed for our curiosity, we went down a large walk, at the sides whereof
several slender streams of water gush out of pipes concealed underneath,
that interchangeably fall into each other's channels, making a lofty and
perfect arch, so that a man on horseback may ride under it, and not receive
one drop of wet. This canopy, or arch of water, I thought one of the most
surprising magnificences I had ever seen, and very refreshing in the heat of
the summer. At the end of this very long walk, stands a woman in white
marble, in posture of a laundress wringing water out of a piece of linen, very
naturally formed, into a vast laver, the work and invention of M. Angelo
Buonarotti. Hence, we ascended Mount Parnassus, where the Muses played to
us on hydraulic organs. Near this is a great aviary. All these waters came from
the rock in the garden, on which is the statue of a giant representing the

Apennines, at the foot of which stands this villa. Last of all, we came to the
labyrinth, in which a huge colosse of Jupiter throws out a stream over the
garden. This is fifty feet in height, having in his body a square chamber, his
eyes and mouth serving for windows and door.
We took horse and supped that night at Il Ponte, passing a dreadful ridge
of the Apennines, in many places capped with snow, which covers them the
whole summer. We then descended into a luxurious and rich plain. The next
day we passed through Scarperia, mounting the hills again, where the
passage is so straight and precipitous toward the right hand, that we climbed
them with much care and danger; lodging at Firenzuolo, which is a fort built
among the rocks, and defending the confines of the Great Duke's territories.
The next day we passed by the Pietramala, a burning mountain. At the
summit of this prodigious mass of hills, we had an unpleasant way to Pianura,
where we slept that night and were entertained with excellent wine. Hence to
Scargalasino, and to bed at Loiano. This plain begins about six miles from
Bologna.
Bologna belongs to the Pope, and is a famous University, situate in one of
the richest spots of Europe for all sorts of provisions. It is built like a ship,
whereof the Torre d'Asinelli may go for the mainmast. The city is of no great
strength, having a trifling wall about it, in circuit near five miles, and two in
length. This Torre d'Asinelli, ascended by 447 steps of a foot rise, seems
exceedingly high, is very narrow, and the more conspicuous from another
tower called Garisendi, so artificially built of brick (which increases the
wonder) that it seems ready to fall. It is not now so high as the other; but
they say the upper part was formerly taken down, for fear it should really fall,
and do mischief.
Next, we went to see an imperfect church, called St. Petronius, showing
the intent of the founder, had he gone on. From this, our guide led us to the
schools, which indeed are very magnificent. Thence to St. Dominic's, where
that saint's body lies richly enshrined. The stalls, or seats, of this goodly
church have the history of the Bible inlaid with several woods, very curiously
done, the work of one Fr. Damiano di Bergamo, and a friar of that order.
Among other relics, they show the two books of Esdras, written with his own
hand. Here lie buried Jac. Andreas, and divers other learned persons. To the
church joins the convent, in the quadrangle whereof are old cypresses, said to
have been planted by their saint.

Then we went to the palace of the Legate; a fair brick building, as are most
of the houses and buildings, full of excellent carving and moldings, so as
nothing in stone seems to be better finished or more ornamental; witness
those excellent columns to be seen in many of their churches, convents, and
public buildings; for the whole town is so cloistered, that one may pass from
house to house through the streets without being exposed either to rain or
sun.
Before the stately hall of this palace stands the statue of Paul IV. and divers
others; also the monument of the coronation of Charles V. The piazza before it
is the most stately in Italy, St. Mark's at Venice only excepted. In the center of
it is a fountain of Neptune, a noble figure in copper. Here I saw a Persian
walking about in a rich vest of cloth of tissue, and several other ornaments,
according to the fashion of his country, which much pleased me; he was a
young handsome person, of the most stately mien.
I would fain have seen the library of St. Savior, famous for the number of
rare manuscripts; but could not, so we went to St. Francis, a glorious pile, and
exceedingly adorned within.
After dinner I inquired out a priest and Dr. Montalbano, to whom I brought
recommendations from Rome: this learned person invented, or found out, the
composition of the lapis illuminabilis, or phosphorus. He showed me their
property (for he had several), being to retain the light of the sun for some
competent time, by a kind of imbibition, by a particular way of calcination.
Some of these presented a blue color, like the flame of brimstone, others like
coals of a kitchen fire. The rest of the afternoon was taken up in St. Michael in
Bosco, built on a steep hill on the edge of the city, for its fabric, pleasant
shade and groves, cellars, dormitory, and prospects, one of the most delicious
retirements I ever saw; art and nature contending which shall exceed; so as
till now I never envied the life of a friar. The whole town and country to a vast
extent are under command of their eyes, almost as far as Venice itself. In this
convent there are many excellent paintings of Guido Reni; above all, the little
cloister of eight faces, painted by Caracci in fresco. The carvings in wood, in
the sacristy, are admirable, as is the inlaid work about the chapel, which even
emulates the best paintings; the work is so delicate and tender. The paintings
of the Savior are of Caracci and Leonardo, and there are excellent things of
Raphael which we could not see.
In the church of St. John is a fine piece of St. Cecilia, by Raphael. As to
other paintings, there is in the church of St. Gregory an excellent picture of a
Bishop giving the habit of St. Bernard to an armed soldier, with several other

FERRARA
figures in the piece, the work of Guerchino. Indeed, this city is full of rare
pieces, especially of Guido Domenico, and a virgin named Isabella Sirani, now
living, who has painted many excellent pieces, and imitates Guido so well,
that many skillful artists have been deceived.
At the Mendicants are the Miracles of St. Eloy, by Reni, after the manner of
Caravaggio, but better; and here they showed us that famous piece of Christ
calling St. Matthew, by Annibal Caracci. The Marquis Magniani has the whole
frieze of his hall painted in fresco by the same hand.
Many of the religious men nourish those lapdogs which the ladies are so
fond of, and which they here sell. They are a pigmy sort of spaniels, whose
noses they break when puppies; which, in my opinion, deforms them.
At the end of the turning in one of the wings of the dormitory of St.
Michael, I found a paper pasted near the window, containing the dimensions
of most of the famous churches in Italy compared with their towers here, and
the length of this gallery, a copy whereof I took.
Braccia32
Piede di Bolognia Canna di Roma.
St. Pietro di Roma, longo 284 473 84
Cupalo del muro, alta 210 350 60
Torre d'Asinello, alto 208 4/5 348 59 pr.mi 6
Dormitorio de St. Mich. a Bologn.
longo 254 423 72 ½
From hence being brought to a subterranean territory of cellars, the
courteous friars made us taste a variety of excellent wines; and so we
departed to our inn.
The city is famous also for sausages; and here is sold great quantities of
Parmegiano cheese, with Botargo, Caviare, etc., which makes some of their
shops perfume the streets with no agreeable smell. We furnished ourselves
with wash balls, the best being made here, and being a considerable
commodity. This place has also been celebrated for lutes made by the old
masters, Mollen, Hans Frey, and Nicholas Sconvelt, which were of
extraordinary price; the workmen were chiefly Germans. The cattle used for
draught in this country (which is very rich and fertile, especially in pasturage)
are covered with housings of linen fringed at the bottom, that dangle about
them, preserving them from flies, which in summer are very troublesome.
From this pleasant city, we proceeded toward Ferrara,
carrying with us a bulletino, or bill of health (customary in

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