02 control of flexible robot

Control of Flexible Manipulators. Theory and Practice 267
x
Control of Flexible Manipulators.
Theory and Practice
Pereira, E.; Becedas, J.; Payo, I.; Ramos, F. and Feliu, V.
Universidad de Castilla-La Mancha,
ETS Ingenieros Industriales, Ciudad Real
Spain
1. Introduction
Novel robotic applications have demanded lighter robots that can be driven using small
amounts of energy, for example robotic booms in the aerospace industry, where lightweight
manipulators with high performance requirements (high speed operation, better accuracy,
high payload/weight ratio) are required (Wang & Gao, 2003). Unfortunately, the flexibility
of these robots leads to oscillatory behaviour at the tip of the link, making precise pointing
or tip positioning a daunting task that requires complex closed-loop control. In order to
address control objectives, such as tip position accuracy and suppression of residual
vibration, many control techniques have been applied to flexible robots (see, for instance,
the survey (Benosman & Vey, 2004)). There are two main problems that complicate the
control design for flexible manipulators viz: (i) the high order of the system, (ii) the no
minimum phase dynamics that exists between the tip position and the input (torque applied
at the joint). In addition, recently, geometric nonlinearities have been considered in the
flexible elements. This chapter gives an overview to the modelling and control of flexible
manipulators and focuses in the implementation of the main control techniques for single
link flexible manipulators, which is the most studied case in the literature.
2. State of the art
Recently, some reviews in flexible robotics have been published. They divide the previous
work attending to some short of classification: control schemes (Benosman & Vey, 2004),
modelling (Dwivedy & Eberhard, 2006), overview of main researches (Feliu, 2006), etc. They
are usually comprehensive enumerations of the different approaches and/or techniques
used in the diverse fields involving flexible manipulators. However, this section intends to
give a chronological overview of how flexible manipulators have evolved since visionaries
such as Prof. Mark J. Balas or Prof. Wayne J. Book sowed the seeds of this challenging field
of robotics. Moreover, some attention is given to main contributions attending to the impact
of the work and the goodness of the results.
In the early 70's the necessity of building lighter manipulators able to perform mechanical
tasks arises as a part of the USA Space Research. The abusive transportation costs of a gram
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Robot Manipulators, Trends and Development268
of material into orbit and the reduced room and energy available inside an spacecraft cause
the imperative need for reducing weight and size as far as possible in any device aboard.
Unfortunately, as the manipulator reduces weight, it reduces also accuracy in its
manoeuvres due to the appearance of structural flexibility (and hence, vibrations) of the
device.
The interest of NASA in creating these manipulators for use in spatial applications
motivated the investment for the research of flexible robots and its associated new control
problems. In 1974, Prof. Wayne J. Book provided the first known work dealing with this
topic explicitly in his Ph. D. Thesis (Book, 1974) entitled as “Modeling, design and control of
flexible manipulators arms” and supervised by Prof. Daniel E. Whitney, who was a professor
at MIT Mechanical Engineering Department. In the same department than Prof. Book, the
very same year Dr. Maizza-Neto also studied the control of flexible manipulator arms but
from a modal analysis approach (Maizza-Neto, 1974). Fruits of their joint labour, the first
work published in a journal in the field of flexible robotics appeared in 1975, dealing with
the feedback control of a two-link-two-joints flexible robot (Book et al., 1975). After this
milestone, Dr. Maizza-Neto quitted from study of elastic arms but Prof. Book continued
with its theoretical analysis of flexible manipulators, e.g. taking frequency domain and
space-state approaches (Book & Majette, 1983), until he finally came up with a recursive,
lagrangian, assumed modes formulation for modelling a flexible arm (Book, 1984) that
incorporates the approach taken by Denavit and Hartenberg (Denavit & Hartenberg, 1955),
to describe in a efficient, complete and straightforward way the kinematics and dynamics of
elastic manipulators. Due to the generality and simplicity of the technique applied, this
work has become one of the most cited and well-known studies in flexible robotics. This
structural flexibility was also intensively studied in satellites and other large spacecraft
structures (again spatial purposes and NASA behind the scenes) which generally exhibit
low structural damping in the materials used and lack of other forms of damping. A special
mention deserves Prof. Mark J. Balas, whose generic studies on the control of flexible
structures, mainly between 1978 and 1982, e.g. (Balas, 1978) and (Balas, 1982), established
some key concepts such as the influence of high nonmodelled dynamics in the system
controllability and performance, which is known as "spillover". In addition, the
numerical/analytical examples included in his work dealt with controlling and modelling
the elasticity of a pinned or cantilevered Euler-Bernoulli beam with a single actuator and a
sensor, which is the typical configuration for a one degree of freedom flexible robot as we
will discuss in later sections.
After these promising origins, the theoretical challenge of controlling a flexible arm (while
still very open) turned into the technological challenge of building a real platform in which
testing those control techniques. And there it was, the first known robot exhibiting notorious
flexibility to be controlled was built by Dr. Eric Schmitz (Cannon & Schmitz, 1984) under the
supervision of Prof. Robert H. Cannon Jr., founder of the Aerospace Robotics Lab and
Professor Emeritus at Stanford University. A single-link flexible manipulator was precisely
positioned by sensing its tip position while it was actuated on the other end of the link. In
this work appeared another essential concept in flexible robots: a flexible robot it is a
noncolocated system and thus of nonminimum phase nature. This work is the most
referenced ever in the field of flexible robotics and it is considered unanimously as the
breakthrough in this topic.
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Point-to-point motion of elastic manipulators had been studied with remarkable success
taking a number of different approaches, but it was not until 1989 that the tracking control
problem of the end-point of a flexible robot was properly addressed. Prof. Siciliano
collaborated with Prof. Alessandro De Luca to tackle the problem from a mixed open-closed
loop control approach (De Luca & Siciliano, 1989) in the line proposed two years before by
Prof. Bayo (Bayo, 1987). Also in 1989, another very important concept called passivity was
used for the first time in this field. Prof. David Wang finished his Ph.D Thesis (Wang, 1989)
under the advisement of Prof. Mathukumalli Vidyasagar, studying this passivity property
of flexible links when an appropriate output of the system was chosen (Wang & Vidyasagar,
1991).
In (Book, 1993), a review on the elastic behaviour of manipulators was meticulously
performed. In his conclusions, Prof. Book remarks the exponential growth in the number of
publications and also the possibility of corroborating simulation results with experiments,
what turns a flexible arm into "one test case for the evaluation of control and dynamics
algorithms". And so it was. It is shown in (Benosman & Vey 2004) a summary of the main
control theory contributions to flexible manipulators, such as PD-PID, feedforward,
adaptive, intelligent, robust, strain feedback, energy-based, wave-based and among others.
3. Modelling of flexible manipulators
One of the most studied problems in flexible robotics is its dynamic modelling (Dwivedy &
Eberhard, 2006). Differently to conventional rigid robots, the elastic behaviour of flexible
robots makes the mathematical deduction of the models, which govern the real physical
behaviour, quite difficult. One of the most important characteristic of the flexible
manipulator models is that the low vibration modes have more influence in the system
dynamics than the high ones, which allows us to use more simple controllers, with less
computational costs and control efforts. Nevertheless, this high order dynamics, which is
not considered directly in the controller designed, may give rise to the appearance of bad
system behaviours, and sometimes, under specific conditions, instabilities. This problem is
usually denoted in the literature as spillover (Balas, 1978).
The flexibility in robotics can appear in the joints (manipulators with flexible joints) or in the
links (widely known as flexible link manipulators or simply flexible manipulators). The joint
flexibility is due to the twisting of the elements that connect the joint and the link. This
twisting appears, for instance, in reduction gears when very fast manoeuvres are involved,
and produces changes in the joint angles. The link flexibility is due to its deflection when
fast manoeuvres or heavy payloads are involved. From a control point of view, the
flexibility link problem is quite more challenging than the joint flexibility.
3.1 Single-link flexible manipulators
Single-link flexible manipulators consist of a rigid part, also denominated as actuator, which
produces the spatial movement of the structure; and by a flexible part, which presents
distributed elasticity along the whole structure. Fig. 1 shows the parametric representation
of a single-link flexible manipulator, which is composed of the following: (a) a motor and a
reduction gear of 1:nr reduction ratio at the base, with total inertia (rotor and hub) J0,
dynamic friction coefficient  and Coulomb friction torque f; (b) a flexible link with
uniform linear mass density , uniform bending stiffness EI and length L; and (c) a payload
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of mass MP and rotational inertia JP. Furthermore, the applied torque is m, coup denotes the
coupling torque between the motor and the link, m is the joint angle and t represents the
tip angle.
Fig. 1. Parametric representation of a single link flexible manipulator with a rotational joint.
The dynamic behaviour of the system is governed by a differential partial equation which
presents infinite vibration modes. The objective is to obtain a simplified model (finite
number of vibration modes) of the differential equation that characterizes the dynamics of
the link. A number of models can be found in the literatures obtained from methods such as
the truncation of the infinite dimensional model (Cannon & Schmitz, 1984); the
discretization of the link based on finite elements (Bayo, 1987); or directly from concentrated
mass models (Feliu et al., 1992).
The hypothesis of negligible gravity effect and horizontal motion are considered in the
deduction of the model equations. In addition, the magnitudes seen from the motor side of
the gear will be written with an upper hat, while the magnitudes seen from the link side will
be denoted by standard letters. With this notation and these hypotheses, the momentum
balance at the output side of the gear is given by the following expression
           0
ˆ ˆˆ ˆ ˆ
m m m m f coupt K u t J t t t t          , (1)
where Km is the motor constant that models the electric part of the motor (using a current
servoamplifier) and u is the motor input voltage. This equation can be represented in a block
diagram as shown in
Fig. 2, where Gc(s) and Gt(s) are the transfer functions from m to coup and t respectively.
Fig. 2. Block diagram of the single-link flexible manipulator system.
dynamics
Km
u
+
– –
+
1/nrJ0
s(s+/J0)
m
joint dynamics
link
1/nr
ˆ
m
ˆ
f ˆ
coup
Gc(s)
Gt(s)
coup
t
MP, JP
EI, L, 
flexible link
m(t)
J0, , f ,
m(t) t(t)
joint
payload
Y
X
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The link model is deduced by considering small deformations, which allows us to use a
linear beam model to obtain the dynamic equations. Based on this hypothesis, in this
chapter we use models derived from the truncation of infinite dimensional model obtained
from concentrated mass model and assumed mode method.
3.1.1 Concentrated mass models
In the concentrated mass models, the link mass is concentrated in several points along the
whole structure (see Fig. 3), where the inertia produced by the point mass rotations is
rejected. An example of this technique can be found in (Feliu et al., 1992). Fig. 3 shows the
scheme of the concentrated mass model. The lumped masses are represented by mi, with 1 i
 n; the distance between two consecutive masses i–1 and i is li, l1 is the distance between the
motor shaft and the first mass; finally, the distance between the mass mi and the motor shaft
is Li. Fn represents the applied external force at the tip of the link. n is the torque applied in
the same location. Assuming small deflections and considering that the stiffness EI is
constant through each interval of the beam the deflection is given by a third order
polynomial:
2 3
,0 ,1 1 ,2 1 ,3 1( ) ( ) ( ) ( )i i i i i i i iy x u u x L u x L u x L         , (2)
where uij are the different coefficients for each interval, and L0=0.
Fig. 3. Concentrated masses model of a single-link flexible manipulator.
The dynamic model of the flexible link is obtained from some geometric and dynamic
equations as follows (see (Feliu et al., 1992) for more details):
 
2
2 m n n
d
M EI A B P QF
dt


      , (3)
where M=diag(m1,m2,…,mn) represents the masses matrix of the system and =[1,2,…, n]T.
On the other hand, Anxn is a constant matrix, B=-A[1,1,…,1]T, Pnx1 and Qnx1 are
constant column vectors, which only depend on the link geometry.
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Finally, the coupling torque affecting the motor dynamics (see Equation (1)) is defined as
coup=–2EIu1,2. Notice that the coupling torque has the same magnitude and different sign to
the joint torque 2EIu1,2. This torque can be expressed as a linear function:
1 2coup n m n nC c c       , (4)
where C=(c1,c2,…,cn), ci, 1 i  n+2, are parameters which do not depend on the concentrated
masses along the structure and cn+1=-C[1,1,…,1]t.
For example, the transfer functions Gc(s) and Gt(s) for only one point mass located in the tip
(m1) are as follows:
                2 2 2 2 2
1 1 13 / / and /c tG s EI L s G s s , (5)
in which   3
1 13 /EI L m . This model can be used for flexible robots with a high
payload/weight ratio.
3.1.2 Assumed mode method
The dynamic behaviour of an Euler-Bernoulli beam is governed by the following PDE (see,
for example, (Meirovitch, 1996))
     , , ,IV
EIw x t w x t f x t  , (6)
where f(x,t) is a distributed external force, w is the elastic deflection measured from the
undeformed link. Then, from modal analysis of Equation (6), which considers w(x,t) as
     
1
, i i
i
w x t x t 


  , (7)
in which i(x) are the eigenfunctions and i(t) are the generalized coordinates, the system
model can be obtained (see (Belleza et al., 1990) for more details).
3.2 Multi-link flexible manipulators
For these types of manipulators truncated models are also used. Some examples are: (De
Luca & Siciliano, 1991) for planar manipulators, (Pedersen & Pedersen, 1998) for 3 degree of
freedom manipulators and (Schwertassek et al., 1999), in which the election of shape
functions is discussed.
The deflections are calculated from the following expression:
          , , 1T
i i i Lw x t x t i n , (8)
(see for example (Benosman & Vey 2004)), in which i means the number of the link, nL the
number of links, i (x) is a column vector with the shape functions of the link (for each
considered mode), i(t)=(1i,…, Ni)T is a column vector that represents the dynamics of each
mode, in which N is the number of modes considered.
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The dynamics equations of the overall system from the Lagrange method are described as
follows:
R
k
k k k
d L L D
u
dt q q q
  
  
   
, (9)
where L is the lagrangian defined as L=E-P, being E the total kinetic energy of the
manipulator and P its potential energy. This expression is similar to the used in rigid robots,
but in this case the potential energy is the sum of the gravity and the elastic deformation
terms. The term DR is the dissipation function of Rayleigh, which allows us to include
dissipative terms like frictions, and uk is the generalized force applied in qk. From Equation
(9) the robot dynamics can be deduced (see for example Chapter 1 of (Wang & Gao, 2003))
       ,I Q Q b Q Q K Q Q D Q g Q F           , (10)
were Q=(1,…, nL|1,…,nL)T is the vector of generalized coordinates that includes the first
block of joint angles i (rigid part of the model) and the elastic deflections of the links i;  is
the vector of motor torques of the joints, I is the inertias matrix of the links and the payload
of the robot, which is positive definite symmetric, b is the vector that represents the spin and
Coriolis forces (  ,b Q Q Q   ) , K is stiffness matrix, D is the damping matrix, g is the
gravity vector and F is the connection matrix between the joints and the mechanism.
Equation (10) presents a similar structure to the dynamics of a rigid robot with the
differences of: (i) the elasticity term (  K Q Q  ) and (ii) the vector of generalized coordinates
is extended by vectors that include the link flexibility.
3.3 Flexible joints
In this sort of systems, differently to the flexible link robots, in which the flexibility was
found in the whole structure from the hub with the actuator to the tip position, the flexibility
appears as a consequence of a twist in those elements which connect the actuators with the
links, and this effect has always rotational nature. Therefore, the reduction gears used to
connect the actuators with the links can experiment this effect when they are subject to very
fast movements. Such a joint flexibility can be modelled as a linear spring (Spong, 1987) or
as a torsion spring (Yuan & Lin, 1990). Surveys devoted to this kind of robots are (Bridges et
al., 1995) and (Ozgoli & Taghirad, 2006), in which a comparison between the most used
methods in controlling this kind of systems is carried out. Nevertheless, this problem in
flexible joints sometimes appears combined with flexible link manipulators. Examples of
this problem are studied in (Yang & Donath, 1988) and (Yuan & Lin, 1990).
4. Control techniques
This section summarizes the main control techniques for flexible manipulators, which are
classified into position and force control.
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4.1 Position Control
The benefits and interests jointly with advantages and disadvantages of the most relevant
contributions referent to open and closed control schemes for position control of flexible
manipulators have been included in the following subsections:
4.1.1 Command generation
A great number of research works have proposed command generation techniques, which
can be primarily classified into pre-computed and real-time. An example of pre-computed is
(Aspinwall, 1980), where a Fourier expansion was proposed to generate a trajectory that
reduces the peaks of the frequency spectrum at discrete points. Another pre-computed
alternative uses multi-switch bang-bang functions that produce a time-optimal motion.
However, this alternative requires the accurate selection of switching times which depends
on the dynamic model of the system (Onsay & Akay, 1991). The main problem of pre-
computed command profiles is that the vibration reduction is not guaranteed if a change in
the trajectory is produced.
The most used reference command generation is based on filtering the desired trajectory in
real time by using an input shaper (IS). An IS is a particular case of a finite impulse response
filter that obtains the command reference by convolving the desired trajectory with a
sequence of impulses (filter coefficients) ((Smith, 1958) and (Singer & Seering, 1990)). This
control is widely extended in the industry and there are many different applications of IS
such as spacecraft field (Tuttle & Seering, 1997), cranes and structures like cranes (see
applications and performance comparisons in (Huey et al., 2008)) or nanopositioners
(Jordan, 2002). One of the main problems of IS design is to deal with system uncertainties.
The approaches to solve this main problem can be classified into robust (see the survey of
(Vaughan et al., 2008)), learning ((Park & Chang, 2001) and (Park et al., 2006)) or adaptive
input shaping (Bodson, 1998).
IS technique has also been combined with joint position control ((Feliu & Rattan 1999) and
(Mohamed et al., 2005)), which guarantees trajectory tracking of the joint angle reference
and makes the controlled system robust to joint frictions. The main advantages of this
control scheme are the simplicity of the control design, since an accurate knowledge of the
system is not necessary, and the robustness to unmodelled dynamics (spillover) and
changes in the systems parameters (by using the aforementioned robust, adaptive and
learning approaches). However, these control schemes are not robust to external
disturbance, which has motivated closed loop controllers to be used in active vibration
damping.
4.1.2 Classic control techniques
In this chapter, the term “classic control techniques” for flexible manipulators refers to
control laws derived from the classic control theory, such as proportional, derivative and/or
integral action, or phase-lag controllers. Thus, classic control techniques, like Proportional-
Derivative (PD) control (De Luca & Siciliano, 1993) or Lead-Lag control (Feliu et al., 1993)
among others, have been proposed in order to control the joint and tip position (angle) of a
lightweight flexible manipulator. The main advantage of these techniques is the simplicity
of its design, which makes this control very attractive from an industrial point of view.
However, in situations of changes in the system, its performance is worse (slow time
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response, worse accuracy in the control task...) than other control techniques such as robust,
adaptive or learning approaches among others. Nevertheless, they can be used in
combination with more modern and robust techniques (e.g. passive and robust control
theories) to obtain a controller more adequate and versatile to do a determined control task,
as a consequence of its easy implementation. Classic control techniques are more convenient
when minimum phase systems are used (see discussions of (Wang et al., 1989)), which can
be obtained by choosing an appropriate output ((Gervarter, 1970), (Luo, 1993) and (Pereira
et al., 2007)) or by redefining it ((Wang & Vidyasagar 1992) and (Liu & Yuan, 2003)).
4.1.3 Robust, Optimal and Sliding Mode Control
It is widely recognized that many systems have inherently uncertainties, which can be
parameters variations or simple lack of knowledge of their physical parameters, external
disturbances, unmodelled dynamics or errors in the models because of simplicities or
nonlinearities. These uncertainties may lead to inaccurate position control or even
sometimes make the closed-loop system unstable. The robust control deals with these
uncertainties (Korolov & Chen, 1989), taking them into account in the design of the control
law or by using some analysis techniques to make the system robust to any or several of
these uncertainties. The output/input linearization added to Linear Quadratic Regulator
(LQR) was applied in (Singh & Schy, 1985). Nevertheless, LQR regulators are avoided to be
applied in practical setups because of the well-known spillover problems. The Linear
Quadratic Gaussian (LQG) was investigated in (Cannon & Schmitz, 1984) and (Balas, 1982).
However, these LQG regulators do not guarantee general stability margins (Banavar &
Dominic, 1995). Nonlinear robust control method has been proposed by using singular
perturbation approach (Morita et al., 1997). To design robust controllers, Lyapunov’s second
method is widely used (Gutman, 1999). Nevertheless the design is not that simple, because
the main difficulty is the non trivial finding of a Lyapunov function for control design.
Some examples in using this technique to control the end-effector of a flexible manipulator
are (Theodore & Ghosal, 2003) and (Jiang, 2004).
Another robust control technique which has been used by many researchers is the optimal
H∞ control, which is derived from the L2-gain analysis (Yim et al., 2006). Applications of this
technique to control of flexible manipulators can be found in (Moser, 1993), (Landau et al.,
1996), (Wang et al., 2002) and (Lizarraga & Etxebarria, 2003) among others.
Major research effort has been devoted to the development of the robust control based on
Sliding Mode Control. This control is based on a nonlinear control law, which alters the
dynamics of the system to be controlled by applying a high frequency switching control.
One of the relevant characteristics of this sort of controllers is the augmented state feedback,
which is not a continuous function of time. The goal of these controllers is to catch up with
the designed sliding surface, which insures asymptotic stability. Some relevant publications
in flexible robots are the following: (Choi et al., 1995), (Moallem et al., 1998), (Chen & Hsu,
2001) and (Thomas & Mija, 2008).
4.1.4 Adaptive control
Adaptive control arises as a solution for systems in which some of their parameters are
unknown or change in time (Åström & Wittenmark, 1995). The answer to such a problem
consists in developing a control system capable of monitoring his behaviour and adjusting
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the controller parameters in order to increase the working accuracy. Thus, adaptive control
is a combination of both control theory, which solves the problem of obtaining a desired
system response to a given system input, and system identification theory, which deals with
the problem of unknown parameters.
For obvious reasons, robotics has been a platinum client of adaptive control since first robot
was foreseen. Manipulators are general purpose mechanisms designed to perform arbitrary
tasks with arbitrary movements. That broad definition leaves the door open for changes in
the system, some of which noticeably modify the dynamics of the system, e.g. payload
changes (Bai et al., 1998).
Let us use a simple classification for adaptive control techniques, which groups them in
(Åström & Wittenmark, 1995):
•Direct Adaptive Control, also called Control with Implicit Identification (CII): the system
parameters are not identified. Instead, the controller parameters are adjusted directly
depending on the behaviour of the system. CII reduces the computational complexity and
has a good performance in experimental applications. This reduction is mainly due to the
controller parameters are adjusted only when an accurate estimation of the uncertainties is
obtained, which requires, in addition to aforementioned accuracy, a fast estimation.
•Indirect Adaptive Control, also called Control with Explicit Identification (CEI): the system
parameters estimations are obtained on line and the controller parameters are adjusted or
updated depending on such estimations. CEI presents good performance but they are not
extendedly implemented in practical applications due to their complexity, high
computational costs and insufficient control performance at start-up of the controllers.
First works on adaptive control applied to flexible robots were carried out in second half of
80’s (Siciliano et al., 1986), (Rovner & Cannon, 1987) and (Koivo & Lee, 1989), but its study
has been constant along the time up to date, with application to real projects such as the
Canadian SRMS (Damaren, 1996). Works based on the direct adaptive control approach can
be found: (Siciliano et al., 1986), (Christoforou & Damaren 2000) and (Damaren, 1996); and
on the indirect adaptive control idea: (Rovner & Cannon, 1987) and (Feliu en al., 1990). In
this last paper a camera was used as a sensorial system to close the control loop and track
the tip position of the flexible robot. In other later work (Feliu et al., 1999), an accelerometer
was used to carry out with the same objective, but presented some inaccuracies due to the
inclusion of the actuator and its strong nonlinearities (Coulomb friction) in the estimation
process. Recently, new indirect approaches have appeared due to improvements in sensorial
system (Ramos & Feliu, 2008) or in estimation methods (Becedas et al., 2009), which reduce
substantially the estimation time without reducing its accuracy. In both last works strain
gauges located in the coupling between the flexible link and the actuator were used to
estimate the tip position of the flexible robot.
4.1.5 Intelligent control
Ideally, an autonomous system must have the ability of learning what to do when there are
changes in the plant or in the environment, ability that conventional control systems totally
lack of. Intelligent control provides some techniques to obtain this learning and to apply it
appropriately to achieve a good system performance. Learning control (as known in its
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beginnings) started to be studied in the 60’s (some surveys of this period are (Tsypkin, 1968)
and (Fu, 1970)), and its popularity and applications have increased continuously since, being
applied in almost all spheres of science and technology. Within these techniques, we can
highlight machine learning, fuzzy logic and neural networks.
Due to the property of adaptability, inherent to any learning process, all of these schemes
have been widely applied to control of robotic manipulator (see e.g. (Ge et al., 1998)), which
are systems subjected to substantial and habitual changes in its dynamics (as commented
before). In flexible robots, because of the undesired vibration in the structure due to
elasticity, this ability becomes even more interesting. For instance, neural networks can be
trained for attaining good responses without having an accurate model or any model at all.
The drawbacks are: the need for being trained might take a considerable amount of time at
the preparation stage; and their inherent nonlinear nature makes this systems quite
demanding computationally. On the other hand, fuzzy logic is an empirical rules method
that uses human experience in the control law. Again, model is not important to fuzzy logic
as much as these rules implemented in the controller, which rely mainly on the experience
of the designer when dealing with a particular system. This means that the controller can
take into account not only numbers but also human knowledge. However, the performance
of the controller depends strongly on the rules introduced, hence needing to take special
care in the design-preparation stage, and the oversight of a certain conduct might lead to an
unexpected behaviour. Some examples of these approaches are described in (Su &
Khorasani, 2001), (Tian et al., 2004) and (Talebi et al., 2009) using neural networks; (Moudgal
et al., 1995), (Green, & Sasiadek, 2002) and (Renno, 2007) using fuzzy logic; or (Caswar &
Unbehauen, 2002) and (Subudhi & Morris, 2009) presenting hybrid neuro-fuzzy proposals.
4.2 Force control
Manipulator robots are designed to help to humans in their daily work, carrying out
repetitive, precise or dangerous tasks. These tasks can be grouped into two categories:
unconstrained tasks, in which the manipulator moves freely, and constrained task, in which the
manipulator interacts with the environment, e.g. cutting, assembly, gripping, polishing or
drilling.
Typically, the control techniques used for unconstrained tasks are focused to the motion
control of the manipulator, in particular, so that the end-effector of the manipulator follows
a planned trajectory. On the other hand, the control techniques used for constrained tasks can
be grouped into two categories: indirect force control and direct force control (Siciliano &
Villani, 1999). In the first case, the contact force control is achieved via motion control,
without feeding back the contact force. In the second case, the contact force control is
achieved thanks to a force feedback control scheme. In the indirect force control the position
error is related to the contact force through a mechanical stiffness or impedance of
adjustable parameters. Two control strategies which belong to this category are: compliance
(or stiffness) control and impedance control. The direct force control can be used when a force
sensor is available and therefore, the force measurements are considered in a closed loop
control law. A control strategy belonging to this category is the hybrid position/force control,
which performs a position control along the unconstrained task directions and a force
control along the constrained task directions. Other strategy used in the direct force control is
the inner/outer motion /force control, in which an outer closed loop force control works on an
inner closed loop motion control.
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There are also other advanced force controls that can work in combination with the previous
techniques mentioned, e.g. adaptative, robust or intelligent control. A wide overview of the
all above force control strategies can be found in the following works: (Whitney, 1987),
(Zeng & Hemami, 1997) and (Siciliano & Villani, 1999). All these force control strategies are
commonly used in rigid industrial manipulators but this kind of robots has some problems
in interaction tasks because their high weight and inertia and their lack of touch senses in
the structure. This becomes complicated any interaction task with any kind of surface
because rigid robots do not absorb a great amount of energy in the impact, being any
interaction between rigid robots and objects or humans quite dangerous.
The force control in flexible robots arises to solve these problems in interaction tasks in
which the rigid robots are not appropriated. A comparative study between rigid and flexible
robots performing constrained tasks in contact with a deformable environment is carried out
in (Latornell et al., 1998). In these cases, a carefully analysis of the contact forces between the
manipulator and the environment must be done. A literature survey of contact dynamics
modelling is shown in (Gilardi & Sharf, 2002).
Some robotic applications demand manipulators with elastic links, like robotic arms
mounted on other vehicles such a wheelchairs for handicapped people; minimally invasive
surgery carried out with thin flexible instruments, and manipulation of fragile objects with
elastic robotic fingers among others. The use of deformable flexible robotic fingers improves
the limited capabilities of robotic rigid fingers, as is shown in survey (Shimoga, 1996). A
review of robotic grasping and contact, for rigid and flexible fingers, can be also found in
(Bicchi & Kumar, 2000).
Flexible robots are able to absorb a great amount of energy in the impact with any kind of
surface, principally, those quite rigid, which can damage the robot, and those tender, like
human parts, which can be damaged easily in an impact with any rigid object. Nevertheless,
despite these favourable characteristics, an important aspect must be considered when a
flexible robot is used: the appearance of vibrations because of the high structural flexibility.
Thus, a greater control effort is required to deal with structural vibrations, which also
requires more complex designs, because of the more complex dynamics models, to achieve a
good control of these robots. Some of the published works on force control for flexible
robots subject, by using different techniques, are, as e.g., (Chiou & Shahinpoor, 1988),
(Yoshikawa et al., 1996), (Yamano et al., 2004) and (Palejiya & Tanner, 2006), where a hybrid
position/force control was performed; in (Chapnik, et al., 1993) an open-loop control system
using 2 frequency-domain techniques was designed; in (Matsuno & Kasai, 1998) and (Morita
et al., 2001) an optimal control was used in experiments; in (Becedas et al., 2008) a force
control based on a flatness technique was proposed; in (Tian et al., 2004) and (Shi & Trabia,
2005) neural networks and fuzzy logic techniques were respectively used; in (Siciliano &
Villani, 2000) and (Vossoughi & Karimzadeh, 2006), the singular perturbation method was
used to control, in both, a two degree-of-freedom planar flexible link manipulator; and
finally in (Garcia et al., 2003 ) a force control is carried out for a robot of three degree-of-
freedom.
Unlike the works before mentioned control, which only analyze the constrained motion of
the robot, there are models and control laws designed to properly work on the force control,
for free and constrained manipulator motions. The pre-impact (free motion) and post-
impact (constrained motion) were analyzed in (Payo et al., 2009), where a modified PID
controller was proposed to work properly for unconstrained and constrained tasks. The
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authors only used measurements of the bending moment at the root of the arm in a closed
loop control law. This same force control technique for flexible robots was also used in
(Becedas et al., 2008) to design a flexible finger gripper, but in this case the implemented
controller was a GPI controller that presents the characteristics described in Section 0
5. Design and implementation of the main control techniques for single-link
flexible manipulators
Control of single link flexible manipulators is the most studied case in the literature (85% of
the published works related to this field (Feliu, 2006)), but even nowadays, new control
approaches are still being applied to this problem. Therefore, the examples presented in this
section implement some recent control approaches of this kind of flexible manipulators.
5.1 Experimental platforms
5.1.1 Single link flexible manipulator with one significant vibration mode
In this case, the flexible arm is driven by a Harmonic Drive mini servo DC motor RH-8D-
6006-E050A-SP(N), supported by a three-legged metallic structure, which has a gear with a
reduction ratio of 1:50. The arm is made of a very lightweight carbon fibre rod and supports
a load (several times the weight of the arm) at the tip. This load slides over an air table,
which provides a friction-free tip planar motion. The load is a disc mass that can freely spin
(thanks to a bearing) without producing a torque at the tip. The sensor system is integrated
by an encoder embedded in the motor and a couple of strain gauges placed on to both sides
of the root of the arm to measure the torque. The physical characteristics of the platform are
specified in Table 1. Equation (5) is used for modelling the link of this flexible manipulator,
in which the value of m1 is equal to MP. For a better understanding of the setup, the
following references can be consulted (Payo et al., 2009) and (Becedas et al., 2009). Fig. 4a
shows a picture of the experimental platform.
5.1.2 Single link flexible manipulator with three significant vibration modes
The setup consists of a DC motor with a reduction gear 1:50 (HFUC-32-50-20H); a slender
arm made of aluminium flexible beam with rectangular section, which is attached to the
motor hub in such way that it rotates only in the horizontal plane, so that the effect of
gravity can be ignored; and a mass at the end of the arm. In addition, two sensors are used:
an encoder is mounted at the joint of the manipulator to measure the motor angle, and a
strain-gauge bridge, placed at the base of the beam to measure the coupling torque. The
physical characteristics of the system are shown in Table 1. The flexible arm is approximated
by a truncated model of Equation (7) with the first three vibration modes to carry out the
simulations (Bellezza et al., 1990). The natural frequencies of the one end clamped link
model obtained from this approximate model, almost exactly reproduce the real frequencies
of the system, which where determined experimentally. More information about this
experimental setup can be found in (Feliu et al., 2006). Fig. 4b shows a picture of the
experimental platform.
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(a) (b)
Fig. 4. Experimental platforms: (a) Single link flexible arm with one significant vibration
mode; (b) Single link flexible arm with three significant vibration modes.
PARAMETER DESCRIPTION
PLATFORM 1
VALUE
PLATFORM 2
VALUE
Data of the flexible link
EI Stiffness 0.37 Nm2 2.40 Nm2
l Length 0.7 m 1.26 m
d Diameter 2.80·10-3 m -
h Width - 5·10-2 m
b Thickness - 2·10-3 m
MP Mass in the tip 0.03 kg 0-0.30 kg
JP Inertia in the tip - 0-5.88·10-4 kgm2
Data of the motor-gear set
J0 Inertia 6.87·10-5 kgm2 3.16·10-4 kgm2
 Viscous friction 1.04·10-3 kgm2s 1.39·10-3 kgm2s
nr Reduction ratio of the motor gear 50 50
Km Motor constant 2.10·10-1 Nm/V 4.74·10-1 Nm/V
usat Saturation voltage of the servo
amplifier
± 10 V ± 3.3 V
Table 1. Physical characteristics of the utilized experimental platforms.
5.2 Actuator position control.
Control scheme shown in Fig. 5 is used to position the joint angle. This controller makes the
system less sensible to unknown bounded disturbances (coup in Equation (1)) and minimizes
the effects of joint frictions (see, for instance (Feliu et al., 1993)). Thus, the joint angle can be
controlled without considering the link dynamics by using a PD, PID or a Generalized
Proportional Integral (GPI) controller, generically denoted as Ca(s). In addition, this
controller, as we will show bellow, can be combined with other control techniques, such as
command generation, passivity based control, adaptive control or force control.
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Fig. 5. Schematic of the inner control loop formed by a position control of m plus the
decoupling term coup/nrKm.
5.3 Command generation
The implementation of the IS technique as an example of command generation is described
herein. It is usually accompanied by the feedback controller like the one shows in Fig. 5.
Thus, the general control scheme showed in Fig. 6 is used, which has previously utilized
with success for example in (Feliu & Rattan, 1999) or (Mohamed et al., 2005). The actuator
controller is decided to be a PD with the following control law:
          *
coup r m p m m v mu t t n K K t t K t        , (11)
where coup/nrKm (decoupling term) makes the design of the PD constants (Kp, Kv)
independent of the link dynamics. Thus, if the tuning of the parameters of the PD controller
(Kp, Kv) is carried out to achieve a critically damped second-order system, the dynamics of
the inner control loop (Gm(s)) can be approximated by
         2* *
1m m m ms G s s s s      , (12)
where  is the constant time of Gm(s). From Equations (11) and (12) the values of Kp and Kv
are obtained as
 2
0 0, 2p r m v r mK J n K K n J K     . (13)
As it was commented in Section 0, the IS (C(s)) can be a robust, learning or adaptive input
shaper. In this section, a robust input shaper (RIS) for each vibration mode obtained by the
so-called derivative method (Vaughan et al., 2008) is implemented. This multi-mode RIS is
obtained as follows:
        1 1
1 1
i
i
N N p
sd
i i i
i i
C s C s z e z
 
     , (14)
in which
 2
1 2
, 1i i
i i i iz e d 
  
   , (15)
pi is a positive integer used to increase the robustness of each Ci(s) and i and i denote the
natural frequencies and damping ratio of each considered vibration mode.
1/nrKm


u Flexible
Robot
coup
Gm(s)

*m
mCa(s)
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Fig. 6. General control scheme of the RIS implementation.
This example illustrates the design for the experimental platform of Fig. 4b of the multi-
mode RIS of Equation (14) for a payload range MP[0.02, 0.12]kg and JP[0.0, 5.88·10-4]kgm2.
Each of one Ci(s) is designed for the centre of three first frequency intervals, which has the
next values: 1=5.16 2=35.34 and 3=100.59rad/s. If the damping is neglected (1, 2 and 3
equal to zero), the parameters of C(s) are z1=z2=z3=1, d1=0.61, d2=0.089 and d3=0.031s. In
addition, if the maximum residual vibration is kept under 5% for all vibration modes, the
value of each pi is: p1=3, p2=2 and p3=2. The dynamics of Gm(s) is designed for =0.01. Then
from Table 1 and Equations (12) and (13), the values of Kp and Kv were 350.9 and 6.9. This
value of  makes the transfer function Gm(s) robust to Coulomb friction and does not
saturate the DC motor if the motor angle reference is ramp a reference with slope and final
value equal to 2 and 0.2rad, respectively. Fig. 7 shows the experimental results for the multi-
mode RIS design above. The residual vibration for the nominal payload (Mp=0.07 kg and
Jp=310-4 kgm2) is approximately zero whereas one of the payload limits (Mp = 0.12 kg and Jp
= 5.8810-4 kgm2) has a residual vibration less than 5%.
0 1 2 3 4 5 6 7
0
0.1
0.2
0.3
0.4
Time (s)
Tipangleandreference(rad)
0 1 2 3 4 5 6 7
0
0.1
0.2
0.3
0.4
Time (s)
Tipangleandreference(rad)
(a) Mp = 0.07 kg and Jp = 310-4 kgm2 (b) Mp = 0.12 kg and Jp = 5.8810-4 kgm2
Fig. 7. Experimental results for the multi-mode RIS. (…) References, (---) without RIS and (−)
with RIS.
5.4 Classic control techniques
This subsection implements the new passivity methodology expounded in (Pereira et al.,
2007) in the experimental platform of Fig. 4b, whose general control scheme is shown in Fig.
8. This control uses two control loops. The first one consists of the actuator control shown in
Section 5.2, which allows us to employ an integral action or a high proportional gain. Thus,
the system is robust to joint frictions. The outer controller is based on the passivity property
of coup(s)/sm(s), which is independent of the link and payload parameters. Thus, if
sC(s)Gm(s) is passive, the controller system is stable. The used outer controller is as
following:
   1 ,cC s K s s  (16)
in which the parameter Kc imparts damping to the controlled system and  must be chosen
together with Gm(s) to guarantee the stability. For example, if Gm(s) is equal to Equation (12),
t(s)
Gm(s)
*m(s)
G(s)C(s)
m(s)*t(s)
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the necessary and sufficient stability condition is 0</2< (see (Pereira et al., 2007) for more
details).
Fig. 8. General control scheme proposed in (Pereira, et al., 2007).
Fig. 9. Tip angle t: ( ). Simulation with MP = 0; ( ) Experiment with MP = 0; ( )
Simulation with MP = 0.3; ( ) Experiment with MP = 0.3; ( ) the reference.
Taking into account the maximum motor torque (i.e., usat in Table 1), the constant time of the
inner loop is set to be  = 0.02. Then, the parameters of the PD controller are obtained: Kp =
83.72 and Kv = 3.35. Next, the nominal condition is taken for MP = 0 and C(s) is designed
( = 0.05 and Kc = 1.8) in such a way that the poles corresponding to the first vibration mode
are placed at 3.8. Notice that  fulfils the condition 0</2< and is independent of the
payload. Once the parameters of the control scheme are set, we carry out simulations and
experiments for MP = 0 and MP = 0.3 kg (approximately the weight of the beam) and
Jp  0 kgm2). Figure 9 shows the tip angle, in which can be seen that the response for the two
mass values without changing the control parameters is acceptable for both simulations and
experiments. Notice that the experimental tip position response is estimated by a fully
observer since it is not measured directly, which is not used for control purpose. Finally, a
steady state error in the vicinity of 1% compared with the reference command arises for in
the tip and motor angle for experimental results. This error is due to Coulomb friction and
can be minimized using a PD with higher gains in the actuator control.
5.5 Adaptive control
Adaptive controller described in this section is based on the flatness characteristic of a
flexible robotic system (see (Becedas, et al., 2009)). The control system is based on two
C(s)


*t
Gm(s)
dynamics
m
link
Gc(s)
Gt(s)
coup
t
*m
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nested loops with two controllers designed for both motor and flexible link dynamics. The
controller is called Generalized Proportional Integral (GPI). This presents robustness with
respect to constant perturbations and does not require computation of derivatives of the
system output signals. Therefore, the output signals are directly feedbacked in the control
loops, then the usual delays produced by the computation of derivatives and the high
computational costs that require the use of observers do not appear. In addition, due to the
fact that one of the most changeable parameter in robotics is the payload, a fast algebraic
continuous time estimator (see (Fliess & Sira-Ramírez, 2003)) is designed to on-line estimate
the natural frequency of vibration in real time. The estimator calculates the real value of the
natural frequency when the payload changes and updates the gains of the controllers.
Therefore, this control scheme is an Indirect Adaptive Control. A scheme of the adaptive
control system is depicted in Fig. 10, where 1e represents the estimation of the vibration
natural frequency of the flexible arm, used to update the system controller parameters.
Fig. 10. Two-stage adaptive GPI control implemented in (Becedas, et al., 2009).
The system dynamics is obtained by the simplification to one vibration mode of the
concentrated mass model (see Section 0). Adding the decoupling term defined in Section 5.2
to the voltage control signal uc allows us to decouple both motor and link dynamics. Thus,
the design of the controllers, one for each dynamics, is widely simplified. By using the
flatness characteristic of the system, the two nested GPI controllers are designed as follows:
Outer control law (Co(s)):
   * *
1 0 2( ) ( )m m t ts s             , (17)
where *
m is now an auxiliary ideal open loop control for the outer loop, *
t represents the
reference trajectory for the payload, and i, i=0, 1, 2, are the outer loop controller gains,
which are updated each time that the estimator estimates the real values of the system
natural frequency.
Inner control law (Ca(s)):
   * 2 *
2 1 0 3( ) ( ) ( )c c mr mu u s s s s             , (18)
1/nrKm


u Flexible
Robot
*t
mCa(s)
1,e Estimator
t
u*
c
*mr


*
m
Co(s)

S1
Inner control loop
Outer control loop
uc coup
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where u*
c represents the ideal open loop control for the inner loop, *
mr represents the
reference trajectory for the motor angle, and i, i=0, 1, 2, 3 are the inner loop controller gains.
The algebraic estimator for the natural frequency is given by the following equation:
 
 
2
1
, 0,
( ) ( ), ,e
e e
arbitrary t
n t d t t

 

  
, (19)
where
2
1
1 2
2
( ) ( )
4 ( )
2 ( )
e t
t
t
n t t t z
z z t t
z t



 
 



3 4
2
4
( )
( ( ) ( ))
e
m t
d t z
z z
z t t t 


 


. (20)
Then, this control technique is implemented in the experimental platform of Figure 4a. The
value of the tip angle is approximated by t=m-L/(3EI)coup, where m and coup are obtained
from the encoder and strain gauges measurements respectively. The desired reference to be
tracked by the flexible robotic system is a two seconds Bezier eighth order trajectory with
1rad of amplitude. The control system starts working with an arbitrary computation of the
tip mass, which is represented by a natural frequency 0i=9rad/s, very different from the
real value 1e=15.2rad/s. In a small time interval =0.5s (dashed line), the algebraic fast
estimator estimates the real value 1e, and updates the inner (u*
c) and outer (*
m, 2, 1 and
0) loop controllers (see details in (Becedas et al., 2009)). After the updating the control
system perfectly tracks the desired trajectory (see Fig. 11).
0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
Time (s)
Angle(rad)
t
*
t
Fig. 11. Trajectory tracking of the reference trajectory with the GPI adaptive controller.
5.6 Force control
The special feature of the force control for flexible robots described here (Payo et al., 2009) is
that the control law is designed to control the force exerted by a robot on the environment,
which surrounds for both free and constrained motion tasks. The flexible robot of one
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degree of freedom used is described in Section 0. The system dynamics of the arm is
obtained by the simplification to one vibration mode of the concentrated mass model (see
Section 0, specifically Equation (5)). The tracking of the desired force is obtained by using a
feedback control loop of the torque at the root of the arm. This control law is based on a
modified PID controller (I-PD controller (Ogata, 1998)), and it is demonstrated the
effectiveness of the proposed controller for both free and constrained motion tasks. The
sensor system used in this control law is constituted by a sole sensor very lightweight (two
strain gauges placed at the root of the arm) to measure the torque, neither the contact force
sensor nor the angular position sensor of the motor are used in the control method, unlike
others methods described in Section 4.2. The controlled system presents robust stability
conditions to changes in the tip mass, viscous friction and environment elasticity. It is also
important to mention the good performance of the system response in spite of the nonlinear
Coulomb friction term of the motor which was considered to be a perturbation. Fig. 12
shows the control scheme used to implement this force control technique, where the control
law is given by the following equation:
 0 1 2
0
coupd
coup coup coup
d
u a dt a a
dt
 
       , (21)
where a0, a1 and a2 are the design parameters of the I-PD and d
coup is the reference signal.
The environment impedance is represented by the well known spring-dashpot model
(Latornell et al., 1998) and (Erickson et al., 2003):
n e e e eF k x b x   , (22)
where ke, be are the stiffness and damping characteristics of the environment and xe is the
local deformation of the environment. The plant dynamics for free and constrained motion
tasks are given respectively by the following equiations:
 
     
0
2 2 2
0 0 0 0
/
/ / /
coup c r
r
s sK J n
U s s s s J c J n s J  


   
, (23)
 
 
  
    
2
0
2 2 2 2 2
0 0 0
/ / /
/ / / / / /
c r e ecoup
e e r e e
K J n s sb m k ms
U s s s J s sb m k m c J n s sb m k m 
 

      
. (24)
Fig. 12. Force control scheme.
u Flexible
Robot
coup
a0+a1s+a2s2
s
Environment
–
Collision detection algorithm
a0
s
No
Yes
coup (Free motion)d
coup (Constrained motion)d
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The proposed strategy needs an online collision detection mechanism in order to switch
between a command trajectory for free motion torque and a contact torque reference for the
case of constrained motion. The collision was detected when the torque exceeded a
threshold () that depends on the amplitude of the reference signal, the Coulomb friction of
the motor (C) and the noise in the measured signal (3) according to the following equation
(a detailed explication of this can be found in (Payo, et al., 2009)):
1 2 3coup f        , (25)
where 1 and 2 are normalized maximum deviations of the measured signal.
Fig. 13 and Fig. 14 show the results obtained in two experimental tests where the robot
carried out both free and constrained motion tasks. The controlled torque is displayed
before and after collision. A small value for the torque in free motion was used to prevent
possible damages to the arm or to the object at the moment of collision. The chosen torque in
these tests for free motion was equal to 0.07Nm. The constrained environment used in these
tests was a rigid object with high impedance. Once the collision was detected, the Control
law changed the reference value of the torque for constrained motion depending on the
particular task carried out. For example, the first experiment matches a case in which the
force exerted on the object was increased; and in the second experiment the force exerted on
the object was decreased to avoid possible damages on the contact surfaces (case of fragile
objects, for instance).
Fig. 13. System response for first experiment.
Fig. 14. System response for second experiment.
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6. Future of flexible manipulators
After the huge amount of literature published on this topic during the last thirty years,
flexible robotics is a deeply studied field of autonomous systems. Even complete books
have been already devoted to the subject (Tokhi & Azad, 2008) and (Wang & Gao, 2003).
Still, new control techniques can be studied due to simplicity of the physical platform,
but, as discussed in (Benosman & Vey, 2004), most of the topics regarding modelling or
controllability have been satisfactorily addressed in the previous literature.
However, some topics are still open and leave a considerable margin for improvement.
Some manipulators with a small rigid arm attached to a large flexible base (called macro-
micro manipulators, see (George & Book, 2003) for instance) have been developed for
precision tasks, but the technological issue of building flexible robots with similar features
to those of actual industrial robots has not been completely solved. While there exists a
real prototype of a 3 dof flexible robot (Somolinos et al., 2002) achieving three
dimensional positioning of the tip, a mechanical wrist still needs to be coupled for giving
the manipulator the ability of reaching a particular position with a particular orientation.
On the control side, the search for the perfect controller is still open and, probably, never
to be closed. All the robust, adaptive, intelligent techniques have their limitations and
drawbacks. Many new controllers have been proposed but there is no standard
measurement of the performance and, hence, no objective classification can be performed.
The creation of a family of ‘benchmark’ problems would provide some objectivity to the
results analysis.
One of the most potential aspects of flexible robots is their recently evolution in the
position and force control. Such a combination provides of touch sensibility to the robotic
system. Thus, the robot does not only have accuracy in the different positioning tasks, but
also has the possibility of detecting whatever interaction with the environment that
surrounds it. This characteristic allows the system to detect any collision with an object or
surface, and to limit the actuating force in order not to damage the robotic arm nor the
impact object or surface. Applications in this sense can be developed for robots involved
in grasping, polishing, surface and shape recognition, and many other tasks (Becedas et
al., 2008).
Nonlinear behaviour of flexible manipulators has been poorly accounted for in literature.
A few works dealing with modelling of geometrical nonlinearities due to large
displacements in the links have been published in (Payo et al., 2005) and (Lee, 2005) and a
solution for achieving precise point-to-point motion of these systems has also been
reported in (O’Connor et al., 2009). But these works are based on single link manipulators,
and the multiple link case still has to be addressed. If we think of applications in which
the robot is interacting with humans, these large displacements structures increase the
safety of the subjects because the system is able to both absorb a great amount of energy
in the impact and control effectively the contact force almost instantaneously (hybrid
position/force controls). Thus, the development of human-machine interfaces becomes a
potential application field for this kind of systems (Zinn, 2004).
Another interesting and not very studied approach to the flexibility of manipulators
consists of taking advantage of it for specific purposes. Flexibility is considered as a
potential benefit instead of a disadvantage, showing some examples with margin of
improvement in assembling (Whitney, 1982), collision (García et al., 2003), sensors (Ueno
et al., 1998) or mobile robots (Kitagawa et al., 2002).
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7. References
Aspinwall, D. M. (1980). Acceleration profiles for minimizing measurement machines.
ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 102 (March of
1980), pp. 3-6.
Åström, K. J. & Wittenmark, B. (1995). Adaptive control, Prentice Hall (2nd Edition), ISBN:
0201558661.
Bai, M.; Zhou, D. & Fu, H. (1998). Adaptive augmented state feedback control. IEEE
Transactions on Robotics and Automation, Vol. 14, No. 6 pp. 940-950.
Balas, M. J. (1978). Active control of flexible systems. Journal of Optimisation Theory and
Applications, Vol. 25, No. 3, pp. 415–436.
Balas, M. J. (1982). Trends in large space structures control theory: Fondest hopes, wildest
dreams. IEEE Transactions on Automatic Control, Vol. 27, No. 3, pp. 522-535.
Banavar, R. N. & Dominic, P. (1995). An LQG/H∞ Controller for a Flexible Manipulator.
IEEE Transactions on Control Systems Technology, Vol. 3, No. 4, pp. 409-416.
Bayo, E. (1987). A finite-element approach to control the end-point motion of a single-link
flexible robot. Journal of Robotics Systems, Vol. 4, No. 1, pp. 63–75.
Becedas, J.; Payo, I.; Feliu, V. & Sira-Ramírez, H. (2008). Generalized Proportional Integral
Control for a Robot with Flexible Finger Gripper, Proceedings of the 17th IFAC
World Congress, pp. 6769-6775, Seoul (Korea).
Becedas, J.; Trapero, J. R.; Feliu, V. & Sira-Ramírez, H. (2009). Adaptive controller for
single-link flexible manipulators based on algebraic identification and
generalized proportional integral control. IEEE Transactions on Systems, Man and
Cybernetics, Vol. 39, No. 3, pp. 735-751.
Belleza, F.; Lanari, L. & Ulivi, G. (1990). Exact modeling of the flexible slewing link,
Proceedings of the IEEE International Conference on Robotics and Automation, pp. 734-
804.
Benosman, M. & Vey, G. (2004). Control of flexible manipulators: A survey. Robotica, Vol.
22, pp. 533–545.
Bicchi, A & Kumar, V. (2000). Robotic grasping and contact: a review, Proceedings of the
IEEE International Conference on Robotics and Automation, No. 1, pp. 348–353.
Bodson, M. (1998). An adaptive algorithm for the tuning of two input shaping methods.
Automatica, Vol. 34, No. 6, pp. 771-776.
Book, W. J. (1974). Modeling, design and control of flexible manipulator arms. Ph. D. Thesis,
Department of Mechanical Engineering, Massachusetts Institute of Technology,
Cambridge MA.
Book, W. J.; Maizza-Neto, O. & Whitney, D.E. (1975). Feedback control of two beam, two
joint systems with distributed flexibility. Journal of Dynamic Systems, Measurement
and Control, Transactions of the ASME, Vol. 97G, No. 4, pp. 424-431.
Book, W. J. & Majette, M. (1983). Controller design for flexible, distributed parameter
mechanical arms via combined state space and frequency domain techniques.
Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME,
Vol. 105, No. 4, pp. 245-254.
Book, W. J. (1984). Recursive lagrangian dynamics of flexible manipulator arms.
International Journal of Robotics Research, Vol. 3, No. 3, pp. 87-101.
Book, W. J. (1993). Controlled motion in an elastic world. Journal of Dynamic Systems,
Measurement and Control, Transactions of the ASME, Vol. 115, No. 2, pp. 252-261.
www.intechopen.com

Bridges, M. M.; Dawson, D. M. & Abdallah, C. T. (1995). Control of rigid-link flexible joint
robots: a survey of backstepping approaches. Journal of Robotic Systems, Vol. 12,
No. 3, pp. 199-216.
Cannon, R. H. & Schmitz, E. (1984). Initial experiments on the end-point control of a
flexible robot. International Journal on Robotics Research, Vol. 3, No. 3, pp. 62–75.
Caswar, F. M. & Unbehauen, H. (2002). A neurofuzzy approach to the control of a flexible
manipulator. IEEE Transaction on Robotics and Automation, Vol. 18, No. 6, pp. 932-
944.
Chapnik, B. V.; Heppler, G.R. & Aplevich, J.D. (1993). Controlling the impact response of
a one-link flexible robotic arm. IEEE Transactions on Robotics and Automation, Vol.
9, No. 3, pp. 346–351.
Chen, Y. P. & Hsu, H. T. (2001). Regulation and vibration control of an FEM-based single-
link flexible arm using sliding mode theory. Journal Vibration and Control, Vol. 7,
No. 5, pp. 741-752.
Chiou, B. C. & Shahinpoor, M. (1988). Dynamics stability analysis of a one-link force-
controlled flexible manipulator. Journal of Robotic Systems, Vol. 5, No. 5, pp. 443–
451.
Choi S. B.; Cheong C. C. & Shin H. C. (1995). Sliding mode control of vibration in a single-
link flexible arm with parameter variations, Journal Vibration and Control, Vol. 179,
No. 5, pp. 737-748.
Christoforou E. G. & Damaren C. J. (2000). The control of flexible-link robots manipulating
large payloads: Theory and Experiments. Journal of Robotic Systems, Vol. 17, No. 5,
pp. 255-271.
Damaren C. J. (1996). Adaptive control of flexible manipulators carrying large uncertain
payloads. Journal of Robotic Systems, Vol. 13, No. 4, pp. 219-228.
De Luca, A. & Siciliano, B. (1989). Trajectory control of a non-linear one-link flexible arm.
International Journal of Control, Vol. 50, No. 5, pp. 1699-1715.
De Luca, A. & Siciliano, B. (1991). Closed form dynamic model of planar multilink
lightweight robots. IEEE Transactions Systems, Man and Cibernetics, Vol. 21, No. 4,
pp. 826-839.
De Luca, A. & Siciliano, B. (1993). Regulation of flexible arms under gravity. IEEE
Transasctions on Robotics and Automation, Vol. 9, No. 4, pp. 463-467.
Denavit, J. & Hartenberg, R. S. (1955). A kinematic notation for lower-pair mechanisms
based on matrices. ASME Journal of Applied Mechanics, (June), pp. 215-221.
Dwivedy, S. K. & Eberhard, P (2006). Dynamic analysis of flexible manipulators, a
literature review. Mechanism and Machine Theory, Vol. 41, No. 7, pp. 749–777.
Erickson, D.; Weber, M. & Sharf, I. (2003). Contact stiffness and damping estimation for
robotic systems. International Journal of Robotic Research, Vol. 22, No. 1, pp. 41–57.
Feliu, V.; Rattan K. S. & Brown, H. B. (1990). Adaptive control of a single-link flexible
manipulator. IEEE Control Systems Magazine, Vol. 10, No. 2, pp. 29-33.
Feliu, V.; Rattan, K. & Brown, H. (1992). Modeling and control of single-link flexible arms
with lumped masses. Journal of Dynamic Systems, Measurement and Control, Vol.
114, No. 7, pp. 59–69.
Feliu V.; Rattan, K. & Brown, H. (1993). Control of flexible arms with friction in the joint.
IEEE Transactions on Robotics and Automation. Vol. 9, No. 4, pp. 467-475.
www.intechopen.com

Feliu V. & Rattan K. S. (1999). Feedforward control of single-link flexible manipulators by
discrete model inversion. Journal of Dynamic Systems, Measurement, and Control,
Vol. 121, pp. 713–721.
Feliu, J. J.; Feliu, V. & Cerrada C. (1999). Load adaptive control of single-link flexible arms
base don a new modeling technique. IEEE Transactions on Robotics and Automation,
Vol. 15, No. 5, pp. 793-804.
Feliu, V. (2006). Robots flexibles: Hacia una generación de robots con nuevas prestaciones.
Revista Iberoamericana de Automática e Informática Industrial, Vol. 3, No. 3, pp. 24-
41.
Feliu, V.; Pereira, E.; Díaz, I. M. & Roncero, P. (2006). Feedforward control of multimode
single-link flexible manipulators based on an optimal mechanical design. Robotics
and Autonomous Systems, Vol. 54, No. 8, pp. 651-666.
Fliess, M.; Lévine, J.; Martin, P. & Rouchon, P. (1999). A Lie-Bäcklund approach to
equivalence and flatness of nonlinear systems. IEEE Transactions on Automatic
Control, Vol. 44, No. 5, pp. 922-937.
Fliess, M. & Sira-Ramírez, H. (2003). An algebraic framework for linear identification.
ESAIM - Control, Optimisation and Calculus of Variations, No. 9, pp. 151-168.
Fu, K. S. (1970). Learning control systems-Review and Outlook. IEEE Transactions on
Automatic Control, Vol. AC-15, pp. 210-221.
García, A.; Feliu, V. & Somolinos, J. A. (2003). Experimental testing of a gauge based
collision detection mechanism for a new three-degree-of-freedom flexible robot.
Journal of Robotic Systems, Vol. 20, No. 6, pp. 271-284.
Gervarter, W. (1970). Basic relations for controls of flexible vehicles. AIAA Journal, Vol. 8,
No. 4, pp. 666-672.
Ge, S. S.; Lee, T. H. & Harris, C. J. (1999). Adaptive neural network control of robotic
manipulators. World Scientific, 981023452X.
George, L. E. & Book, W. J. (2003). Inertial vibration damping control of a flexible base
manipulator. IEEE/ASME Transactions on Mechatronics, Vol. 8, No. 2, pp. 268-271.
Gilardi, G. & Sharf, I. (2002). Literature survey of contact dynamics modelling. Mechanism
and Machine Theory, 2002, Vol. 37, No. 10, pp. 1213–1239.
Green, A. & Sasiadek, J. Z. (2002). Inverse dynamics and fuzzy repetitive learning flexible
robot control, 15th Trienial World Congress, IFAC, Barcelona (Spain).
Gutman, S. (1999). Uncertain Dynamical Systems-A Lyapunov Min-Max Approach. IEEE
Transactions on Automatic Control, Vol. 24, No. 3, pp. 437-443.
Huey, J. R.; Sorensen, K. L. & Singhose W. E. (2008). Useful applications of closed-loop
signal shaping controllers. Control Engineering Practice, Vol. 16, No. 7, pp. 836-846.
Jiang, Z. H. (2004). End-Effector Robust Trajectory Tracking Control for Flexible Robot
Manipulators. IEEE International Conference on Systems, Man and Cybernetics, Vol.
5, pp. 4394-4399.
Jordan, S. (2002). Eliminating vibration in the nano-world. Photonics Spectra, Vol. 36, pp.
60-62.
Kitagawa, H.; Beppu, T.; Kobayashi, T. & Terashima, K. (2002). Motion control of
omnidirectional wheelchair considering patient comfort, Proceedings of the 2002
IFAC World Congress, Barcelona, (Spain).
www.intechopen.com

Koivo, A. J. & Lee K. S. (1989). Self-tuning control of planar two-link manipulator with
non-rigid arm, Proceedings of the IEEE International Conference on Robotics and
Automation, pp. 1030-1035.
Korolov, V. V. & Chen, Y.H. (1989). Controller design robust to frequency variation in a
one-link flexible robot arm. ASME Journal of Dynamic Systems, Measurement, and
Control, Vol. 111, No. 1, pp. 9–14.
Landau, I. D.; Daniel Rey, J. L. & Barnier J. (1996). Robust Control of a 360o Flexible Arm
Using the Combined Pole Placemen/Sensitivity Function Shaping Method. IEEE
Transactions on Systems Technology, Vol. 4, No. 4, pp. 369-383.
Latornell, D. J.; Cherchas D. B. & Wong R. (1998). Dynamic characteristics of constrained
manipulators for contact force control design. International Journal of Robotics
Research, Vol. 17, No. 3, pp. 211-231.
Lee, H. H. (2005). New Dynamic Modeling of Flexible-Link Robots. Journal of Dynamic
Systems Measurement and Control-Transactions of the ASME, Vol. 127, No. 2, pp.
307-309.
Liu, L. Y. & Yuan, K. (2003). Noncollocated passivity-based PD control of a single-link
flexible manipulator. Robotica, Vol. 21, No. 2, pp. 117-135.
Lizarraga I. & Etxebarria V. (2003). Combined PD-H∞ approach to control of flexible
manipulators using only directly measurable variables. Cybernetics and Systems,
Vol. 34, No. 1, pp. 19-32.
Luo, Z. H. (1993). Direct strain feedback control of flexible robot arms: new theoretical and
experimental results. IEEE Transactions on Automatic Control, Vol. 38, No. 11, pp.
1610-1622.
Maizza-Neto, O. (1974). Modal analysis and control of flexible manipulator arms. Ph. D. Thesis,
Department of Mechanical Engineering, Massachusetts Institute of Technology,
Cambridge MA.
Matsuno, F. & Kasai, S. (1998). Modeling and robust force control of constrained one-link
flexible arms. Journal of Robotic Systems, Vol. 15, No. 8, pp. 447–464.
Meirovitch, L. (1996). Principles and techniques of vibration. Prentice Hall, 0023801417
Englewood Cliffs, New Jersey.
Moallem, M.; Khorasani, K. & Patel, R.V. (1998). Inversion-based sliding control of a
flexible-link manipulator. International Journal of Control, Vol. 71, No. 3, pp. 477-
490.
Mohamed, Z.; Martins, J. M.; Tokhi, M. O.; Sà da Costa, J. & Botto, M.A. (2005). Vibration
control of a very flexible manipulator system. Control Engineering Practice, Vol. 13,
No. 3, pp. 267-277.
Morita, Y.; Ukai, H. & Kando, H. (1997). Robust Trajectory Tracking Control of Elastic
Robot Manipulators. Transactions on ASME, Journal of Dynamic Systems,
Measurement and Control, Vol. 119, No. 4 pp. 727-735.
Morita, Y.; Kobayashi, Y.; Kando, H.; Matsuno, F.; Kanzawa, T. & Ukai, H. (2001). Robust
force control of a flexible arm with a nonsymmetric rigid tip body. Journal of
Robotic Systems, Vol. 18, No. 5 pp. 221–235.
Moser A. N. (1993). Designing controllers for flexible structures with H-nfinity/μ-
synthesis. IEEE Control Systems Magazine, Vol. 13, No. 2, pp. 79-89.
Moudgal, V. G.; Kwong, W. A. & Passino, K. M. (1995). Fuzzy learning control for a
flexible-link robot, IEEE Transactions of fuzzy systems, Vol. 3, No. 2, pp. 199-210.
www.intechopen.com

O’Connor, W. J. (2007). Wave-based analysis and control of lump-modeled flexible robots.
IEEE Transactions on Robotics, Vol. 23, No. 2, pp. 342-352.
O’Connor, W. J.; Ramos, F.; McKeown, D. & Feliu, V. (2007). Wave-based control of non-
linear flexible mechanical systems. Nonlinear Dynamics, Vol. 57, No. 1-2, pp. 113-
123.
Ogata, K. (2001). Modern control engineering. Prentice Hall, 0130609072.
Onsay, T. & Akay, A. (1991). Vibration reduction of a flexible arm by time-optimal open-
loop control. Journal of Sound and Vibration, Vol. 147, No. 2, pp. 283–300.
Ozgoli, S. & Taghirad, H. D. (2006). A survey of the control of flexible joint robots. Asian
Journal of Control, Vol. 8, No. 4, pp. 332-334.
Palejiya, D. & Tanner, H. (2006). Hybrid velocity/force control for robot navigation in
compliant unknown environments. Robotica. Vol. 24, No. 6, pp.745–758.
Park, J. & Chang, P. H. (2001). Learning input shaping technique for non-LTI systems.
ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 123, No. 2, pp.
288-293.
Park, J.; Chang, P. H.; Park, H.S. & Lee, E. (2006). Design of learning input shaping
technique for residual vibration suppression in an industrial robot. IEEE/ASME
Transactions on Mechatronics, Vol. 11, No. 1, pp. 55-65.
Payo, I.; Ramos, F.; Feliu, V. & Cortázar, O. D. (2005). Experimental validation of
nonlinear dynamic models for single-link very flexible arms, Proceedings of the
44th IEEE Conference on Decision and Control and European Control Conference,
December 2005, Seville, (Spain).
Payo, I.; Feliu, V. & Cortazar, O. D. (2009). Force control of a very lightweight single-link
flexible arm based on coupling torque feedback. Mechatronics, Vol. 19, No. 3, pp.
334-347.
Pedersen, N. L. & Pedersen, M. L. (1998). A direct derivation of the equations of motion
for 3-D flexible mechanical systems. International Journal of Numerical Methods in
Engineering, Vol. 41, No. 4, pp. 697-719.
Pereira, E.; Díaz, I. M.; Cela, J. J. & Feliu V. (2007). A new methodology for passivity based
control of single-link flexible manipulator, IEEE/ASME International Conference on
Advanced Intelligent Mechatronics, 978-1-4244-1263-1, Zurich (Switzerland).
Ramos, R. & Feliu, V. (2008). New online payload identification for flexible robots.
Application to adaptive control. Journal of Sound and Vibration, Vol. 315, No. 1-2,
pp. 34-57.
Renno, J. M. (2007). Inverse dynamics based tuning of a fuzzy logic controller for a single-
link flexible manipulator. Journal of Vibration and Control, Vol. 13, No. 12, pp.
1741-1759.
Rovner, D. M. & Cannon, R. H. (1987). Experiments toward on-line identification and
control of a very flexible one-link manipulator. The International Journal of Robotics
Research, Vol. 6, No. 4, pp. 3-19.
Sanz, A. & Etxebarria V. (2005). Composite Robust Control of a Laboratory Flexible
Manipulator. Proceedings of the 44th IEEE Conference on Decision and Control, pp.
3614-3619.
Schwertassek, R.; Wallrapp O. & Shabana A. (1999). Flexible multibody simulation and
choice of the shape functions. Nonlinear dynamics, Vol. 20, No. 4, pp. 361-380.
www.intechopen.com

Shimoga, K. B. (1996). Robot grasp synthesis algorithms: a survey. International Journal of
Robotic Research, Vol. 15, No. 3, pp. 230–266.
Shi, L. & Trabia, M. (2005). Comparison of distributed PD-like and importance based
fuzzy logic controllers for a two-link rigid-flexible manipulator. Journal of
Vibration and Control, Vol. 11, No. 6, pp. 723–747.
Siciliano B.; Yuan B. S. & Book W. J. (1986). Model reference adaptive control of a one link
flexible arm. Proceedings of 25th Conference on Decision and Control, 3, pp. 91-95,
Athens, (Greece).
Siciliano, B. & Villani, L. (1999). Robot force control, Springer, ISBN: 0792377338.
Siciliano, B. & Villani, L. (2000). Parallel force and position control of flexible
manipulators. IEE Proceedings: Control Theory Applications, Vol. 147, No. 6, pp.
605–612.
Singer, N. C. & Seering, W. C. (1990). Preshaping command inputs to reduce system
vibration. ASME Journal of Dynamic System, Measurement and Control, Vol. 112, No.
1, pp. 76-82.
Singh, S. & Schy, A. (1985). Robust torque control of an elastic robotic arm based on
invertibility and feedback stabilization. Proceedings of the 24th IEEE Conference on
Decision and Control, pp. 1317-1322.
Spong, M. W. (1987). Modeling and control of elastic joints robots, ASME Journal of
Dynamic Systems, Measurement, and Control, Vol. 109, No. 6, pp. 310-319.
Smith, O. J. M. (1958). Feedback Control Systems. McGraw-Hill Book Co., Inc. New York.
Somolinos, J. A.; Feliu, V. & Sánchez, L. (2002). Design, dynamic modelling and
experimental validation of a new three-degree-of-freedom flexible arm.
Mechatronics, Vol. 12, No. 7, pp. 919-948.
Subudhi, B. & Morris, A. S. (2009). Soft computing methods applied to the control of a
flexible robot manipulator. Applied soft computing journal, Vol. 9, No. 1, pp. 149-
158.
Su, Z. & Khorasani, K. (2001). A neural-network-based controller for a single-link flexible
manipulator using the inverse dynamics approach. IEEE Transactions on Industrial
Electronic, Vol. 48, No. 6, pp. 1074-1086.
Talebi, H. A.; Khorsani, K. & Patel, R.V. (2009). Control of flexible-link manipulators using
neural networks, Springer, 978-1-85233-409-3.
Theodore, R. J. & Ghosal A. (2003). Robust control of multilink flexible manipulators.
Mechanism and Machine Theory, Vol. 38, No. 4, pp. 367–377.
Thomas S. & Mija S. J. (2008). A practically implementable discrete time sliding mode
controller for flexible manipulator. International Journal of Robotics and Automation,
Vol. 23, No. 4, pp. 235-241.
Tian, L.; Wang, J. & Mao, Z. (2004). Constrained motion control of flexible robot
manipulators based on recurrent neural networks, IEEE Transactions on Systems,
Man, and Cybernetic, part B:, Vol. 34, No. 3, pp. 1541-1551.
Tokhi, M. O. & Azad, A. K. M. (2008). Flexible robot manipulators: modelling, simulation and
control, Springer-Verlag, 0863414486, London.
Tuttle, T. & Seering, W. (1997). Experimental verification of vibration reduction in flexible
spacecraft using input shaping. Journal of Guidance, Control, and Dynamics, Vol. 20,
No. 4, pp. 658-664.
www.intechopen.com

Tsypkin, Y. (1968). Self-learning: What is it? IEEE Transactions on Automatic Control, Vol.
AC-13, pp. 608-612.
Ueno, N.; Svinin, M. M. & Kaneko M. (1998). Dynamic contact sensing by flexible beam.
IEEE/ASME Transactions on Mechatronics, Vol. 3, No. 4, pp. 254-264.
Vaughan, J.; Yano A. & Singhose W. (2008). Comparison of robust input shapers. Journal of
Sound and Vibration, Vol. 315, No. 4-5, pp. 797-815.
Vossoughi, G. R. & Karimzadeh, A. (2006). Impedance control of a two degree-of-freedom
planar flexible link manipulator using singular perturbation theory. Robotica, Vol.
24, No. 2, pp. 221–228.
Wang, D. (1989). Modelling and control of multi-link manipulators with one flexible link. Ph. D.
Thesis, Department of Electrical Engineering, University of Waterloo, Canada.
Wang, W.; Lu, S. & Hsu, C. (1989). Experiments on the position control of a one-link
flexible robot arm. IEEE Transactions on Robotics and Automation, Vol. 5, No. 3, pp.
131-138.
Wang, D. & Vidyasagar, M. (1991). Transfer functions for a single flexible link.
International Journal of Robotics Research, Vol. 10, No. 5, pp. 540-549.
Wang, D. & Vidyasagar M. (1992). Passive control of a stiff flexible link, The International
Journal of Robotics Research, Vol. 11, No. 6, pp. 572-578.
Wang, Z.; Zeng, H.; Ho, D. W. C. & Unbehauen H. (2002). Multi-objective Control of a
Four-Link Flexible Manipulator: A Robust H∞ Approach. IEEE Transactions on
Systems Technology, Vol. 10, No. 6, pp. 866-875.
Wang, F. & Gao Y. (2003). Advanced studies of flexible robotic manipulators, modeling, design,
control and applications, World Scientific, ISBN: 978-981-279-672-1, New Jersey.
Whitney, D. E. (1982). Quasi-static assembly of compliantly supported rigid parts. Journal
of Dynamic Systems, Measurement and Control, Transactions of the ASME, Vol. 104,
No. 1, pp. 65-77.
Whitney, D. E. (1987). Historical perspective and state of the art in robot force control.
International Journal of Robotic Research, Vol. 6, No. 1, pp. 3–14.
Yamano, M.; Kim, J.; Konno, A. & Uchiyama, M. (2004). Cooperative control of a 3D dual-
flexible arm robot. Journal of Intelligent and Robotic Systems, Vol. 39, No. 1, pp. 1–
15.
Yuan, K. & Lin, L. (1990). Motor-based control of manipulators with flexible joints and
links, Proceedings of the IEEE International Conference on Robotics and Automation,
pp. 1809–1814, 0-8186-9061-5, Cincinnati, OH, USA .
Yang, G. B. & Donath, M. (1988). Dynamic model of a one link robot manipulator with
both structural and joint flexibility, Proceedings of the IEEE International Conference
on Robotics and Automation, pp. 476–481.
Yim, J. G.; Yeon, J. S.; Lee, J.; Park, J. H.; Lee, S. H. & Hur, J. S. (2006). Robust Control of
Flexible Robot Manipulators. SICE-ICASE International Joint Conference, pp. 3963-
3968.
Yoshikawa, T.; Harada, K. & Matsumoto, A. (1996). Hybrid position/force control of
flexible-macro/ rigid-micro manipulator systems. IEEE Transactions on Robotics
and Automation, Vol. 12, No. 4, pp. 633–640.
Zeng, G. & Hemami, A. (1997). An overview of robot force control. Robotica, Vol. 15, No. 5,
pp. 473-482.
www.intechopen.com

Zinn, M.; Khatib, O.; Roth, B. & Salisbury, J. K. (2004). Playing it safe. IEEE Robotics and
Automation Magazine, Vol. 11, No. 2, pp. 12-21.
www.intechopen.com

Robot Manipulators Trends and Development
Edited by Agustin Jimenez and Basil M Al Hadithi
ISBN 978-953-307-073-5
Hard cover, 666 pages
Publisher InTech
Published online 01, March, 2010
Published in print edition March, 2010
InTech Europe
University Campus STeP Ri
Slavka Krautzeka 83/A
51000 Rijeka, Croatia
Phone: +385 (51) 770 447
Fax: +385 (51) 686 166
www.intechopen.com
InTech China
Unit 405, Office Block, Hotel Equatorial Shanghai
No.65, Yan An Road (West), Shanghai, 200040, China
Phone: +86-21-62489820
Fax: +86-21-62489821
This book presents the most recent research advances in robot manipulators. It offers a complete survey to
the kinematic and dynamic modelling, simulation, computer vision, software engineering, optimization and
design of control algorithms applied for robotic systems. It is devoted for a large scale of applications, such as
manufacturing, manipulation, medicine and automation. Several control methods are included such as optimal,
adaptive, robust, force, fuzzy and neural network control strategies. The trajectory planning is discussed in
details for point-to-point and path motions control. The results in obtained in this book are expected to be of
great interest for researchers, engineers, scientists and students, in engineering studies and industrial sectors
related to robot modelling, design, control, and application. The book also details theoretical, mathematical
and practical requirements for mathematicians and control engineers. It surveys recent techniques in
modelling, computer simulation and implementation of advanced and intelligent controllers.
How to reference
In order to correctly reference this scholarly work, feel free to copy and paste the following:
Pereira E., Becedas J., Payo I., Ramos F. and Feliu V. (2010). Control of Flexible Manipulators. Theory and
Practice, Robot Manipulators Trends and Development, Agustin Jimenez and Basil M Al Hadithi (Ed.), ISBN:
978-953-307-073-5, InTech, Available from: http://www.intechopen.com/books/robot-manipulators-trends-and-
development/control-of-flexible-manipulators-theory-and-practice

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