Control of Flexible Manipulators. Theory and Practice

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Control of Flexible Manipulators. Theory and Practice267
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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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Control of Flexible Manipulators. Theory and Practice269
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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Robot Manipulators, Trends and Development270
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

   
mmm
tK u tJt

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.

ˆ
f
coup



 
t
 
t
 
t
0ˆˆ

ˆˆˆ
mfcoup
   

,
(1)

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
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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Control of Flexible Manipulators. Theory and Practice 271
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:
( )()(
iiiii
y xuuxLu


23
,0,11,21,31
)()
iii
xLuxL


,
(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
MEI AB
dt

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.

2
mnn
d
PQF


  
,
(3)