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932 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 4, AUGUST 2000
A Nonlinear Disturbance Observer for
Robotic Manipulators
Wen-Hua Chen, Member, IEEE, Donald J. Ballance, Member, IEEE, Peter J. Gawthrop, Senior Member, IEEE,
and John O’Reilly, Senior Member, IEEE
Abstract—A new nonlinear disturbance observer (NDO)
for robotic manipulators is derived in this paper. The global
exponential stability of the proposed disturbance observer (DO)
is guaranteed by selecting design parameters, which depend
on the maximum velocity and physical parameters of robotic
manipulators. This new observer overcomes the disadvantages of
existing DO’s, which are designed or analyzed by linear system
techniques. It can be applied in robotic manipulators for various
purposes such as friction compensation, independent joint control,
sensorless torque control, and fault diagnosis. The performance of
the proposed observer is demonstrated by the friction estimation
and compensation for a two-link robotic manipulator. Both
simulation and experimental results show the NDO works well.
Index Terms—Friction, nonlinear estimation, observers, robots.
I. INTRODUCTION
D
ISTURBANCE observers (DO’s) have been used in
robotic manipulator control for a long time. In general,
the main objective of the use of DO’s is to deduce external
unknown or uncertain disturbance torques without the use of
an additional sensor. The use of DO’s in robotic manipulators
can be divided into the following categories.
1) Independent joint control is widely used in industrial
robots where a multilink manipulator is divided into
several independent links with linear dynamics. The per-
formance of these kind of controllers can be improved by
the use of a DO. This is accomplished by considering the
coupling torques from other links as an unknown external
disturbance and using a DO to estimate and compensate
for it [1]. This technique has also been extended to deal
with parameter variations and unmodeled dynamics
whereby it improves the robustness of robots [2];
2) Friction is a common phenomenon in mechanical sys-
tems and plays an important role in system performance.
Many friction models and compensation methods have
been proposed [3]. One of the most promising methods
Manuscript received December 22, 1998; revised March 23, 2000. Abstract
published on the Internet April 21, 2000. An earlier version of this paper was
presented at the 38th IEEE Control and Decision Conference, Phoenix, AZ,
December 7–10, 1999. This work was supported by the U.K. Engineering and
Physical Sciences Research Council under Grant GR/L 62665.
W.-H. Chen and J. O’Reilly are with the Centre for Systems and Control,
Department of Electronics and Electrical Engineering, University of Glasgow,
Glasgow G12 8QQ, U.K. (e-mail: w.chen@elec.gla.ac.uk).
D. J. Ballance and P.J. Gawthropare with the Centre for Systems and Control,
Department of Mechanical Engineering, University of Glasgow, Glasgow G12
8QQ, U.K. (e-mail: D.Ballance@mech.gla.ac.uk).
Publisher Item Identifier S 0278-0046(00)06805-2.
is observer-based control where a DO is used to estimate
friction [4];
3) DO’s have been used in robotic manipulators for force
feedback and hybrid position/force control where the DO
works as a torque sensor [5]–[7]. In this case, it is sup-
posed that the friction and other dynamics are well mod-
eled and compensated for. The use of a DO, rather than
several torque sensors (for each link, at least one torque
sensor is required), simplifies the structure of the system,
reduces the cost, and improves the reliability. With this
technique, sensorless torque control can be implemented
[5]–[7].
4) Robotic manipulators work in a dynamic highly uncertain
environment. In this application, DO’s provide signals for
monitoring and trajectory planning rather than forcontrol.
For example, in robotic assembly when the component is
misinserted, the reaction torque/force is greatly increased
and could damage the robotic manipulator. A DO can es-
timate the reaction torque online and transmit this infor-
mation to the monitoring or planning level. Chan [8] uses
a DO inelectronic component assembly, while Ohishi and
Ohde [9] give an example of the use of a DO in collision.
Although the DO technique has been widely used in robotic
manipulator control for various purposes, in almost all cases,
the analysis or design is based on linearized models or using
linear system techniques. Since a multilink robotic manipulator
is a highly nonlinear and coupled system, the validity of using
linear analysis and synthesis techniques may be doubtful. Many
important properties of existingDO’s have not been established,
e.g., unbiased estimation or even global stability. There are,
however, some recent results using nonlinear disturbance ob-
servers (NDO’s). A variable structure DO has been proposed
[10] and a nonlinear observer for a special kind of friction, i.e.,
Coulomb friction, has been proposed by Friedland and Park
[11]. This nonlinear observer is designed based on the model of
Coulomb friction, and the global convergence ability has been
shown.It has been further modified and implemented on robotic
manipulators by Tafazoli
et al. [12]. However, a specific model
of friction will not be used in this paper, and the whole design
is based on the DO concept. That is, similar to other unknown
torques, the friction is considered as a disturbance on the control
torque.
It should be stressed that the DO rather than a velocity ob-
server of a manipulator is considered in this paper. A velocity
observer has been considered by many authors. Bona and Indri
have compared and implemented several linear and nonlinear
velocity observers on a robotic manipulator [13].
0278–0046/00$10.00 © 2000 IEEE
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CHEN et al.: NDO FOR ROBOTIC MANIPULATORS 933
An NDO for multilink robotic manipulators will be presented
in this paper. By carefully selecting the observer gain function, it
will be shown that global convergence is guaranteed. This result
is based on Lyapunov theory.
II. P
ROBLEM FORMULATION AND A BASIC OBSERVER
A. Problem Formulation
For the sake of simplicity, a two-link robotic manipulator is
considered in this paper. The main idea is readily extended to the
more general case. The model of a two-link robotic manipulator
can be represented by
(1)
where
, , and are displacement,
velocity, and control vectors, respectively. When only the dy-
namics of the links areconsidered in the model, the control input
is either the torque or force. is a disturbance torque
or force vector. It should be noted that
has different meanings
in different observer applications. For example, it can be fric-
tion in friction compensation, reaction torque or force in force
control, and unmodeled dynamics in independent joint control.
Since a general observer will be derived in this paper, all of them
are consideredas “disturbances.” When thefirst-order dynamics
of dc motors are included in the above model,
is the voltage
vector imposed on the motors instead of the torque vector. As
a result, the torque disturbance is also equivalent to the distur-
bance on the voltage on the motors. Hence,
is the disturbance
voltage in this case.
The objective of this paper is to design an observer such
that the estimation
yielded by the observer exponentially
approaches the disturbance
under any , , and
.
B. Initial Observer
A basic idea in the design of observers/estimators is to modify
the estimation by the difference between the estimated output
and the actual output. Since (1) can be written as
(2)
a DO is proposed as
(3)
Since, in general, there is no prior information about the
derivative of the disturbance
, it is reasonable to suppose that
(4)
which implies that the disturbance varies slowly relative to the
observer dynamics. However, it will be illustrated by simulation
and experiment that the observer developed in this paper can
also track some fast time-varying disturbances.
Define the observer error
(5)
Combining (4) with the observer (3) yields
(6)
That is, the observer error is governed by
(7)
It can be shown that the observer is globally asymptotically
stable by choosing
(8)
where
. More specifically, the exponential convergence
rate can be specified by choosing
.
III. NDO
The acceleration signal
is not available in many robotic ma-
nipulators, and it is also difficult to construct the acceleration
signal from the velocity signal by differentiation due to mea-
surement noise. However, although the observer (3) is not prac-
tical to implement, it provides a basis for the further nonlinear
observer design.
A. Modified Observer
Define an auxiliary variable vector
(9)
where
. The designed function vector is to be
determined.
Let the function
in (3) be given by the following non-
linear equation:
(10)
Invoking (9) and (10) with (3) yields
(11)
Hence, the NDO is given by
(12)
and
(13)
where
is given by (10).
B. Stability of the NDO
It follows from (5), and (11)–(13) that the observer error
equation is given by
(14)
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934 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 4, AUGUST 2000
The estimation approaches the disturbance if is
chosen such that (14) is asymptotically stable. Hence, the func-
tion
must be selected such that the function
given by (10) satisfies this condition. In general, it is not easy
to select such a nonlinear function
especially for multi-
variable systems. However, for a multilink robotic manipulator,
with the help of its characteristics, a systematic method for se-
lecting the nonlinear function
, such that the observer
with
given by (10) is asymptotically stable, is devel-
oped in this paper.
The inertial matrix
for a two-link manipulator is given
by [14]
(15)
where
, , , and are inertial parameters, which depend
on the masses of the links, motors and tip load, and the lengths
of the links.
Theorem: For the two-link robotic manipulator (1), when the
function
in the observer (12) and (13) is chosen as
(16)
and
satisfies
(17)
where
denotes the maximum velocity of the second link,
then the observer (12) and (13) is globally asymptotically stable.
Proof: Since
is given by (16), it yields
(18)
Combining the above equation with (10) gives
(19)
since
is positive definite for all and and, therefore,
invertible.
From (15),
can be written as
(20)
where
(21)
Hence,
(22)
Since
is also positive definite for all , a Lyapunov func-
tion candidate for the observer (12) and (13) can be chosen as
(23)
Differentiating the Lyapunov function with respect to time
along the observer trajectory gives
(24)
Hence,
for all and if
(25)
That is,
(26)
Since
satisfies (17), the above inequality is met. The equi-
librium point (
) is then globally asymptotically stable.
From the Theorem, the stability of the observer depends on
the maximum velocity of the second link and other physical
parameters. By choosing the design parameter
satisfying the
inequality (17),the global stability ofthe observer isguaranteed.
C. Convergence Rate of the NDO
It can be shown the minimum eigenvalue of the matrix in (25)
for all
is given by
(27)
Thus,
(28)
Let
denote the maximum eigenvalue of the matrix for
all
. It then follows from (24) that
(29)
The speed of convergence is bounded by
. For a robotic
manipulator,
and the prescribed maximum velocity are fixed.
By choosing the parameter
, the desired exponential conver-
gence rate can be achieved.
IV. S
IMULATION AND EXPERIMENTAL RESULTS
The proposed NDO is tested in this section. As stated in Sec-
tion I, the NDO can be used in robotic manipulators for various
purposes. In what follows, the NDO is designed as a friction
observer. That is, the NDO is used to estimate the friction for
a two-link robotic manipulator. The reason for designing a fric-
tion observer here is that friction varies rapidly, even discontin-
uously. It is a challenging task in observer design. The simula-
tion and experiment are based on a two-link manipulator with
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CHEN et al.: NDO FOR ROBOTIC MANIPULATORS 935
Fig. 1. Structure of the revised friction model.
dc motor actuators. The dynamic model of the manipulator in-
cluding the first-orderdc motor dynamics is governed by (1) and
is detailed in [15].
A. Friction Simulation
The friction considered is Coulomb and Viscous friction,
given by
(30)
The parameters for first and second links in the simulation are
given by
N m
N m/rad/s (31)
and
N m
N m/rad/s (32)
respectively.
There are some problems in using the friction model (30) in
simulation directly. One is due to the discontinuity of the fric-
tion characteristics at zero velocity—a very small step size is
required for testing zero velocity. The other is that when the ve-
locity is zero, or the system is stationary, the friction is indefinite
and depends on the controlled torque. In the simulation, to im-
prove the numerical efficiency, a revised friction model, which
is modified from [16], is adopted. The structure for the revised
friction model is given by Fig. 1. It can be described by the fol-
lowing mathematical model:
(33)
where
is given by (30), is the revised friction and is a
small positive scalar, and
is given by
(34)
When the velocity is within a very small area near zero, de-
fined by
, the friction is equal to the applied torque . When
the velocity is greater than this, the second term in the above ex-
pression vanishesand the friction
given by this revisedmodel
is equal to the friction givenby (30). To compare the accuracy of
the revised frictionmodel (33), the frictiondefined bythe classic
Coulomb and viscous friction model (30) and the revised model
Fig. 2. Comparison of the Columb and Viscous friction model and the revised
friction model.
Fig. 3. First-link velocity and friction time histories.
(33) for first link is shown in Fig. 2. In this figure, is chosen
as 0.001. The frictions given by these two models are almost in-
distinguishable. However, experience has shown that using the
revisedmodel greatly improves the computational efficiency for
simulation.
B. Simulation Results
A controller is designed by computed torque control where
the disturbance is not taken into account. When square-wave
command references are givenfor first and second links, respec-
tively, the velocity and friction histories are shown in Figs. 3 and
4. It can be seen that the friction varies very rapidly with the ve-
locity.
A friction observer is designed by the NDO technique de-
veloped in this paper. The design parameter
is chosen as 50.
The estimation given by the observer (12) and (13) is shown
by Figs. 5 and 6. The observer exhibits excellent tracking per-
formance. It successfully estimates the friction on-line and no
friction model is required.
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