2001 Santibañez - Global asymptotic stability of bounded output feedback tracking control for robot manipulators.pdf

Proceeding 01 the 40th IEEE
Conference on Decision and Control
Orlando, Florida USA, December 2001
WeA03-5
Global Asymptotic Stability of Bounded Output Feedback
Tracking Control for Robot Manipulators'
Victor Santibacez
Rafael Kelly
Instituto Tecnolbgico de la Laguna Divisibn de Fisica Aplicada
Apdo. Postal 49 Adm. 1
CICESE
Torrebn Coahuila, 27001 Apdo. Postal 2615, Adm. 1
MEXICO Ensenada, B. C., 22800
e-mail: vsantiba@itla6aguna.edu.mx
MEXICO
e-mail: rkelly@cicese.mx
Abstract
This note shows that global asymptotic stability of
the closed-loop system formed by an output feedback
tracking hounded controller (previously reported in the
literature) for robot manipulators in presence of sufficiently
large viscous friction, can be achieved provided
that a feedforward compensation term of the viscous
friction is added to the controller.
Keywords: TYacking, Robot control, Bounded output
feedback control, Lyapunov's second method.
1 Introduction and robot dynamics
In contrast with the bounded torque inputs set-point
control problem of robot manipulators -which has
been the subject of several researches [1]-[8]- the
bounded tracking control problem has been treated by
few authors, namely; [9], and [lo]. For the latter prob
lem, to the best of the authors' knowledge, until now
only semi-global asymptotic stability has been ensured.
This note considers the problem of designing a global
tracking controller for robot manipulators under the
constraints that only joint position measurements are
available (output feedback) and the input torques are
bounded by prescribed limits. To this end we resort
to the tracking controller proposed in [lo]. We prove,
under the only requirement of having sufficient viscous
friction in the robot manipulator joints, such a controller
is effective to achieve global asymptotic stability
provided that viscous friction is feedforward compensated
by the controller.
Throughout this note, we use the notation X,{A}
and Xy{A} to indicate the smallest and largest eigenvalues,
respectively, of a symmetric positive defi-
'Work partidly supported by COSNET and CONACyT
grants No. 31948-A and 32613-A.
0=1803=106i-9mi~io.cm e, 2001 IEEE 1370
nite bounded matrix A(z), for any I E R". The
norm of vector I is defined as 11111 = a.
The vector function tanh(z) E R" is defined as:
tanh(z) = [tanh(zl) ... tanh(zn)lT, where I =
[.I z* '' ' z,]T E R".
In presence of viscous friction, the dynamics of a serial
n-link rigid robot can be written as [ll]:
M(q)ii + C(q, d9 + Fil+ g(n) = T (1)
where q is the n x 1 vector of joint displacements, q
is the n x 1 vector of joint velocities, T is the n x 1
vector of applied torques, M(q) is the n x n symmetric
positive definite manipulator inertia matrix, C(q, q) is
the n x n matrix of centripetal and Coriolis torques, F
is the n x n constant, diagonal positive definite, viscous
friction coefficient matrix, and g(q) is the n x 1 vector
of gravitational torques.
We assume that the links are jointed together with revolute
joints. Some important and well known properties
of dynamics (1) are the following: 1.-) There exists
a positive constant kc such that: IlC(z,p)xll <
kcllyll llzll V z,p, z E R". 2.-) There exists a positive
constant kl such that: 11g(z)11 5 kl for all I E R".
2 Control problem formulation
We shall denote the desired joint trajectory by qd(t)
which is chosen twice continuously differentiable such
that 4d(t) satisfies Il@d(t)ll, Ikd(t)ll> Ilqd(t)ll 5 Bdr with
Bd being a finite constant. The position and velocity
errors will he denoted by q = qd - q and qd = qd - Q,
respectively.
For the system (1) assume that only position measurements
are available and that the robot inputs are constrained
to
[Til 5
Vi = l,.. . ,n
(2)

where ri stands for the i-th entry of vector r and
7rnax IS . the allowed maximum torque. Then, the goal
is to find an output feedback controller which renders
the closed-loop system globally asymptotically
stable. It implies that for all initial conditions we have
limt,, q(t) = 0.
To solve this problem we present the following:
Proposition. Consider the robot dynamics (1) under
input constraints (2) and assume
7”’- > Bd [XM{M} + kcBd + XM{F)] + ki. (3)
Consider the control law
r = Kp tanh(q) + K, tanh(t9) (4)
+M(q)@d + C(q,qd)@d +
+g(q)?
i = -Atanh(x +Bq), (5)
t9 = %+Bq, (6)
where A, B, Kp and K” E IR”’“ are diagonal positive
definite matrices. Then, the robot dynamics
(1) in closed-loop with (4)-(6) is uniformly globally
asymptotically stable provided that the viscous friction
is enough large in the sense that matrix F satisfies
Xrn{F} > 3kCBd.
Outline of the proof. The proof follows the same
steps used in [lo]. The feedforward friction viscous
compensation term Fq,, in (4) is the only difference
with respect to controller introduced in [lo]. This modifies
the corresponding closed-loop equation adding the
term -F$, which is the key to obtain globality of the
asymptotic stability.
3 Conclusions
0
controls”, in Proc. IEEE Int. Conf. Robotics and Automation,
Albuquerque, NM, Apr. 1997, pp. 1148-
1155.
(21 R. Gorez, “Globally stable PID-like control of
mechanical systems”, Systems €4 Control Letters, vol.
38, pp. 61-72, 1999.
[3] R. Kelly and V. Santibafia, “A class of global
regulators with bounded control actions for robot manipulators”,
in Pmc. IEEE Conf. Decision and Control,
Kobe, Japan, Dec. 1996, pp. 3382-3387.
[4] A. Laib, “ Adaptive output regulation of
robot manipulators under actuator constraints”, IEEE
Trans. Robot. Automat., Vol. 16, pp. 29-35, Feb. 2000.
[5] A. Loria, R. Kelly, R. Ortega and V. Santibafiez,
‘’ On global output feedback regulation of Euler-
Lagrange systems with bounded inputs”, IEEE Tmns.
Automat. Contr., vol. 42, pp. 113&1143, Aug. 1997.
[GI V. Santibaiiez and R. Kelly, “On Global regulation
of robot manipulators: saturated linear state feedback
and saturated linear output feedback”, European
Journal of Control, vol. 3, pp. 104-113,1997.
[7] V. Santibafiez and R. Kelly, “A New set-point
controller with bounded torques for robot manipulators’’,
IEEE Transactions on Industrial Electronics,
vol. 45, pp. 126-133, Feb. 1998.
[8] E. Zergeroglu, W. Dixon, A. Behal and D. Dawson,
“Adaptive set-point control of robotic manipulators
with amplitude-limited control inputs”, Robotica,
vol. 18, pp. 171-181, 2000.
[9] W. E. Dixon, M. S. de Queiroz, F. Zhang and D.
M. Dawson, “’Backing control of robot manipulators
with bounded torque inputs”, Robotica, vol. 17, pp.
121-129, 1999.
[lo] A. Loria and H. Nijmeijer, “Bounded output
feedback tracking control of fully actuated Euler-
Lagrange systems”, Systems # Control Letters, vol. 33,
pp. 151-161, 1998.
[ll] Spong M. and M. Vidyasagar, Robot Dynamics
and Control. John Wiley and Sons, New York, 1989.
We have shown that considering the robot manipulator
dynamics in presence of sufficiently large viscous
friction, it is possible to ensure uniform global
asymptotic stability of the closed-loop system formed
with the bounded output feedback tracking controller,
introduced in [lo], if it is added to aforementioned
controller a feedforward viscous friction compensation
term. That means, that having access only to position
measurements, and independently of the initial conditions,
the tracking position error may go to zero and
the torque inputs remain bounded by prescribed limits.
References
[l] R. Colbaugh, E. Barany and K Glass, “Global
regulation of uncertain manipulators using bounded
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