-
The original version of this paper was presented at the
IFACWorldCongress which was held in San Francisco, USA during 1996.
Thispaper was recommended for publication in revised form by
AssociateEditor J.Z. Sasiadek under the direction of Editor
Mituhiko Araki.
*Corresponding author. Tel.: #1-405-744-6579; fax:
#1-405-744-7873.E-mail address: [email protected] (P. R.
Pagilla).
Automatica 37 (2001) 983}995
An adaptive output feedback controller for robot arms:stability
and experiments
Prabhakar R. Pagilla*, Masayoshi TomizukaSchool of Mechanical
and Aerospace Engineering, Oklahoma State University, Stillwater,
OK 74078-5016, USA
Department of Mechanical Engineering, University of California,
Berkeley, CA 94720-1970, USA
Received 8 May 1998; revised 24 December 1999; received in "nal
form 3 January 2001
An adaptive output feedback controller for robot arms is
developed in this paper. A nonlinear observerbased on desired joint
velocities and bounded joint position error is used to estimate
joint velocities.Experimental results validate the ewectiveness of
the proposed adaptive output feedback controller.
Abstract
An adaptive output feedback controller for robot arms is
developed in this paper. To estimate the joint velocities, a simple
nonlinearobserver based on the desired velocity and bounded
position tracking error is proposed. The closed-loop system formed
by theadaptive controller, observer and the robot system is shown
to be semi-global asymptotically stable. Extensive experiments
conductedon a two link robot manipulator con"rm the e!ectiveness of
the proposed controller}observer structure. To highlight the
performanceof the proposed scheme, it is compared via experiments
with a well-known passivity based control algorithm. 2001
ElsevierScience Ltd. All rights reserved.
Keywords: Adaptive control; Robot control; Observers; Lyapunov
stability
1. Introduction
Adaptive control of robot arms based on completestate
measurements has been dealt in great detail in theliterature. The
feed-forward and passivity based algo-rithms for robot arms
proposed in Sadegh and Horowitz(1990), Slotine and Li (1991) and
Ortega and Spong(1989), and the references therein, have been
extensivelyused. A comparative experimental study of the
standardand new algorithms has been done in Whitcomb, Rizzi,and
Koditschek (1993). Most of these algorithms needcomplete state
measurements. A major drawback of suchschemes is that both joint
position and joint velocitymeasurements of the robot are required
for feedback
control. Sensors for measuring robot joint velocities
areexpensive. Further, measurements from these sensors areoften
contaminated by noise. Velocity estimated feed-back control of
robot arms can be used instead and therequirement of robots to be
equipped with velocity sen-sors can be eliminated. Most of the
robot adaptiveschemes use velocity errors or modi"ed velocity
errors todrive the parameter adaptation algorithms. When theactual
velocities are not available, estimated velocitiesand position
errors have to be used to drive the para-meter adaptation
algorithms. This leads to an addeddi$culty in proving the stability
of these algorithms.Considerable research is being conducted in the
area of
output feedback control of nonlinear systems. Outputfeedback
control of robot arms has been studied by manyresearchers. In
Berghuis and Nijmeijer (1993), theauthors consider passivity based
controller}observer de-sign for robots. A linear observer is
designed to estimatethe velocities. It is shown that the
closed-loop systemformed by the controller}observer and the robot
is lo-cally exponentially stable. A linear velocity observer
isdesigned assuming complete knowledge of the structuralparameters
of the robot. A robust variable structure
0005-1098/01/$ - see front matter 2001 Elsevier Science Ltd. All
rights reserved.PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 0 4 8 - 6
-
controller and a nonlinear observer is designed in Zhu,Chen, and
Zhang (1992). Berghuis and Nijmeijer (1994)proposes a linear
controller and a linear observer forrobust control in the presence
of parameter uncertainties.In Canudas de Wit and Fixot (1992),
tracking control ofrobot manipulators is proposed by combining a
passivitybased controller and a nonlinear sliding observer.
Localasymptotic convergence of the position tracking errorsand the
velocity estimation errors was shown.A nonlinear observer based on
the robot error dynam-
ics was designed in Nicosia and Tomei (1990), and a con-trol
design that uses joint position measurements andestimated velocity
is proposed. Repetitive and adaptivecontrol of robot manipulators
with velocity estimation ispresented in Kaneko and Horowitz (1997).
In the case ofrepetitive control, the robot achieves tracking of
thedesired periodic trajectory through repeated learningtrials. An
adaptive controller is also designed. A linearobserver is designed
to estimate the velocities. Localasymptotic stability is shown for
both the repetitive andthe adaptive cases.In this work, an adaptive
feedback controller for robot
arms is designed using partial state feedback, i.e., onlyjoint
position measurements are used to design the adap-tive controller.
A simple nonlinear observer is designed toestimate the robot joint
velocities. The closed-loop sys-tem formed by the adaptive
controller, observer and therobot system is shown to be semi-global
asymptoticallystable, i.e., the region of attraction can be
increasedarbitrarily by increasing the controller and the
observergains.Convergence of the estimated parameters to the
true
parameters depends on whether the regressor matrixsatis"es the
persistence of excitation condition. In theproposed adaptive
controller the regressor matrix entire-ly depends on the desired
trajectory. Hence, the persist-ence of excitation condition is
satis"ed by choosinga persistently exciting desired trajectory.
Experimentswere conducted on a two link planar arm for the
pro-posed controller}observer. Successful experimental re-sults
show the validity of the proposed controller andobserver. The
proposed scheme is compared, via experi-ments, with a well-known
passivity based controller. Thepassivity based controller used for
this comparisonassumes that the parameters are exactly known anda
"rst-order numerical di!erentiation of joint positionmeasurements
is used to estimate velocities.The remainder of this paper is
organized as follows. In
Section 2, robot dynamics and problem formulation isgiven.
Section 3 gives the proposed adaptive controllerand observer.
Closed-loop error dynamics is also derivedin Section 3. Stability
of the closed-loop system is shownin Section 4. Section 5 discusses
the experimentalplatform and the experimental results. Some
concludingremarks with a summary of this paper are given inSection
6.
2. Robot dynamics and problem formulation
Consider the dynamics of an n degree of freedom robotarm
x"x
,
M(x)x
#C(x
,x
)x
#g(x
)", (1)
where x3, x
3 are the generalized position and
velocity, respectively, M(x)3 is the inertia matrix,
C(x,x
)3 is the matrix composed of Coriolis and
centrifugal terms, g(x)3 is the gravity vector, and
3 is the vector composed of joint torques. The struc-ture of the
robot dynamics satis"es the properties givenin Appendix A.Given a
desired trajectory of the robot, the objective is
to design a stable tracking controller that only requiresjoint
position measurements for feedback. To achievethis objective, an
adaptive controller together witha simple nonlinear observer to
estimate joint velocities isproposed. Let x
(t) and x
(t) be the desired position and
velocity, respectively. It is assumed that the desired
statetrajectory is twice continuously di!erentiable. Let x(
(t)
and x((t) denote the estimated position and estimated
velocity, respectively. Let 3 denote the actual para-meter
vector as given by property (iv) in Appendix A. LetK (t) denote the
estimate of . De"ne the tracking and theestimation errors by
e(t) :"x
(t)!x
(t), e(
(t) :"x(
(t)!x
(t),
e(t) :"x
(t)!x
(t), e(
(t) :"x(
(t)!x
(t),
e(t) :"e
(t)#e
(t), I (t) :"K (t)!,
e(t) :"
(e
)e
(t),
where e(t) and e
(t) are the position and velocity track-
ing errors, respectively, e((t) and e(
(t) are the estimated
position and estimated velocity errors, respectively, e(t)
is the reference velocity error, I (t) is the parameter
es-timation error, e
(t) is an auxiliary bounded position
tracking error, and(e
(t)) is a positive de"nite diagonal
matrix given by
(e
(t))"diag
1#e(t),2,
1#e(t), (2)
where e(t),2, e(t) are the components of the position
error vector e(t) and
is a positive gain. Notice that this
choice of (e
) renders e
(t) to be bounded by
. In the
remainder of the paper, whenever it is clear from thecontext,
explicit dependence of variables on time isnot shown. Also,
throughout the paper A denotes the2-norm of A. The following
section gives the adaptivecontroller, observer, and the closed-loop
error dynamics.
984 P. R. Pagilla, M. Tomizuka / Automatica 37 (2001)
983}995
-
3. Adaptive controller and observer
The following control scheme is proposed:
"Y(x
,x
,x
)K !K
(e
#e(
)!K
e, (3)
where K, K
are positive de"nite gain matrices and K is
the estimated parameter vector of the robot. Note thatthe second
term in the control law is a function ofestimated velocity, desired
velocity, and actual positionerror, i.e., e
#e(
"x(
!x
#e
. The desired regressor
matrix, Y(x
, x
, x
), is given by
Y(x
, x
, x
)K "MK (x
)x
#CK (x
, x
)x
#g( (x
),
where MK (x), CK (x
,x
), and g( (x
) are the estimates of
M(x), C(x
,x
), and g(x
), respectively. The desired re-
gressor matrix depends only on the desired trajectoryand can be
pre-computed. The parameter adaptation lawis chosen as follows:
K (t)"K (0)!Ye (t)!
YQ e() d, (4)
where K (0) is the initial estimate of the unknownparameter
vector, is a positive de"nite gain matrix, andeis given by
e(t)"e
(t)!e(
(t)#
e() d!
e(() d.
The following observer is proposed to estimate the states:
x("!
e(#x(
, (5)
x("x
!
e(#e
, (6)
where and
are positive gains. Thus, by rearranging
terms, the observer error dynamics is given by
e("!
e(#e(
,
e("!
e(#
e(#2e
!e
.
(7)
The closed-loop error dynamics is derived in the follow-ing
section.
3.1. Error dynamics
Noting that e"e
#e
and !e
#e
#e
"0,
M(x)e
can be expressed as
M(x)e
"!C(x
,x
)e
#M(x
)e
#C(x
, x
)e
!g(x)#M(x
)e
#C(x
,x
)e
. (8)
From (1) and (8), we obtain
M(x)e
"!C(x
,x
)e
#!M(x
)x
#C(x,x
)x
#M(x
)e
#C(x
, x
)e
. (9)
Using the control law and noting that e"E
e, where
E"
, the error equation is
M(x)e
"!C(x
,x
)e
#YdI !W!K(e#e( )
!Ke#M(x
)E
e!M(x
)E
e, (10)
where W is given by
W"[M(x)!M(x
)]x
#g(x
)!g(x
)
# [C(x,x
)(x
!e
)!C(x
, x
)x
].
Using Eq. (7), the observer error equation can be derivedas
follows:
M(x)e(
"!
M(x
)e(
#
M(x
)e(
# 2M(x)e
!M(x
)e
"!C(x, x
)e(
!M(x
)e
#C(x
,x
)e
! M(x
)e(
#C(x
, x
)(e(
#e
)
# 2M(x)e
#
M(x
)e(
. (11)
On substitution of the robot error dynamics (10) andusing x
"e
!e
#x
, we obtain
M(x)e(
"!C(x,x
)e(
!Y
I #W#K
(e
#e(
)#K
e
#M(x)E
e!M(x
)E
e!
M(x
)e(
# M(x
)e(
#C(x
, e
)!C(x
, e
)(e(
#e
)
#C(x, x
)(e(
#e
). (12)
4. Stability
First, de"ne an extended vector z given byz :"[e
, e
, e(
, e(
, I ]. The following theorem gives the
stability of the closed-loop system with the
proposedcontroller}observer structure.
Theorem 1. For the robot dynamics given in (1), using
theadaptive controller (3) together with the update law (4) andthe
observer (6), it is always possible to choose feedbackgains K
, K
and
and the observer gains
and
such
that z"0 is locally uniformly stable, e, e
, e(
and
e(
locally asymptotically converge to zero. Further, theclosed-loop
system is semi-globally asymptotically stable,i.e., the region of
attraction can be arbitrarily increased byincreasing the controller
and the observer gains.
Proof. Consider the following Lyapunov function candi-dates,