W. L. Brogan, Modern Control Theory. Englewood Cliffs, NJ: ... recently received great attention in the robot manipulator control design (see, e.g., [l] for a recent ...
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341
An Adaptive Variable Structure Model Following Control Design for Robot Manipulators
(6.4) is relatively easy and only lower dimensional matGx computations are involved. 2) Step 2 and Step 4 can be implemented simultaneously, which makes the algorithm attractive for parallel implementation.
T i n - h i h u n g , Qi-Jie Zhou, and Chun-Yi Su VII. CONCLUSION The notion of partial eigenstructure assignment (PEA), which is a natural extension of eigenstructure assignment and partial eigenvalue assignment, for linear multivariable systems has been introduced. Theoretical basis for PEA has been presented. Algorithms both for PEA and for eigenstructure assignment have been developed and analyzed. These algorithms have been shown to be especially effective for large scale system applications.
Abstract- An adaptive variable structure model following control design is presented for the nonlinear robot manipulator system. The controller does not require any knowledge of a nonlinear robotic system and does not necessarily need the occurrence of a sliding mode at each individually stable discontinuity surface. It thus greatly reduces the complexity of design. In the closed loop, the joint angles asymptotically converge to the reference trajectory with a prescribed transient response. The problem of chattering is reduced by the introduction of a boundary layer.
I. INTRODUCTION
REFERENCES
Adaptive model following control (AMFC) methodologies have recently received great attention in the robot manipulator control design (see, e.g., [l] for a recent review). Among developed Prentice-Hall, 1985. AMFC algorithms, several approaches have been considered. Some H. Kimura, “Pole assignment by gain output feedback,” ZEEE use the Lyapunov stability method [2], [3], others do use the Trans. Automat. Contr., vol. AC-20, pp. 509-516, Aug. 1975. B. C. Moore, “On the flexibility offered by state-feedback in multi- hyperstability theory [4], [5], and the deterministic approach [6], variable systemsbeyond closed eigenvalue assignment,” ZEEE Trans. [7]. But the strict positive realness is invariably required. Furthermore, some of them can only guarantee the error between the states Automat. Contr , vol. AC-21, pp. 689-692, Oct. 1976. G. Klein and B! C. Moore, “Eigenvalue-generalized eigenvector of the model and those of the controlled plant going to zero, the assignment with state feedback,” ZEEE Trans. Automat. Contr., transient behavior of this error is not prescribed. A variable strucvol. AC-22, pp. 140-141, Feb. 1977. ture model following control (VSMFC) was proposed in [SI, [9] as V. Sinswat and F. Fallside, “Eigenvalue/eigenvector assignment by an alternative to adaptive model following control. The advantage of state-feedback,” Znt. J. Contr., vol. 26, no. 3, pp. 369-403, 1977. the VSMFC design lies in its ability to prescribe transient response B. Porter and J. J. D’Azzo, “Closed-loop eigenstructure assignment by state feedback in multivariable linear systems,” Znt. J. Contr., requirements as well as providing a robust controller. But the control law, resulting from the method of control hierarchy [ l l ] in vol. 27, no. 3, pp. 487-492, 1978. M. M. Fahmy and J . O’Reilly, “On eigenstructure assignment in order to force every trajectory to eventually come in contact with linear multivariable systems,” ZEEE Trans. Automat. Contr., vol. and remain on the intersection of n surfaces in joint space, is AC-27, pp. 690-693, June 1982. defined implicitly by a set of fairly complicated algebraic inequali-, “Eigenstructure assignment in linear multivariable systems-A ties. Moreover, the control law is based on the restrictive assumpparametric solution,” ZEEE Trans. Automat. Contr., vol. AC-28, tion that the ranges of the variation of parameters are known and pp. 990-993, Oct. 1983. that the resulting control torques are excessive. G. Roppeneck, “Minimum norm output feedback design under speIn this note, an adaptive variable structure model following cific eigenvalue areas,” Syst. Contr. Lett., vol. 3, pp. 101-103, control (AVSMFC) design is proposed for accomplishing trajectory 1983. -, “On parametric state feedback design,” Znt. J. Contr., vol. tracking in a nonlinear robot system which ensures the stability of 43, no. 3, pp. 793-804, 1986. the intersection of the surfaces without necessarily stabilizing each M. M. Hassan and M. H. Amin, “Recursive eigenvalue assignment individual one. Moreover, unlike [SI, [9], we assume here that the in linear system,” Znt. J. Contr., vol. 45, no. 1, pp. 291-310, system matrix is completely unknown and no information on its 1987. possible size is given. This has important implications. The involved M. A. Khan and N. Sreenivasulu, “A systematic method for eigencomputations of [SI, [9] to obtain the feedback gains are not structure assignment and application to response shaping,” Znt. J . required here, instead, the feedback elements are generated as a Contr., vol. 43, no. 2, pp. 117-735, 1986. J. W. Howze and R. K. Cavin, III, “Regular design with modal function of state of the dynamics and the trajectory error. The insensitivity,” ZEEE Trans. Automat. Contr., vol. AC-24, pp. approach avoids the difficulties linked to the strict positive realness requirement in traditional AMFC by taking advantage of the inher466-469, Oct. 1979. M. M. Fahmy and J. O’Reilly, “Use of the design freedom of ent positive definiteness of manipulators inertia matrix, and is easily time-optimal control,” Syst. Contr. Lett., vol. 3, pp. 23-30, 1983. extendable to a higher number of links. Chattering is eliminated by Y. Saad, “Projection and deflation methods for partial pole assign- restricting the state of the system to slide within a boundary layer ment in linear state feedback,” ZEEE Trans. Automat. Contr., vol. rather than along the intersection of the surfaces. 33, pp. 290-296, Mar. 1988. M. J. Balas, “Trends in large structure control theory: Fondest 11. ROBOTIC SYSTEM dreams, widest hopes,” ZEEE Trans. Automat. Contr., vol. AC-27, A . Manipulator Model pp. 522-535, Oct. 1982. G. H. Golub and D. F. Vanloan, Matrix Computation. Baltimore, The dynamic equations of motion for a general rigid link manipuMD: Johns Hopkins University Press, 1989. Ami Arbel, “Controllability mea’sures and actuator placement in Manuscript received May 23, 1989; revised April 13, 1990. This work oscillatory systems,” Znt. J . Contr., vol. 33, no. 3, pp. 565-574, was supported in part by the Beijing-Hong Kong Academic Exchange Center 1980. R. Fletcher, Practical Method of Optimization, Vol. 2 . New under Grant 89014/R. T.-P. h u n g is with the Department of Mechanical and Marine EngineerYork: Wiley, 1987. J. E. Dennis, Jr., Numerical Methods for Unconstrained Opti- ing, Hong Kong Polytechnic, Hong Kong, People’s Republic of China. Q.-J. Zhou and C.-Y. Su are with the Department of Automation, South mization and Nonlinear Equations. Englewood Cliffs, NJ: PrenChina University of Technology, Guangzhou 510641, People’s Republic of tice-Hall, 1983. J. J. More, B. S . Garbow, and K. E. Hillstorm, User Guide for China. IEEE Log Number 9040499. Minpack-I.
r11 W . L. Brogan, Modern Control Theory. Englewood Cliffs, NJ: 121 r31 r41
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01991 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 3, MARCH 1991
348
lator having n degrees of freedom can be described as follows:
D(q)ii + F ( q , 4)q + G(q) = u(f) (1) where q E Rfl is joint displacement, U E W" is applied joint torque (or forces), D( q ) = D'( q ) > 0 , D ( q ) E Rflx" is the inertia matrix, F ( q , q ) q E W" is the centripetal and Coriolis torques, and G( q ) E W" is the gravitational torque. Defining x E W2" to be the state vector
x=
[
[;I. ] [
9
D - ' ( q ) ( - F ( q , 414 -
io-'(q)]'
0 G;'G,
u =K,x
(2a)
where E
(2b) = AX
(I
+ G;'G,)
(8)
Rnx2n
$2
+ K,r + K 3 e + $ , x + $,r + g3e
(9)
K 2 ~ W f l XK" 3, ~ R f l x 2areconstantmatrices, ' E R n , $3 E R n " are discontinuous matrices.
Differentiating s with respect to time gives sliding mode equation
i = Ge
+ Bu.
(2c)
= G(A,x,
In AMFC design, the desired behavior of the plant is expressed through the use of a reference model driven by a reference model of the form
+Bmr - ( A = -S
(3a)
=
+ B,r
- A x - Bu)
+ (A, +A,)x, + A , ) x Bu)
= G ( A , ( x - x,)
-[ ] [A",, L2][ I: [ill. d qm dt 4.m
(7c)
where
and B+ is pseudo inverse of B given by B + = [0 B ; ' ] . Remark: We refer to (7a)-(7c) as matching conditions, which can always be satisfied due to the special structures of B , B,, A , A , , A , , and B + . Define the AVSMFC law as the following:
Equation (1) can be written in a state variable form
x=
( I - BB+)(A, +A,) = 0
-
+ GB[B+(A, - A )
-
K, -
$1).
+
+ B,r
= Amxm
+(B+B, - K,
(3b)
r ~ n ER"^" ~ " are where x , = (q: q z ) T ~ R 2 nA, , E W ~ 9 ~B constant matrices, r E R" is an external input.
B. A VSMFC Law The considered tracking problem is stated as follows. Knowing a reference input r , determine a control law u and a sliding surface such that sliding mode occurs on the sliding surface, the tracking error e = x, - X E R2, has a prescribed transient response and it goes to zero asymptotically as t W . We define the sliding surface s E W" as a hyperplane
- $,)r
+ ( B + ( A , + A , ) - K 3 - $,)e].
(10)
Remark: The role of A , is twofold. First, it satisfies G = C A , , which makes it possible to use the Lyapunov function. Second, the matching condition (7c) can be satisfied. We assume that the following condition is satisfied for all q, q.
Assumption A2:
+
(4)
s=Ge=O
where G E Rflx2' is a constant matrix and will be determined as follows. Let e = (E' ,')'i E = qm - q, i = q, - q, then the sliding hyperplane becomes
s ( E , ~=)Ge
=
+
( G I G,)e = G , E G 2 i = 0.
(5)
Without losing generality, suppose G , E W n x n is nonsingular. If the sliding mode exists on s = 0, then from the theory of variable structure system [lo], the sliding mode is governed by the following linear differential equation whose behavior is dictated by the sliding hyperplane design matrices G I and G,
i = -G-'G 2
le.
(6)
Obviously, the tracking error transient response is then determined entirely by the eigenvector structure of the matrix -G;'G,. The approach of the determination of G G , that E has desired property is given in [ l l ] , [12]. Thus, if the control law is designed such that the sliding mode exists on s = 0, the tracking error transient response is completely governed by the linear dynamic equation (6). In order to derive the AVSMFC law, we require the following assumption. Assumption AI: For all ( x , t ) E W2" X E?!+, satisfy
( I - BB+)B, = o
IIB'(Am + A n ) - K3II < Q5
Lin(G2BIG:) 2
-
A) =0
Grs
$1
= [ ~i =~l j l l e ~ l i - lII-ssIIg n ( x ) '
\Is11+ IISII
10
(7a) (7b)
(12)
PI.
Remark: Assumption A3 can always be held for suitable PI. Now consider the control law (9), if we take $ias
$2
( I - BB+)(A,
(114
where ai > 0, Pi > 0 are some positive numbers. Remark: Due to the mechanical characteristics of robotic manipulators and the boundness of the reference model, the assumption is valid. We also make the following assumption. Assumption A3: The minimum eigenvalue of matrix G , BIG: satisfies
=
o
=0
(13a)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 3, MARCH 1991
xi
(14a)
i= 1
n
= g4 II s II
c I ri I
(14b)
i= 1
2n
~ ~ = ~r = ~ IeiI IIsIIC l
(14c)
where g; > 0, i = 1 * 5 are arbitrary constant numbers. We have the following theorem. Theorem: Consider robotic system (1) with sliding surface (4) and control laws (9), (13), (14), satisfying Assumptions A l , A2, and A3, then the tracking error e converges to the sliding surface and is restricted to the surface for all subsequent times. Proof: Consider the following Lyapunov function:
v ( s ) = -sTs 2
c
C
+ ~ ~ I I S I I I ~ ; I+ G I I S IiI= 1 lei1
2n
? j = g j ~ / e ~ ~ j - Ilx~i I~, s ~i /=~ 1 , 2 , 3
24
I
I
+-
(c; - c ; ) 2 / g i
P1
(15)
where ci are numbers satisfying ci2 ai& /PI and 2; is its estimation. Differentiating V ( s ) with respect to time t and using (10)-(14), we can observe that
(sTG2B1GZTS)2 )hun(G2B1G2T) IlsIl2 then yields
i= 1
I-sTs
2n
< 0.
I
Hence, the theorem is proved. This, in turn, implies that q q,, q q, as t 00. Hence, the AVSMFC defined by (9), (13), and (14) is globally asymptotically stable and guarantees zero tracking error. +
+
+
Remark: 1) In the theorem, we avoid the difficulties linked to the strict positive realness requirement in traditional AMFC by taking advantage of the inherent positive definiteness of the manipulators' inertia matrix. 2) The matrices K; can be chosen in such a way so as to reduce the lower bounds on ai. This, in turn, reduces the magnitudes of c;. Consequently, the energy required by the adaption mechanism in generation $; is minimized.
C. Insertion of the Boundary Layer While assuring the desired behavior, the control law (9) is discontinuous across the sliding surface s, which leads to control chattering. Chattering is, in general, highly undesirable in practice, since it involves extremely high control activity and, further, may excite high-frequency dynamics neglected during modeling [ 131. We can remedy this situation by smoothing out the control discontinuities in a boundary layer neighboring the sliding surface. Consider the vector 6 = (6, 6, 6,) where 6; > 0 for i = 1 , 2 , 3 . Let us replace $; in (13) by &(6) where
By applying the AVSMFC laws (9), (16), (14), we will guarantee the attractiveness of the boundary layer. For the region inside the boundaries, it can be proved that this will guarantee the ultimate boundedness of the system to within any neighborhood of the boundary layer 1141.
III. SIMULATION EXAMPLE j= 1
i= 1
n
i= 1
Fig. 1 shows a 2-link robotic manipulator model used by [15]. The dynamic equation is given by
j= 1
2n
+~4ll~llC IriIPl + G I l s l l C i= 1 i= 1
e; I P1
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 3, MARCH 1991
350
Y
Is
.s W
'2
0
0.6
0*4
'H
os2 0.0 -0 ,z -0.4
c
-066 -0.8
'n
0
.A
Fig. 1. Two-link robotic manipulator model.
-1.0 -1.2 -1.4 -1.6 2.5
IL
where
oil(+)
+ m2)r: + m2r; + 2 m 2 r , r 2 cos (4)
= (m,
%(+)
= m2r2 =
s k
m2r,'
v
9 2 .o, c,
q,(e,+) = - ( ( m l + m 2 ) r l c 0 s ( + ) + m 2 r 2 c o s ( + + f l ) )
+ 4).
.d
(18)
r,
r2 = 0 . 8 m
= 1m,
m,
c .4
J2=5kg*m
.$
m2 = 6.25 kg.
= 0.5 kg,
i = 1,2
-
::;1 0.8
0.7 0.6
0.5
0.2
0*1 0-,o
.-o. 1
B
A reference model is chosen such that the desired behavior of the arm motion is expressed by two decoupled linear systems
0,; = Arieri+ B r j r j ,
1.2 1.1
0.4 0.3
The parameter values used are the same as those in [15].
J, = 5 k g . m ,
1 1 . 0
h
+ m 2 r l r 2cos (4)
q 2 ( e . + )= - m 2 r 2 c o s ( 8
7.5
Fig. 2. Time response of joint one.
+ J,
+ J~ F12(+) = m 2 r l r 2sin ($1 %(+)
5.0
Tire (5Sz)
-
: 0.0
2.5
7.5
5.0
10.
Tile
(L
Fig. 3. Time response ofjoint two.
(19)
where
c 0
0,935-
c,
We select the following set of reference model parameters:
0.025-
3
0,015..
0
all = U I 2 = u2, = u22 = -1
0.03-
.A
0.02-
b, = b, = 1 . The rise time and the settling time of the reference model are 1.9 s and 8 s, respectively. We choose driving inputs to the reference model to be constants. The goal of the AVSMFC design is to force the tracking errors E , = O r , - t9 and c2 = Or2 to slide along the sliding surfaces. We choose these sliding surfaces:
+
s, = Cl,€, s2
=
C2IE2
+ i, + i2
(21)
where c,, = , c 2 , = 5. The resulting sliding mode equations are two decoupled first-order systems i = -c;E;, i = 1,2. (22) Since we are interested in trajectoxy tracking, we consider a situation characterized by the same initial condition on the reference model state and on the plant state. For this example, we pick the initial displacements and velocities to be
e,,(t0)
=
e ( t o ) = -1.57, e,,(t0)
=
+ ( t o )= o
e,,(to) = e r z ( t o )= b ( t o ) = d ( t o ) = 0. (23) The K ; ( i = 1 , 2 , 3 ) in (9) are chosen so as to reduce the lower bounds of a!; in (1 1). Here, due to the complexity of the matrix A ,
we simply choose
K,=
[I: K 2 = [
K3 = we took as
[
-' -1
-
67.5 17
4 91
15.75 4 17 40
75
20
-' I -1
20 45
L?;( to)( i = 1 * * 5) in control law (14) and reference inputs
?,(to) = z 2 ( t 0 )= L?3(t0)= 25, ? 4 ( t o )= 10, P 5 ( t 0 ) rl
= r2 = 1.
= 30 (25)
(26)
~
35 I
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s
4J
c1
a:;,0 . 0 0 8 0,007
.d 0 , 0 0 6 0 0.005 .n 0,004
3
0.003 0.002 0.001
k 0 k
-0.001
0.0
-0.002 -0.003
-01004
x
-
n L
J
--
-I””
-120
E”
4 5 .o
2.5
0.0
7.5
4
.140 0.0 *
“11,
F‘ 2.5
,,,,1.,,.1
5.0
7.5
10.0
Time (sec)
10.0
Fig. 9. Torque developed at joint two
Time (sec) Fig. 5. Tracking error of joint two.
I
0.06’
0.14 0.12 0-1
I
4 IY I
0-oa
!
-0.06 0
.o
2.5
5
.o
1 7.5
10.0
*d
-5
Time ( s e c ) Fig. 6. Sliding surface s,.
b
-o.or!0.0
2.5
5.0
7.5
10.5
Time (sec)
M
Fig. 10. Tracking error of joint one with boundary layer. h
-0
rd v
I
1
-0.12 0
.o
2.5
5.0
7.5
10.0
Time (sec) Fig. 7. Sliding surface s2. 140 120
I
5L
__
Time (sec) Fig. 11. Tracking error of joint two with boundary layer.
1
420 -140 , 0 .o
-
w 2.5
5 .o
Time (sec) Fig. 8. Torque developed at joint one.
7.5
1
Using control laws (9), (13), (14), Figs. 2 and 3 show the time responses. Figs. 4 and 5 show the tracking errors. Figs. 6 and 7 show the sliding surface which imply the AVSMFC law achieves its objective successfully. Figs. 8 and 9 show torques developed at manipulator joints which result in undesirable robot chattering. To reduce the chattering, we will implement the boundary layer AVSMFC scheme given in (9), (16), and (14). Here, we took 6, = 6, = 0.05. Figs. 10 and 11 show the tracking errors. Figs. 12 and 13 show sliding sectors. Figs. 14 and 15 show the torques exerted at manipulator joints. As can be seen from these figures, chattering is almost eliminated.
352
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 36, NO 3, MARCH 1991
0.28-
0.26 0.26 0.220.20.1s0.160.16. 0.120 .10.08-
Time (sec) Fig. 12. Sliding surface s, with boundary layer. 0.03
IV. CONCLUSION
-0.01
-
-0.02
-
-0.03
+
0 .o
2.5
5
.o
7.. 5
o.
An adaptive variable structure model following control design methodology is presented by using the theory of VSS. The major contribution of this methodology lies in the introduction of a special matrix, which makes it possible for using the Lyapunov function such as V = sTs instead of the requirement siSi< 0, and the derivation of the VSS controller does not require any knowledge of nonlinear robotic systems by adaptation of scalar gain. Chattering during the transient phase can be reduced by using the boundary layer technique. Simulation results show the good performance of the robot system.
Time (sec)
REFERENCES
Fig. 13. Sliding surface s2 with boundary layer.
[ l ] T. C. Hsia, “Adaptive control of robot manipulators-A review,” in Proc. IEEE Int. Conf. Robot. Automat., 1986, p. 183. S. Dubowsky and D. T. DesForses, “The application of model referenced adaptive control to robot manipulators,” Trans. ASME J. Dynam. Syst., Measur., Contr., vol. 101, pp. 193-200, Sept.
h
5
v
a,
9 0 4
C
1979.
H. Durrant-Shyte, “Practical adaptive control of actuated spatial mechanisms,” in Proc. IEEE Int. Conf. Robot. Automat., 1985, p. 650. A. Balestrino, G. DeMaria, and L. Sciavicco, “An adaptive model following control for robot manipulators,” Trans. ASME J . Dynam. Syst., Measur., Contr., vol. 105, pp. 143-152, Sept. 1983. S. Nicosia and P. Tomei, “Model reference adaptive control algorithms for industrial robots,” Automafica, vol. 20, pp. 635-644,
0 U
k
.d -n 0
w
0
--
-10 -20
-
-30 -40 -50
.
0.0
2.5
5.0
7.5
Time (sec) Fig. 14. Torque developed at joint one with boundary layer.
10.0
1984. S. N. Singh, “Adaptive model following control of nonlinear robotic systems,” IEEE Trans. Automat. Contr., vol. AC-30, pp. 1099-1100. 1985. [7] Y. H. Chen, “Adaptive robust model-following control and application to robot manipulators,” Trans. ASME J . Dynam. Syst., Measur.. Contr.. vol. 109. DD.209-215. Seut. 1987.
-
CI .d C
0
3
20100-
-10-
0
-20-
$k
-30-
-40-
-50
.
[9] L. Guzzela and H. P. Geering, “Model following variable structure control for a class of uncertain mechanical systems,” in Proc. 25th Conf. Decision Contr., 1986, p. 312. [lo] V. I. Utkin, Sliding Modes and Their Applications. Moscow: Mir, 1978. [ l l ] 0. M. E. El-Ghezawi, A. S. Zinober, and S. A. Billings, “Analysis
and design of variable structure system using a geometric approach,”
Int. J. Contr., vol. 38, pp. 657-671, 1983. [12] C. M. Dorling and A. S. I. Zinober, “Robust hyperplane design in multivariable variable structure control systems,” Int. J . Contr., vol. 48, pp. 2043-2054, 1988. [13] J.-J. E. Slotine and S. S. Sastry, “Tracking control of nonlinear
