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Exponentially stable trajectory following of robotic manipulators under a class of adaptive controls
0005-1098/92$5.00+ 0.00 PergamonPress Ltd © 1992InternationalFederationof AutomaticControl
Automatica, Vol. 28, No. 3, pp. 579-586,1992
Printedin GreatBritain.
Brief Paper
Exponentially Stable Trajectory Following of Robotic Manipulators Under a Class of Adaptive Controls* ZHIHUA Q U , t DARREN M. DAWSON$ and JOHN F. DORSEY§ Key Words--Lyapunov methods; robots; tracking systems; torque control; robust control; adaptive control; stability.
Ortega and Spong (1989) illustrates that adaptive control of a robot is stable in terms of either asymptotic stability or exponential stability, the existing results have some significant shortcomings. First, the condition of (semi) persistent excitation was required to obtain exponential stability. Second, in order to achieve exponential or asymptotic stability, it was always assumed that the manipulator be subjected to no disturbance or friction. In the case where there exist disturbances, the only available result is BIBO stability (Sadegh and Horowitz, 1990b). For the majority of robot control applications there wiII be unknown friction and/or other disturbance. The requirements to overcome these difficulties, and to make adaptive control a viable method for the tracking control of robots, can be summarized as follows. (1) No requirement such as persistent excitation should be imposed. (2) Friction, disturbance, and other possible u n k n o w n dynamics should be considered as a part of the overall adaptive control system. (3) Explicit expressions for the feedback gains required to achieve stability, that may be based on the initial conditions and bounds on the nonlinear dynamics, are required. (4) The most desirable performance, exponential stability of the tracking error under possible disturbance, must be guaranteed by using an adaptive control scheme. (5) Because disturbances and other unknown dynamics are considered, asymptotic or exponential convergence of the estimation error is not achievable in general. But, the adaptation law for estimates should be stable and a n analytical expression on the bound of the estimation error should be developed. In this note, a new class of adaptive controls is introduced to achieve the above goals. Every choice in the proposed class is a hybrid robust/adaptive control, namely, contains an adaptive part and a robust part. The fundamental idea is that a robust control part is necessary to guarantee desired performance under disturbances. The adaptive control part is a general class itself and includes existing results as special cases. It is shown that the proposed control always guarantees exponential stability of the tracking error and meanwhile keeps parameter estimates bounded under any bounded disturbance. The major difference between the present work and existing results is that unknown, but bounded, functionals are not excluded from the dynamics equation of robotic manipulators. Moreover, the results require neither persistent excitation nor any kind of approximation of nonlinearities.
Abstract--For the trajectory following problem of a robot manipulator, a new class of adaptive controls is introduced. A control in this class consists of a robust control part and an adaptive part. It is shown that the adaptive control part can be chosen to be any of existing adaptation laws and that consequently existing results can be viewed as special cases of the proposed result. The robust control part is added to make the whole adaptive control system be robust with respect to possible u n k n o w n functional dynamics, estimation error, and disturbances. The choice of robust control part depends only on bounding functions of the uncertainties. The global and exponential stability of the position and velocity tracking errors is guaranteed under every control in the proposed class. 1. Introduction
ADAPTIVE CONTROL of robot manipulators has been the subject of considerable research over the last decade. We give a brief synopsis of the primary results. Craig et al. (1986) first introduced a nonlinear adaptive method that guarantees asymptotic stability without any approximation of nonlinear dynamics. The drawback (i.e. the method requires acceleration measurements and invertibility of the estimate of the inertia matrix) was overcome by the latter work of Slotine and Li (1987). The so-called composite control proposed by Siotine and Li (1989) presents a complete solution to the asymptotic adaptive control problem of disturbance-free and friction-free robotic manipulators. Recently, transient performance such as exponential stability has been studied. The works by Sadegh and Horowitz (1990a,b) addressed local, exponential stability of a disturbance-free manipulator under the assumption of (semi) persistent excitation and the condition that a feedforward control is used to cancel possible frictions perfectly. Song et al. (1991) proposed a direct adaptive control to achieve exponentially stable tracking for a friction-free and disturbance-free robotic manipulator. Although the work cited here as well as summarized by *Received 29 January, 1991; revised 9 August, 1991; received in final form 3 October, 1991. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor C. C. Hang under the direction of Editor P. C. Parks. tProfessor Z. Qu is with Department of Electrical Engineering, University of Central Florida, Orlando, FL 32816, U.S.A. Author to whom all correspondence should be addressed. e Professor D. M. Dawson is with Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 29634, U.S.A. § Professor J. F. Dorsey is with School of Electrical Engineering, Georgia Institute of Technology, Atlanta, G A 30332, U.S.A.
2. Problem formulation
We shall consider the dynamics of a robot to be described by the following nonlinear differential equation: T = M(q)~t + N ( q , il),
N(q, ~I)= Vm(q, il)il+ G(q) + Fd(t)t~ + Fs(t])+ rd, 579
(1)
580
Brief Paper
where q is an n x 1 vector of joint variables, T is n x 1 vector of input torques, M(q) is an n x n inertia matrix (symmetric and positive definite), Vm(q,4) is an n x n matrix of centripetal and Coriolis terms, G(q) is an n x 1 vector of gravity terms, Fd(t) is an n x n diagonal matrix of dynamic friction coefficients, F,(4) is an n x 1 vector of static friction terms, and Td is an n x 1 vector of any unknown but bounded disturbances. It is worth noting that To is in general an unknown nonlinear functional of q, q, and time t. Let qd, qd, and /Jd characterize the desired trajectory that the robot should track. It is assumed without loss of any generality that qd, 4a, tla are bounded functions of time. Adaptive control is primarily used for systems with unknown parameters. As shown by Craig (1988), the inertia, Coriolls, centripetal, and gravity terms in (1) can be represented into a regression form, namely,
The following are the important properties of robot dynamics that will be used in this paper. (1) It has been shown by Craig (1988) that the inertia matrix satisfies
_ml.
where it is assumed that _m and n~ be known. Note that this is true in general only for revolute joint robots. (2) The unknown parameters have known upper and lower bounds, namely,
~i<--~i<--~pi,
r}.
llW(q, q, #d)@ -- W(qd, qd, #d)@ll
where W is an n x r known matrix function of q, 4, and #d, and ~ is a r x I vector of unknown constant parameters such as payload mass. The parameterization of robot dynamics, shown in (2), is of vitalimportance for adaptive control of a robotic manipulator. Unlike existing results on adaptive control of robots, we admit there exist unknown functionals in dynamics equation (I). As we can see from (I), possible unknown functionals include dynamic and static friction, bounded disturbances, etc. Let the unknown functional be defined by = Fdq + Fs(q) + Td-
Vie{1 .....
(3) As extension to the work of Sadegh and Horowitz (1990a), it can be shown that W(q, q, qd)~ is a function at most quadratic in q and 4. Thus, its bounding function can be assumed without loss of any generality to be
M(qd)4d -I-Vra(qd, qd)4d + G(qa) = W(qd, 4d, #o)¢P, (2)
H(q, q)
V q e R "×1,
-< P, Ilxll +/~2 Ilxll ~ ~
pl(x),
where I1" II denotes Euclidean norm., x is the vector of the system states defined by x = [E r E r ] r, and E is an n x 1 vector representing the trajectory error, i.e. E = qd -- q(4) It has been shown by Craig (1988) that V,,(q, 4) is a function at most of first order in q and q. Therefore,
IIVm(q, 4)11-
IIFaq + Fjq)ll -
(3)
Position errors under Exponential HARC
O. 0.02 tad 0 0
4 el e2 ~ 2
4
d
d
secs
P o s i t i o n e r r o r s u n d e r U U B RC
O.0 rad
0
4
~
0.02
el e2 sees
e
2
.~
Position errors under D C C L o, ]
A
el e~
secs 0
2
4
6
B
e. 2 P o s i t i o n e r r o r s u n d e r D C C L w i t h s a t u r a t i o n of e s t i m a t e s
rad
0"11[~~ 0
el e2 2
4
6
FIG. 1. Position errors under different controls.
8
581
Brief Paper Velocity errors under Exponential HARC O.S
et e~
rad/s 8
Velocity errors under U U B RC 0.5
el dz
rad/s
o
~
,i
d
d
,i
fi
d
Velocity errors under D C C L
~ad/s
°{(IIi I o
~
2 Velocity errors under D C C L with saturation of estimates
8
2
4
6
8
FIG. 2. Velocity errors under different controls.
parameterized, namely if H(q, il)= O, an adaptive control scheme can be used to yield asymptoticai stability of the tracking error (Sadegh and Horowitz, 1990a). However, due to imperfect knowledge of the robot dynamics, H(q, ~1) is generally non-zero. In this case, a robust control in addition to adaptive control should be used. The control problem addressed in this paper is to combine adaptive and robust control as a whole to make the robot track exponentially any given trajectory under bounded, unknown dynamics. Moreover, we will show that the proposed hybrid robust/adaptive control law is well defined in the sense that both the control and parameter estimates are always bounded. The convergence of parameter estimation error fails in general because of unknown friction and other dynamic functionals in the dynamics of the robot. The proposed class of hybrid adaptive/robust controls has the following formulation:
(6) It is assumed that there exists a known and non-negative function p4(x, t) such that IITdll--
i = Ax + B(AA - T),
(4)
where a=[00
~],
0 B--[M-,(q)]'
AA = H(q, el) + (W(q, Cl, #d) -- W(qd, qd, #d)}O + W(qd, #d, #d)O' It is well known that, if the dynamics of a robot can be fully
(5)
T= U 1 Jr- U2, ul = ko~ + kpe + u~, U2 = W ( q d '
qd' qd)~' •
^
_
-~
If ~i - ~ and iff/(qd, ¢]d,#d, X, t) > 0 ~i =
(qd, qd, #d, X, t)
f eP,=~: but fl(qd, ild, f~d,x,t)
582
Brief Paper
where 4", ~'_are constants (they are_usually chosen such that _~" = ~ a n d 4" = ~i), 4" -< ~'(to) -< q~', kp, k v > 0 are control gains, u~ is a class of rt3bust control which will be given later, and fi(qd, qd, #d, X, t), i = 1. . . . . r, can be chosen to be any well-defined functions. It is easy to see that any choice in the above class contains a robust control part u I and an adaptive control part u 2. For now, the parameter adaptation law is chosen to be those such that the estimates are bounded. A n extension will be made in Section 5 such that no saturation requirement on ~, and consequently no switching in q~, is imposed. Substituting control (5) into (4) yields the error system .~ = / i x + B(AA - u; - u2),
The following Lemma guarantees that P is positive definite. A brief outline of the proof is given in the Appendix.
Lemma 1. The matrix P is positive definite if kp + txko > O. If, in addition, 0 < o : < 1 and kp+Olk~>(1-tr)a-th, the Lyapunov function (7) satisfies the following inequality. 0 -< _~p Ilxll 2 -< V(x) <- Xp Itxll 2, where
_Ap=min{~_m,
kp+°lk~'2a~(1-°l)},
(6)
~o=max{ke+°&~+a'Fa(l+°O
where
2
'
1 + °:rh } 2 "
Remark. The stronger conditions on o:, ;tp, and k, in the above lemma make easier for us to find lower and upper bounds on the eigenvalues of P. These bounds will be used in stability analysis. The requirement that 0 < o~< 1 can be moved if the matrix P is chosen to be
The stability and tracking performance of the robotic manipulator will be investigated in terms of this error system.
3. Choice of Lyapunov function To show that system (6) is stable, consider the Lyapunov function candidate defined by
V(x) = xrPx,
1
(7)
olM(q) ] oCM(q)J'
4. Stability of the error system
=l[(kp + ock~)l+ otZM(q) a'M(q)] 2/ teM(q) M(q) r
We first introduce the following theorem that will facilitate the subsequent discussion.
Parameter estimates under Exponential
6
aM(q)
with 0 < o~< o:'.
where
P
[(kp + te/%)l + a-ZM(q)
P = 2 [_
HARC
rhl r~2 I
Parameter estimates under UUB RC
6
I
il-
rhl ~2
d
d
6
8
2%,Parameter estimates under DCCL Kg
-tell II IJ
.
el
2
.
.
.
4
~ c ~
6
Parameter estimates under DCCL with saturation of estimates
4
rhl rh~
Kg
8
2
4
6
FIG. 3. Torques required under different controls.
8
Brief Paper 40o.
583
Torque outputs under Exponential HARC
~o~
TtT2
NI'I
I
i 2
e
f~
~1
[ sec~
6
8
6
8
6
8
4oo] Torque outputs under UUB RC 2eo~_
Tx T2
bll'l
0
2
~1
Torque outputs under
4oo
~ool~
DCCL
TiT2
bll'l
0
2
'4
2oe~ Torque outputs under DCCL with saturation of estimates
Nil
-
1
B
-200
~
0
2
O
~
~
I
.4
6
~
~Pcs
8
Fro. 4. Parameter estimates under different controls.
Theorem 1. Let V be a Lyapunov function candidate for any given continuous time system with the following properties:
It follows that co(t) -< 0. Then, we have
V (x, t)= V (x(to), to)e -~ct-t°) + [t e-~('-')( e e - # " + co(r))dT (1)
_~ Ilxllml -< V(x, t) <- :, Ilxll ~1 '¢(x, t) e R ~ x R
(2)
V(x, t) -< -;-1 Ilxll "~ + e e - m
V(x, t) ¢ R" X R,
where n ~ > m l > O , ~->~.>0, ;t~,fl, E > 0 are constants. Then, the system is exponentially convergent to the origin in the large. Moreover, the exponential convergence of V(x, t) satisfies the following inequality:
(V(x(to), to)e -x('-'°) J + e(t - to)e -xt, if k = fl V(x, t) <--] V(X(to), to)e -~<'-'°) +
~/(X- #)e-#'°
(e -#(t-t°)-e-x(t-t°)),
~to
< V(x(to), t0)e -~c'-'°) + e e - ~ [ ' e c~-#)" dT, -,o which gives the exponential convergence of V(x,t). Consequently so is ]]xl[ for any initial condition X(to)= Xo. [] The following lemma illustrates the property of the derivative of Lyapunov function V(x) along every trajectory of system (6). /,emma 2. The Lyapunov function (7) satisfies the following inequality for every possible solution of system (6).
if ~.~#,
V(x) -< - Z 1 Ilxll 2 + IIz(x)ll p(x. t) - zr(x)u;.
where Z = 21/~.
Z, ffinfln {~kp, ko} - IIR,Il th, Proof. It follows that
(:(x. t) < - ~ e(x, t) + Ee -#t ~=-~.V(x, t) + Now, define a new variable co(t) as
z(x) ~ [od l]x,
po(t) = tlW(qd, qd, #d)ll " (~ + ~'), ~ e - I it.
~3 ~_.
- 2 _~,}. 2 max {O,.
0'=
max {(~.)2. (_¢;)2}.
.o(x. t) = IIz(x)ll .o2(x) + pl(x) + .o3(x) + pa(x. t) + Po(t). 1
co(t) = V(x, t) + XV(x, t) - ~e-#'.
(8)
where
0
584
Brief Paper
Proof. It follows from (6) and (7) that f'(x) = x r ( A r P + PA + P)x + 2 x ~ P B ( a A - u~ - u2) _
-
_xr, x + !xrr l (q) otM(q)q ~
2
[ atilt(q)
i rl0 + ~x [a, ZM(q)
~4(q)
JX
a'ZM(q) ]
2trM(q)]X
+ x ~ [ : t ] ( a a - u~- .~) = -xrQx + ~zr(x)l(I(q)z(x) 0 + lxrRl[M~ q) M(q)]X + Zr(X)(AA-u~-u2) = - x r a x + zr(x)V,,,(q, :l)z(x) 0 + IxrRI[M~ q) M(q)]x + zT(x)[H(q, il)
function of the state) to compensate for the uncertainties. Therefore, if the ultimate control objective is to obtain a global exponential stability result while ensuring the control bandwidth is finite at any finite instant of time, it is believed that the controller given in this paper is the best trade-off between the designer's ability to achieve desired stability result and his ability to realistically implement the controller. Furthermore, it can be easily shown that the proposed control will not become discontinuous at t = o0 if p(x, t) = p'(x, t)Ilxll. The following theorem shows exponential stability of system (6) under control (5) and (10).
Theorem 2. Consider system (6) under control (5) and (10). For a given 0 < ot < 1, if Zl defined in Lemma 2 is greater than zero and if kp + otk~ > (1 - ot)a,rh, the tracking error x converges to zero exponentially for any initial condition. Moreover, if ~p? ~ Zl, we have Vt -> to IIx(t)ll~ ~ l l x ( t o )
+ (W(q, (1, #a) - W(qd, ild, # a ) ) ¢ + W(qa, (Id, #a)(dP -- (b) -- ul] ,
(9)
where
+_ZplZl_y,gpl e-v'('-'°),
(12)
where ),* = ~min {y, ).l/:tp}.
Proof. Applying control (10) to (8), we have two cases. First, if I1~11> Ee -r', it follows that
The first two steps in the derivation of (9) contain only basic matrix computations. The last step follows from the fact that yrM(q)y=2yrVm(q, il)y holds for any vector y (Craig, 1988). Thus, we have
I?(y, t) -< -).1 Ilxll 2. Secondly, if IIt~ll -< ~e -r', we have f'(x, t ) = -).1 Ilxll 2 + I1~11+ IIz(x)ll" Ilu*(x, t)ll
f'(x) -< - Z l Ilxll 2 + IIz(x)ll 2 P2(x) + IIz(x)ll (P1(x)
-< - Z l Ilxll + 2 I1#11 -< -).1 Ilxll 2 + 2 e e - r ' .
-~-P3(x) -~-P4(x, t) + Po) - zr(x)u;
Consequently, we have that for all (x, t) • R" X R,
-).1 Ilxll 2 + IIz(x)ll p(x, t ) - z r ( x ) u ; .
[]
f'(x, t) -< -).1 Ilxll 2 + 2 E e - : -
Remark. From the proof of the above lemma, we can see easily that, if the adaptive control part in (5) is changed to be u 2 = W(q, il, #d)¢, the bounding function pl(x) can be set to be zero. We let u~ to be chosen from the following class of state feedback controls. For any ~ > 0, the class is defined by
f p(x, t)]l~-~
ul tu*(x, t)
V(x, t) ¢ N
V(x, t) • N'
(13)
Thus, it follows from Theorem 1 that V(x) and consequently every solution x(t; Xo, to) converge to zero exponentially and globally. To prove inequality (12), we note from Lemma 1 and Theorem 1 that _2p Ilzll 2 < V <- V(x(to))e -x'/xpO-'°)
(10)
+~ ).1 - Y).p"
where u*(x, t) is any function satisfying Ilu*(x, t)ll - p(x, t), • , ? > 0 are constants, and
(e-V~t-to) _ e-X¢£p(t-to) L -
"
<--(~p Ilxq x l l 2 + ~ ] e -2v't'-t°) ol I).1 - ~,:tpl}
[]
g(x, t) ~=z(x)p(x, t), N =~ {(x, t): II#(x, t)ll -< Ee-r').
Remark. The existing results on robust controls (Corless and
It is worth noting that control (5) is well defined by noting Ilu~ll <-p(x, t) for all (x, t) • R" × R. It is also worth noting that continuity of control (5) can be guaranteed by choosing u*(x, t) properly. A special, continuous choice in the class is
Leitmann, 1981; Dawson eta/., 1990; Ou and Dorsey, 1991; Qu et al., 1991) guarantee only uniform ultimate boundedness. The result in the above theorem guarantees exponential convergence. The reason is because robust control law (10) is a modified version of existing ones.
f Ul
[ la(x, t ) ~ ] l p(x, t) ~-~
V(x, t) ¢ N V(x, t) • N"
(11)
Remark. The proposed continuous control (10) or (11) may approach a discontinuous (switching) control as time approaches infinity. This implies that, practically, the control requires larger and larger bandwidth (or quicker speed of response) of the actuators as time goes by. At first glance, this observation seems to show that the proposed control has only theoretical importance; however, requiring infinite bandwidth is indeed a necessary condition to guarantee exponential stability. This fact can be seen from the following argument. If the system is exponentially stable, the state of the system is zero at time infinity. At this point in time, any feedback control has three possibilities: that is, the control is equal to zero or a constant; or is a switching function of the state. In all cases, if there are particular uncertainties bounded by constants (and therefore the origin is not an equilibrium point of the system), the state of the system will not remain zero unless the state feedback control has infinite bandwidth (i.e. the control must be a switching
5. Extension of adaptation laws In the last section we have proven that, as long as the parameter adaptation law is chosen such that the parameter estimates are bounded, the tracking error is always made to converge to zero exponentially and globally under control (10). It should be noted that the control, more specifically, the function Po, depends on the constant upper bound of the parameter estimates. Although the adaptation law in (5) can be made to be continuous by choosing the function f~(qd, ild, #d, X, t) properly for all i, the adaptation law does not have a well-defined derivative itself. In many applications, one would like to use the following class of non-switching adaptation laws because it is easy to implement. d~(t)= dt
d ( O - ~) dt
f(qd, ild,#o,X, t),
(14)
where f(qd, ild, #d, X, t) is a well-defined function, and ~(to) is given. We have the following lemma, which shows the boundedness of ~(t).
Brief Paper
Lemma 3. Suppose that x converge to zero exponentially, namely, that there exist a known function s ( - ) > 0 and a constant y* > such that IIx(t)ll-
(15)
Then, there exists a known non-negative function s'(-) such that II~(t)ll -< s'(E, Ilxoll, q~(to), y*) for all t > to.
585
law. The manipulator was modeled as two rigid links of lengths I 1 and 12 with point masses mt and m 2 at the distal ends of the links. The dynamic equations of the manipulator can be found in the book of Craig (1988). The simulation was carried out using SIMNON~ with the following choices. (1) The actual values of the parameters were chosen to be m I = 0.5 kg, mE = 2 kg, and Ii =12= 1 m. (2) An adaptive law was used to identify the parameters m~ and m 2. The initial values of the estimates, rh~(0) and rh2(0), were set to be zero. The specific choice of the adaptation law was chosen to be
= 20Wr(qd, qd' qd)Z(X)' Z(X) = [Of/ 1],
Proof. It follows from (14) that ~(t) = ~(to) + ½
where ~ = [~, 6~2], cr-~0.5, and W is the regression matrix with respect to ~b. The upper and lower bounds for the estimates were chosen to be rh~ =rfi2=5 and
f(qd(t), Od(~'), Od(r), x(r), l') dr.
It follows from the boundedness convergence of Ilxll, and (15) that
of
qd, dld, qa,
the
IIf(qd, qd, qd, X, t)ll -< C(llx(to)ll, qd, £1d, #d) IIx(t)ll, where C > 0 is a constant. Thus, we have II~(t)ll -< II~(to)ll + ~i f(qd(r), ¢ld(r), #d(t'), X(r), T) d~ - II~(to)ll +
I: o
-< II~(t0)ll + C
IIf(qd(r), ¢~d(r), #d(r), X(I~), r)ll dr
~ 1 = -~2 = 0.
(3) The initial conditions and the desired trajectory were set to be qt(0) = q2(0) = t~l(0) = t~2(0) = 0 and qdl = qd2 = sin t. (4) The robot was assumed to be subject to unknown friction F(q, tl) and disturbance Td where r5 sin (3003 Td = [5 sin (30t)J
[2~, + 0.5 sgn (q,)]
F(q, 4) -- L2 2 + O.5 sin (t~2) J'
(5) The hybrid adaptive/robust control was chosen to be u 1 = 25e + 10~ + W(qd, qd, qd)~ + U~,
IIx(r)ll dr
--- II~(t0)ll "~ s(E, I1~o11,Y*) C
~=s'(e, IIx(to)ll, ~(to), ~'*). [] We can now state the following theorem about exponential convergence of the tracking error under a class of non-saturation adaptation laws. The proof can be proceeded by combining the proofs of Theorem 2 and Lemma 3; it is omitted for briefness.
Theorem 3. Consider system (6) under control T = koe + kpe + W(qd, £Id,#d)~I~
[p'(x, t ) ~ + tu**(x, t)
(16)
V(x, t) q N' V(x, t)e N"
=f(qd, qd' qd' X, t), where z(x) is defincd in Lcmma 2, f(.) is a function satisfying (15), u**(x, t) is any function satisfying Ilu**(x, 011 < p'(x, t), E, ~, > 0 are constants, A~ is defined in Lemma 2, y* is defined in Theorem 2, and
p~(t) = IlW(qd, t~a,/]d)ll" (~ + s'(e, IIx(to)ll, ~(t0), Y*)), p'(x, t) = IIz(x)ll p2(x) + p~(x) + p3(x) + p4(x, t) + p~(t), g'(x, t) ~=z(x)p'(x, t), N' ~ {(x, t): II/~'(x, t)ll--< Ee-~'}.
(17)
where u~ is defined in (11) with 1,=0.2, E=0.5, z(x, t) - 0 . 4 e +~, and p = 50+ 10 Ilell + 10 Ilell2. It follows from the discussion in the previous sections that the designer has a lot of freedom in choosing the parameters and the adaptation algorithm in the hybrid adaptive/robust control law. There are several requirements on the choices of the parameters. First, the proportional gains kp and k~ should satisfy Lemma 1. Second, it follows from Theorems 1 and 2 that the parameter y should be chosen to be small. Finally, the bounding function p(x) is at most quadratic in IIxll and has large coefficients to satisfy Lemma 2. The adaptation law (16) was set in simulation to be DCAL (Sadegh and Horowitz, 1990a) for the convenience of comparative study. Simulations were carried out for three controllers: hybrid adaptive/robust control (Exponential HARC) proposed in this paper; robust control guaranteeing uniform ultimate boundedness (UUBRC) suggested in Dawson et al (1990), which can be obtained by letting y = 0 in (17); desired compensation control law (DCCL) given in Sadegh and Horowitz (1990a) with nonlinear feedback term f(z, e ) = 50 Ilell2z(x). The results of the simulation, shown in Figs 1-4, demonstrate that the proposed HARC works effectively.
7. Conclusion
Choose the gains kp and k~ such that A I > 0 and kp + trk~ > ( 1 - o~)trgt. Then, for a given 0 < ~r< 1 and for any initial condition Ilx(t0)ll and q~(to), there exists a constant s'(E, IIx(to)ll, ¢(t0), y*)), which depends only on the initial condition, such that the tracking error x converges to zero exponentially. Moreover, if 3.py =/=AI we have Vt ~>to
A new class of adaptive controls is proposed for robot trajectory following problem. Every one in the class is a hybrid robust and adaptive control and always guarantees that the tracking error converges to zero exponentially even under unknown functionals of disturbances or any other nonlinear dynamics. The only a priori information required is bounding functions of possible unknown and nonlinear dynamics. No other requirement such as persistent excitation is needed.
4E).p e_y.(t_t0). IIx(t)ll - ~ _ ~ IIx(t0)ll2 +_Zp IX1 - Y),pl
Acknowledgement--The authors are grateful to the reviewers for their helpful comments on this paper.
Remark. The constant s'(E, IIx(t0)ll, ~(t0), I'*)) can be found in a similar fashion as the proof of Lemma 3. The difference between Theorems 2 and 3 is that Theorem 2 is a global result but requires switching in adaptation law while Theorem 3 does not require switching and works for any initial conditon but, strictly speaking, is a local result.
6. Simulation example A simple two-degree-of-freedom manipulator was simulated to test the proposed hybrid adaptive/robust control
References Corless, M. J. and G. Leitmann (1981). Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems. IEEE Trans. Automat. Contr., 26, 1139-1144. Craig, J. J. (1988). Adaptive Control of Mechanical Manipulators. Addison-Wesley, New York. Craig, J. J., P. Hsu and S. S. Sastry (1986). Adaptive control of mechanical manipulators. Proceedings of the 1986 IEEE
586
Brief Paper
International Conference on Robotics and Automation, San Francisco, CA. Dawson, D. M., Z. Qu, F. L. Lewis and J. F. Dorsey (1990). Robust control for the tracking of robot motion. Int. J. Control, $2, 581-595. Ortega, R. and M. W. Spong (1989). Adaptive motion control of rigid robots: A tutorial. Automatica, 25, 877-888. Qu, Z. and J. F. Dorsey (1991). Robust tracking control of robots by a linear feedback law. IEEE Trans. Automat. Contr., 38, 1081-1084. Qu, Z., J. F. Dorsey, X. Zhang and D. M. Dawson (1991). Robust control of robots by computed torque law. Systems & Control Letters, 16, 25-32. Sadegh, N. and R. Horowitz (1990a). Stability and robustness analysis of a class of adaptive controllers for robotic manipulators. Int. J. of Robotics Res., 9, 74-92. Sadegh, N. and R. Horowitz (1990b). An exponentially stable adaptive control law for robotic manipulators. IEEE Transactions on Robotics and Automation, 6, 491-496. Slotine, J. J. E. and W. Li (1987). On the adaptive control of robot manipulators. Int. J. Robotics Res., 6, 49-59. Slotine, J. J. E. and W. Li (1989). Composite adaptive control of robot manipulators. Automatica, 24, 509-520. Song, Y. D., R. H. Middieton and J. N. Anderson (1991). Study on the exponential path tracking control of robot manipulators via direct adaptive methods. 1991 American Control Conference, Boston, MA. Stewart, G. W. (1973). Introduction to Matrix Computations. Academic Press, New York. Appendix Proof of Lemma 1. It is easy to see that xrpx = Er(kp + ¢k~)E + (E + ¢E)rM(/~ + ~E) -> O, Vx ¢ R 2~.
To see that P is positive definite if kp + otk~ > O, we notice that xrPx = O:> E r (kp + od%)E = O, and (/~ + ~ E ) r M ( k + ~E) = O ~ x = O, since M is positive definite. To develop bounds on the eigenvalues of P, let T be a transformation such that M = T-IAT, where A = diag {al . . . . . an} and the ai are the eigenvalues of M. It follows that [To-1
0 ]piT
+trkv)l+a'2A
AA]
(18)
o^
By the Gershgorin Theorem (Stewart, 1973), we know that the eigenvalues z/, j = 1 . . . . . 2n, of the matrix on the right-hand side of (18) satisfy the following inequalities: - ota, + kp + o&~ + ~ a i <-2zi <- eta, + kp + ~k~ + ~a~, (1 -- o[)ai<2Zn+ i ~--(1 "t- oOai, for i -- 1 , . . . , n. The constant upper and lower bounds on z/, ] ffi 1. . . . . 2n, can be easily found by using the fact that < a~ < th for all time and all i. []
Automatica, Vol. 28, No. 3, pp. 579-586,1992
Printedin GreatBritain.
Brief Paper
Exponentially Stable Trajectory Following of Robotic Manipulators Under a Class of Adaptive Controls* ZHIHUA Q U , t DARREN M. DAWSON$ and JOHN F. DORSEY§ Key Words--Lyapunov methods; robots; tracking systems; torque control; robust control; adaptive control; stability.
Ortega and Spong (1989) illustrates that adaptive control of a robot is stable in terms of either asymptotic stability or exponential stability, the existing results have some significant shortcomings. First, the condition of (semi) persistent excitation was required to obtain exponential stability. Second, in order to achieve exponential or asymptotic stability, it was always assumed that the manipulator be subjected to no disturbance or friction. In the case where there exist disturbances, the only available result is BIBO stability (Sadegh and Horowitz, 1990b). For the majority of robot control applications there wiII be unknown friction and/or other disturbance. The requirements to overcome these difficulties, and to make adaptive control a viable method for the tracking control of robots, can be summarized as follows. (1) No requirement such as persistent excitation should be imposed. (2) Friction, disturbance, and other possible u n k n o w n dynamics should be considered as a part of the overall adaptive control system. (3) Explicit expressions for the feedback gains required to achieve stability, that may be based on the initial conditions and bounds on the nonlinear dynamics, are required. (4) The most desirable performance, exponential stability of the tracking error under possible disturbance, must be guaranteed by using an adaptive control scheme. (5) Because disturbances and other unknown dynamics are considered, asymptotic or exponential convergence of the estimation error is not achievable in general. But, the adaptation law for estimates should be stable and a n analytical expression on the bound of the estimation error should be developed. In this note, a new class of adaptive controls is introduced to achieve the above goals. Every choice in the proposed class is a hybrid robust/adaptive control, namely, contains an adaptive part and a robust part. The fundamental idea is that a robust control part is necessary to guarantee desired performance under disturbances. The adaptive control part is a general class itself and includes existing results as special cases. It is shown that the proposed control always guarantees exponential stability of the tracking error and meanwhile keeps parameter estimates bounded under any bounded disturbance. The major difference between the present work and existing results is that unknown, but bounded, functionals are not excluded from the dynamics equation of robotic manipulators. Moreover, the results require neither persistent excitation nor any kind of approximation of nonlinearities.
Abstract--For the trajectory following problem of a robot manipulator, a new class of adaptive controls is introduced. A control in this class consists of a robust control part and an adaptive part. It is shown that the adaptive control part can be chosen to be any of existing adaptation laws and that consequently existing results can be viewed as special cases of the proposed result. The robust control part is added to make the whole adaptive control system be robust with respect to possible u n k n o w n functional dynamics, estimation error, and disturbances. The choice of robust control part depends only on bounding functions of the uncertainties. The global and exponential stability of the position and velocity tracking errors is guaranteed under every control in the proposed class. 1. Introduction
ADAPTIVE CONTROL of robot manipulators has been the subject of considerable research over the last decade. We give a brief synopsis of the primary results. Craig et al. (1986) first introduced a nonlinear adaptive method that guarantees asymptotic stability without any approximation of nonlinear dynamics. The drawback (i.e. the method requires acceleration measurements and invertibility of the estimate of the inertia matrix) was overcome by the latter work of Slotine and Li (1987). The so-called composite control proposed by Siotine and Li (1989) presents a complete solution to the asymptotic adaptive control problem of disturbance-free and friction-free robotic manipulators. Recently, transient performance such as exponential stability has been studied. The works by Sadegh and Horowitz (1990a,b) addressed local, exponential stability of a disturbance-free manipulator under the assumption of (semi) persistent excitation and the condition that a feedforward control is used to cancel possible frictions perfectly. Song et al. (1991) proposed a direct adaptive control to achieve exponentially stable tracking for a friction-free and disturbance-free robotic manipulator. Although the work cited here as well as summarized by *Received 29 January, 1991; revised 9 August, 1991; received in final form 3 October, 1991. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor C. C. Hang under the direction of Editor P. C. Parks. tProfessor Z. Qu is with Department of Electrical Engineering, University of Central Florida, Orlando, FL 32816, U.S.A. Author to whom all correspondence should be addressed. e Professor D. M. Dawson is with Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 29634, U.S.A. § Professor J. F. Dorsey is with School of Electrical Engineering, Georgia Institute of Technology, Atlanta, G A 30332, U.S.A.
2. Problem formulation
We shall consider the dynamics of a robot to be described by the following nonlinear differential equation: T = M(q)~t + N ( q , il),
N(q, ~I)= Vm(q, il)il+ G(q) + Fd(t)t~ + Fs(t])+ rd, 579
(1)
580
Brief Paper
where q is an n x 1 vector of joint variables, T is n x 1 vector of input torques, M(q) is an n x n inertia matrix (symmetric and positive definite), Vm(q,4) is an n x n matrix of centripetal and Coriolis terms, G(q) is an n x 1 vector of gravity terms, Fd(t) is an n x n diagonal matrix of dynamic friction coefficients, F,(4) is an n x 1 vector of static friction terms, and Td is an n x 1 vector of any unknown but bounded disturbances. It is worth noting that To is in general an unknown nonlinear functional of q, q, and time t. Let qd, qd, and /Jd characterize the desired trajectory that the robot should track. It is assumed without loss of any generality that qd, 4a, tla are bounded functions of time. Adaptive control is primarily used for systems with unknown parameters. As shown by Craig (1988), the inertia, Coriolls, centripetal, and gravity terms in (1) can be represented into a regression form, namely,
The following are the important properties of robot dynamics that will be used in this paper. (1) It has been shown by Craig (1988) that the inertia matrix satisfies
_ml.
where it is assumed that _m and n~ be known. Note that this is true in general only for revolute joint robots. (2) The unknown parameters have known upper and lower bounds, namely,
~i<--~i<--~pi,
r}.
llW(q, q, #d)@ -- W(qd, qd, #d)@ll
where W is an n x r known matrix function of q, 4, and #d, and ~ is a r x I vector of unknown constant parameters such as payload mass. The parameterization of robot dynamics, shown in (2), is of vitalimportance for adaptive control of a robotic manipulator. Unlike existing results on adaptive control of robots, we admit there exist unknown functionals in dynamics equation (I). As we can see from (I), possible unknown functionals include dynamic and static friction, bounded disturbances, etc. Let the unknown functional be defined by = Fdq + Fs(q) + Td-
Vie{1 .....
(3) As extension to the work of Sadegh and Horowitz (1990a), it can be shown that W(q, q, qd)~ is a function at most quadratic in q and 4. Thus, its bounding function can be assumed without loss of any generality to be
M(qd)4d -I-Vra(qd, qd)4d + G(qa) = W(qd, 4d, #o)¢P, (2)
H(q, q)
V q e R "×1,
-< P, Ilxll +/~2 Ilxll ~ ~
pl(x),
where I1" II denotes Euclidean norm., x is the vector of the system states defined by x = [E r E r ] r, and E is an n x 1 vector representing the trajectory error, i.e. E = qd -- q(4) It has been shown by Craig (1988) that V,,(q, 4) is a function at most of first order in q and q. Therefore,
IIVm(q, 4)11-
IIFaq + Fjq)ll -
(3)
Position errors under Exponential HARC
O. 0.02 tad 0 0
4 el e2 ~ 2
4
d
d
secs
P o s i t i o n e r r o r s u n d e r U U B RC
O.0 rad
0
4
~
0.02
el e2 sees
e
2
.~
Position errors under D C C L o, ]
A
el e~
secs 0
2
4
6
B
e. 2 P o s i t i o n e r r o r s u n d e r D C C L w i t h s a t u r a t i o n of e s t i m a t e s
rad
0"11[~~ 0
el e2 2
4
6
FIG. 1. Position errors under different controls.
8
581
Brief Paper Velocity errors under Exponential HARC O.S
et e~
rad/s 8
Velocity errors under U U B RC 0.5
el dz
rad/s
o
~
,i
d
d
,i
fi
d
Velocity errors under D C C L
~ad/s
°{(IIi I o
~
2 Velocity errors under D C C L with saturation of estimates
8
2
4
6
8
FIG. 2. Velocity errors under different controls.
parameterized, namely if H(q, il)= O, an adaptive control scheme can be used to yield asymptoticai stability of the tracking error (Sadegh and Horowitz, 1990a). However, due to imperfect knowledge of the robot dynamics, H(q, ~1) is generally non-zero. In this case, a robust control in addition to adaptive control should be used. The control problem addressed in this paper is to combine adaptive and robust control as a whole to make the robot track exponentially any given trajectory under bounded, unknown dynamics. Moreover, we will show that the proposed hybrid robust/adaptive control law is well defined in the sense that both the control and parameter estimates are always bounded. The convergence of parameter estimation error fails in general because of unknown friction and other dynamic functionals in the dynamics of the robot. The proposed class of hybrid adaptive/robust controls has the following formulation:
(6) It is assumed that there exists a known and non-negative function p4(x, t) such that IITdll--
i = Ax + B(AA - T),
(4)
where a=[00
~],
0 B--[M-,(q)]'
AA = H(q, el) + (W(q, Cl, #d) -- W(qd, qd, #d)}O + W(qd, #d, #d)O' It is well known that, if the dynamics of a robot can be fully
(5)
T= U 1 Jr- U2, ul = ko~ + kpe + u~, U2 = W ( q d '
qd' qd)~' •
^
_
-~
If ~i - ~ and iff/(qd, ¢]d,#d, X, t) > 0 ~i =
(qd, qd, #d, X, t)
f eP,=~: but fl(qd, ild, f~d,x,t)
582
Brief Paper
where 4", ~'_are constants (they are_usually chosen such that _~" = ~ a n d 4" = ~i), 4" -< ~'(to) -< q~', kp, k v > 0 are control gains, u~ is a class of rt3bust control which will be given later, and fi(qd, qd, #d, X, t), i = 1. . . . . r, can be chosen to be any well-defined functions. It is easy to see that any choice in the above class contains a robust control part u I and an adaptive control part u 2. For now, the parameter adaptation law is chosen to be those such that the estimates are bounded. A n extension will be made in Section 5 such that no saturation requirement on ~, and consequently no switching in q~, is imposed. Substituting control (5) into (4) yields the error system .~ = / i x + B(AA - u; - u2),
The following Lemma guarantees that P is positive definite. A brief outline of the proof is given in the Appendix.
Lemma 1. The matrix P is positive definite if kp + txko > O. If, in addition, 0 < o : < 1 and kp+Olk~>(1-tr)a-th, the Lyapunov function (7) satisfies the following inequality. 0 -< _~p Ilxll 2 -< V(x) <- Xp Itxll 2, where
_Ap=min{~_m,
kp+°lk~'2a~(1-°l)},
(6)
~o=max{ke+°&~+a'Fa(l+°O
where
2
'
1 + °:rh } 2 "
Remark. The stronger conditions on o:, ;tp, and k, in the above lemma make easier for us to find lower and upper bounds on the eigenvalues of P. These bounds will be used in stability analysis. The requirement that 0 < o~< 1 can be moved if the matrix P is chosen to be
The stability and tracking performance of the robotic manipulator will be investigated in terms of this error system.
3. Choice of Lyapunov function To show that system (6) is stable, consider the Lyapunov function candidate defined by
V(x) = xrPx,
1
(7)
olM(q) ] oCM(q)J'
4. Stability of the error system
=l[(kp + ock~)l+ otZM(q) a'M(q)] 2/ teM(q) M(q) r
We first introduce the following theorem that will facilitate the subsequent discussion.
Parameter estimates under Exponential
6
aM(q)
with 0 < o~< o:'.
where
P
[(kp + te/%)l + a-ZM(q)
P = 2 [_
HARC
rhl r~2 I
Parameter estimates under UUB RC
6
I
il-
rhl ~2
d
d
6
8
2%,Parameter estimates under DCCL Kg
-tell II IJ
.
el
2
.
.
.
4
~ c ~
6
Parameter estimates under DCCL with saturation of estimates
4
rhl rh~
Kg
8
2
4
6
FIG. 3. Torques required under different controls.
8
Brief Paper 40o.
583
Torque outputs under Exponential HARC
~o~
TtT2
NI'I
I
i 2
e
f~
~1
[ sec~
6
8
6
8
6
8
4oo] Torque outputs under UUB RC 2eo~_
Tx T2
bll'l
0
2
~1
Torque outputs under
4oo
~ool~
DCCL
TiT2
bll'l
0
2
'4
2oe~ Torque outputs under DCCL with saturation of estimates
Nil
-
1
B
-200
~
0
2
O
~
~
I
.4
6
~
~Pcs
8
Fro. 4. Parameter estimates under different controls.
Theorem 1. Let V be a Lyapunov function candidate for any given continuous time system with the following properties:
It follows that co(t) -< 0. Then, we have
V (x, t)= V (x(to), to)e -~ct-t°) + [t e-~('-')( e e - # " + co(r))dT (1)
_~ Ilxllml -< V(x, t) <- :, Ilxll ~1 '¢(x, t) e R ~ x R
(2)
V(x, t) -< -;-1 Ilxll "~ + e e - m
V(x, t) ¢ R" X R,
where n ~ > m l > O , ~->~.>0, ;t~,fl, E > 0 are constants. Then, the system is exponentially convergent to the origin in the large. Moreover, the exponential convergence of V(x, t) satisfies the following inequality:
(V(x(to), to)e -x('-'°) J + e(t - to)e -xt, if k = fl V(x, t) <--] V(X(to), to)e -~<'-'°) +
~/(X- #)e-#'°
(e -#(t-t°)-e-x(t-t°)),
~to
< V(x(to), t0)e -~c'-'°) + e e - ~ [ ' e c~-#)" dT, -,o which gives the exponential convergence of V(x,t). Consequently so is ]]xl[ for any initial condition X(to)= Xo. [] The following lemma illustrates the property of the derivative of Lyapunov function V(x) along every trajectory of system (6). /,emma 2. The Lyapunov function (7) satisfies the following inequality for every possible solution of system (6).
if ~.~#,
V(x) -< - Z 1 Ilxll 2 + IIz(x)ll p(x. t) - zr(x)u;.
where Z = 21/~.
Z, ffinfln {~kp, ko} - IIR,Il th, Proof. It follows that
(:(x. t) < - ~ e(x, t) + Ee -#t ~=-~.V(x, t) + Now, define a new variable co(t) as
z(x) ~ [od l]x,
po(t) = tlW(qd, qd, #d)ll " (~ + ~'), ~ e - I it.
~3 ~_.
- 2 _~,}. 2 max {O,.
0'=
max {(~.)2. (_¢;)2}.
.o(x. t) = IIz(x)ll .o2(x) + pl(x) + .o3(x) + pa(x. t) + Po(t). 1
co(t) = V(x, t) + XV(x, t) - ~e-#'.
(8)
where
0
584
Brief Paper
Proof. It follows from (6) and (7) that f'(x) = x r ( A r P + PA + P)x + 2 x ~ P B ( a A - u~ - u2) _
-
_xr, x + !xrr l (q) otM(q)q ~
2
[ atilt(q)
i rl0 + ~x [a, ZM(q)
~4(q)
JX
a'ZM(q) ]
2trM(q)]X
+ x ~ [ : t ] ( a a - u~- .~) = -xrQx + ~zr(x)l(I(q)z(x) 0 + lxrRl[M~ q) M(q)]X + Zr(X)(AA-u~-u2) = - x r a x + zr(x)V,,,(q, :l)z(x) 0 + IxrRI[M~ q) M(q)]x + zT(x)[H(q, il)
function of the state) to compensate for the uncertainties. Therefore, if the ultimate control objective is to obtain a global exponential stability result while ensuring the control bandwidth is finite at any finite instant of time, it is believed that the controller given in this paper is the best trade-off between the designer's ability to achieve desired stability result and his ability to realistically implement the controller. Furthermore, it can be easily shown that the proposed control will not become discontinuous at t = o0 if p(x, t) = p'(x, t)Ilxll. The following theorem shows exponential stability of system (6) under control (5) and (10).
Theorem 2. Consider system (6) under control (5) and (10). For a given 0 < ot < 1, if Zl defined in Lemma 2 is greater than zero and if kp + otk~ > (1 - ot)a,rh, the tracking error x converges to zero exponentially for any initial condition. Moreover, if ~p? ~ Zl, we have Vt -> to IIx(t)ll~ ~ l l x ( t o )
+ (W(q, (1, #a) - W(qd, ild, # a ) ) ¢ + W(qa, (Id, #a)(dP -- (b) -- ul] ,
(9)
where
+_ZplZl_y,gpl e-v'('-'°),
(12)
where ),* = ~min {y, ).l/:tp}.
Proof. Applying control (10) to (8), we have two cases. First, if I1~11> Ee -r', it follows that
The first two steps in the derivation of (9) contain only basic matrix computations. The last step follows from the fact that yrM(q)y=2yrVm(q, il)y holds for any vector y (Craig, 1988). Thus, we have
I?(y, t) -< -).1 Ilxll 2. Secondly, if IIt~ll -< ~e -r', we have f'(x, t ) = -).1 Ilxll 2 + I1~11+ IIz(x)ll" Ilu*(x, t)ll
f'(x) -< - Z l Ilxll 2 + IIz(x)ll 2 P2(x) + IIz(x)ll (P1(x)
-< - Z l Ilxll + 2 I1#11 -< -).1 Ilxll 2 + 2 e e - r ' .
-~-P3(x) -~-P4(x, t) + Po) - zr(x)u;
Consequently, we have that for all (x, t) • R" X R,
-).1 Ilxll 2 + IIz(x)ll p(x, t ) - z r ( x ) u ; .
[]
f'(x, t) -< -).1 Ilxll 2 + 2 E e - : -
Remark. From the proof of the above lemma, we can see easily that, if the adaptive control part in (5) is changed to be u 2 = W(q, il, #d)¢, the bounding function pl(x) can be set to be zero. We let u~ to be chosen from the following class of state feedback controls. For any ~ > 0, the class is defined by
f p(x, t)]l~-~
ul tu*(x, t)
V(x, t) ¢ N
V(x, t) • N'
(13)
Thus, it follows from Theorem 1 that V(x) and consequently every solution x(t; Xo, to) converge to zero exponentially and globally. To prove inequality (12), we note from Lemma 1 and Theorem 1 that _2p Ilzll 2 < V <- V(x(to))e -x'/xpO-'°)
(10)
+~ ).1 - Y).p"
where u*(x, t) is any function satisfying Ilu*(x, t)ll - p(x, t), • , ? > 0 are constants, and
(e-V~t-to) _ e-X¢£p(t-to) L -
"
<--(~p Ilxq x l l 2 + ~ ] e -2v't'-t°) ol I).1 - ~,:tpl}
[]
g(x, t) ~=z(x)p(x, t), N =~ {(x, t): II#(x, t)ll -< Ee-r').
Remark. The existing results on robust controls (Corless and
It is worth noting that control (5) is well defined by noting Ilu~ll <-p(x, t) for all (x, t) • R" × R. It is also worth noting that continuity of control (5) can be guaranteed by choosing u*(x, t) properly. A special, continuous choice in the class is
Leitmann, 1981; Dawson eta/., 1990; Ou and Dorsey, 1991; Qu et al., 1991) guarantee only uniform ultimate boundedness. The result in the above theorem guarantees exponential convergence. The reason is because robust control law (10) is a modified version of existing ones.
f Ul
[ la(x, t ) ~ ] l p(x, t) ~-~
V(x, t) ¢ N V(x, t) • N"
(11)
Remark. The proposed continuous control (10) or (11) may approach a discontinuous (switching) control as time approaches infinity. This implies that, practically, the control requires larger and larger bandwidth (or quicker speed of response) of the actuators as time goes by. At first glance, this observation seems to show that the proposed control has only theoretical importance; however, requiring infinite bandwidth is indeed a necessary condition to guarantee exponential stability. This fact can be seen from the following argument. If the system is exponentially stable, the state of the system is zero at time infinity. At this point in time, any feedback control has three possibilities: that is, the control is equal to zero or a constant; or is a switching function of the state. In all cases, if there are particular uncertainties bounded by constants (and therefore the origin is not an equilibrium point of the system), the state of the system will not remain zero unless the state feedback control has infinite bandwidth (i.e. the control must be a switching
5. Extension of adaptation laws In the last section we have proven that, as long as the parameter adaptation law is chosen such that the parameter estimates are bounded, the tracking error is always made to converge to zero exponentially and globally under control (10). It should be noted that the control, more specifically, the function Po, depends on the constant upper bound of the parameter estimates. Although the adaptation law in (5) can be made to be continuous by choosing the function f~(qd, ild, #d, X, t) properly for all i, the adaptation law does not have a well-defined derivative itself. In many applications, one would like to use the following class of non-switching adaptation laws because it is easy to implement. d~(t)= dt
d ( O - ~) dt
f(qd, ild,#o,X, t),
(14)
where f(qd, ild, #d, X, t) is a well-defined function, and ~(to) is given. We have the following lemma, which shows the boundedness of ~(t).
Brief Paper
Lemma 3. Suppose that x converge to zero exponentially, namely, that there exist a known function s ( - ) > 0 and a constant y* > such that IIx(t)ll-
(15)
Then, there exists a known non-negative function s'(-) such that II~(t)ll -< s'(E, Ilxoll, q~(to), y*) for all t > to.
585
law. The manipulator was modeled as two rigid links of lengths I 1 and 12 with point masses mt and m 2 at the distal ends of the links. The dynamic equations of the manipulator can be found in the book of Craig (1988). The simulation was carried out using SIMNON~ with the following choices. (1) The actual values of the parameters were chosen to be m I = 0.5 kg, mE = 2 kg, and Ii =12= 1 m. (2) An adaptive law was used to identify the parameters m~ and m 2. The initial values of the estimates, rh~(0) and rh2(0), were set to be zero. The specific choice of the adaptation law was chosen to be
= 20Wr(qd, qd' qd)Z(X)' Z(X) = [Of/ 1],
Proof. It follows from (14) that ~(t) = ~(to) + ½
where ~ = [~, 6~2], cr-~0.5, and W is the regression matrix with respect to ~b. The upper and lower bounds for the estimates were chosen to be rh~ =rfi2=5 and
f(qd(t), Od(~'), Od(r), x(r), l') dr.
It follows from the boundedness convergence of Ilxll, and (15) that
of
qd, dld, qa,
the
IIf(qd, qd, qd, X, t)ll -< C(llx(to)ll, qd, £1d, #d) IIx(t)ll, where C > 0 is a constant. Thus, we have II~(t)ll -< II~(to)ll + ~i f(qd(r), ¢ld(r), #d(t'), X(r), T) d~ - II~(to)ll +
I: o
-< II~(t0)ll + C
IIf(qd(r), ¢~d(r), #d(r), X(I~), r)ll dr
~ 1 = -~2 = 0.
(3) The initial conditions and the desired trajectory were set to be qt(0) = q2(0) = t~l(0) = t~2(0) = 0 and qdl = qd2 = sin t. (4) The robot was assumed to be subject to unknown friction F(q, tl) and disturbance Td where r5 sin (3003 Td = [5 sin (30t)J
[2~, + 0.5 sgn (q,)]
F(q, 4) -- L2 2 + O.5 sin (t~2) J'
(5) The hybrid adaptive/robust control was chosen to be u 1 = 25e + 10~ + W(qd, qd, qd)~ + U~,
IIx(r)ll dr
--- II~(t0)ll "~ s(E, I1~o11,Y*) C
~=s'(e, IIx(to)ll, ~(to), ~'*). [] We can now state the following theorem about exponential convergence of the tracking error under a class of non-saturation adaptation laws. The proof can be proceeded by combining the proofs of Theorem 2 and Lemma 3; it is omitted for briefness.
Theorem 3. Consider system (6) under control T = koe + kpe + W(qd, £Id,#d)~I~
[p'(x, t ) ~ + tu**(x, t)
(16)
V(x, t) q N' V(x, t)e N"
=f(qd, qd' qd' X, t), where z(x) is defincd in Lcmma 2, f(.) is a function satisfying (15), u**(x, t) is any function satisfying Ilu**(x, 011 < p'(x, t), E, ~, > 0 are constants, A~ is defined in Lemma 2, y* is defined in Theorem 2, and
p~(t) = IlW(qd, t~a,/]d)ll" (~ + s'(e, IIx(to)ll, ~(t0), Y*)), p'(x, t) = IIz(x)ll p2(x) + p~(x) + p3(x) + p4(x, t) + p~(t), g'(x, t) ~=z(x)p'(x, t), N' ~ {(x, t): II/~'(x, t)ll--< Ee-~'}.
(17)
where u~ is defined in (11) with 1,=0.2, E=0.5, z(x, t) - 0 . 4 e +~, and p = 50+ 10 Ilell + 10 Ilell2. It follows from the discussion in the previous sections that the designer has a lot of freedom in choosing the parameters and the adaptation algorithm in the hybrid adaptive/robust control law. There are several requirements on the choices of the parameters. First, the proportional gains kp and k~ should satisfy Lemma 1. Second, it follows from Theorems 1 and 2 that the parameter y should be chosen to be small. Finally, the bounding function p(x) is at most quadratic in IIxll and has large coefficients to satisfy Lemma 2. The adaptation law (16) was set in simulation to be DCAL (Sadegh and Horowitz, 1990a) for the convenience of comparative study. Simulations were carried out for three controllers: hybrid adaptive/robust control (Exponential HARC) proposed in this paper; robust control guaranteeing uniform ultimate boundedness (UUBRC) suggested in Dawson et al (1990), which can be obtained by letting y = 0 in (17); desired compensation control law (DCCL) given in Sadegh and Horowitz (1990a) with nonlinear feedback term f(z, e ) = 50 Ilell2z(x). The results of the simulation, shown in Figs 1-4, demonstrate that the proposed HARC works effectively.
7. Conclusion
Choose the gains kp and k~ such that A I > 0 and kp + trk~ > ( 1 - o~)trgt. Then, for a given 0 < ~r< 1 and for any initial condition Ilx(t0)ll and q~(to), there exists a constant s'(E, IIx(to)ll, ¢(t0), y*)), which depends only on the initial condition, such that the tracking error x converges to zero exponentially. Moreover, if 3.py =/=AI we have Vt ~>to
A new class of adaptive controls is proposed for robot trajectory following problem. Every one in the class is a hybrid robust and adaptive control and always guarantees that the tracking error converges to zero exponentially even under unknown functionals of disturbances or any other nonlinear dynamics. The only a priori information required is bounding functions of possible unknown and nonlinear dynamics. No other requirement such as persistent excitation is needed.
4E).p e_y.(t_t0). IIx(t)ll - ~ _ ~ IIx(t0)ll2 +_Zp IX1 - Y),pl
Acknowledgement--The authors are grateful to the reviewers for their helpful comments on this paper.
Remark. The constant s'(E, IIx(t0)ll, ~(t0), I'*)) can be found in a similar fashion as the proof of Lemma 3. The difference between Theorems 2 and 3 is that Theorem 2 is a global result but requires switching in adaptation law while Theorem 3 does not require switching and works for any initial conditon but, strictly speaking, is a local result.
6. Simulation example A simple two-degree-of-freedom manipulator was simulated to test the proposed hybrid adaptive/robust control
References Corless, M. J. and G. Leitmann (1981). Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems. IEEE Trans. Automat. Contr., 26, 1139-1144. Craig, J. J. (1988). Adaptive Control of Mechanical Manipulators. Addison-Wesley, New York. Craig, J. J., P. Hsu and S. S. Sastry (1986). Adaptive control of mechanical manipulators. Proceedings of the 1986 IEEE
586
Brief Paper
International Conference on Robotics and Automation, San Francisco, CA. Dawson, D. M., Z. Qu, F. L. Lewis and J. F. Dorsey (1990). Robust control for the tracking of robot motion. Int. J. Control, $2, 581-595. Ortega, R. and M. W. Spong (1989). Adaptive motion control of rigid robots: A tutorial. Automatica, 25, 877-888. Qu, Z. and J. F. Dorsey (1991). Robust tracking control of robots by a linear feedback law. IEEE Trans. Automat. Contr., 38, 1081-1084. Qu, Z., J. F. Dorsey, X. Zhang and D. M. Dawson (1991). Robust control of robots by computed torque law. Systems & Control Letters, 16, 25-32. Sadegh, N. and R. Horowitz (1990a). Stability and robustness analysis of a class of adaptive controllers for robotic manipulators. Int. J. of Robotics Res., 9, 74-92. Sadegh, N. and R. Horowitz (1990b). An exponentially stable adaptive control law for robotic manipulators. IEEE Transactions on Robotics and Automation, 6, 491-496. Slotine, J. J. E. and W. Li (1987). On the adaptive control of robot manipulators. Int. J. Robotics Res., 6, 49-59. Slotine, J. J. E. and W. Li (1989). Composite adaptive control of robot manipulators. Automatica, 24, 509-520. Song, Y. D., R. H. Middieton and J. N. Anderson (1991). Study on the exponential path tracking control of robot manipulators via direct adaptive methods. 1991 American Control Conference, Boston, MA. Stewart, G. W. (1973). Introduction to Matrix Computations. Academic Press, New York. Appendix Proof of Lemma 1. It is easy to see that xrpx = Er(kp + ¢k~)E + (E + ¢E)rM(/~ + ~E) -> O, Vx ¢ R 2~.
To see that P is positive definite if kp + otk~ > O, we notice that xrPx = O:> E r (kp + od%)E = O, and (/~ + ~ E ) r M ( k + ~E) = O ~ x = O, since M is positive definite. To develop bounds on the eigenvalues of P, let T be a transformation such that M = T-IAT, where A = diag {al . . . . . an} and the ai are the eigenvalues of M. It follows that [To-1
0 ]piT
+trkv)l+a'2A
AA]
(18)
o^
By the Gershgorin Theorem (Stewart, 1973), we know that the eigenvalues z/, j = 1 . . . . . 2n, of the matrix on the right-hand side of (18) satisfy the following inequalities: - ota, + kp + o&~ + ~ a i <-2zi <- eta, + kp + ~k~ + ~a~, (1 -- o[)ai<2Zn+ i ~--(1 "t- oOai, for i -- 1 , . . . , n. The constant upper and lower bounds on z/, ] ffi 1. . . . . 2n, can be easily found by using the fact that < a~ < th for all time and all i. []