Journal of Intelligent and Robotic Systems 14: 263-282, 1995. © 1995 Kluwer Academic Publishers. Printed in the Netherlands.
263
A Robust Control Scheme for Asymptotic Tracking of Robot Motion* MAGDI S. MAHMOUD1 and SAMIR KOTOB2 1Department of Electrical and Computer Engineering, Kuwait University, P. O. Box 5969, 13060 Safat, Kuwait 2Systems and Control Department, Engineering Division, Kuwait Institute for Scientific Research, P. O. Box 24885, 13109.Safat, Kuwait
(Received: 17 September 1993; in final form: 17 October 1994) Abstract. In this paper, a controller structure is developed to provide for asymptotic tracking of robot motion. The design tool is the theory of hyperstability and the analysis has led to a simple and an easy-to-implement robust version of the inverse dynamics. Simulation studies are worked out to demonstrate the controller performance. A comparison with other methods is done to show the merits of the developed scheme vs. other recently developed schemes. The implementation and computational requirements of the control schemes are determined and shown to be within the capabilities of new control hardware. Key words. Robust control, robot motion, hyperstability theory, PUMA 560 robot arm.
1. Introduction
The problem of trajectory tracking control design for robot manipulators has been the subject of extensive research effort over the last decade; [1-12, 19] and their bibliographies. Many controller structures have been proposed and shown to have acceptable performance by simulation or experimental implementation [5, 6]. Robot control schemes can be broadly classified into direct adaptive, indirect adaptive and robust [15]. Direct adaptive control, also known as model-following adaptive control, uses a model to specify the desired performance which is to by attained by the robot manipulator. An error signal is utilized to generate the adaptive control law so that this error tends to zero asymptotically. No attempt is usually made to estimate the parameters of the manipulator. Design methods include local parametric optimization, Lyapunov functions and hyperstability theory [7]. Indirect adaptive control refers to the class of schemes that explicitly estimate the manipulator parameters and then incorporates these estimates in the computation of the controller parameters [3, 6]. Quite often in the course of * This work was supported by the Kuwait University Research Administration under Grant No. EE063.
264
M.S. MAHMOUD AND S. KOTOB
control design, a trade-off exists between the computational load and tracking accuracy. The work of [1, 3] has concentrated on designing a controller that forces a dynamic model of the manipulator to follow a preselected model. By representing the robot manipulator as a rigid body and allowing parameter uncertainties and load disturbances, different controllers have been developed in [1-4, 6, 8-11] to ensure acceptable trajectory tracking. In this paper, our methodology follows the technical trends in [6, 8-11] in modeling the robot manipulator as a rigid body, subject to parameter uncertainties and load disturbances, and in developing nonlinear feedback controllers to regulate the behavior of n-DOF robot manipulator. In particular, we present a simple control structure and prove that under its action, asymptotic tracking of the robot motion is attained. The controller is essentially a version of the inversedynamics approach that would achieve exact linearization and decoupling if the dynamic parameters of the manipulator were exactly known. Since in practice such parameters are only known within some uncertainty bounds, an additional signal is required to robustify the controller against this parameter uncertainty. This additional signal is chosen in this work to be of the 'unit vector' type, and the overall system is established to be asymptotically hyperstable. In some way, our work illuminates that hyperstability theory represents as equivalent stability framework to that provided by Lyapunov analysis [7]. Two case studies are considered for computer simulations. The first is a two-degrees of freedom planar manipulator and the second is the first three joints of the Stanford PUMA 560. The obtained results illustrate the excellent performance of the developed control scheme.
Notation. We use W t, W -1, A(W) to denote, respectively, the transpose of, the inverse of, and the eigen values of any square matrix W. The vector norm is taken to be Euclidian and the matrix norm is the corresponding induced one; that 1/2 is IIW[[ = AM (W t W), where AM(m)(W) stands for the operation of taking the maximum (minimum) eigenvalue of W. We use W > 0 (W < 0) to denote a positive- (negative-) definite matrix W. 2. Robot Control by Linearization and Decoupling The dynamics of an n degrees of freedom (n-DOF) robot manipulator is characterized by [5]:
D(q)d] + C(q, (1)0 + g(q) = T
(1)
where q, q,/i are n-vectors of joint positions, velocities and accelerations; D(q) C 9l nxn is the symmetric positive-definite inertia matrix; C(q,q)(1 E 9l n is the vector of centrifugal and Coriolis loading torques and defined by C(.,- ) = [Cij],
ASYMPTOTIC TRACKING OF ROBOT MOTION
265
with
n 1 {ODij__+ODik Cij = Z Ckji(tk; ckdi = -~ Oqk Oqj
OD~j} Oqi
(2)
k=I
are the Christoffel symbols of first kind. Also, 9(q) E 9~n and T E Yln are the vector of gravitational loading torques and the vector of input torques, respectively. A fundamental approach to robot control [2] starts by converting system (1) into the standard state-variable representation. Let the state vector be defined by r
i t
t t
(3)
x l = q, X2 =_O, X = - [ Z l Z 2 ]
Using (3) into (1), we obtain x2
0
;~= [-D-l(xl)[C(xl,x2)x2 + 9(xl)]] + [ D - ' ( x l ) l T
(4)
Then an exact linearizing control [2] is introduced as
T = D(xl)a + C(xl, Z2)X2 + g(Xl)
(5)
where a is the reference acceleration vector. From (4) and (5), we obtain the closed-loop robot dynamics as I
Equation (6) signifies the concept of exact linearization in the sense that while the original system is highly nonlinear, the closed-loop control input (5) yields a linear decoupled system. Assuming a desired trajectory qd, which is continuously-differentiable and expressed in terms of desired robot position q, the joint tracking error e0 is defined by
eo=q
-
qd
(7)
The stable robot control problem developes into finding a linearizing control torque T such that the error such that the error e0 decays to zero, given any bounded initial condition. Now, we postulate a second-order dynamics of the error to be in the form go + A14o + Aoeo = 0
(8)
where the matrices A0,A1 are selected to be diagonal matrices with positive elements. As a linear second-order set of equations, model (8) represents a
266
M . S . MAHMOUD AND S. KOTOB
decoupled state error with negative real eigen values. Then the error e0 and its derivative will converge to zero. By manipulating (6) and (7) we find that q -- qd
q- A14o q- A0eo = 0
Substituting for q its value from (2) and (3) and manipulating, we get
a = qd -- A14O - A0eo
(9a)
Thus if a linearizing torque T computed according to (5) is developed, where a satisfies (9a), then the closedqoop system is bounded for any initial condition. The closed-loop law is given by
T
= D ( x ) [ q d -- A140 -
Aoe] +
C(x)x2
(9b)
q- g ( x l )
3. A Hyperstable Controller
Despite the fact that the exact linearization and decoupling scheme presented in Section 2 yields asymptotically stable tracking error, it requires exact knowledge of the dynamic parameters of the manipulator summarized by D(q), C(q, (1) and 9(q); see (9b). Since these parameters contain uncertain factors, the exact linearization and decoupling scheme would not be the best to apply in practice. In this section, we develop a modified version of the control law (5) and subsequently establish its asymptotic behavior using hyperstability theory [7]. Let p E ~ s be a constant vector containing the unknown elements in the suitably selected set of equivalent dynamic parameters. Let ~ E 9~s be its time-varying estimate and let Do(q), Co(q, (1) and go(q) be the matrices obtained from the original D(q), C(q, (1) and 9(q) by substituting the estimated ~" for the actual p. The selection of Do(q), Co(q, (t) and go(q) depends on results of a priori simulation and robotic experience. Based on this, the following controller is proposed
(lO)
T = Do(q)[a+uo]+Co(q, q)
It is interesting to note that controller (10) has similar structure like (5) with two basic differences: (1) The dynamic parameters {D(q), C(q, (1) and 9(q)} of are now replaced by their estimates (Do(q), Co(q, (1) and 90(q)}. (2) An additional signal u0 is used which represents a design parameter and will be determined later. The substitution of (6) and (10) into (3) yields the closed-loop dynamics
[ {c =
x2 -D-I(cx2
+ g) + D-I(Cox2 + go) + D-1Do( a + uo)
]
(11)
ASYMPTOTIC TRACKING OF ROBOT MOTION
267
Define the error terms =Co-C,
~=go-g,
(12)
E=D-1Do-I
Using (11) in (10), it becomes x =
(13)
D - I ( C x 2 + 9) + (a + uo) + E(a + uo)
In terms of the desired position qd, the desired velocity 46, and letting x~ = [q~ 0~], the error vector e is defined by: e:
-= X - -
X d --~
[el I e2
(14)
Using (8) and (14) in (13) we get X 2 -- X d 2
]
=
D - I ( C x 2 + g~ + (a + uo - ?]d) + E(a + uo) J
=
D - I ( C z 2 + g) + (u0 - A0el - Ale2)
(15)
In view of (2) and (14), system (15) describes the dynamics of the error in robot state trajectories. Simple arrangement of (15) gives e=
[o-A0 ,] -A1
e+
[01
uo +
d(e, uo)
(16)
where the model uncertainty is given by the vector d(e, uo) = D -1 [Cz2 + g] + E(a + uo)
(17)
It should be observed that the dynamic model (15) in nominally linear in the sense that in the absence of uncertainties d(e, u0) = 0, the model becomes linear and time-invariant. Now, our objective is to design the control signal u0 so as to drive the robot state error to zero. This problem had been studied by Corless [10], Kim and Shin [1], Koivo and Guo [31, Ortega and Spong [21, Ryan et al. [81, Shoueshi et al. [9], Slotine and Li [6], Spong [11] and Spong and Vidyasagar [5, 12]. The basis for the result was the assumption that the class of the uncertainties under consideration is cone-bounded. In the present work, we enlarge this class and generalize their results by invoking the following: Assumption 1. The error term E is bounded," that is max II~( )fl = ~3 < 1 t
(18)
268
M. S. MAHMOUD AND S. KOTOB Assumption 2. The residual error d(e, uo) is bounded; that is lid(e, uo)ll
elllell + ezllel] 2 + c311uoll + c4
(19)
As delineated in Mahmoud and Hajeer [15], Dawson et al. [18], the physical properties of the robot manipulator can be used to show that the dynamics in the form (15)-(17) can be bounded. This takes place regardless of the large variations in payload mases, link masses, coefficients of friction and disturbances. In our case, the validity of expression (19) stems from the fact that the matrix C(q, (1) of (2) is a linear function of q and so is Co(q, (1). Thus it follows that the term C(q, (1)X2 appearing in (17) is quadratically dependent on the error e via (3) and (14). Expression (19) can be obtained from (8) and (17) with Cl = c5 + max [[E[-Ao - A1]I[, c4 = C6 q- mtax [[Eqd + D-lgI[
(20)
for some positive constants el, • • •, c6 such that IlD-1Cx2ll <~ c5[[ell + c21lell2 + c6
(21)
and Assumption 1 holds. Further technical details can be found in [13]. Looking at the existing literature [2], expression (19) represents a new element added to the theory of robust control of robot arms. Proceeding further, we define
-Ao
-A1
'
Now the error model (16) can be put into the standard-feedback structure [7]" = Ae - Bw
(23)
w = - u o - d(e, uo)
Hyperstability theory can then be conveniently applied to the error model (23). According to this theory, the basic idea is to constrain the inner product of the output w and a projected error signal v (to be defined shortly) in a way to satisfy stability properties. In this regard, the signal v is called the output generalized error and is given by v = Ge
(24)
where G E fj~nx2n is a constant selected such that the transfer matrix H ( s ) from w to --v H(s) = G[sI - A]-I B
(25)
ASYMPTOTIC TRACKING OF ROBOT MOTION
269
is strictly positive real [7]. The strict positive realness of H(s) is ensured by selecting
G = BtP
(26)
where 0 < P = pt E 912nx2n is the solution of the Lyapunov equation
PA + AtP = -Q
(27)
where 0 < Q = Qt E 912nx2r~ is an arbitrary matrix. Now the additional control signal u0 is selected such that
foot vt w dt > -7~)
Vt
(28)
9l
This can be assured, whenever
vtw >~ 0
(29)
Using (23), we get:
vtw = - v t u o - vtd(e, uo)
(30)
Now we choose
,~o =
~
[5111511+ c211~[[2 + c4]/(1
--
53)
(31)
53)
(32)
and observe in view of (18) c3 < 1 such that
-v%
= I1~11[5111~l[+ c211~ll2 + ca]/(1 = IIVII I1~011 >/ 0
--
But from (31) it follows that
fl~*oll ~< ~111511+ 52115112+ 5311~011+ c4
(33)
On substituting (33) into (32), it becomes
-vt~o = IlvllEcl 1[511+ cz[[ell 2 + c311~01[ + c4] N~II lid(5, ~o)11 ~ [ - v~d(5, uo)l
(34)
Finally we reach
vtw = - v t u o - vtd(e, uo) ) 0
(35)
Therefore, we conclude that the additional signal u0 defined by (3 l) guarantees that the error system is asymptotically hyperstable. In short, by combining (8), (1 O) and (3 l) the hyperstable controller guarantees the desired stability properties and therefore we get e ~ 0 as t --4 oe (sufficiently large processing time).
270
M. S. MAHMOUD AND S. KOTOB
4. Simulation Studies To demonstrate the performance of the developed controller, two case studies were considered for computer simulation. The first is a 2-DOF planar manipulator and the second is a 3-DOF revolute joint robot. 4.1. CASE 1 A schematic of the two-link manipulator is shown in Figure 1. The physical properties of link j are specified by 4 parameters: mass rrzj, moment of inertia about center of gravity Jj, length lj and the distance of the center of gravity Y~
!
"-X w
Fig. 1. Two-linkplanar manipulator in a vertical plane. 1.6 1.4
1.2 1 2: .< 0.8
Case
I At
= I m.s.
,< 0.6 0.4 0.2
o~3
~.L0
I.'5
z'.o
TIME (sec)
Fig. 2a. Desired trajectory of first joint.
~is
3.0
ASYMPTOTIC TRACKING OF ROBOT MOTION
271
1.6
1.4 1.2 1
Z o
Case 1 ~ t = 1 rrl.s.
0,8
r~
0.6 0.4 0.2 0
o15
,'0
,'5
TIME (sec)
2!0
2;
3.0
Fig. 2b. Desired trajectory of second joint.
!
x 10-3
o
Case I At =I
°i 0
OlS
1.o
I.t5 TIME ( s e c )
zlo
2.'s
m,s.
3.0
Fig. 2c. Trackingerror of joint 1. from the previous joint Icj. The dynamic equation of type (1) is expressed as q=[ql
q2]t,
D(q)
[71 -4- T2 + 2T3 cosq2 [ 72 + T3 COSq2
9=9,81m/s
C(q, q) = [-T302 sinq2 ~-302 sin q2
2 7"2 4- T3 COSq2 ] 7-2 J
-7-3(02 + q3)sinq2] 0
9(q) = [TegCOSql + TsgCOS(ql + q2)] 7-5gcos(q1 + q2)
272
M. S. MAHMOUD AND S. KOTOB Xl0-~
-2
-4 z <(
l:r.
CaseI =Im.s, At
-S
-8 -10 _12 -14
0
015
1.10
1,15
210
215
3.0
TIME ( s e c )
Fig. 2d.
Tracking error of joint 2.
200
150II C a s e 1 5 t = 1 re.s,
50
:; '°° t.,,/
-50 0
0.5
Fig. 2e.
1,0
I
2.0
2.5
3.0
Input torque for joints 1 and 2.
T1 = m l ~ 1 + J1 + m 2 ~2, T3 = Tr~2~c2~l,
I
t.5 TIME (sec)
"7-2 = m2ec22 -{- J2
7-4 = 7Ttl~ci q- m2~l,
7-5 = m2~c2
The desired joint trajectory is considered to be a critically damped second-order response to a step input having the form: qd] = (re/2)[1 -- (1 + toni t)exp(--COnit)] qd2 = (7r/2)[1 - (1 + (.,on2t) exp(-COn2t)]
273
ASYMPTOTIC TRACKING OF ROBOT MOTION Z X 10-3
0
-2
-4
Case 2 A t = 5 m.s.
-6
-8
-10
015
Fig. 3a.
1.JO
1,15 T I M E (see)
210
2~S
3.0
2.s
3o
Tracking error of first joint.
XlO-3
0 cc -
I
0
~
-15 -20o
o.s
Fig. 3b.
Io
1.5 TIME (sec)
2.o
Tracking error of second joint.
200
150 I00
=
. .
50 0 50
o!s
,.'o
,.'s
21o
215
TIME (sec)
Fig. 3c.
Input torque for joints 1 and 2.
To
274
M.S. MAHMOUD AND S. KOTOB
where w~ 1 = co=2 = 5 rad/sec were used for simulation purposes. The desired trajectories for both joints are shown in Figure 2. The weighting matrix Q in (25) was taken as Q = diag{lO00, 100000, 10, 100} and two cases of sampling rate were considered: (1) At = 1 ms. (2) At = 5 ms. The simulation results are depicted in Figures 2c-2e for Case 1 and Figures 3a3c for Case 2. The results of extensive simulation runs have indicated that the behavior of the proposed controller is fast and high under worst-case changes in the robot parameters. This reveals the robust performance of the proposed controller.
4.2. CASE 2 Here we consider a 3-DOF robot model of the Stanford PUMA 560 robot arm (excluding the wrist). The dynamic model is specified in Appendix A together with the actual no-load parameters. For simulation purposes, the available fixed estimate of the parameters was set at 80% of the actual values. In terms of (25), the weighting matrix Q was selected as Q = diag{1000, 1000, 1000, 10, 10, 10} This selection reflects more weight on position error than on velocity error. The reference (desired) trajectories of the three joints are taken identical qdl = qd2 = qd3
= (re/2)[1 -- (1 + 5t)exp(--5t)]
1.6 1.4
1.2
1 Z 0.8 n-" 0.6
0.4
0.2 0
0
015
1.i0
1.w5
TIME (sec)
210
215
Fig. 4a. Actual and desired trajectory of joint 1.
3.0
ASYMPTOTIC TRACKING OF ROBOT MOTION
275
1,6
1,4
1.2
1 Z 0.8 CC 0.6
0,4
0.2 0 0.5
1.0
,.'s
210
2~5
3°
TIME ( s e e )
Fig. 4b. Actual and desired trajectory of joint 2. 1.6 1.4 ~.~ I
Z 0.8 rr 0.6 0.4 0.2 0
0.5
1.0
I.Ls
21o
2~5
3.0
TIME ( s e c )
Fig. 4c. Actual and desired trajectory of joint 3. with sampling period of 5 ms. The resulting joint trajectories, superimposed on the desired ones, are depicted in Figures 4a-4c. The plots show excellent performance of the developed controller where the actual and the desired trajectories are almost coincident over the entire control period. In Figure 5, the tracking error for the three joints are presented. As expected, the tracking error converges asymptotically to zero. The maximum tracking error was 0.004 radian in joints 1 and 3. The control inputs for the three joints are given in Figure 6. As can be seen, the input torque is smooth and approaches zero in the steady-state.
276
M. S. MAHMOUD AND S. KOTOB 2 1 AMFC 2 HC
1
0
-3
-4
-50
0'.5
L
1.'0 TIME ( s e c )
Fig. 5. Tracking error. 200 I 2 3
~50
AMFC HC RNC
100 2 50 3
1
0
-50
-100
0
-.
L
0.5
I
1.0
i 1.5
2 iO
2.15
3.0
TIME ( s e e )
Fig. 6. Control input. It has noticed that the tracking error exhibits a very small variation around the steady-state. The main source for this is the accumulation of numerical errors due to the stiffness of the differential equations involved.
5. A Comparison In the course of developing reliable control schemes for robot arms, it was recently shown by Mahmoud and Hajeer [15] that the resulting controllers can be broadly categorized into: direct adaptive, indirect adaptive and robust. An
ASYMPTOTIC TRACKING OF ROBOT MOTION
277
adaptive model-following controller (AMFC) was designed by Mahmoud and Bahnasawi [16], based on hyperstability theory which guarantees global asymptotic stability of the closed-loop robot system. Alternatively, a robust nonlinear controller (RNC) was constructed by Mahmoud [13, 14] for the stabilization of robot arms including motor dynamics and using the deterministic methodology of uncertain dynamics systems pioneered by Corless and Leitmann [17]. In the sequel, we will compare the three schemes: adaptive model-following controller, robust nonlinear controller and hyperstability controller, particularly when applied to the Stanford PUMA 560 robot arm. Simulation studies for the three schemes have indicated that the best performance is obtained with the HC scheme. However, it is also the most computationally-demanding controller. The RNC scheme has required less computations but it can only guarantee uniform ultimate boundedness of the position and velocity errors. The associated control signal exhibits a high-frequency chattering component. When the controller is modified to cope with the actuator dynamics, the chattering has disappeared yielding smooth control signal and smooth joint trajectories. The AMFC also exhibits the similar high-frequency chattering characteristic of the unit-vector control. In general, the choice of a controller depends on the available hardware, computational capabilities and the desired tracking precision. Therefore a trade-off exists between processor cost and tracking accuracy. For the purpose of comparisons, we have estimated the number of computations taken to generate the control input for the three controllers. Let m be the number of robot parameters, n be the number of degrees of freedom and L0 be the amount of multiplications and divisions per one sample. The result is summarized below: For the AMFC scheme, L0 = 14n 2 + 4n + 2, for the HC scheme (the present work), L0 = 8 n 2 + 2 m n + m 2 and for the RNC scheme L0 = 6 n 2 + 3 n + 2 . Since m > n, the HC scheme demands more computations. However, this does not represent an obstacle in practice in view of the modern advancement in microtechnology and the explosive growth in computing power. On the other hand, the three controllers differ in their requirements. For the AMFC, a rather good fixed estimate of the actual time-varying inertia matrix is needed for the auxiliary signal and for an estimate of some bounding factors. It also requires a suitably selected reference model. In the present work, the acceleration vector is needed in order to be able to generate the torque prediction error and to evaluate the control law. As for the RNC, a rather good estimate of the inertia matrix is needed and in addition, the regressor matrix must be known. To complete the comparison, we have simulated the first three joints of the Stanford PUMA 560 robot arm under the action of the three controllers and have evaluated the maximum error in the transient response and steady-state performance (asymptotic behavior). It was found that the maximum error with the AMFC (for the three joints) was nearly 0.007 radian, but with the HC and the RNC the errors were 0.0016 and 0.0042, respectively. The asymptotic behavior
278
M.S. MAHMOUD AND S. KOTOB x~( s
1 AMFC 2 HC 3 RNC
1 z < <
0
ec
-1 -2 -3
-5
o!s
L
,.o
,)5
~Io
i
2.s
3.0
TIME (see)
Fig. 7a. Tracking error of joint 1. X~6 3 6
I
1 AMFC 2 HC 3 RNC
2
Z < I
< rv
0 -1 -2 -3
-%
o'.s
,.'o
,)s
~'.o
2.'S
3.o
TIME ( s e c )
Fig. 7b. Tracking error of joint 2. was best with the HC, then came the AMFC in the second place but suffering from chattering. As indicated by theory, the robot arm with the RNC do not converge to the desired ones but stay in a close neighbourhood of them. Thus it yields a small but finite tracking error. For purpose of illustration, the tracking errors with the three controllers for each joint are plotted in Figures 7a-7c. 6. Conclusions A hyperstable controller structure has been developed to provide asymptotic tracking of robot motion. The controller consists of two terms: the first is based
ASYMPTOTIC TRACKING OF ROBOT MOTION 8
279
xO l-~ ~
6
AMFC HC
RNC
4
2 Z <
0 < rr" -2
-4
-6 -8
ols
,.b
TIM,.'sE(sec)
~!o
~.'s
~.o
Fig. 7c. Trackingerror of joint 3. on inverse-dynamics using available estimate off the robot model parameters and the second is an additional signal to make the controller robust against parameter uncertainties. The simulation results of the developed controller greatly enhances the practical utility of robust control of robot arms. It should be stressed that the structure of the controller is simple and its realization is quite systematic. Computer simulations have been undertaken to illustrate the design procedure.
Appendix The dynamical model of the first three joints of PUMA 560 robot arm (excluding the wrist) is given by: O(q)tl + C(q, ~l)~l+ g(q) = ~-
(A1)
where D(q) =
I
C(q,~l)= with
Dll D12 D13
DÂ; D22 D23
D131 D23 D33
iq
~1:r C 2 q:r C 3
(A2)
(A3)
[ch C1:{C]2 C12 C13 L %
(A4)
280
M. S. MAHMOUD AND S. KOTOB
and
g(q)=
(AS)
g2 93
In the sequel, we use the following notation
Si
Ci = cos(q0,
Cij
= cos(q/+
= sin(q0,
qj),
Sij
= sin(qi + qd).
The elements of the inertia matrix D are given by D l l = 81 Jr- 82(C2) 2 q- 83($23) 2 - 84C2C23 - 85C23S23 + 86C2S23, D12 =- 87S2 - 88S23 - 89C23,
D13 = - 8 8 S 2 3 - 89C23,
D22 = 810 -}- 06S 3 - 84C3,
D23 = 811 + 812~q3 - 813C3,
D33 = 814.
The elements of C 1 are given by C~1 = 0 C~2 = 815 -- 816C2~ 2 q-
83C3S
3 -
817~2~q3~23 q- 84C2S23q-
q- 812(C2C23 - S2S23) - 818C22 - 819(C23) 2 - 820S3
C~3 = 815 q- 83C2S2 if- 8 3 C 3 S 3 - 817S2S3S23 if- 820C2S23qq- 812C2C23 - 819(C23) 2
C122821C 2 q- 89S23 =
~=
88C23 ,
C213 = 819S23 - 88C23 ,
G.
The elements of C 2 are given by
c, ~, = c~2,
c1~2 = o,
C22 = 0,
C23 = 0 ,
C23 = 812C 3 q- 813S 3,
The elements of C 3 are given by C~, = C~3,
C132 = 0,
c3~ = -c~3,
c~3 : o,
C~3 = 0, C333 = 0
The elements of g are given by gl = 0,
g2 = -822C2 q- 823S2 -~ 824C23 - 825S23,
g3 = 024(5'23 -
025Sz3
ASYMPTOTIC TRACKING OF ROBOT MOTION
281
The actual parameters are taken as 01= 2.4975
07 = 2.4492
013= 0.0165
019 = 0.0405
02= 2.1007
08 = 0.0070
014= 1.1295
02o = 0.0166
03----- 0.5323
09 = 0.1596
015= 0.0203
02t = 1.9686
04= 0.0330
010 = 0.4190
016 = 1.5685
022 = 52.1060
05= 0.0405
011 = 0.5468
017 = 1.0643
023 = 1.0972
06= 0.9161
012 = 0.4581
018= 0.0483
024 = 0.3761 025 = 10.4068
References l. 2. 3. 4. 5. 6. 7. 8.
9.
10. 11. 12. 13. 14. 15. 16. 17.
Kim, B. K., and Shin, K. G.: An adaptive model following control of industrial manipulators, IEEE Trans. Aerospace and Electronics Systems AES-19(6) (1983), 805-813. Ortega, R., and Spong, M. W.: Adaptive motion control of rigid robots: a tutorial, Automatica 25(6) (1989), 877-888. Koivo, A. J., and Guo, T. H.: Adaptive linear controller for robotic manipulators, IEEE Trans. Auto. Contr. AC-28 (1983), 162-171. Young, K.-K. D.: Controller design for a manipulator using theory of variable structure systems, IEEE Trans. Sys. Man and Cyb. SMC--8(8) (1978). Spong, M. W., and Vidyasagar, M.: Robust linear compensator design for nonlinear robotic control, IEEE J. Robotics Automation RA-3 (1987) 345-351. Slotine, J.-J. E., and Li, W.: Adaptive manipulator control: A case study, IEEE Trans. Automatic Control 33(11) (1988), 995-1003. Landau, I. D.: Adaptive Control: The Model Reference Approach, Marcel Dekker, New York, 1979. Ryan, E. E, Leitmann, G. and Corless, M.: Practical stabilizability of uncertain dynamical systems: applications to robotic tracking, J. Optimization Theory and Applications 47 (1985), 235-252. Shoureshi, R., Corless, M. J., and Roesler, M. D.: Control of industrial manipulators with bounded uncertainties, ASME J. Dynamic Systems, Measurement and Control 109 (1987), 53-59. Corless, M.: Tracking controllers for uncertain systems: Application to a Manutec r 3 robot, ASME J. Dynamic Systems, Measurement and Control 111 (1989), 609-618. Spong, M.: On the robust control of robotic manipulators, IEEE Trans. Automatic Control AC-37 (1992). Spong, M., and Vidyasagar, M.: Robot Dynamic and Control, Wiley, New York, 1989. Mahmoud, M. S.: Robust control of robot arms including motor dynamics, Int. J. Control 58 (1993), 853-873. Mahmoud, M. S.: Robustness of a class of uncertain nonlinear systems with applications to robot control, Int. J. Math. Modelling and Scientific Computing (1994), in press. Mahmoud, M. S., and Hajeer, H. Y.: Robust and Adaptive Robot Arm Control, Kuwait University Technical Report ECE-92-6--05, 1992. Mahmoud, M. S., and Bahnasawi, A. A.: Adaptive model-following controller for robotic manipulators arms, Int. J. Control 59 (1994), 1465-1483. Corless M., and Leitmann, G.: Continuous state feedback guaranteeing uniform ultimate boundedness of uncertain dynamic systems, IEEE Trans. Automatic Control AC-26 (1981), 1139-1144.
282 18. 19.
M. S. MAHMOUD AND S. KOTOB Dawson, D. M., Qu, Z., Lewis, E L., and Dorsey, J. E: Robust control for the tracking of robot motion, Int. J. Control 52 (1990), 581-595. Wu, S., and Cetinkunt, S.: Model reference adaptive inverse control of a single link robot, Computer and Structures 47 (1993), 213-223.