THE CATHOLIC UNIVERSITY OF AMERICA 

DEPARTMENT OF ELECTRICAL ENGINEERING 

JOINT-SPACE ADAPTIVE CONTROL OF A 

6 DOF END-EFFECTOR 

WITH CLOSED-KINEMATIC CHAIN MECHANISM 

Charles C. Nguyen 

Principal Investigator and Associate Professor 

Zhen-Lei Zhou 

Graduate Research Assistant 

and A,/; 3-- .I,’& 

,//to/ -37 I ‘J) fL- 

, _-I^ > /I 

;I- * ,’-.9@ 

submitted to 

Mr. David E. Provost 

Code 735.1 

Goddard Space Flight Center (NASA) 

Greenbelt, Maryland 

August 1989 
SUMMARY 

This semi-annual report presents the msearrh results from the research gmnt entitled ”Active 

Control of Robot Manipulators,” funded by the Goddard Space Flight Center, under the Grant 

Number NAG 5-780, for the period between February 1, 1989 and August 1, 1989. 

In this report, we present the development of a joint-space adaptive scheme that controls the 

joint position of a six-degree-of-freedom (DOF) robot end-effector performing fine and precise 

motion within a a very limited workspace. The end-eflector was built at NASA to study autonomous assembly of NASA hardwares in space. The design of the adaptive controller is based 

on the concept of model reference adaptive control (MRAC) and Lyapunov direct method. In 

the development, we assume that the end-eflector performs slowly varying motion. Computer 

simulation is performed to investigate the performance of the developed control scheme on position contml of the end-eflector. Simulation results manifest that the adaptive control scheme 

provides excellent tracking of several test paths. 
1 INTRODUCTION 

Recognizing the fact that performing operations in space is dangerous, NASA has paid its 

attention to the research of telerobotics which is the combination of two different concepts, 

teleopemtion and robotics [l]. Telerobotic operations can be executed either in a traded control 

mode (serial operation) or in a shared control mode (parallel operation). In the traded control 

mode, using teleoperation the human operator performs some portion of a task and then let 

the telerobot perform some other portion of the task autonomously while on the other hand, 

the human operator and the telerobot perform portions of the task simultaneously in a shared 

control mode. In either modes, successful robotic tasks require that the motion of the telerobot 

be controlled precisely. A telerobotic system generally consists of a master arm and a slave arm. 

It has been proposed [2] that a light and compact 6 DOF end-effector be built and mounted to 

the slave arm to autonomously perform fine and precise motion during the traded mode of the 

telerobotic operation. In this report we present the development of a joint-space adaptive control 

scheme for controlling the end-effector motion by using the concept of MRAC and Lyapunov 

theorem. 

When a dynamic model can accurately represent the real end-effector dynamics, then computed torque [3] scheme whose development is mainly based on the dynamic model, can be 

employed to control the end-effector motion. Using the above scheme, time-varying controllers 

can be designed so that disturbances are minimized and excellent tracking performance can be 

achieved. Since it is relatively difficult, if not possible to derive an accurate dynamic model for 

a robot end-effector, the computed torque scheme is not feasible. The urgent need of a control 

scheme that is able to effectively react to the presence of nonlinearities and uncertainties in 

robot end-effector dynamic model and payloads has motivated the research of adaptive control 

schemes, according to a recent survey [4]. MRAC method and Lyapunov theorem function were 

employed by several researchers to design adaptive controllers for Cartesian- and joint-space trajectory control, which was proved to provide global stability [5,6]. Lim and Eslami [7] considered 

the design of robust adaptive controllers. Adaptive force control problem was investigated by 

Daneshmend and Pak [8] for a cutting problem. In [9] Houshangi and Koivo designed an adaptive force-position controller with self-tuning in Cartesian space by using eigenvalue assignment 

method and minimization of a quadratic performance criterion. Recently, Seraji [ 101 presented 

the implementation of adaptive force and position controllers for robot manipulators within the 

hybrid control structure using an improved MRAC. The problem of hybrid control of force and 

position was also considered by Nguyen and Pooran [ll]. 

In this report, we first describe the structure of a 6 DOF end-effector built at NASA to 

study telerobotic assembly. We then present the derivation of a joint-space adaptive control for 

controlling the end-effector motion. After that, the developed control scheme will be applied to 

control the planar motion of a 2 DOF end-effector built in our robotic laboratory. Discussion 

of the simulation results and recommendation for future research direction will conclude the 

report. 

Notations used in this report are listed below 

0 MT: transpose of the matrix M 

0 0,: (nxn) matrix whose elements are all zero 

0 I,,: (nxn) identity matrix 

1 
2 THE CKCM ROBOT END-EFFECTOR 

Most telerobotic assembly of parts such mating or fastening can be accomplished in a traded 

control male[l] in which using the master arm, the human operator remotely moves the slave arm 

into the assembly workspace and then let a robot end-effector, mounted to the end of the slave 

arm perform the assembly task autonomously. In addition to the requirements of compactness 

and lightweight, the end-effector must be able to perform very precise motion within a very 

limited workspace. In order to study the feasibility of autonomous assembly of parts in a 

telerobotic operation, an end-effector whose size is about ten times that of the telerobot endeffector was designed and built at the Goddard Space Flight Center (GSFC) [12] and is currently 

located at the Center for Artificial Intelligence and Itobotics (CAIR)'. As shown in Figure 1, 

the end-effector resembles the structure of the Stewart platform [13], and mainly consists of an 

upper payload platform, a lower base platform and six linear actuators. The movable payload 

platform is supported above the stationary base platform by six axially entensible rods where 

in order to provide the extensibility, the system uses recirculating ballscrews that are driven 

by stepper motors. The motion of the payload platform is produced by the combination of 

extending and shortening the actuator lenghts. Each end of the actuator links is mounted to 

the platforms by 2 rotary joints with intersecting and perpendicular axes. The end-effector has 

24 rotary joints, 6 prismatic joints, and 14 links including the 2 platforms and therefore has 6 

DOF, which can be proved by applying the number synthesis[2]. 

3 THE JOINT-SPACE ADAPTIVE CONTROLLER 

The robot end-effector described in previous section assumes a closed-kinematic chain mechanism 

(CKCM) which has been showed to possess high precision positioning capability [2]. Since 

CKCM end-effectors generally have a closed-form solution for its inverse kinematic problem, 

joint-space trajectory control scheme is proposed to control the end-effector motion so that 

time-consuming iterative computation of the forward kinematics in a Cartesian-space control 

scheme of CKCM end-effectors can be avoided. In other words, the error actuating signal 

to the joint actuators is computed based on the error between the desired variable and the 

actual variable. In addition, adaptive controllers are utilized instead of fixed-gain controllers in 

order to compensate the presence of nonlinearities and uncertainties in end-effector dynamics 

and payloads. Figure 2 presents the joint-space adaptive control scheme proposed to control the 

end-effector motion. As the figure shows, position sensors mounted on the end-effector actuators 

provide feedback information of the actual lenghts of the six actuators. The actual lenghts are 

then compared to the desired lenghts that are computed by the inverse kinematics from the 

desired Cartesian variables specified by the user or some path planner. The lenght difference 

will then serve as inputs to the adaptive controllers which in turn produce required joint forces 

for the actuators to track the end-effector along a desired trajectory. 

If we denote a (6x1) vector 1 composed of six actuator lenghts Z; for i=1,2,. . . ,6 such that 

1 = (11 12 . . .IS)* 

'To test control schemes developed under a research grant with NASA/GSFC 

2 
as the joint variable vector, then the end-effector dynamics can be written as [ll]: 

7(t) = ~(1, i) &t) t NO, i) i(t) + ~(1, i) i(t) (2) 

where T(t) denotes the (6x1) joint force vector, M(l,i), the manipulator mass matrix is a 

symmetric positive-definite matrix of order (6x6), N(l, 1) and G(l, i) are (6x6) matrices whose 

elements are highly complex nonlinear functions of 1 and 1. In the right-hand side of (2), if we 

neglect joint friction, the second term represents the centrifugal and Coriolis forces, and the 

third term the gravity forces. 

Consider now a PD time-varying controller defined by 

represents the deviation of the actual joint vector l(t) from the desired lenght vector ld(t). 

Furthermore, Kp(t) and Kd(t) denote the proportional and derivative adaptive controller gain 

matrices, respectively. 

Substituting (3) into (2) we obtain 

M le + (N t Kd) 1, t (G + KP) 1, = M ld + N 1, + G Id (5) 

where the dependent variables of the matrices and vectors have been dropped for simplicity. 

Defining a (12x1) state vector z(t) such that 

converts (5) into the state space representation described by 

A3 A4 I6 

where 

Ai = M-'(G t Kp), A2 = M-'(N + Kd), 

and 

A3 = M-' G, A4 = M-' N, 

and 

(7) 

In the framework of the model reference adaptive control (MRAC), the adjustable system is 

represented by Equation (7). The reference model specifies the desired performance of the endeffector in terms of le(t) = [Zel(t) Z,,(t). . .Z,(t)lT, which is the tracking error vector. Suppose the 

tracking errors Ze;(t) for i=1,2,. . . ,6, are decoupled from each other, and satisfy the relationship 

iei(t) + 2 wi iei(t) + W: Zei(t) = o (11) 

3 
~~ ~ ~~~ ~ ~~~ ~ 

for i=1,2,. . . ,6, where & and ai denote the damping ratio and the natural frequency of lei, 

respectively. Then the reference model can be described by 

where D1 =diag(wT) and D2=diag(2&;) are constant (6x6) diagonal matrices, and 

with 

= (le1 le2 - - -l~6)~- 

The solution to (12) can be found as 

zm(t) = exp(Dt) zm(0) 

which under the assumption that the initial values of the actual and reference lenghts are the 

same, i.e., zm(0) = 0, yields zm(t) = 0. 

Now if the adaptation error vector e(t) is defined as 

e(t) = zm(t) - z(t), (16) 

then from (7) and (12), we obtain an error system defined as 

We proceed to select a Lyapunov function candidate v(t) such that 

v(t) = eTPe + tr [(AI - D1)*n1(A1 - Dl)] 

+tr [(A2 - D2)T172(A2 - Dd] 

+t~[&~nsA3] t tr[A4Tn4A4], (18) 

where tr[M] is the trace of matrix M, P and n; for i=1,2,. . . ,4, axe positive definite matrices 

to be determined later. 

Taking the time derivative of (18) and simplifying the resulting expression yield 

i)(t) = eT(PD t DTP)e 

+2tr [(AI - D1)T(J2LT + ~IAI)] 

+2tr [(A2 - D~?)~(nl? + n2A2)] 

-2tr [A3T(J21dT - n3A3)] 

-2tr [AsT(J21z - n4A4)] 

4 
where 

n = [Pz P3]z(t) = -Pzl, - P34 

and P is denoted by 

p= [ p1 Pz] PZ p3 

and it is remarked that e(t) = -z(t) since zm(t) = 0. 

We note that & and w; can be selected so that D is a Hunuitz matrix defined as a matrix 

whose eigenvalues all have negative real parts [14]. Therefore according to Lyapunov theorem, 

there exists a positive definite symmetric matrix P that satisfies the Lyapunov equation 

PD i- DTP = -Q, (22) 

for any given positive-definite symmetric matrix Q. 

Now in (19), if we set 

file' + nlAl= nif + 172~2 = o 

and 

.T nldT - n3A3 = nld - n4A4 = 0, 

then (19) becomes 

6(t) = -eTQe 

which is a negative definite function of e(t). Furthermore, from (23)-(24), we otain 

Al = -n,lni,~; Az = -n;lni:, (26) 

and 

(27) 

*T A3 = ni1nbT; A4 = nhlnld. 

We already showed that P is a positive definite matrix. Now if we could show that 17; 

for i=1,2,. . . ,4, are also positive definite matrices, then the error system described in (17) is 

asymptotically stable, i.e., e(t) -+ 0, or z(t) -+ zm as t + 00. 

As specified in previous section, since the telerobot end-effector will perform slow and precise 

motion, M, N and G are slowly time-varying matrices which can be considered as nearly 

constant matrices. In this case, from (8) and (9) we obtain 

A1 N M-lk,; N M-'kd (28) 

and 

A3 N 0; A4 N 0. 

Next substituting (28)-(29) into (26)-(27) results in 

(30) 

T M-'K, = -n,'nleT; M-'Kd = --n;'ni,, 

5 
Now in (30), if we let 

(32) 

8, = a&-, (33) 

1 1 

a1 a2 

I71= --M; I72 = --M, 

where a1 and a2 are arbitrary positive scalars, then solving for K, and Kd, we get 

We observe that in (32), 171 and I72 are positive definite matrices that can be considered as 

nearly constant because M, the end-effector mass matrix is positive definite and slowly timevarying. To satisfy (31), I73 and I74 should be chosen such that their determinants approach 

infinitive in addition to the positive definite property. Obviously I73 and I74could be selected 

such that they are diagonal matrices whose main diagonal elements are all positive and very 

large. 

We proceed to integrate both sides of (33) and (34) to obtain 

t 

Kp(t) = Kp(0) + ai / 0 (P2le t Psie)lrdt (35) 

where Kp(0) and Kd(0) are initial conditions of Kp(t) and Kd(t), respectively and can be set 

arbitrarily. 

Equations (35) and (36) provide the solutions for the controller gain matrices of the adaptive 

controller, which are based on the lenght errors and their derivatives. 

4 COMPUTER SIMULATION 

In order to examine the performance of the developed joint-space adaptive control scheme, we 

implement it on a 2 DOF end-effector that represents a special case of the 6 DOF end-effector. 

As Figure 3 illustratest the structured of the 2 DOF end-effector that is mainly composed of 2 

ball-screw linear actuators driven by dc motors and hung below a stationary platform via pin 

joints. Position feedback is accomplished by 2 linear voltage differential transformers mounted 

along the actuator links, Based on the diagram given in Figure 4, the Cartesian position x and 

y expressed with respect to a reference coordinate system &xed to the stationary platform are 

related to the joint positions 11 and 12 as follows: 

and 

1i2 - 1z2 + d2 

2d 5= 

y=-. 2d 

(37) 

6 
where d is the distance between the pin joints hanging the actuators. Using Lagrangian formulation, we derive the following dynamic model of the end-effector: 

~(t) = ~(1, i) I(t) -+ ~(1, i) i(t) + ~(1, i) (39) 

where 

T(t) = (71 72)T; 1 = (11 12y 

where 7; and 1; denote the joint force to and the lenght of the ith actuator for i=1,2, respectively. 

In addition 

with 

M=[ml 0 ml '1 

and 

u1 = 1; - 1; -+ d2; 112 = If - 1; t d2; u = JG, (46) 

where m is the mass of the moving part of the link, m the total mass of the link, and Zm the 

fixed lenght of the actuators and g the gravitational acceleration. 

In this study, we implement the developed adaptive control scheme to control the Cartesian 

position of the above 2 DOF end-effector tracking a desired path, which is shown in Figure 5. 

Three study cases are considered below where the performance of the adaptive control scheme 

will be compared with that of a fixed-gain control scheme developed earlier [15]. For the graphs 

given in Figures 6-8, solid line presents the desired path, dashed line and dashed-dotted line 

present the actual paths obtained from the adaptive and fixed-gain control schemes, respectively. 

4.1 Tracking a Straight Line 

The end-effector is controlled to track a straight line specified by y = z + 42 [in cm]. Computer 

simulation results as shown in Figure 6 indicates that a steady-state error of 4mm in both 

horizontal and vertical axes exists in the case of fixed-gain control scheme while in the case of 

adaptive control scheme, the steady-state error is reduced to 1 mm in horizontal axis and to 0.1 

mm in vertical axis. In the case of the adaptive control scheme, it is interesting to note that 

at the beginning of the path, some minor deviation from the desired path occurs because the 

adaptive controller was trying to adapt to the end-effector dynamics. 

7 
4.2 Tracking a Sinusoidal Path 

In this case we study the tracking of a sinusoidal path described by the equation y = sin(2x - 

50) - 83 [in cm]. Simulation results presented in Figure 7 show that using fixed-gain control 

scheme, the robot tracks the desired path with a maximum deviation of 1.5 mm and 3 mm in 

horizontal and vertical directions, respectively, while with adaptive control scheme, the tracking 

quality is improved in the sense that the maximum deviation along the horizontal and vertical 

axes are reduced to 0.8 mm and 0.1 mm, respectively. Unlike the straight line case, faster 

adaptation occurs in the current study case as showed in the beginning of the path. 

Tracking a Circular Path 

Figure 8 presents the simulation results of tracking a circular path defined by the equaiton 

(x - 34)2 + (y + 83)2 = 16 [in cm]. Comparative evaluation of the results of the two applied 

control schemes shows that the steady-state errror is much smaller in the case of adaptive 

control scheme compared to the case of fixed-gain control scheme. In particular, the fixed gain 

and adaptive control schemes have a steady-state error of 3mm and O.lmm, respectively in both 

horizontal and vertical axes. 

In the above simulation study, the control scheme parameters were set as follows: 

e Fixed-Gain Control Scheme: Kpl = Kp2 = 3000N/m and Kdl = Kd2 = SON.sec/m. 

e Adaptive Control Scheme: & and w; for i=1,2 were selected so that 2 characteristic roots 

are both located at -10. Thus D1=10012 and D2=2012. 

5 CONCLUSION 

A joint-space adaptive control scheme was developed in this report to control the motion of a 6 

DOF end-effector mounted to the slave arm of a telerobotic system to perform assembly tasks in 

the traded mode of a teleoperation. The adaptive control scheme consists of a proportional and 

a derivative time varying controllers designed by employing the concept of model reference adaptive control and Lyapunov theorem. The joint-space adaptation scheme was derived under the 

assumption that the end-effector performs slow motion so that the end-effector mass matrix can 

be considered as nearly constant. Unlike other nonlinear control schemes for robot manipulators, 

the developed control scheme does not require the computation of the end-effector dynamics. 

Therefore this computationally efficient control scheme can be implemented in red-time control 

applications without the requirement of a super fast computer. Implementation of the developed control scheme on a 2 DOF end-effector was investigated using computer simulation. We 

considered three different cases of path tracking: tracking a straight line, tracking a sinusoidal 

path, and tracking a circular path and compared the performance of the adaptive control scheme 

with a fixed-gain controller scheme. Simulation results showed that the adaptive control scheme 

provides better tracking performance with smaller steady-state errors than the fixed-gain control scheme. Future research should be focused on the development of Cartesian-space adaptive 

control schemes and hybrid adaptive control schemes [15] and extend the developed adaptive 

control scheme to handle fast robot motion. 

8 
References 

[l] JPL, "Telerobotica Project Plan," Jet Propulsion Laborntory, JPL D-5692, August 1988. 

[2] Nguyen, C.C., Pooran F.J., "Kinematic Analysis and Workspace Determination of a 6 DOF CKCM 

Robot End-Effector," Journal of Mechanical Working Technology, Vol. 20, pp. 283-294, June 1989. 

[3] Arimoto, S., Miyazaki, F., "Stability and Robustness of PID Feedback Control for Robot Manipulators of Sensory Capability," Third Internaiional Symposium of Robotics Research, Gouvieux, 

France, July 1985. 

[4] Hsia, T.C., "Adaptive Control of Robot Manipulators: A Review," Proc. IEEE Conf. on Roboiics 

and Automation, San F'rancisco, pp. 183-189, 1986. 

[5] Craig, J.J., Adaptive Control of Mechanical Manipulators, Addison-Wesley, Reading, Massachusetts, 

1988. 

[6] Seraji, H., "A New Approach to Adaptive Control of Manipulators,", ASME Journal of Dynamic 

Systems, Measurement, and Control, Vol. 109, pp. 193-202, 1987. 

[7] Lim, K.Y., and Eslami, M., "Robust Adaptive Controller Designs For Robot Manipulator Systems," 

Pmc. IEEE Conf. on Robotics and Automation, San Francisco, pp. 1209-1215, 1986. 

[8] Daneshmend, L.K. and Pak, H.A., "Model Reference Adaptive Control of Feed Force in Turning," Buns. ASME, Journal of Dynamic Sysiems, Measurement, and Control, Vol. 108, pp. 215222, 

September, 1986. 

[9] Houshangi, N. and Koivo, A.J., "Eigenvalue Assignment and Performance Index Based ForcePosition Control with Self-Tuning For Robotic Manipulators," Proc. IEEE Con5 on Robotics and 

Auiomation, pp. 1386-1391, 1987. 

[lo] Seraji, H., "Adaptive Force and Position Control of Manipulators," Journal of Robotics Systems, 

4(4), pp. 551-578, 1987. 

[ll] Nguyen, C.C., Pooran, F.J., "Adaptive Force/Position Control of Robot Manipulators with ClosedKinematic Chain Mechanism," in Robotics and Manufacturing: Recent lhnds in Research, Education, and Application, Chapter 4, edited by M. Jamshidi et al, ASME Press, New York, pp. 177-186, 

1988. 

[12] Premack, Timothy et al, "Design and Implementation of a Compliant Robot with Force Feedback 

and Strategy Planning Software," NASA Technical Memornndum 86111, 1984. 

[13] Stewart, D., "A Platform with Six Degrees of Freedom," Pmc. Institute of Mechanical Engineering, 

VO~. 180, part 1, NO. 5, pp. 371-386, 1965-1966. 

[14] Landau, Y.D., Adaptive Control: The Model Reference Approach, Marcel1 Dekker, New York, 1979. 

[15] Nguyen, C.C., Pooran, F.J. and Premack, T., "Trajectory Control of Robot Manipulator with 

Closed-Kinematic Chain Mechanism," Pmc. 20th Southeastern Symposium on System Theory, North 

Carolina, pp. 454-458, March 1988. 

[16] Nguyen, C.C., Pooran F.J., "Joint-Space Adaptive Control of Robot End-Effectors Performing Slow 

and Precise Motions," Pmc. 21th Southeastern Symposium on System Theory, Florida, pp. 547-552, 

March 1989. 

9 
. 

ORIGINAL PAGE 

BLACK AND WHtTE PHOTOGRAPH 

Figure 1: The NASA Robot End-Effector 

10 
TRANS FO Rbf ATIO N 

Xd 

DESIRED DESIRED JOINT JO I" 

CARTESIAN JOINT VARIABLE FORCE ACTUAL 

NASA ROBOT "RAJ"ORY 

END- EFFECTOR 

TRAJECToRy, ERRORS ADAPTIVE * * CONTROLLEX ' + 

1. 

5 

'd 4 

I , I 

Figure 2: The joint-space adaptive control scheme 

I 

t' 

Figure 3: The 2 DOF end-effector 

11 

-ty 

T 

Figure 4: Free body diagram 

ORIGINAL PAGE IS 

OF POOR QUAlrrV 
RY 

I I' 

id 

1 

SENSOR9 

Figure 5 Implementation of the joint-space adaptive control scheme 

-77 - 

-78 - 

- 

E 2 -79- 

b 

-80 - 

-81 - 

'. - : Desired -. 

o : Adaptive 

--. . Fixed-gain 

'. *. 

'. 

1 

-82 L 

37 38 39 40 34 35 36 

Figure 6: Computer simulation result for tracking a linear path 

12 
30 32 34 36 38 40 24 26 28 

x (cm) 

Figure! 7: Computer simulation result for tracking a sinusoidal path 

x (em) 

Figure 8 Computer simulation msult for tra- a circular path 

13