THE CATHOLIC UNIVERSITY OF AMERICA 

DEPARTMENT OF ELECTRICAL ENGINEERING 

DESIGN OF AN ADAPTIVE CONTROLLER FOR A 

TELEROBOT MANIPULATOR 

Charles C. Nguyen 

Principal Investigator and Associate Professor 

‘# 

()L, 4.1 )/ -,‘ and 

Zhen-Lei Zhou 

Graduate Research Assistant /’l?J 

submitted to 

Mr. Gary E. Mosier 

Code 712.1 

Goddard Space Flight Center (NASA) 

Greenbelt, Maryland 

August 1989 
SUMMARY 

In this report, we present the research results from the research gmnt entitle "Development of 

Advanced Control Schemes for Telembot Manipulators, funded by the Goddad Space Flight 

Center, under the Gmnt Number NAG 5-1124 for the period between Mamh 1, 1989 and August 

31, 1989. 

This report presents the design of a joint-space adaptive control scheme for controlling the 

slave arm motion of a dual-arm telembot system developed at Goddad Space Flight Center 

(GSFC) to study telembotic opemtions in space. Each slave arm of the dual-arm system is a 

kinematically redundant manipulator with 7 degrees of freedom (DOF). Using the concept of 

model refemnce adaptive control (MRAC) and Lyapunov direct method we derive an adaptation algorithm which adjusts the PD contmller gains of the contml scheme. The development 

of the adaptive control scheme assumes that the slave arm motion is non-compliant and slowlyvarying. The implementation of the derived control scheme does not need the computation of the 

manipulator dynamics, which makes the contml scheme sufficiently fast for real-time applications. Computer simulation study performed for the 7-DOF slave arm shows that the developed 

control scheme can efficiently adapt to sudden change in payloads while tmcking various test 

trajectories such as mmp or sinusoids with negligible position errors. 
1 INTRODUCTION 

Telerobotics, a concept combined by two principles, tefeoperntion and robotics [l] has attracted 

tremendous research attention because it has potential applications in hazardous environments 

such as the future NASA space station, nuclear power plants, or undersea. Extravehicular 

activities such as assembly, maintenance or inspection in space, can be made less dangerous by 

cooperating astronauts and telerobots in a telerobotic operation. Depending on the complexity 

of the task to be performed, the astronaut can be either assisted or replaced by the telerobot. 

Teleoperations can be performed either in a tekopemted control mode or in a autonomous control 

mode [2]. In the teleoperated control mode, the astronaut remotely controls the motion of the 

slave arm of the telerobot using a master arm system. On the other hand, the slave arm 

performs its motion autonomously under the supervision of the astronaut, in the autonomous 

control mode. In any mode, to achieve a successful task, precise control of the slave arm motion 

should be ensured. 

Goddard Space Flight Center (GSFC) is developing a Flight Telerobotic Servicer (FTS) 

to perform a variety of tasks on the NASA space station, which include assembly, inspection, 

servicing, and maintenance. To support the FTS development, a dual-arm telerobot system 

consisting mainly of a pair of 6-DOF mini-master controllers and a pair of 7-DOF slave arms 

has been built at GSFC to serve as a testbed for studying a variety of issues such as zero-g 

operation, dual-arm robot control, advanced control schemes, and hierarchical control etc. [3]. 

This report concerns with the motion control of the slave arm which is a kinematically redundant 

robot manipulator. 

Control of redundant robot manipulators, defined as manipulators having more DOF than 

necessary to perform a specified tasks has been an active research area [4]-[7] because of its many 

advantages including the use of extra DOF to maneuver in a congested workspace and avoid 

collisions with obstacles. Most conventional control schemes such as Computed Toque Technique [8], are derived based on the dynamic modeling the robot manipulator. Since in addition 

to the massive computation requirement in evaluating dynamic models in real-time control, it 

is almost impossible to derive an accurate dynamic model for a robot manipulator, the above 

developed control schemes are generally not feasible. In an attempt to alleviate the problem 

of uncertainties in manipulator dynamics and payloads, considerable research effort has been 

focused on studying Adaptive Control [9]. Lyapunov method and MRAC were used to design 

adaptive controllers for trajectory control, which was proved to provide global stability [lo]. 

The authors in [ll] considered the design of robust adaptive controllers. Adaptive force control 

problem was investigated in [12] for a cutting problem. In [13] an adaptive force-position controller with self-tuning was designed in Cartesian space by using eigenvalue assignment method 

and minimization of a quadratic performance criterion. Research work in [14] resulted in the 

development of a decentralized adaptive control scheme based on independent joint control concept. A joint-space adaptive control scheme was developed in [15] to control the motion of a 

robotic end-effector with closed kinematic chain mechanism [16]. The problem of hybrid control 

of force and position was considered in [17]. Employing the Kalman filter, the authors in [18] 

proposed an approach to adaptive control of manipulators based on an identification scheme. 

Recently, MRAC and Lyapunov method were employed in [19] to design a joint-space adaptive 

control scheme for a telerobot manipulator. 

The organization of this report is as follows. In the next section, we first describe the struc1 
ture and operations of the GSFC telerobot testbed. Next a joint-space adaptive control scheme 

is developed using MRAC and Lyapunov method. We then present the computer simulation 

performed to study the performance of the developed adaptive control scheme which is implemented to control the motion of the 7 DOF slave arms, and evaluate the simulation results. 

Finally in the conclusion, we discuss the results of the report and outline some future research 

direct ions. 

The following notations are used in this report: 

0 MT: transpose of the matrix M 

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

0 z: (nxn) identity matrix. 

2 THE NASA TELEROBOT TESTBED 

Figure 1 presents the dual-arm telerobot system recently developed at GSFC to serve as a testbed 

for investigating the feasibility of telerobotic operations in space. The system mainly consists 

of a pair of slave arms, each of which is a 7-DOF kinematically redundant robot manipulator, 

manufactured by Robotics Research Cooperation (RRC) and a pair of 6-DOF Kraft Mini-Master 

(KMM) hand controllers from Kraft Telerobotics, Inc. At this time, the RRC controllers have 

been modified by GSFC engineers to allow implementation of advanced control schemes, and to 

incorporate other advanced features such as high-speed parallel processing. 

The operations of the telerobot system can be best explained using the block diagram given 

in Figure 2. The RRC slave arm system receives actual data of joint force and joint positions/velocities via force sensors and joint position/velocity sensors mounted on the slave arms. 

Joint forces that are transmitted to the KMM system for force reflection can be determined 

using the slave arm joint forces. In a teleoperated control mode, the human operator residing 

in an operator control station such as Space Shuttle, Space Station, or on ground, controls the 

motion of the slave arms via the mini-master controllers using familiar hand and arm movements while observing the slave arm motion and the task space from a cupola, a window or a 

TV monitor. The force/torque applied by the human operator produces 6 joint positions in the 

master arm system, which then are converted to 7 corresponding joint positions in the slave arm 

system through an appropriate coordinate transformation. The 7 joint positions of the slave 

arms will then serve as the desired variables to be supplied to a control system in the RRC system. Based on the difference between the desired joint variables and the actual joint variables 

of the slave arms, provided by feedback data, the controller then employs a control scheme to 

provide necessary control signals to the actuators of the slave arms so that it tracks the desired 

motion as closely as possible while maintaining a desired contact force on the task environment. 

In an autonomous control mode where the slave arms perform the task autonomously while the 

human operator plays only a supervisory role, the desired inputs to the slave arm control system 

can be generated by a Path Planner. 

2 
3 DESIGN OF THE ADAPTIVE CONTROLLER 

Since the desired inputs to the slave arm control system are expressed as joint variables, as 

explained in previous section, a joint-space control scheme should be considered to avoid the 

time-consuming evaluation of the forward and inverse kinematics when a Cartesian-space control 

scheme is employed in real-time control. Also to effectively react to the nonlinearity of the 

manipulator dynamics, errors in dynamic modeling, and sudden change in payloads, adaptive 

controllers should be developed instead of fixed-gain controllers which work well only when the 

manipulator stays within the linearized operating region [lo]. 

The following development of the adaptive control scheme focuses on controlling the motion 

of a single slave arm performing non-compliant motion, i.e. the slave arm moves freely in space 

without being in contact with the environment. In this case, pure position control scheme can be 

employed to control the motion of the slave arm, since contact force control is not needed. Hybrid 

position/force control scheme can be applied in the case of compliant motion [17]. Furthermore, 

we assume that the arm motion is slowly varying because rapid motion of the slave arms is 

considered to be dangerous during a telerobotic operation, especially in obstacle avoidance. 

The dynamics of the 7-joint slave arm can modeled by the following nonlinear vector differential equation [19]: 

T(t) = M(q, ii(t) + N(q, 4) q(t) + G(q, 4) q(t) (1) 

where q(t) denotes (7x1) joint positions of the slave arm, T(t), the (7x1) joint force vector, 

M(q,q), the slave arm mass matrix is a symmetric positive-definite matrix of order (7x7), 

N(q, q) and G(q, q) are (7x7) matrices whose elements are highly complex nonlinear functions 

of q and q. In the right-hand side of (l), if we neglect joint friction, the second term represents 

the centrifugal and Coriolis forces, and the third term the gravity forces. 

An adaptive controller consisting of a Proportional (P) and a Derivative (D) terms is defined 

by 

T(t) = Kp(t) qe(t) t Kd(t) qe(t) (2) 

denotes the joint error vector between the actual joint vector q(t) and the desired joint vector 

qd(t), Kp(t) and &(t) are the matrices of the proportional and derivative terms, respectively, 

of the adaptive controller. 

Substituting (2) into (1) yields 

M & + (N+Kd) t (G+ Kp) Qe = M 4 +N& +G Qd (4) 

where the dependent variables of the matrices and vectors were omitted for readability. 

z(t) such that 

In order to transform (4) into a state space form, we proceed to define a (14x1) state vector 

(5) 

which rewrites (4) as 

3 
where 

and 

A1 = M-'(G + Kg), A2 = M-'(N + Kd), 

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

and 

(7) 

(9) 

In the framework of MRAC, Equation (6) represents the adjustable system. The desired 

performance of the slave arm motion can be specified by a reference model in terms of the 

tracking error vector qe(t) = [qel(t) qe2(t). . . qe7(t)lT. Suppose the tracking errors qei(t) for 

i=1,2,. . . ,6, are decoupled from each other, and satisfy 

qei(t) + 2 ti wi qei(t) + w: qei(t) = o (10) 

for i=1,2, ..., 6, where ti and wi denote the damping ratio and the natural frequency of qei, 

respectively. Then the dynamics of the reference model can be represented by 

where Dl=diag(w?) and Dz=diag(Xiwi) are constant (7x7) diagonal matrices, and 

Solving (ll), we obtain 

We note from (14), that if zm(0) = 0, i.e. the initial dues of the actual and reference joint 

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

zm(t) = exp(Dt) zm(0). (14) 

vectors are identical, then zm(t) = 0. 

e(t) = zm(t) - ~(t), (15) 

then from (6) and (ll), we obtain an error system described by 

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

v(t) = eTPe i- tr [(A1 - D1)TIT1(A1 - Dl)] 

+it [(A2 - D2IT172(A2 - Da)] 

t tr [ A,TIT3A3] + tr[ATn4A4], 

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

to be determined later. 

Taking the time derivative of (17) and simplifying the resulting expression, we obtain 

6(t) = eT(PD+DTP)e 

+2tr [(Ai - DI)~(~$ + niAi)] 

t2tr [(A2 - D2)'(nG + nzAz)] 

-2tr [A3*(n9$ - n&)] 

-2tt [A4=(nq$ - n,A4)] (18) 

and P is given by 

pi p2 

p= [ p2 P3] 

and it is noted that e(t) = --J(t) since zm(t) = 0. 

In (10) (; and w; can be selected so that D is a matrix having stable eigenvalues, which 

is also called a Aurwitz matrix [20]. Therefore according to Lyapunov theowm, for any given 

positive-definite symmetric matrix Q, there exists a positive definite symmetric matrix P that 

satisfies the Lyapunov equation 

PD + DTP = -Q. (21) 

then (18) becomes 

which is a negative definite function of e(t). Furthermore, from (22)-(23), we obtain 

(24) 

(25) 

T 6(t) = -e Qe 

A1 = -n;lnqT; A2 = -n,'saqT, 

and 

A3 = n,'n& A4 = n:'n& 

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

for i=1,2,. . . ,4, are also positive definite matrices, then the error system described in (16) is 

asymptotically stable, i.e., e(t) -+ 0, or z(t) 4 z, as t -+ 00. 

As mentioned earlier, since we assume that the slave arm performs slowly varying motion, M, 

N and G are slowly time-varying matrices which can be considered as nearly constant matrices. 

Consequently from (7) and (8) we obtain 

Ai II M-'K,; A2 M-'Kd (27) 

5 
and 

A3zO; A4cO. 

Next substitution of (27)-(28) into (25)-(26) yields 

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

Kp = a1S24, (32) 

In (31), we note that nl and I72 are positive definite matrices that can be considered 

as nearly constant because the slave arm mass matrix M is positive definite and slowly timevarying. To satisfy (30), n3 and n4 should be chosen such that their determinants approach 

00 in addition to the positive definite property. To achieve this, we can select XI3 and n4 such 

that they are diagonal matrices whose main diagonal elements assume very large positive values. 

Now integrating both sides of (32) and (33) yields 

t 

Kp(t) = Kp(0) t ai / (P2qe t P34,)q:dt (34) 0 

t 

and 

Kd(t) = Kd(0) t a2 / (p2qe t P3q,)$dt (35) 0 

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

arbitrarily. 

Solutions for the controller gain matrices of the adaptive controller are given in (34) and 

(35). The adaptation scheme, as illustrated in Figure 3 is mainly based on the errors of the joint 

variables of the slave arm and the submatrices of P. Since P is a constant matrix and qe can 

be easily computed from the desired and actual joint variables available from the master arm 

system and feedback data, respectively, the computation time required to calculate the adaptive 

control law given in (2) is relatively small. Consequently, high sampling rates up to 1 KHz 

can be utilized in the implementation of the proposed control scheme. Thus the manipulator 

dynamics can be considered almost constant during each sampling interval of typically about 1 

ms. This feature of the proposed scheme is very attractive to real-time control implementation. 

We also observe that unlike other conventional adaptive control schemes designed using MRAC, 

the implementation of the developed adaptive control scheme does not require the slave arm 

dynamics. 

6 
4 COMPUTER SIMULATION STUDY 

In this section, using computer simulation we examine the performance of the developed adaptive 

control scheme which is applied to control the motion of the 7-DOF RRC slave arm illustrated 

in Figure 4. A complete list of the slave arm parameters can be found in [21]. In the following 

we will report two study cases. In the first case, the slave arm is controlled so that the 7 joints 

variables follow a set of desired ramp trajectories while suffering from step change of payload. In 

the second case, the 7 joint variables are controlled to track a set of desired sinusoidal trajectories 

during step changes in payload. In the second case, we will also compare the performance of 

fixed-gain controller versus the developed adaptive controller in terms of tracking capability. 

The computer simulation study is performed using a modified version of Manipulator Simulation 

Program (MSP) [21]. 

Controller Parameters 

0 Adaptive Controller: si and wi for i=1,2 were selected so that 2 characteristic roots of 

(10) are -1 and -2. In this case, we have D1 = 217 and D2 = 317. Selecting Qi = 114 for 

i=1,2,. . . ,6 and solving (21) using MATLAB we obtain P2 = P3 = 0.2517. The adaptation 

law can be now implemented by substituting the derived values of P2 and P3 into (34)-(35) 

where Kp(0) and Kd(0) can be arbitrarily set. The scalars a1 and cy2 can be adjusted to 

improve the tracking quality provided that they always assume positive values. 

0 Fixed-Gain Controller: In the case of fixed-gain controller, we use K, = diag(2925,1912.5, 

630, 360, 607.5, 24.4125, 5.67) and Kd = diag(416, 272, 89.6, 51.2, 86.4, 3.472, 0.8064), 

which are identical to Kp(0) and Kd(0) of the adaptive controller, as specified above. 

4.1 Payload Change 

Step change in payload occurs when the slave arm drops a load it is carrying, performs some 

tasks, and picks up the same load later. To study the adaptability of the control schemes in the 

above situation, we first set the payload at 70 lb. and drop it to 1/3 of the initial payload at 

1/3 of the simulation time and then increase it to full payload at 2/3 of the simulation time. 

Sudden payload changes are modeled by delayed step functions. 

4.2 Tracking Ramp Trajectories 

In this case the adaptive control scheme is applied to control the motion of the slave arm so that 

its joint variables follow a set of desired ramp trajectories. For each joint, the initial (I) and final 

(F) desired joint angles (A) are specified by the user, and the desired velocity is simply obtained 

by dividing the difference between the final and initial desired joint angles by the simulation 

time which is 50 seconds in all cases. Initial and final joint angles used in this case are given in 

the following vectors showing the data in the order starting with Joint 1 and ending with Joint 

7: IA = (0, 90, 0, 0, 0, 0, 0); FA = (195, 250, 305, 128, 200, 95.5, 270) with all data in degrees. 

Simulation results presented in Figure 5 show that the joint angles track their corresponding 

desired trajectories very closely with some negligible off-track errors in Joints 3, 4 and 6 at the 

transition of sudden payload changes. 

7 
4.3 Tracking Sinusoidal Trajectories 

In this case the slave arm is controlled so that its joint angles track a set of desired sinusoidal 

trajectories. The desired trajectory is specified by q;(t) = LA + 229sin(Vt) where IA denotes the 

initial joint angle, V = and with tf = 50 seconds, the simulation time. The desired velocity 

is obtained as q(t) = 229Vcos(Vt). In this case we use IA = (0, 30, 60, 0, 30, 60, 90) and 

from qi the desired final angle at the end of the simulation is equal to IA. Figure 6 presents the 

simulation results which show excellent tracking quality in all joints despite sudden change in 

payloads. Finally, to make a comparative evaluation of the developed adaptive control scheme, 

we now control the slave arm using the fixed-gain controller as specified before to track the same 

sinusoidal trajectories as in the case of the adaptive controller. As Figure 7 illustrates, tracking 

quality is deteriorated in Joints 6 and 7 because the controller is unable to react to the dynamic 

and static coupling between the joints and to sudden change in payload. 

t f 

5 CONCLUSION 

In this report, employing the concept of model reference adaptive control and Lyapunov method, 

we have developed a joint-space adaptive control scheme which is suitable for controlling noncompliant motion of robot manipulators performing slowly varying motion. The developed 

adaptive control scheme was applied to control the motion of the kinematically redundant slave 

arm of a telerobot system built at Goddard Space Flight Center to study teleoperation. Computer simulation of the control scheme implementation showed that the adaptive control scheme 

provided good tracking quality in spite of dynamic and static coupling between joints and sudden 

change in payload. Simulation results also showed that fixed-gain controller failed to effectively 

react to above disturbances. Current research activities are focusing on the development of 

an adaptive control scheme in Cartesian space for redundant manipulators and on compliant 

control of the slave arm motion using a hybrid position/force control [17]. 

References 

[l] JPL, ”Telerobotics Project Plan,” Jet Propulsion Labomtory, JPL D-5692, August 1988. 

[2] Kan, E.P., Austin, E., ”The JPL Telerobot Teleoperation System,” Robotics and Manufacturing: 

Recent ?).ends in Researth, Education, and Application, edited by M. Jamshidi et al, ASME Press, 

New York, pp. 577-585, 1988. 

[3] Schnurr, R., O’brien, M., Cofer, S., ”The Goddard Space Flight Center (GSFC) Robotics Technology 

Teatbed,” NASA Confewnce on Space Telerobotics, Pasadena, January 1989. 

[4] Klein, C.A., Huang, C.H., ”Review of Pseudoinverse Control for Use with Kinematically Redundant 

Manipulators,” IEEE %ns. Sys., Man, and Cyber., pp. 245250, March 1983. 

[5] Hanafusa, H., Yoshikawa, T., Nakamura, Y., ”Analysis and Control of Articulated Robot Arms with 

hdundancy,” Proc. 8th IFAC %ennial World Congress, Kyoto, Japan, pp. 1927-1932, 1981. 

[6] Yoshikawa, T., ”Analysis and Control of Robot Manipulators with redundancy,” Proc. 1st Infern. 

Symp. on Robotics Researth, New Hampshire, pp. 735-747, 1983. 

8 
[7] Baillieul, J., Hollerbach, J., Brockett, R., ”Programming and Control of Kinematically Itedundant 

Manipulators,” Proc. 23d IEEE Conf on Decision and Control, pp. 768-774, 1984. 

[8] Luh, J.Y.S., “Conventional Controller Design for Industrial Robote-A Tutorial,” IEEE hns. Systems, Man, and Cybern., Vol. SMC-13, No. 3, pp. 298-316, 1983. 

[9] Hsia, T.C., ”Adaptive Control of Robot Manipulators: A Review,” Proc. IEEE Conf. on Robotics 

and Automation, San fiancisco, pp. 183-189, 1986. 

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

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

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

Pmc. IEEE Conf. on Robotics and Aufomation, San Francisco, pp. 12041215, 1986. 

[12] Daneshmend, L.K. and Pak, H.A., ”Model Reference Adaptive Control of Feed Force in Turning,” hns. ASME, Journal of Dynamic Systems, Measurement, and Control, Vol. 108, pp. 215222, 

September, 1986. 

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

Automation, pp. 1386-1391, 1987. 

[14] Seraji, H., ”Decentralized Adaptive Control of Manipulators: Theory, Simulation, and Experimentation,” IEEE hns. Robotics and Automation, Vol. 5, No. 2, pp. 183-201, April 1989. 

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

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

March 1989. 

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

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

[17] 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. 

[18] Nemec, B., Zlajpah, L., Matko, D., ”Adaptive Control of Robots Using the Kalman Filter,” Int. 

Journal of Robotics and Automation, Vol. 4, No.1, pp. 19-26, 1989. 

[19] Nguyen, C.C., Mosier, G.E., ”Model Reference Adaptive Control of a Telerobot System,” Pmc. 

ISMM Intern. Symp. Computer Applications in Design, Simulation and Analysis, Nevada, pp. 282- 

285, Feb. 1989. 

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

[21] Chen, J., Ou Y.J., ”Dynamic Formulation for Efficient Digital Simulation of Telerobotic Manipulation,” Final Report, Grant No. NAG 5-1019 NASA/GSFC, December 1988. 

9 
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Figu~ 1: The CSFC Tclembot -bod 

\ 

I 1 

+: Jt 4; 0: Jt 5; *: Jt 6: x: Jt 7; --: Payload 

=O0 

,501 

1 

- 50 

0 10 20 30 40 50 

time in seconds 

+: Jt 1: 0: Jt 2; *: Jt 3; --: P8yload 

300 

250. 

200 . 

0 10 20 30 40 50 

time in seconds 

rSgure Sb 

ORIGINAL PAGE 

BLACK AND WHITE PJI-IOTOGRAPH 

10 
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I 

I 

+: Jt 1; 0: Jt 0; *: Jt 3; --: Payload 

Ol' 

0 10 20 30 40 50 

time in secondr 

Ficurr 6. 

360 

300 

250 

200 

I50 

100 

PL 

50 

PL/3 

0 

-50 

+: Jt 4: 0: Jt S: *: Jt 6; I: Jt 7; --: Payload 

3% I 

-60' 

20 30 40 50 0 10 

time in seconds 

+: Jt 4; 0: Jt 5; *: Jt 8: X: Jt 7: --: Paylobd 

1 

0 10 20 30 40 50 

time in seconds 

Figure 'fa 

+: Jt 1; 0: Jt 2; *: Jt 3; --: Payload 

300 1 

+ 

IO 20 30 40 50 0 

time in seconds 

Pigum 7b 

Figure 7: Packing using Fixed-Cain Controller 

11'