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FEEDBACK

CONTROL OF A FLEXIBLE MANIPULATOR K.

CYLINDRICAL

and M.-S. Lu

KRISHNAMURTHYf’

Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla, Rolla, MO 654014249, U.S.A. (Received 5 September 1989)

Abstract-Flexible robotic manipulators have been the subject of numerous studies in the past few years. Past studies, however, have concentrated on flexible manipulators with only revolute joints. This paper considers the case of a structurally flexible manipulator with prismatic and revolute joints; specifically, an algorithm for controlling a structurally flexible three-degree-of-freedom cylindrical manipulator is presented. The control algorithm involves two steps. Using nonlinear feedback, the equations of motion are first decoupled into three subsystems representing the three rigid degrees of freedom together with their

associated ‘flexibleequations’, if any. Then, using linear optimal control theory, controllers are designed for the three subsystems independently. Computer simulated results for an example system are presented.

1. INTRODUCTION

Lately there has been a great deal of interest in using high-speed, lightweight robotic manipulators for automated manufacturing and for space applications. A number of different methods have been presented to control single-link and multi-link flexible manipulators. Although past studies have concentrated on manipulators with only revolute joints, to date, only a very small number of studies have been conducted on flexible manipulators with revolute and prismatic joints. The problem with prismatic joints is that they increase the complexity of the dynamic model many-fold because the vibrating length of the link changes as it translates in the joint. This problem does not arise in the case of manipulators with only revolute joints, as the links will be of a fixed length. The dynamics of a spherical coordinate robot with one prismatic and two revolute joints (RRP) were considered by Chalhoub and Ulsoy [I]. The equations of motion were derived by the assumed mode/ Lagrangian approach and include the distributed mass and elasticity of the portion of the last link which extends beyond the prismatic joint. Also, the interrelationship between structural flexibility and controller design was investigated by implementing a feedback controller based on the rigid model of the robot on the rigid and flexible model. Chalhoub and Ulsoy [2] later presented limited experimental results for controlling the rotation of the spherical coordinate robot in only the horizontal plane. The partial results showed that by including the flexible motion in the controller design the positional accuracy of the end effector could be improved significantly, as was expected. By considering the time rate of change of the total vibrational energy, Wang and Wei [3] tTo whom correspondence should be addressed.

presented preliminary simulated results for the feedback control of motion-induced vibration during rest-to-rest maneuvers of a flexible cylindrical manipulator with one revolute and two prismatic joints (PRP). The equations of motion were developed by considering the forces and moments on a differential segment of the arm and neglecting the vibration of the arm’s tail section. In a different study, Krishnamurthy [4] presented a dynamic model of a flexible cylindrical manipulator similar to the one considered in [3]. The model included the vibration of the arm’s tail section and therefore improved on the model presented in [3]. It was shown that the motioninduced vibration could be considerable and it would be necessary to design sophisticated controllers to dampen out the motion-induced vibration so that a specified maneuver could be executed efficiently. The limited results presented in [l&4] draw our attention to the fact that control of flexible manipulators with prismatic and revolute joints requires further consideration. The objective of this paper is to present an efficient method for controlling the structurally flexible cylindrical manipulator considered in [4]. A feedback control algorithm for rest-to-rest maneuvers without large starting or stopping transients and with minimum residual vibration is presented. Computer simulated results for an example system are presented. The flexible cylindrical manipulator considered in this paper is representative of manipulators with prismatic and revolute joints, a configuration that has received very little attention in the past. 2. FEEDBACK CONTROL ALGORITHM

The equations of motion for the cylindrical manipulator shown in Fig. 1 can be obtained as 987

K. KRISHNAMURTHY and M.-S. Lu

988

I

Front End of the

FJt) Fig. I. Sketch of a flexible cylindrical manipulator

feedback is investigated. Rewriting the equation in r as

(see [4] for details) M(x)% + K(x)x + FNL(x, ic) = QF,

(1) (m,L +mrY

wherex=(q,,,q,,,r,0,p,,,p2,,z)r,j=1...n,isthe coordinate vector (4n + 3) x 1; q,, and y2, are the time-dependent generalized coordinates in the modal expansion of the in-plane deflection of the tail end and front end of the flexible arm, respectively; p,, and pz, are the time-dependent generalized coordinates in the modal expansion of the out-of-plane deflection of the tail end and front end of the flexible arm, respectively; r, 0, and z are the rigid-body degrees of freedom; n is the number of vibratory modes retained in the modal expansion; M is the mass matrix (4n + 3) x (4n + 3); K is the stiffness matrix (4n + 3) x (4n + 3); FNL is the vector containing the nonlinear terms (4n + 3) x 1; Q is the control distribution matrix (4n + 3) x 3; F = (F,, u, F,Jr is the input vector (3 x 1); F,, u, and Fz are the input forces/torque driving the rigid-body degrees of freedom r, 8, and Z, respectively; and the dots refer to time derivatives. To accomplish rest-to-rest maneuvers without large starting or stopping transients and with minimum residual vibration, a two-step feedback control algorithm is presented. In the first step, the equations of motion are decoupled into three subsystems representing the three rigid degrees of freedom together with their associated ‘flexible equations’, if any. In the second step, controllers for the three subsystems are designed independently. By inspecting eqns (3a)-(3g) of [4] one can see that the equations of motion can be decoupled by decoupling the equation for radial motion from the remaining equations. In this study decoupling by

+f(r, r, 8,& qll 421..

. . en,

. q2n3 42,

”

4,,

. 42n)

.

.4,.,

=

F,,

(2)

where mh is the mass per unit length of the arm, 15is the length of the arm, and mT is the tip mass, one can see that decoupling can be accomplished by letting F,=F:+F,**,

(3)

F:* =f(r, i, 8, 0, 411 . 4ln7 411. . q1,,3 where q2, q2n, cj2, cj2,,).Substituting eqn (3) into eqn (2) yields (m,L + m,)L’ = F,*,

(4)

which is decoupled from 0 and z. The constant coefficient linear differential equation (4) can now be used to design the controller for r without any difficulty. In other words, the radial control force is separated into two parts; one part will equal the nonlinear terms of the equation and the remaining part will control a linear radial motion. Once the controller for r is designed, the remaining equations become time-varying linear differential equations. Also, 0 and z subsystems are no longer coupled. The equations of motion for the 0 and 2 rigid degrees of freedom together with their associated ‘flexible equations’ can be written as .. M,,(r(r))x,, + C&(r), i(r))% + K&(r), i(r). i(f))+, = QGJ

(5)

989

Feedback control of a flexible cylindrical manipulator M,(r(r))&

+ C,(i(r))k,

+

K,(r(t), i(t), i(t))x, = Q,Fz position and velocity of the three rigid degrees of (6) freedom r, fl, and z. The time-dependent generalized

where xU=

(4lj?

42jv e)T

j=l...n.

xI=(p,,,p2,‘z)T,

The key aspect here is to decouple the equation for radial motion from the remaining equations. The nature of the dynamic model together with the procedure outlined breaks down the control problem into designing controllers for the three rigid degrees of freedom (together with their associated ‘flexible The procedure equations’, if any) independently. outlined is very appealing for two reasons: (a) As the controllers for the three subsystems given by eqns (4)-(6) can be designed independently, the computational difficulties associated with large order models, typical of flexible structures, can be avoided. (b) From a practical standpoint, the control algorithm can be implemented easily by using dedicated microprocessors for each controller. As the real-time computations are done in-parallel by different microprocessors, the procedure precludes the need for a large and/or faster computer to implement the control algorithm. A schematic of the control algorithm is shown in Fig. 2. The decoupling algorithm [eqn (3)] and the controllers for the three subsystems (to be presented in the next section) require full state feedback and the revolute joint acceleration. These requirements do not pose any particular problem from a practical standpoint. Various sensors are available to measure

xer t

&T ’

input)

coordinates qii and pij (i = 1,2, j = 1 . . . n) and their velocities can be obtained by analog integration of accelerometer signals [2]. The revolute joint acceleration can be obtained by differentiating the joint angular position/velocity in hardware or software [5]. The use of nonlinear state feedback to decouple has been investigated in the past (see, for example, Freund [6] and Fournier and Schilling [7]). Past work, however, has been restricted to robotic manipulators with rigid links. The decoupling procedure presented here is similar to that of [6] and [7]. However, it is important to note that the control scheme presented in this paper, i.e. decoupling the equations of motion so that the problem reduces to designing controllers for subsystems composed of the rigid degrees of freedom together with their ‘flexible equations’ independently, has not been previously investigated for flexible manipulators. 3. OPTIMAL CONTROL FORMULATION SOLUTION PROCEDURE

The actual control laws for the three subsystems, alluded to in the previous section, are presented here. Again, the control objective is to accomplish rest-torest maneuvers without large starting or stopping transients and with minimum residual vibration. As it is necessary to minimize any deviation from the specified final position of the arm and also suppress the motion-induced vibration, the control problem will be formulated as a linear quadratic regulator problem.

e-subsystem -

AND

u

flexible

d manipulator

F

Fig. 2. Schematic of control system.

e

%* %I xz’

;( z

990

Consider

K.

KRISHNAMURTHYand

the linear system

where g and 6 are (b x 1) vectors, Q. /?, and y are (b x b) matrices, and U is the input (1 x 1). The three subsystems eqns (4)-(6) have a form similar to eqn (7). Therefore, the control formulation and solution procedure will be outlined for the system given in eqn (7). Note that the r subsystem is timeinvariant, whereas the (3 and z subsystems are time-varying. To design the controller, eqn (7) is first rewritten in state space form as

X= AX+-BU,

(8)

M.-S. Lu

Solutions for the feedback gains and state trajectory equations for the r subsystem were obtained by utilizing the procedure outlined by Junkins and Turner [8]. They have presented closedform solutions for the feedback gains and state trajectory equations for time-invariant linear systems. Solutions for the time-varying 0 and z subsystems were obtained by adapting the procedure obtained in [8]. Solutions for the feedback gains and state trajectory equations were obtained by applying the procedure outlined in [8] in a piece-wise manner as follows. The total maneuver time was divided into m small, convenient intervals. Let t, be the time at the end of the ith interval. In the time interval t,_ , < t < t,, the time-varying system was approximated by the timeinvariant system

where X = A,X + B, U,

x = (g’, g’)’ A=

0

1

I

-a-‘y

-a-‘/I

where A, = A(t = t,) and B, = B(t = t,). The solutions for the feedback gains and state trajectories in the time interval t, _, < t < ti were then obtained by applying the procedure outlined by Junkins and Turner to the time-invariant system (11).

B=

and I is the identity matrix. The control law to drive the state vector X to zero (regulator problem) is picked so as to minimize the standard quadratic performance index J = ; XQ)S,X(t,)

+ ;

where S, is the terminal weight matrix, Q is the state weight matrix, and R is the control weight matrix. It is well known that for a controllable system the optimal feedback control is given by u(t) = -R

-~‘Brp(t)X(t),

where p(t) is obtained from differential Riccati equation

the

(9)

solution

P(t) + P(t)A + A’p(t) - p(t)BR -‘Brp(t)

(11)

of the

+ Q = 0,

P(f,) = S,.

(10)

In this study it was assumed that all states will be available for feedback. If in reality this is not true, the approach presented will still be valid as a state estimator can be designed. The regulator design will drive the state vector to zero. Therefore, to move the flexible arm to a specified final position, the initial position state variable is set to the starting position minus the final position. Then, as the regulator drives the state vector to zero, the flexible arm is moved to the specified final position.

4. COMPUTER SIMULATED RESULTS

The numerical values used for simulation purpose in this study are the same ones used in [4] with one exception. A tip mass of 2.268 kg was used in this study. The tip mass was increased to increase the flexibility effects the controller had to deal with. Figures 3-6 show simulated results for an example rest-to-rest maneuver where it is desired to change r, 0, and z from 0.5588, 0, and 0.2286 to 0.254m, I .5708 rad, and 0.5334 m, respectively, in 1.5 sec. Even though the maneuver time is only 1.5 set, the time histories have been plotted for 1.75 set to show the residual vibration. Figure 3 shows the time histories of the rigid-body degrees of freedom r, 0. and 1. The time histories clearly show that the maneuver is accomplished smoothly with very little residual vibration. Figure 4 shows the in-plane and out-of-plane tip deflections of the front end of the robot arm. The front end deflects about 2 (in-plane) and 0.6 mm (out-of-plane) during the initial part of the maneuver; however, the deflections are smoothly damped out. As the robot arm moves radially inwards, the length of the tail section increases, and so do the tail end deflections as shown in Fig. 5. However, the magnitude of the deflections is very small and the maneuver is completed with very little residual vibration in the tail section. Results presented in Figs 4 and 5 show that the controller damps out the motion-induced vibration very effectively. These figures also show the frequencies changing as the vibrating lengths of the front and tail sections change when the arm translates radially.

Feedback control of a flexible cylindrical manipulator

991

0.7 (a) 0.6

s e L

-2.5 0

0.38

0.7

1.0s

1.4

1.75

I.4

1.75

Time (d

5 0.2 *I :: E -0.1 d ,o -0.4 c 0 G -0.7

0

0.38

0.7

1.05

1.4

2 _,I tk 0

I.75

0.35

0.7

1.05

Time

(II)

Tina (11

Fig. 4. Tip deflection of the front end of the robot arm. (a) In-plane; (b) out-of-plane.

0.7 (d 0.6 0.5

3

3 0.4 N

s

0.3

0.25 (2) 0.15

3 0.2 0.1

L

0

f 0.05 L d P -0.05 If 0 5 -0.15

I

0.35

0.7 Time

1.05

1.4

I.75

(al

Fig. 3. Time histories of the rigid-body degrees of freedom. (a) r displacement; (b) B rotation; (c) z displacement.

The time histories of the input forces/torque are shown in Fig. 6. The jump discontinuities in the input forces/torque at the initial and final times are the result of formulating the problem without any penalty on the control rate. The small oscillations in the u and F, profiles are contributed by the bending modes being controlled. The results presented were obtained by including the first two bending modes for the front and tail sections. This resulted in the 0 and z subsystems being 10th order. The control weight matrix for the three subsystems was chosen to be 1 and the state weight matrix was chosen to be a diagonal matrix with the diagonal terms to be equal to 1. The terminal weight matrix was also chosen to be a diagonal matrix with the diagonal terms equal to IO6 for r, and IO’* for 0 and r subsystems.

=:

z

-0.25 1 0

0.35

0.7

i.05

1.4

1.75

Time bl

P

0.035 (W

s

!

0.025 -

z m E

0.015 -

6 ,o e 1 hi

i .I

1

0.005 -

-0.005 -

I

I

e -0.015 OX

1.75 Time

(d

Fig. 5. Tip deflection of the tail end of the robot arm. (a) In-plane; (b) out-of-plane.

K.

992

KRISHNAMURTHY

and M.-S. Lu 5. CONCLUDING

’ Ita)

-3o----L [email protected]

1.78 (9)

REMARKS

The control of a structurally flexible three-degreeof-freedom cylindrical manipulator was considered in this paper. The approach to solve the control problem was to decouple the equations of motion so that the problem reduced to designing controllers for the three rigid degrees of freedom (together with their associated ‘flexible equations’, if any) independently. Such an approach is advantageous from a practical standpoint and has not been previously investigated for flexible manipulators. Computer simulated results for an example rest-to-rest maneuver without large starting or stopping transients and with minimum residual vibration of the flexible cylindrical manipulator were presented. REFERENCES

N. G. Chalhoub and A. G. Ulsoy, Dynamic simulation of a leadscrew driven flexible robot arm and controller. ASME, (1986).

.l. dynam. Syst. Meas. Control 108, 119-126

_. N. G. Chalhoub and A. G. Ulsov. . Control of a flexible robot arm: experimental

and theoretical

results. In

Modeling and Control of Robotic Manipulators and Manufacturing Processes (Edited by R. Shoureshi, K. -2 ' 0

0.35

0.7 Tim

1.08

1.4

I 1.75

(d

Youcef-Toumi and H. Kazerooni), pp. 287-295. ASME, New York (1987). P. K. C. Wang and J. -D. Wei, Feedback control of vibrations in a moving flexible robot arm with rotary and prismatic joints. In Proceedings of IEEE International

11

Conference

on

Robotics

and

Automation,

pp. 1683-1689. IEEE Computer Society Press, Washington, D.C. (1978). K. Krishnamurthy, Dynamic modelling of a flexible cylindrical manipulator. J. Sound Vibr. 132, 143-154

7

(1989). .I. J. Craig, P. Hsu and S. Shankar, Adaptive control of mechanical manipulators. In Proceedings of IEEE International Conference on Robotics and Automation, pp. 190-195. IEEE Computer Society Press, -3

-13 ' 0

0.31

0.7 Tima

1.05

1.4

I 1.7s

(~1

Fig. 6. Input forces/torque. (a) Force for r; (b) torque for 8; (c) force for z.

Washington, D.C. (1986). E. Freund, Fast nonlinear control with arbitrary poleplacement for industrial robots and manipulators. ht. J. Robotics Res. 1, 65-78 (1982). S. J. Fournier and R. J. Schilling, Decoupling of a two-axis robotic manipulator using nonlinear state feedback: a case study. ht. J. Robotics Res. 3,76-86 (1984). J. L. Junkins and J. D. Turner, Optimal Spacecraft Rotational Manewers, pp. 41&411. Elsevier Science, Amsterdam (1986).