Compliant control of a two-link flexible manipulator by constraint Hamiltonian system

Mech. Mach. Theory Vol. 33, No. 3, pp. 293-306. 1998

Pergamon PII: S0094-114X(97)00041-4

© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-114X/98 $19.00 + 0.00

COMPLIANT CONTROL OF A TWO-LINK FLEXIBLE MANIPULATOR BY CONSTRAINT HAMILTONIAN SYSTEM SEUNG-BOK CHOI and HYUC-BUM LEE Department of Mechanical Engineering, Smart Structures and Systems Laboratory, Inha University, Incheon 402-751, Korea

and

BRIAN S. THOMPSON Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, U.S.A.

(Received 15 January 1996)

Abstract--Two essential tasks of robotic manipulators are characterized by position-to-position movement and contact with external environment. The contact usually results in the generation of external forces in the end-effector of the manipulator, which always modify the dynamic behavior. Consequently, issues of appropriate modeling techniques and effective compliant control strategies arise. In this paper, a new approach employing the framework of constraint Hamiltonian system is proposed for the compliant control of a two-link flexible manipulator with surface constraints. Two non-linear controllers consisting of force part and position part are derived from a constrained Hamiltonian system, followed by the formulation of corresponding linear feedback controllers that satisfy the Lyapunov stability of the total Hamiltonian system which possesses the non-linear controllers. The compliant control strategy is accomplished by steering the end-effector of the flexible manipulator onto the constraint surface with the linear controllers, and subsequently by executing imposed desired motion with the non-linear controllers. Computer simulations are undertaken in order to demonstrate the analytical formulation and the effectiveness of the proposed control methodology. © 1997 Elsevier Science Ltd

NOMENCLATURE A---cross-sectional area of the flexible link °d~--position vector of any point on the link i in the reference coordinate system E--Young's modulus of the flexible link H(a,p)--free Hamiltonian system H~(a,p)---constraint Hamiltonian system with Lagrangian multiplier H~.(a, p)--constraint Hamiltonian system with desired Lagrangian multiplier H~.(a,p)---constraint Hamiltonian system with force controller Hr(a, p)---total Hamiltonian system /.-,--inertia moment at the center of mass along the direction z~ /.-A:--area inertia moment along the neutral axis of the flexible link l~--length of the ith link M(a)--inertia matrix m~--mass of the /th link mr--tip mass °Pt--position vector from the reference coordinate to the tip mass p--momentum qj(t)~generalized coordinate of the j t h vibrational mode rj--length from the /th joint coordinate system to the center of the ith link ~j(x)---eigenfunction of the j t h vibrational mode p--mass density of the flexible link (,)--Poisson bracket [(,)]--matrix of Poisson brackets

INTRODUCTION

Most industrial robots depend on a bulky design. The bulkiness is mainly due to inherent design requirement to minimize structural vibration by increasing the mechanical stiffness of each component. This massive structural design makes the robot slow and heavy; hence, the system 293

294

Scung-BokChoi and Hyuc-BumLee

requires large actuators and a high mounting strength and rigidity, which consequently leads to high energy consumption and high overall operation cost. Therefore, modeling and active control methods for robot manipulators featuring flexible links are needed. Two essential tasks of the manipulators are characterized by position-to-position movements and contacts with external environment. The former tasks involves pick-and-place and spray painting, while the latter includes deburring, grinding and assembling. The contact usually results in the generation of external forces in the end-effector of the manipulator, which always modify the dynamic behavior. Consequently, issues of appropriate modeling techniques and of effective compliant control strategies arise. Compliant motion of the manipulator calls for the input torque to achieve accurate tracking for a desired path on the constraint surface and with a specified contact force. In principle such tracking is possible because the movement of the end-effector is limited to a submanifold on the constraint surface and consequently frees some components of the input torque to control the contact force upon the surface. However, the non-linearity of the governing dynamics as well as the presence of singularities at some points on the constraint surface could potentially make the control process difficult if not impossible. Recognizing the importance of compliant motion for robots, numerous researchers have investigated this problem in the past decade. Some compliant control strategies for robotic manipulators featuring rigid-links include Lyapunov stability [1], optimal control [2], sliding mode control [3, 4], and adaptive control [5, 6]. However, up to now, the researches on dynamic modeling and compliant control schemes for robotic manipulators featuring flexible-links are relatively rare. The flexible manipulators, though having some inherent advantages over the conventional rigid robots, have posed more stringent requirements on the control system design such as fast suppression of transient vibration. Matsuno et al. [7] proposed a method for the hybrid position/force control of the two degree-of-freedom manipulator with flexible second link which moves in a horizontal plane. By introducing some assumptions, a quasi-static model was derived and the hybrid control law was designed from the model. Yim and Singh [8] treated the question of position and force control of three-axis elastic robotic systems on a constraint surface based on non-linear inversion of an input-output map and linear feedback stabilization. Yoshikawa et al. [9] proposed a hybrid position/force control scheme utilizing the so-called M - m system which consists of the large flexible arm (M-part) and the small rigid manipulator (m-part). This paper presents a new approach for compliant motion control of a two-link flexible manipulator. A governing constraint Hamiltonian system is formulated using the Lagrange and Hamiltonian methods. The constraint imposed to the surface is transformed to the joint angles. This makes it convenient for the control action which manipulates the torques to control the angles directly. Another advantage is that the constraint surface to be recovered from the imposed joint constraint can be accurately divided into two sectors; force control part and position control part. It is shown that the Lagrangian multiplier vector can be uniquely determined for a given set of constraint equations. This property directly leads to the derivation of two non-linear controllers; the control of the contact force and the control of compliant motion. The former is the natural response of the surface to the manipulator. Therefore, the end-effector is acted upon by the force control input to exert the desired contact force at any point on the constraint surface. The latter steers the end-effector to track a prescribed desired path on the constraint surface, while maintaining a desired contact force profile. Consequently, this scenario describes the compliant motion control. The last step to achieve successful compliant control is to formulate linear feedback controllers corresponding to the non-linear controllers. These controllers should be designed to ensure the stability of the overall compliant controlled robot system which possesses the two non-linear controllers. The Laypunov stability theory is herein adopted to design such linear feedback controllers. Computer simulations are undertaken in order to demonstrate the effectiveness of the proposed control methodology. MODELING AND IMPOSITION OF CONSTRAINT CONDITIONS Modeling of a two-link flexible manipulator Consider a planar robot manipulator which consists of one rigid link and one flexible link as shown in Fig. 1. In this figure, O X Y is a reference coordinate system and 02 represents the rotation

Compliant control of a two-linkflexiblemanipulator

295

y/ _\

constraint

6 ( x , i i ~

Yl

~

Y2~ _ ~ _ ~ x l

-°2 02

S:>

O,o,

X

Fig. 1. A two-link planar flexiblemanipulator. of the elastic link in the undeformed position which lies along o2x2 axis. The component 6(x, t) represents the elastic deformations of the flexible link at a distance x from the second joint coordinate system about z2 axis which goes out the paper. This can be given by

6(x, t)= ~ c~j(x)qj(t)

(l)

j=l

Here, n denotes the number of significant vibration modes to be considered in the control system. The free Hamiltonian of a general multi-degrees of freedom mechanical manipulator is achieved by summing the total kinetic energy and the potential energy. The kinetic energy T~of the ith link, the kinetic energy 7", of the tip mass, and the total kinetic energy T(a, p) are given respectively, as follows: [10] 1 r" 0dT.0d, din,+ 1 L,O~ T'= 2Jo TI

-~-

2

(2)

1. 0ltlff0~ IFt

(3)

~¥Flt r t

1 *

T(a, p) = ~, 7", + 7", = ~ a"T M(a)a

"

(4)

i=1

where, a = [0j, 02, q~(t),..., q,(t)] T and superscript T stands for the transpose. In the mean time, using the momentum p, equation (4) can be written by

[aT(a,p)m ah I = M(a)a

P= L

(5)

T(a, p) = ~TM(a)- ~p

(6)

The potential energy V(a) is described by the strain energy stored in the flexible link. Thus, if the extensional stiffness and the torsional stiffness are neglected, then the potential energy is given by the following expression:

'.[o ox,"y

V(a) - ~

"

-,A [a26 (x,

dx

(7)

Therefore, the free Hamiltonian and its equations of motion are given as follows:

H(a, p) = T(a, p) + V(a)

(8)

296

Seung-Bok Choi and Hyuc-Bum Lee

OH(a, p)

0,-

~

ap~

= -

i = 1, 2

OH(a, p) O0,

(9)

i = 1,2

(10)

Imposition of constraint conditions T h e type o f constraint equations depends u p o n environmental working conditions at which the r o b o t can operate. In this study, a circle with a radius of R is a d o p t e d as a surface constraint as shown in Fig. 2. Thus, the constraint equations are given by

(11)

4~(p) = X~ + Y ~ - R2 = 0 X = /lCl -'l- 12C12

-

-

~ESI2

Y = llsl + 12s~2+ 6EC12

(12)

Here fie = f(12, t), Ct = COS 0t, C~z = COS(0~ + 02), St = sin 01 and s~2 = sin(0~ + 02). The surface constraint (11) with (12) is to be transformed to the joint constraints in order to formulate the constraint H a m i l t o n i a n system as follows: 02 = cos-t z 02 =

(13)

~ x/1 -- z 2

(14)

where, -12(g + ~ - R 2 + 6~) - ~ / ( - ~ z =

+ 2gl,~ - l4 + 2 ~ R 2 + 2 ~ R 2 - R')6~ + 2(~ - ~ + R2)6~ - fit 21t(~ + 6~)

Therefore, the constraint equations ~i~(a,p), k = 1, 2 and constraint surface [2 restricting the end-effector o f the m a n i p u l a t o r are expressed by • t(a,p) = 02 - cos -~ z

(15)

• 2(a, p) = 02 + ~ ~/1 - z 2

(16)

[2 = {(a, p ) l ~ ' ( a , p) = ~ ( a , p) = 0}

(17)

F r o m the constraint conditions defined by equations (15) and (16), the constraint force f a t joint o f the robotic m a n i p u l a t o r can be expressed in terms o f the L a g r a n g i a n multiplier vector 2 as Y

f

X

Fig. 2. Constraint surface of the manipulator.

Compliant control of a two-linkflexiblemanipulator

297

2

f=

Z 2, d ~

(18)

k=l

Here d ~ k denotes the differential of the ~*. If this constraint force is added to the free Hamiltonian system using the d'Alembert principle, then equations (8)-(10) are converted to following constraint Hamiltonian system and its equations of motion in which the end-effector of the robotic manipulator is constrained to the constraint surface. 2

H~,(a,p) = H(a,p) + ~ 2k~*(a,p)

(19)

k~l

O, - t~H(a, p) ~ "~ ~ Off~k(a, p) ~p-----7 + ~

i=l,2

(20)

k=l

P' = -

OH(a, p) ~'. J.k O~(a' p) 00, c30,

i = 1, 2

(21)

k=l

The Lagrangian multiplier vector 2 can be determined from above constraint Hamiltonian system as the following lemma. Lemma. If the matrix of Poisson brackets associated with the constraint equations (15) and (16) is chosen to be non-singular, then the Lagrangian multiplier vector 2 is given by 2 = f~g.~E~H*

(22)

where, ). = [21, 22] T t'~

= [~*] = [ ( ~ , ~,~)]

On. = [(H, q~'), (H, ~2)]T Proof

Using the concept of symplectic manifold [12], the vector field Xn~ of the constraint Hamiltonian is achieved by

Xn; = J grad H;, where

Xn~ must be tangential to the constraint surface f~. Thus, from the properties of Poisson bracket operation; the bilinear and the skew-symmetric properties, the following relationship between H~(a,p) and cI~(a,p), j = 1, 2 is obtained [11, 12].

2

(H, tl~) = ~ 2k(t/V, ~,) k=l

Now, since the matrix of Poisson brackets f ~

is non-singular, the lemma follows.

298

Seung-BokChoi and Hyuc-BumLee DESIGN OF N O N - L I N E A R CONTROLLERS

Force controller The Lagrangian multiplier vector ). given in the lemma is proportional to the amplitude of the force along the gradient of the constraint function evaluated at every point (a, p) on the constraint surface. Therefore, the force controller must generate the force such that the Lagrangian multiplier vector 2 defined on the constraint surface has a certain value. A desired Lagrangian multiplier 2", which takes into account the vibrational modes of the flexible link, is to be defined as a function of the constraint surface. And consequently, the corresponding desired normal force at the point of contact on the constraint surface is to be specified so that the force controller and the constraint system converse each other. To do this, let the constraint surface and the constraint equations be F and Ok(a, p), respectively. Then, the constraint Hamiltonian system and the constraint surface are expressed as follows: 2

H;.(a, p) = H(a, p) + ~ 2k*¢bk(a, p)

(23)

k=l

F = {(a,p}lO'(a,p) = ®2(a,p) = 0}

(24)

t o o = [OJkl = [ ( ~ , Ok)l

(25)

The conditions (24) and (25) specify allowable class of F surface. When 2" is defined as a function of the constraint surface F, 2" can be realized by the force controller h. Thus, if the force controller is substituted into equation (23), the constraint Hamiltonian is converted to 2

H~.(a,p) = H~.(a,p) + ~ ~kOk(a,p)

(26)

k=l

Now, the force controller ~ can be determined in the following theorem. Theorem 1. If the matrix of Poisson brackets Foo is non-singular at every point upon the constraint surface, then the force controller ~ is given by = F~r,o

(27)

where, = [~,, ~2] T

Fno = [(Hz., ®'), (H~., ®2)]r Proof Let the vector field of H~. at any point y on the constraint surface F be Xn~.(Fy). Then, for any vector x e ~Fy, the following relationship is satisfied [11, 13]. i~w(Xn~.) = (x, dH~.) = (x, dH~.lr) Here, T stands for the tangent bundle, i~w(Xno.) and (,) denote the interior and inner product, respectively, and dH~. denotes the differential of the H,.. In the above equation, dH,,Ir implies that the vector field of H~. is restricted to the constraint surface r, Thus, from the point of view of the lemma, the following relationship between the H~.(a, p) and OJ(a, p),j = 1, 2 is established. (H~.,~)=

Ha.+

~k®k,O j = ( H ~ . , O j ) k=l

~ ~k(Oj,O k ) = 0 k=l

Therefore, using the non-singular property of the matrix of Poisson brackets Fee, the theorem follows. The Ok(a, p) and F represent arbitrarily defined constraint equation and constraint surface corresponding to the constraint equation (l~(a, p) and constraint surface f~, respectively. If the

Compliant control of a two-link flexiblemanipulator

299

orthogonal submanifold of D is a subset of the orthogonal submanifold of F, consequently we can choose any 2" defined on D. On the other hand, the constraint surface F is a subset of the constraint surface D. This leads to that the flexibility of steering the position and velocity on D is restricted to the subset F. The reverse is also true. Thus, F is the same as the fl and equation (27) can be expressed as follows: Dg~[(H;., ~'), (H;,,

=

(I)2)] T

= Dg,~[Dm, - £~¢,®2"] = Dg~D.. - 2*

(28)

Thus, from equations (15), (16) and (28), the force controller about the given constraint conditions is derived as

D~. =

I

mll%/01 -- Z21

_rnll~/l _ zZ

(29)

1

m,, i x / ~ _ z 2 (H, • 2) - 2*

7

=

,2)j =

(30)

1

m, r------=~/1-z~(H, • ~) - 2*

Position controller On the contrary to the force controller which is derived as a function of the constraint surface defined by the constraint equations, the position controller is defined by the unconstrained degrees of freedom. Letting Wk(a, p) = 0, k = 1, 2 as constraint equations for the degrees of freedom in which the position controller is to be chosen, the corresponding constraint surface S can be expressed by S = {(a, p)lqJk(a, p) = 0,

k = 1, 2}

(31)

S+, = [~t'~k] = [(q~', ~*)]

(32)

In order to achieve compliant motion control, the force controller needs to be augmented by the position controller ~7onto the constraint surface. By doing this, the total Hamiltonian system and its equations of motion become as follows: 2

HT(a, p) = H(a. p) +

2

(2k + ~,)~*(a,p) + ~ ~kWk(a,p) k=l

O~= OH(a.p)op, + ~ (2k + ~*) O~k(a'P) Op, + ~ tTk0Wk(a, Ope p) k=l

P, = _3H(a,p)00,

- ~ k=l

(33)

k=l

i = 1, 2

(34)

k=l

(1, + ~,)

O~k(a'p) ~ " 0Wk(a,p) 80i -uk 00, k=l

i = 1, 2

(35)

Consequently, the position controller is obtained in the following theorem. Theorem 2. If the matrix of Poisson brackets Sv+ is non-singular, then the position controller t7 is given by

= S~SHv, where a=

[a~, a~]~

Sin. = [(H - HT, W'),

(H - HT, W2)]T

(36)

300

Seung-BokChoi and Hyuc-BumLee

The proof can be done similar to Theorem 1. ~Fk(a,p) can be arbitrarily chosen for the unconstrained degrees of freedom so that Svv is always non-singular. In this study, ~Fl(a, p) and W:(a, p) are chosen to be Wl(a, p) = 0~ and W2(a, p) = 01, respectively. Thus, the matrix of Poisson brackets Svv and the position controller ~ are constructed as follows:

Ul]

-_l l(H

-

S v v = [ 0-m22 n~22]

(37)

n~, v') 7

(38)

~-~22(H DESIGN

OF LINEAR

FEEDBACK

HT, Wt) J

CONTROLLERS

The stability of the compliant control system (34) and (35) with the two non-linear controllers (30) and (38) is not guaranteed. Therefore, the design of linear feedback controllers, which ensure the stability of the total Hamiltonian system and also steer the end-effector to a position onto the constraint surface in order to generate the specified force must be accomplished. For this, the non-linear controllers (30) and (38) are firstly modified by connecting the corresponding linear controllers 2" and if* as follows: 1

ftt=

m,t

fi2 -

lx/-f--~-~_ z2 (H, cD2) - ~* + e~, + f~l* 1

mn 1~:i-~_z 2

(H, ~1) _ 22" + e~ + fi~

(39)

a, = - ~ -1~ (H, 'e:)+ a r 1

ti2 -- --m22(H, W1) + a2*

(40)

where, % = 2k* -- 2k, k = 1, 2. The substitution of equations (39) and (40) into equations (34) and (35) yields the following augmented control dynamic systems O, = m22a~'

02 = mnx/1

Z 2 U2~

(41)

'~a'~ ^* a02 u2

(42)

-

b, = -a,*

/,~ = -2,*

-

To design the 2" and a* which guarantee the stability of the control systems (41) and (42), let the Lyapunov candidate function L be L = ½(01- 0~*): + ½(0: - 0~*): + ½(p~ - pt*): + ½(/9: - p2*)2 + ½/~,e~+ ½/~ae~+ ½/~ae~+ ½/~e~ (43) H e r e , / ~ , k = 1. . . . . 4 is any positive number, 0* and p~*, i = 1, 2 are the desired joint angle and momentum, respectively. The error variable e~ is obtained from the following relationships.

~, =

0, -

0~'

b: = 02 -

0~*

b3 = p, - p,*

b, = p2 - p:*

Compliant control of a two-linkflexiblemanipulator

301

Thus, from equations (41) to (43), the linear feedback controllers which satisfy the condition of the Lyapunov stability are designed as follows: 8¢b 2 r

u*=--~2

L

1 m H ~

{ @ -- k,2e2 -- kp2(02 - 02")}] + kae, + k~(p~ - p2*)

l fi2* - m , lx/]--S~_ z 2 [02* - k,:e2 - kp2(02 - 02*)] a* = ki3e3 + kp3(p, - p~*) a~' = ~

1

, [0, - ki, e, - kp,(O, - 0")]

(44)

Here,/~pk, k = 1. . . . ,4 is any positive number. Using equations (41), (42) and (44), the time derivative of L given by (43) becomes /~ = -.~p,(O, - Or*)2 - ~:p2(02 - 02*)2 - .~p3(p, - p , ) 2 _ ~ ( p 2 - p ? ) 2

(45)

It is obviously observed that equation (45) is zero if and only if 0~ = 0* and p~ = p*, i = I, 2, otherwise is negative. Therefore, the condition of Lyapunov stability is satisfied and hence the stability of the augmented control systems is guaranteed. From the practical point of view, the control inputs (39) and (40) should be represented by the joint torques. To do this, equation (34) is modified to be 2 O~k(a' P) ~ OqAk(a' P) M(a)O, = M(a) OH(a, - -ap, p) + M(a) ~ (2k + ~k) ~p, + M ( a) ak ~p, k=l k=l 8.00

6.00 4.00


//

--

:desired

:actual

3:o

e:o

9:o

J2:o

15.o

Time(see) 2.50 2.00 1.50v

¢q ¢3:) 1.00

.....

0.50

0.0%

310

~i0

9~0

:desired :actual

~2'.0

Time(see) Fig. 3. Controlled trajectories of the joint angles.

15.0

(46)

Seung-Bok Choi and Hyuc-Bum Lee

302

1.0 08

0.0 /~

0.4

0.2 0.0

-0.2 -0.4 -0.0

- - -:desired :actual

-0.8

-1.0 -1.0-0.8-0,0-0.4-0.20.O 0.2 0.4 0.6 0.8 1.0

X(m) (a) end point trajectory 1.20. [.00-

z:

0.800.60-

0.400.20 5.00 4.00 3,00 z t~

2.00 i,O0 O.O0-t.O0

o.o

,,

3:0

elo

91o

12".o ts.o

Time(see)

(b)

contact force

Fig. 4. End point trajectory on constraint surface and contact force response.

Then, substituting equations (9) and (10) into (46) and also differentiating equation (5) with respect to time yield following dynamic equations.

Jf/1(a)O, + M(a)t)~. = ~t~

M(a)O~ + M(a)O, +

~H(a,p)

O0-----~= ~2

(47)

(48)

Compliant control of a two-link flexiblemanipulator 4,00

303

'

3.00 ~'.,->

2.00

1.oo,< 0.00

"~ -1.00 -2.00 -3.00

o.o

3.'o

8;o

9;0

12;0

15.0

Time(see) Fig. 5. Tip deflectionof the flexible link. Here the ~ and ~2 are given in Appendix A. Therefore, eliminating the terms involving the Lagrangian multipliers associated with constraint forces gives following joint torques.

Ti

p) - k~ OgPk(a'p) + ~(a) ~ ~cbk(a, @, 00,

~:,\

k=l

@, ) J

k=l

k

*

(49)

6.00 5.00 4.00 3.00 2.00

1.00 ~.__ 0.00

-t.O0 -2.00

0.0

io

8:o

o:o

IZ;O

t5.o

1~io

15.0

Time(sec) 0.00 0.40

'

""

0.20

Z ¢,

o.oo ~ -0.20 -0.40

o.o

31o

81o

olo

Time(see) Fig. 6. Time histories of the joint torques.

304

Seung-Bok Choi and Hyuc-Bum Lee Table 1. Specifications of a two-link flexible manipulator Link 1 Link 2 m~: 10kg m2: 0.2kg l,: 0.5m 12:0.Sin /_-,: 1.6 kgm: A: 1.43 e - 4 m2 L~: 4.77 e - l 1 m4 p: 2800 kg/m3 E: 70 GPa

Here the superscript * denotes the time derivative of the corresponding term. S I M U L A T I O N RESULTS AND D I S C U S S I O N S In order to demonstrate the effectiveness of the proposed control scheme, the two-link robotic manipulator, which has the link specifications in Table 1, is considered. For the flexible link (2 m m thickness), the first and the second vibration modes were considered as the primary modes to be controlled, and the tip mass of 0.02 kg was attached. In this case, the inertia matrix M ( a ) is given by Appendix B. F r o m the material and geometrical properties of the flexible link, the first and second mode natural frequencies were calculated by co~ = 40.63(rad/s) and co2 = 254.65(rad/s). In the mean time, the desired joint motions and the force components were chosen as follows: 0* = 0.05t(rad) 2~* = 1.0

0* = cos -~ z(rad) 22" = 0.0

A n d / ~ and ~pk were chosen to have 5.0 for all k. The initial conditions of 0i were imposed by 0, = 1.5(rad), 02 = 0.0(rad), respectively. Figure 3 presents the tracking responses of the joint angles. It is clearly observed that accurate tracking performance is achieved within a favorable time (about 4 s). Figure 4 shows X - Y coordinates of an end point trajectory on constraint surface and contact force response. The actual path tracks very well the circular reference trajectory without oscillatory motion. And, as shown in Fig. 5, the tip deflection of the flexible link was well suppressed after initial switching motion to have certain steady-state value. Figure 6 shows the time histories of the joint torques. As expected, the joint torque x2 for the flexible-link is much smaller than the joint torque zt for the rigid-link. CONCLUSIONS In this paper, compliant motion control of a two-link flexible manipulator was accomplished. The equations of motion were derived by Lagrange and Hamiltonian methods, and the constraint force corresponding to the surface constraint condition was related to the free Hamiltonian system by the Lagrangian multiplier. The two non-linear controllers, the force and position controller, were derived as a function of the constraint surface defined at the constrained degrees of freedom and of the constraint surface defined at the unconstrained degrees of freedom. This implies that the constraint surface could be accurately divided into the force control surface and the position control surface. Hereafter, the linear feedback controllers which ensure the stability of the total Hamiltonian system possessing the two non-linear controllers were derived from the Lyapunov stability theory. It has been shown through computer simulations that favorable compliant performance featured by joint tracking without oscillation was achieved by employing the proposed control methodology. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8.

Takegaki, M. and Arimoto, S., Robotics Research, exi. M. Brady and R. Paul. MIT Press, 1984, p. 783. You, L. S. and Chen, B. S., International Journal of Control, 1993, 58(2), 253. Su, C. Y., Leung, T. P. and Zhou, Q. J., IEEE Transactions on Automatic Control, 1992, 37(5), 668. Fedele, A., Fioretti, A. and Ulivi, G., IEEE International Conference on Robotics and Automation, 1992, p. 2126. Yu, H., Seneviratne, L. D. and Earles, S. W. E., Second IEEE Conference on Applications, September 1993, p. 505. Arimoto, S., Liu, Y. H. and Naniwa, T., IEEE International Conference on Robotics and Automation, 1993, p. 618. Matsuno, F., Asano, T. and Sakawa, Y., IEEE Transactions on Robotics and Automation, 1994, 10(3), 287. Yim, W. and Singh, S. N., IEEE International Conference on Robotics and Automation, 1994, 3, 2113.

Compliant control of a two-link flexible manipulator

305

9. Yoshikawa, T., Hosoda, K., Harada, K., Matsumoto, A. and Murakami, H., IEEE International Conference on Robotics and Automation, 1994, 2125. 10. Li, C. J. and Sankar, T. S., IEEE Transactions on Systems, Man, and Cybernetics, 1993, 23(1), 77. 11. Martin, D., Manifold Theory: An Introduction for Mathematical Physicists. Ellis Horwood, 1991. 12. Meirovitch, Methods of Analytical Dynamics. McGraw-Hill, 1970. 13. McClamroch, N. H. and Bolch, A. M., Dynamical Systems Approaches to Nonlinear Problems in Systems and Circuits, ed. F. M. A. Salam and M. L. Levi. SIAM, 1988, p. 394.

APPENDIX A ~ in equations (47) and (48)

~, = p , + A:/(a) ~~ , (2k+ fi~)~ Op,

+ M(a)

+ M(a) -- (2k + fi,O

+

k=l

k~l

e¢I~(a, SO, p)

{

--k = I

+ M(a)2.,__I ukt

a, ~

OOi

~(3;

k =I

'P)

~ ' i

Op, )

+ JJ/(a)i<=i ~" acl#(a, ~P' p) + M(a),=,~

+ - k~'=, cgr}'(a,P)~Oi + s(.I(a)k~O=,

+

u, k=l

+ M(a) k.I ~

~2---" --,=l

(2~ + ~ ) ~

Op, fi, + M(a)k=l ~" O@t(a'P) "

2 fd~Pffa, v)'~').

,'P) + M(a),t~l It~Op,

pi

2 dtl~'(a,p) •

) ~uk + M(a) k~l

t= l

k~ I

Op, fi,t

~pi

APPENDIX B Inertial matrix and its elements Frail mu m13 rot41 M(a)= [ m22 m33m23~n24 L sym

m44 ]

mH=mlr~+l~fot2dm2+fo'262dm2+f[x~dm2+21,c2fo'2xdm2-21~s2~'26dm2

+ mt~ + m,l~ + m~ ~2 + 2m,ld2c2 - 2mtlI~ES2 + I=~

ml2 =

m~3 =

m14

----"

f

123 2

dm2 ~-

fo 12

x 2 din2 + ltc2

x~bl din2 + l, c2

fo

t2 X

din2 - lls2

fo

3 dm2 + mt~52E+ mt~ + mdd2c2 - ratlinES2

fo

~btdin2 + m,12$ll + m,l, $uc2

fot2X¢#2din, + llc2 fo12$2 din, + m,12521 + mill Szw,

Uk

306

Seung-Bok Choi and Hyuc-Bum Lee