A new control strategy for robotic systems in tracking tasks

Mechatronics Vol, 1, No. 3, pp. 353-366, 1991 Printed in Great Britain

0957-4158/91 $3.0(/+0.00 © 1991 Pergamon Press plc

A NEW C O N T R O L S T R A T E G Y FOR ROBOTIC SYSTEMS IN T R A C K I N G TASKS WEIBING GAO and MIAN CHENG

The Seventh Research Division, Beijing University of Aeronautics and Astronautics, Beijing 100083, P. R. China

(Received 18 May 1990; accepted 5 December 1990) Abstract--The control issue of robotic systems when accomplishing tracking tasks is studied. It is assumed that a multi-robot system consisting of several robots is employed only when one, or some of the robots, involved in the system is unable to achieve the given task. Thus assumption is implied by the fact that the control activities are constrained owing to the limitations from the power source. A new type of " m a s t e r - h e l p e r " control strategy, which solves the problem of task distribution a m o n g the robots in a simple and cooperative way, is investigated with the aid of force and velocity transition chains established in this paper.

1. I N T R O D U C T I O N Multi-robot systems are expected to accomplish far more complex tasks than one robot. Of such tasks, moving an object to track a desired trajectory is the simplest and, at the same time, the major one. This problem is attractive at present and quite a few interesting results have been obtained [1-32]. Nevertheless, the control problem of multi-robot systems is still challenging in general. In comparison with a single robot, there are some specific characteristics inherent only for multi-robot systems. The most striking one is the indeterminacy of control variables. This is because the number of control variables is far more than that of the degrees of freedom (d.o.f.) of the object-robots system. Consequently there is non-uniqueness in the determination of control variables. Furthermore this nonuniqueness implies that an internal force conflict may occur as illustrated in Fig. la, where a heavy body M is pivoted at one end, and supported by two robots on the other side to maintain M in a horizontal position. Any increase of reaction N I from robot 1 will cause an increase of N 2 from robot 2 and then may grow larger and larger. Obviously these increases not only offer no contribution to our task, but also induce high strain in our b o d y - r o b o t s system and cause a waste of energy. Hence, it is important to prevent any force conflict, in other words to require the robots to work cooperatively. Other specific characteristics may also be pointed out, e.g. motion coordination, internal force regulation, complex environmental constraints, constraints on control, etc. Furthermore great complications will be encountered in assembly tasks, hand-finger systems, and manufacturing systems. To solve the control problem of robotic systems several techniques and approaches have been suggested, e.g. master-slave type control mode [1-4], closed-chain 353

354

WEIBING GAO and MIAN CHENG

dynamic modelling [5-8], load distribution technique [10-14], hybrid control schemes [15-19], motion coordination [20-23], augmentation technique [24-26], Lyapunov's direct method [27-29] and others [30-31]. These techniques are used to construct a variety of control strategies for robotic systems. Unfortunately, a large part of the previous results were devoted only to two-robot systems and generally require a lot of computations. In this paper a ~ m a s t e r - h e l p e r " type of control strategy is investigated [32] based on a realistic assumption: a multi-robot system is employed only when fewer robots cannot accomplish the task. This assumption implies that the ability of each robot is limited. Constraint on the control activity is chosen as one of the possible measures of the ability of robots. The idea of the " m a s t e r - h e l p e r " control strategy is illustrated in Fig. lb: if robot 1 can accomplish the supporting task by exerting N 1 o n the beam, but this is not our case; if not, that means the maximum effort NT of N1 is not enough to support the body, let robot 2 join in and it will exert a force N2 such that the supporting task is accomplished. In this scheme internal conflict can never occur. The " m a s t e r - h e l p e r " control strategy is realized through a self-organizing algorithm which not only performs load-distribution but also eliminates any internal conflict and ensures a specific cooperation among the robots. This algorithm works equally well no matter how many robots are involved in the system. Matrix representation of force systems and velocities of rigid bodies [33, 34] is used in dynamic analysis of robots by some researchers [23, 35]. In this paper it is shown that it can be developed as a unified tool, which is effective and simple, to deal with statics, kinematics and dynamics of multi-robot systems holding a common object or even to more complex cases such as constrained tracking and assembly [36, 37]. But here we require all entries of the matrices to be projections on local coordinate axes fixed with the object. For example, the load-distribution task may be easily carried out through a transition chain of force systems all represented by 6 × 1 matrices. Similarly, the velocity transition chain gives relations between velocities of the robots and the object, and also another form of kinematic constraint between robots [20, 21]. This paper is organized as follows. In Section 2 specification, equivalence and transition of force systems are given. Kinematic specifications of rigid bodies and their relations to motions of manipulators are stated in Section 3. After the dynamic equation of multi-robot systems are established in Section 4, the main result " m a s t e r - h e l p e r " control strategy is depicted in Section 5. In the last section simulation results are presented.

2. REPRESENTATION AND TRANSITION OF FORCE SYSTEMS

Suppose a given force system: F -- {(fl, P J . . . . .

(f,,, P,,)},

is applied on a body, where ]i is the ith force and Pi is the radius vector of the point of application Pi of ~, and this point may be replaced by any point on the force line if the body is rigid. If we transfer this force system to some point O, we get a simplified force system consisting of a force f (principal force) acting at 0 and a

Control strategy for robotic systems

355

moment (torque) m o (principal moment). Now the force system may be represented by: m

m

F = { f , m o } , f = ~ , f i , m o = ~]r, × fi, i=i

i

1

where r i is the radius vector of point Pi with respect to O. Similarly transferring the force system to point 0 ' , we have: m

tn

X = {f, mo,}, f = •fi,

mo' = ~ r c

i=1

x fc,

i-I

where r c is the radius vector of Pi relative to point O'. Let O x y z and O ' x ' y ' z ' be two coordinate systems fixed with the body, then we have the following equivalent representations to the given force system: Fo = [L, L , fz, mox, moy, moz] r, Fo' = [ k ' , L ' , f~', mox,, moy,, moz,] r

The components of the last two vectors are projections of ( f , m o ) and ( f , m o , ) on the corresponding coordinate axes. Denoting the 3 × 3 orientation matrix of O x y z relative to O ' x ' y ' z ' by R °' and its inverse by R°,, which are both functions of Euler's angles, then we have an identity: R O ' R O , = I.

The above two representations of the same force system are related by: F o, = [f,.,L,,f~,mo.,,mo,] T

(1)

= S[fdc~f:moxmoymo:] T = S F o,

where =

PR

R

,

R = R oo',

P =

E ~

0

0~

,

(2)

and (~, r/, ~) is the coordinate of point O in system O ' x ' y ' z ' . It is obvious that S is non-singular since R is non-singular. Similarly, resulting F o of n force systems F i (i = 1 . . . . . n) is given by tl

Fo = ~,SiF,. i=I

The force systems FI and F 2 are said to be statically equivalent if, and only if, there exists a 6 × 6 matrix S with the structure specified in Eqn (2) such that:

Fig. i. Two-robot supporting system.

356

WEIBING GAO and MIAN CHENG F1 = SF2, S

=

SllS2

(3)

.

It will be adequate here to mention the concept of equivalence of force systems on constrained bodies such as n-link manipulators. Suppose on the robot manipulator a joint force/torque system FI = r = [rt, r2 . . . . . ro] r is applied and on the end-effector a second force system F 2 = [f~, fv, fz, rex, my, mz] 7 is applied to balance F 1. Clearly force systems F1 and - F 2 are not statically equivalent because the reaction forces acting on the manipulator from the basement have not been taken into account. But F1 and - F 2 cause the same effect on the manipulator, so we have: Ft = - G T F 2 o r F l + G I F 2 = O,

where G will be defined in the next section. Now we are ready to formulate a transition chain of force systems, which implies that these force systems are either statically equivalent or cause the same effect on the object. This transition chain plays an important role in the analysis of robots, especially in the formulation of a " ' m a s t e r - h e l p e r " control strategy. The chain is shown in Fig. 2 and written as: 1:i+i =

T -1 - G i + l f E i + l , fEi+l = S,+,fi, fl = SfE,, fl:., = - - G i l T i ,

(4)

where fEi is the force system acting at the ith end-effector from the object, fi its equivalence at mass center of the object, G~ and Si defined above.

3. REPRESENTATION AND TRANSITION OF KINEMATIC QUANTITIES The position vector of a rigid body is determined by the position of some point O on it and its orientation with respect to a given base coordinate system O X Y Z :

Po = [Xo, yo, zo, CPo, 0o, ~o] r, where x o , Yo, Zo are the coordinates of the point O and (q~o, 0o, ~Po) are the orientation angles which may be chosen, for example, as Euler's angles. The drawback of this representation is that its time derivative has no direct meaning in sense of "velocity". As is well known, Euler's kinematic equations:

~r.. Fi --~-~/FEi +1 F ~ ~" ~:==~/~L.~ \

r J r r l . .

Fig. 2. Force transition chain.

Control strategy for robotic systems

357

w, = qbsin 0 sin ~p + 0 cos ~p, coy = ~ s i n 0 c o s ~ p ~o. = ~,cos 0 +

bsin q~,

(5)

#,,

give relations between Euler's angles and angular velocities about the x, y and z axes. Equation (5) may also be rewritten as: ~o =

wt,

~o = [~o,, ~o~, (o~1 ~, e = [,p. 0, ~ ] r ,

(6)

where W =

I s i n 0 sin ~p sin 0 cos ~p cos 0

cos q~ - s i n ~p 0

0] 0J . 1

Now we have two representations:

Po = [~o,, ~o~, ~o=, ~o, ~,o, 0o17,

(7)

Vo = [vo,, roy, ~oz, o , , ~o~, o~1 ~,

(8)

and

to describe compactly the "velocity" of a body. Clearly, either of them is adequate in all circumstances. It is easy to show that they are related by

where B denotes the base coordinate frame and O the frame fixed on the body at point O on it. Now let us turn to consider the transformation between any two velocities of the same body at O and 0 ' , i.e. between Vo and Vo,. It is not difficult to derive:

Vo, = YVo,

(10)

where the 6 x 6 matrix Y is defined by:

y =~R 0

PR~,R=RO, R

Related to multi-robot systems, suppose we have m six-link manipulators firmly grasping a moving body, then the position of each manipulator is uniquely determined by the position of the body and vice versa. Denoting the joint vector of the ith manipulator by qs = [qil, qi2 . . . . . qs6]T and the velocity of the ith end-effector by VEi, we have: V e i = Gigl i.

(11)

In this equation the physical meanings of Vei = [veix, v~0., Veiz, m~ix, ~0~., ~Oei:]r and qi = [qil . . . . . qi6] r are explicit and simple, but in order to get the trajectory qi(t) from any of them, performing an integration of Euler's Eqn (7) is dispensible.

358

WEIBING GAO and MIAN CHENG

Now we have a transition chain of velocities quite similar to that for force systems, as shown in Fig. 3: ¢1,+1 =

G~+llVei,-l,

V~,i÷l : Yi+lVo,

V~.i =Gi[t,.

Vo = Y / I V c i .

(12)

4. DYNAMIC EQUATIONS OF ROBOTIC SYSTEMS The dynamic equations of a robotic system consists of equations of motion of a body and rn manipulators. The equations of motion of the body are given by Newton's law of motion and Euler's equations of m o m e n t u m : l)'., = L,. + m - I f . , ,

(.o,. = H,.(m) + J,.lrn.~,

]/v = L,. + m - l L ,

(0 v = H.~.(m) + J ~ l m ) ,

9 : = L_ + m 1L ,

&. = H:(~o) + J z ~ m : ,

(13)

where m is the mass of the moving body, O x y z a coordinate system fixed to the body with origin O at the center of gravity and coordinate axes along the principal axes of inertia, while scalars f,., f,. and f:, and m,, my and m. are the components of the principal vector and the principal m o m e n t of the force system of all forces acting on the body, including reactive forces form all end-effectors of the manipulators and other external forces such as damping and reactions from the environments. It is to be noted that O x y z is a moving coordinate system, therefore non-linear terms such as Lx, Ly and L: appear in Eqn (13). For simplicity we assume temporarily that x , v and z are axes of symmetry. Then we have: L~ = so:V ,, - ~o,.v:, H,. = J T l ( J y -

J=)o3.#o z,

--

L~ = (o x V : H~ = j~l(j:

_

( o z V r,

J,)~o:(o~.,

L. = ( o , V , H-

= J:l(J

~,J,V,, v -

(14a)

J,.)(ox~o,.. (14b)

Now we are ready to put the equations of motion of the moving body presented by Eqns (13) and (14) into a compact form:

~'~

Fig. 3. Velocity transition chain.

VE i+1

Control strategy for robotic systems

359

d --V=K+F, dt

(15)

Hz] T,

where K = [Lx, Ly, L z, Hx, Hy, m

F = E(fo + f),

f = - ~Sfzi, i=1

E = diag[m -I, m 1, rn -1, j ~ l , j ~ l , j71]T, and f o denotes only the external force system. The equations of the robot manipulators are given conventionally by: M ( q ~ ) ~ + U(qi, ?li) = ri + G~fi

(i = 1, 2 . . . . .

m),

(16)

where qi = [qs~, qi2 . . . . . qi6] r is the joint vector; M(qi) is the inertia matrix: N(q~, ?li) is a vector including gravity, centrifugal and Coriolis forces; r i = [r~l. . . . . ri,,]r is the joint force/torque vector: fi = [ft,i, f, yi, fi~i, mixi, mo.i, mi~i]r the reaction force system exerted on the ith end-effector by the moving body; Gi is the transformation matrix as defined in (11). In case all fi = 0, Eqn (16) is reduced to: Mi(q,)i]i + Ui(qi, ~/i) = r ° ,

(17)

where r~9 is called zero payload control which steers the ith robot itself without contact with any object.

5. " M A S T E R - H E L P E R " CONTROL STRATEGY

Suppose the moving body is to be controlled by a multi-robot system consisting of m robots. We will establish the control strategy for each of the robot manipulators in the following constructive procedure. First suppose the moving body is steered by only one r o b o t - - t h e master robot. While this system is in action, all other robots are controlled independently by their own zero payload controls r ° given by Eqn (17). The equations of motion of the o b j e c t - r o b o t system are not given by: (/ = K + E f o -

Ef,

M l ( q l ) q l + N ~ ( q t , dl~) = r, + G ( f ~ .

(18) (19)

On account of the last two kinematic relations in Eqn (12): V = y I 1 V E 1 , VE1 = Glcll, we get: V = Y1(11, Y I = Y1G1,

f / = ?~ql + 7~01. From Eqn (18) we have:

(20)

360

WEIBING GAO and MIAN CHENG f = f o q- E - I K

-

-

E-I'Ylql

-- E

1~-~1(~1

(21)

Eliminating the unknown f , which identifies with f~ and r e x , and from Eqns (19) and (21), we obtain: M i ( q , ) q l + N~(ql, q,) = rl + G ( f o where M l ( q l ) = M l ( q l ) + G ( E

+ G(E-1K,

(22)

I Y 1,

/~l(ql, ql) = N ~ ( q t , ql) "+- GTIE-I~]I. Let control r~ on the first robot equal: r I = / ~ l ( q , , 01) - G r f o

- G ( E - ~ K + M , ( q l ) [ q ~ ' - K ~ ( q , - qd) _ K2(q 1 _ qd)]

(23) then the closed-loop system will be:

where el = q l ( t ) - q ~ ( t ) , q ~ ( t ) is the desired motion. The above result is obtained by utilizing the well-known computed torque approach. By properly choosing matrices K 1 and K 2 the tracking will be asymptotically stable, and in particular when K~ and K 2 are chosen diagonal a decentralized control is achieved. Consider now the effect of the constraints on the control activities. Assuming the constraint on 17i to be r*,, we have: Iril ~< r*, and this inequality should be considered to hold by components. In our case ~-~ does not satisfy this inequality. Define a function T r ( . ) by: Tr(~:i-

Tr(~; 0 -

r*) = [ T r ( t i l - r*) . . . . . r*) =

Tr(ti,,-

r*,)] r

(24)

t i j - r*, when ~-, > r*, 0, w h e n [rol ~< ri~, ~-~ + r*, when ~ii < - r * .

Taking the constraint on rl into account, we have the realizable, actual control on the master robot: T1 = ~1 -

Tr(~q - r~).

(25)

Obviously the obtained control rl is unable to achieve the tracking task, because there is a lack of control activity on the master robot of amount Tr(~:l - r*). The second robot, as the first helper, should help the master by generating a control activity: ~'2 = r~ + /-2,

(26)

where r ° is the zero payload control and 1-2 is what is needed to be implemented to help the master robot. Now we have the following four pairs of equivalent force systems: (1) force system: T r ( ~ - r~) acting on the master robot and its reactive counterpart on its end-effector

Control strategy for robotic systems

361

from the body F:; (2) forces system - F : on the body and its equivalence at mass center Fo; (3) F o and its equivalence - F 2 , where F2 is the force system on end-effector of the helper; and (4) F2 and ~2. Thus we obtain a force transition chain: Tr(~l -

r~) = - G T F : ,

F: = S l X F o ,

F o = $2F2,

F 2 = - G 2 : t " 2.

Now the control needed to act on the second robot (the first helper) ~2, which is equivalent to the lack of control on the master robot Tr(~l - r*) is represented by: ~'2 = G f S 2 I S 1 G 1 T T r ( ~ ' :

-

r~).

Going on in this way, we obtain: (27)

r2 = r2° + i'2 - r r ( r ° + r2 - r~). Generally, we have: ri = T : i - T r ( ' ? i -

r*),

Ti = T ? -J- 17i,

T, •

~ u i T c° i-- T l ' J~i - T T1 r ( i 3 (i = 2 . . . . .

:

r*:),

(28)

m - 1),

and finally arrive at: o r,,, = rm + L,,.

(29)

We have constructively established the control strategy with Eqns (25)-(29), which is unique and automatically eliminates the possibility of internal-force conflict among the robots and the tracking goal will be satisfactorily guaranteed. In this " m a s t e r - h e l p e r " control strategy, the load distribution is accomplished in an intuitive way by the self-organizing algorithm Eqn (28). The block diagram of the closed loop is shown in Fig. 4. Remark 1: the " m a s t e r - h e l p e r " control strategy is also valid for non-symmetric bodies, where the inertia products l~,,, Jyz and J:x do not vanish. Remark 2: the constraint r* on r i may be chosen at will, for example, smaller than the extreme value. Remark 3: in the case when the computed-torque method is employed, attention should be paid to robustness which can be ensured by various techniques, for example, the variable-structure control.

4

/

Fig. 4. Closed-loop block diagram.

362

WEIBING GAO and MIAN C H E N G 6. SIMULATIONS

Consider the t w o - r o b o t system shown in Fig. 5, where the desired motion of the object is described by: xa = 0.5 sin [~r(t - 1)/2](m), y j = 0.5sin [~r(t - 1)/2](m), 0 d = 0. P a r a m e t e r s of each r o b o t with three d.o.f, are indicated in Fig. 6, w h e r e m , L~, I~, Ic~ and q, are the mass, length, m o m e n t of inertia, distance from the center of mass and the absolute joint angle coordinate of the ith link (i = 1, 2, 3). N o w the dynamic equation of the robot is written as: M(q)O

+ N ( q , gl) + G ( q ) -

G~f~,,

r +

where:

M(q) = mllcl+m2L~+m2L~+ li (m2 L e1~2+ m3 L1L2) cos ( q2-qi)m2 l~_+rn3 L ~+ I~ rn3L1 ld cos (q3--ql)

(m2Lllc~+m~LiL~)cos(q,-ql) .

.

.

m~L~lc~cos(qa-ql)- [

,,,;L,4: COS(q:_q~) i

.

m3L2lc)cos(q3_q2 )

" ;n~lc:~+13

-(rn~Ltla + m~L,L,)sin(q~ - q,)gl~ - m~L,lc~sin(qa - q,)q~-I N(q, il) = (m2Lil~2 + m3LiL2)sm(q2 - q t ) q ~ - m:,L2lc~sm(q3 - q 2 ) q 3 I m3Lllc.~sin(q3 -- q~)il~ + m3L21c3sin(q3 - q2)O~

I

G(q) =

G =

(mll~q + t 1 7 2 L 1 + m3Ll)gcosql i (m2lc2 + m3L2)gcosq2 , m~gl~ cos q3

I

Llsin(q~L1 cos(q3

I

ql) ql)

L, sin (q3 L 2 sin (q,~ 0

f~ = lf,~', J;~, m:~] T, r

=

q2) q2)

0 -] L3~ , 1

[rl, r > r3] J.

Note that G is the matrix defined by relations UE = Ggl, where UE is the velocity and (x E, YE) are the coordinates of the end-effector. D e n o t e the mass and m o m e n t of inertia of the body by rn and 1, respectively, then the e q u a t i o n of m o t i o n of the body will be:

~t t b -

~°v,-

Im 1

-" L~o:~ where

(x,y,O) are the coordinates of the body.

-~

I -'2 L'":/

~-gsinO-

I

-J

C o n t r o l strategy for robotic systems

Controller

- - v

M a s t e r robot

v

363

t_

Controller

°>1

-- ÷ ]

Helper robot 1

__

I Controller Helper robot

•m y I

m-1

Fig. 5. Tracking task by a planar robotic system.

Fig. 6. Parametrization of the robots.

In our case, the velocities of the end-effectors [t~,., u,., (0] r , t h e v e l o c i t y o f t h e b o d y , b y :

UE1,

t~E, a r e

related

l~ = g l U E 1 , l~ = Y2/~E2,

YI =

1 0

1

, Y: =

1 0

.

to

v =

364

WEIBING G A O and MIAN CHENG

The values of the parameters are the same for both robots: m l = m2 = 10, m 3 2, I1=i~=0.8, I , ~ = 0 . 5 , E l = L , = 1.0, L 3 = 0 . 2 , lcl = l ( . : = 0 . 5 , 1(,3=0.1, m=30, I = 2.5, rl = r2 = 0.5, d = 3.4, where length is given in m, mass in kg, m o m e n t of inertia in k g m 2 and force in N. The constraints on the control variables to r o b o t 1 are chosen as: =

Jr, I ~< r* = [350, 300, 150] r (N m). Simulations results are sketched on Figs 7 - 9 . The time history of the control variables of r o b o t 1 ( r , , r,., rn:) is given in Fig. 7, while that of r o b o t 2 is given on Fig. 8. Tracking errors: e1 = x ( t )

-

Xd(t),

e 2 = y(t)

--

yd(t),

e3 = O ( t )

are drawn on Fig. 8. These figures show that the " m a s t e r - h e l p e r " control strategy established in this paper for a robotic system works successfully, as all the tracking errors are negligibly small in accomplishing tracking tasks.

'T1

4OO

g

0

-200

I 0.5

I 1

I 1.5

2

Fig. 7. T o r q u e s on the master robot.

2oo

g

o E

o 1"2 -200

I 0.5

I 1

I 1.5

Fig. S. T o r q u e s on the helper robot.

2

Control strategy for robotic systems

365

5-

2.5-

0

e3

0.5

1

1.5

2

Fig. 9. Tracking errors.

7. C O N C L U S I O N A n e w type of c o n t r o l strategy n a m e d " m a s t e r - h e l p e r " c o n t r o l strategy is studied. It is a s s u m e d that the ability of i n d i v i d u a l robots is limited. T h e c o n s t r a i n t o n control activity m a y be c h o s e n as m e a s u r e of ability of robots. By using the unified 6 × 1 matrix r e p r e s e n t a t i o n s of static, k i n e m a t i c a n d d y n a m i c q u a n t i t i e s , e q u i v a l e n t force a n d velocity t r a n s i t i o n chains are established. T h e s e relations n o t o n l y show the c o n s t r a i n t s o n the forces a n d m o t i o n s of all m robots, b u t also p r o v i d e tools to realize the "masterh e l p e r " c o n t r o l strategy. H e r e the c o n t r o l issues such as load d i s t r i b u t i o n , m o t i o n c o o r d i n a t i o n , i n t e r n a l force conflict e l i m i n a t i o n are solved s i m u l t a n e o u s l y . This c o n t r o l scheme is also a p p l i e d to tracking tasks in the p r e s e n c e of e n v i r o n m e n t a l c o n s t r a i n t a n d assembly tasks by m u l t i - r o b o t systems.

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