Proceedings of the 32nd ISR(International Symposium on Robotics), 19-21 April 2001

Dynamic Control of Multi ngered Hands for Pivoting Operation
Tetsuyoh Watanabe and Tsuneo Yoshikawa
Department of Mechanical Engineering
Kyoto University
Kyoto, 606-8501 Japan(E-mail : ft60x0113@ip.media.,yoshi@mech.gkyoto-u.ac.jp)

Abstract

In this paper, we propose a dynamic control methodof multi ngered hands for pivoting an object in contact
with the environment. This pivoting operation is often
observed when a human moves a large or heavy object
such as furniture on the oor. Di erent from the conventional manipulation of the object by only ngers, thecharacteristics of the pivoting operation is that we canuse the reaction force from the environment. By using
this reaction force, we can expect the magnitude of theforces applied to the object by the ngers is smaller thanthe conventional manipulation of the object by only ngers. In this paper, taking this characteristics of thereaction force into consideration, we propose a dynamic
control method for pivoting. To verify our approach,simulation results are also presented.

1. Introduction

Many researchers have studied the manipulation ofan object which contacts with an environment such asa table. While the environment constrains the motionof the object, we can use the reaction force from theenvironment in the manipulation of the object. Then,
we can expect that the needed number of the ngers
to manipulate the object and the magnitudes of contact forces applied by the ngers are smaller than theconventional manipulation of the object by only ngers
such as Li et al.[1], Cole et al.[2], and Yokokohji et al.[3].
In this paper, the goal is to establish a control method
for forming force closure[4] grasp and manipulating theobject in contact with the environment by small contact forces even when we cannot form force closure grasp
by only ngers. Especially, we are interested in pivoting operation. Pivoting operation is a rotation of theobject around an axis through a contact point between
the object and the environment. This manipulation isoften observed when a human moves a large or heavy
object such as furniture. Aiyama et al[5] have proposed
the quasi-static motion planning for manipulatinga rectangular object, but we cannot apply the planning to a
general object. On the other hand, some manipulations
similar to the pivoting operation have been studied. Han
et al.[7] proposed a control method for the quasi-static
manipulation of the spherical object on a plane by one
nger which has 6 D.O.F. and a at ngertip. Trinkle etal.[8] quasi-statically analyzed the manipulation of lifting up the object on the support. In another paper[9],Trinkle et al. proposed a quasi-static manipulation planning for enveloping the slippery work piece on the palm
and manipulating it. But the dynamic control method
for these manipulations has been studied only by a few

researchers (for example, Yoshikawa et al.[6], but, thecontrol method is proposed under the assumption that
we can form force closure grasp by only ngers ). In thispaper, we propose a dynamic control method of multingered hands for pivoting an object in contact with theenvironment.
This paper is organized as follows. In section 2, we
describe the target system of this paper for pivoting operation, and in section 3, we develop a dynamic control
method for pivoting operation. In section 4, we consider the derivation of desirable magnitude of internal
forces by which the magnitude of the contact forces of
the ngers can become as smallas possible. To verify ourapproach, we show simulation results in section 5.

2. Target System

The target system of this paper is shown in Fig.1.
This system consists of two ngers and a rigid object.
each nger has three joints and a spherical soft ngertip, the object contacts the environment at one vertex,
and each contact surface is smooth. The given task is
to pivot the object in contact with the environment atthe vertex by the two ngers along with a desired trajectory. We make the following assumption for the analysisof the soft ngertips; (i) the contact area is so small thatthe contact between the soft ngertip and the object isregarded as a point contact, (ii) the tangential frictional
force and frictional twist/spin moment around the contact normal are independent and they are approximatedby the Coulomb model, and (iii) the energy dissipation
due to the ngertip deformation is negligible.

3. Control Scheme

In this section, we propose the control scheme basedon the formulations by Li et al.[1], Cole et al.[2], Yokokohji et al.[3], and Yoshikawa et al.[6]. Note that we permit
a rotational slip at the contact point between the ngers
and the object.

3.1. Kinematics Constraints

In this subsection, we formulate the kinematic constraints between the ngers and the object and between
the environment and the object. As shown in Fig.2, R ,
B , Fi (i = 1; 2), and C (i = 0; 1; 2) denote a reference
coordinate frame, an object coordinate frame placed at
the gravity center, a ngertip coordinate frame placed at
the center of the spherical ngertip of the nger i, anda frame xed on the object surface at the contact point
Ci , respectively. Let the position of the origin and theorientation of a with respect to c (R ) be given byiwhere[(RFi Fi pFiCi )]D Fi =(6)
I8
3< I 3 O3 (not spinning)
T
H Ci =(7): [I 30 On3 ]Ci (spinning)
Here, vCi denotes a vector of the contact point velocity,where the components of the contact point movement
due to the rolling are excluded, and ! Fi denotes an angular velocity of Fi .
Finally, the relation between the ngertip velocity ofnger i and the joint velocities of nger i is given by
 
p_ Fi
= J q_(8)

Figure 1: Target System



I3
O3

!Fi

Figure 2: Objet, ngers, and Environmentc pa (pa ) and c Ra (Ra ), respectively. Let a vector directing from the origin of ac to the origin of b with respect
to c (R ) be given by pab (pab).
First, the constraint at the xed contact point between the environment and the object is given by
 
 
p_ B
p_ C 0H D=H= p_ = 0 (1)C 0 B0 ! B

C0 !C0

C0

where



B
= OI 3 [(RB I pBCi )](2)
3
3
H C 0 = [I 3 O3 ](3)Here, !B and !Ci denote an angular velocities of theobject(B ) and Ci , [] denotes a skew symmetricequivalent to the cross product operation, and I i andOi denote the i-order identity and null matrices, respectively. Here, instead of the angular velocity of the object!B , we use roll, pitch and yaw angles for expressing theobject orientation. The relation between !B (=! C 0) andthe velocity of roll, pitch and yaw angles, _ B (=_ C 0), is
given by !B =T r _ B where T r (B ) denotes the matrix
for the transformation. Then, from (1), we get    
p_ B
p_ C 0DB0= !
= T _ 0
(4)
!B
C0
C0
DBi

where T = diag[I 3 T r (C 0 )] ("diag" means a block diagonal matrix).
Second, since the contact between the nger and theobject is a soft- nger type contact, the constraint at thecontact point between the nger i(i = 1; 2) and the object is given by
 
 
p_ B
vCi
H Ci DBi
=
H Ci
!B
!Ci
 
= H Ci DFi !p_ Fi
(5)
Fi

Fi i

where J Fi denotes the Jacobian matrix of nger i andq i denotes the joint vector of nger i.
From (4) (5) (8), we get
 
 
0
1
H CiD Bi DB0 T _= H Ci !vCi
C 0
Ci
= H CiD Fi J Fi q_ i = J CFMi q_ i
(9)3.2. Dynamics of Fingers and Object

In this subsection, we formulate the dynamics of thengers and the object. Let tB be the resultant force andmoment applied to the object by the ngers at the originof the frame B , and f f be the force applied to the environment by the object at the contact point (note thatf f expresses the reaction force). Then, from (1), theresultant force applied by both the ngers and the environment is given by tB D TB0 H TC 0f f . So, the dynamics
of the object is given by 
 B + h = t DT H T f
p
MB
(10)
B
B
B0 C 0 f!_ B
where M B denotes an inertia tensor of the object
and hB denotes centrifugal, Coriolis, and gravitational
forces.
Next, the dynamics of the ngers is given by
M F (q)
q + hF (q ; q_ ) = 

JT fS

(11)
where  = col [ 1  2 ] 2 R 6 ("col" means a column vector or matrix formed by the following vector
or matrix) ( i 2 R 3 denotes a joint torque of nger i), f S = col [f S 1 f S 2 ] (f Si denotes the force
and the moment applied to the object by nger i atthe contact point Ci expressed by a vector representing the magnitudes of the components in the directions expressed by a column vector of H TCi in (7)),
MF= diag [M F 1 M F 2 ](M Fi denotes an inertia matrix of nger i), hF = diag [hF 1 hF 2 ](hFidenotes centrifugal, Coriolis, gravitational forces,
and so on), and J = J CFM + J sm where
J CFM = diag [J CFM 1 J CFM 2](refer to (9)) and

J sm

by

= diag [J sm1 J sm2 ]. Here, J smi is represented





O3 O3
J Tsmi = J TFi
H TCi
N i O3N i = ii nCi nTCi

(12)

(13)
where nCi denotes the contact normal, i (= 1; 0; or1)
shows the direction of the frictional twist/spin momentat the contact point Ci , and i is determined by thesame law proposed by Yokokohji et al.[3] and denotes
the dynamic rotational frictional coeÆcient of nger i
if a rotational slip occurs at Ci. Note that J Tsmi f Si
expresses a joint torque equivalent to the dynamic frictional twist/spin moment, [O3 N i ]H TCif Si .

3.3. Contact Forces and Moment

In this subsection, we formulate the force and the moment applied to the object by the ngers at the contact
points, f S , and the reaction force, f f .
First, we formulate f S . If a rotational slip doesn't
occur at Ci between nger i and the object, the contact
force and moment f Si has 4 D.O.F. On the other hand,nger i has 3 D.O.F. Then, we express f S as follows

f S = J J + f S + (I JJ + )f S = Jf C + J~ f~ C (14)where f C 2 R6 , f~ C 2 R2, J~ is an arbitrary matrix
which satis es J T J~ = 0, and J + denotes the pseudoinverse matrix of J . Note that the former term in (14)denotes the controllable contact force and moment by
the joint torques, and the latter term in (14) denotes theuncontrollable contact force and moment. Then, from(5) (14) and the principle of virtual work, the relation
between tB and f S is given by (Jf C + J~ f~ C ) = Af C + Ae f~ C
tB = A(15)
 = (DTB + O3 O3 O3 O3 )H TC (16)A
N 1 O3 N 2 O3

where D B = col [DB1 DB2 ], H C = diag [H C 1 H C 2].
Note that the second term in (16) relates the dynamic
rotational friction.
From (15), we get
e f~ C ) + f~
f C = A+ (tB A(17)where  is an arbitrary matrix which satis es A =0 , f~ express the magnitude of the component in the+direction expressed by a column vector of , and A
denotes the pseudo-inverse matrix of A.
Next, we formulate f f . Now we hope that the magnitude of the contact force and moment applied to theobject by the ngers can be as small as possible. But,
we can control only the former term in (14). Then, we
consider the following problem.
minf TC J T J f C(18)
 
subj. to. G f C = ^tB
ff

where

G = (AD TB0 H TC 0 )
 h
^tB =
B
 
 
+M B fDB10 T 0 (D_ B10 T + DB10 T_ ) _0 g
C0

C0

Note that the constraint in the problem is in order tocomplement the dynamics of the object by Jf C and f f .
By using (17), if we solve the problem with respect tof f , f~ , and f~ C , we can get f~ C = 0, f~ = 0, and thefollowing equation.
 
fC= G~ ^tB
(19)
f
where
 +
~G = A (I 6f

DTB0 H TC 0B 1 )
B1



T T
T +
+T T +
B 1 = B 3B +
4 (I 6 DB0H C 0B 3B 2 JA )+ B 3(A ) J JA
B 2 = JA+ DTB0 H TC 0
B 3 = (B T2 B 2 ) 1 H C 0D B0
B 4 = (I AA+ )D TB0 H TC 0B 3 (I AA+ )T

3.4. Internal ForceIn this subsection, we consider an internal force forforming force closure grasp. In this paper, we consider
not only an internal force between the ngers but also aninternal force between the nger and the environment.
Then, if let [(Jf IC )T (f If )T ]T be the internal force, theinternal force satisfy the following equation.
 I
fC
G
=0
(20)
fI
f

Now, we set the component of the internal force[(f IC )T (f If )T ]T as follows.
 IfC= diag(J t11 J t21 I 3 )~ k = k
(21)
f If
0
1
e12 + 11 en
e10 + 12 en
13en
~ = @ e21 + 21 en
22en
e20 + 23 en A
2i=1 i1 en e10 +2i=1 i2en e20 +2i=1 i3 en
where eij denote an unit vector which directs from thecontact point Ci to the contact point Cj , k=[k1 k2 k3]T
denote the magnitude of the components of the internal
force in the directions expressed by a column vector of
, en denotes an arbitrary vector orthogonal to bothnC 1 and nC 2 , and J ti(i = 1; 2) is a component of J
expressing the following equation.
 T T
J t1 J r1 O3 O3
T
H TC
J =
O3 O3 J T J T
t2

r2

We determine ij (i = 1; 2 j = 1; 2; 3) in (21) by substituting (21) to (20) . Then, [(Jf IC )T (f If )T ]T given by

Figure 3: Internal Forces(21) represent the internal force. This internal forces includes both the term(white arrows in Fig.3(a)) given bya conventional method to determine the internal forcesin the manipulation of an object by three ngers andthe term (black arrows in Fig.3) which include thestatic/dynamic frictional twist/spin moment at the contact point Ci(i = 1; 2) and the force whose direction isen and which cancels the moment.

3.5. Controller for Pivoting

In this subsection, we derive a controller for pivoting
operation.
From (1) (9) (10) (11) (15), the dynamics of the system for pivoting is given by
W x = b = col[b1 b2 b3 b4]
(22)
0
1
Mf
B
O6
W =B
@ J CFM
O
b1

=



O6
JT J
MB
A
H C DBO
H C 0 DB0O

hf ; b2 =OO
eA DTB0 H TC 0 C
C
A
O
O
O
O3
 
p_ B
_
hB ; b4 = H C 0 DB0
!

 
p_ B
_
_
b3 = (H C DB + H C DB )J_ CFM q_!

where x = [qT p TB !_ TB f TC f~ TC f Tf ]T .
By supposing
the desired value of x is xd =[qd p Bd
 T~!_ Bd f Cd f C f f ] (note that f~ C cannot be controlled
and that f f include both the desired value related with
f Cd and uncontrollable value related with f~ C ), we canget the following controller.
 = M f qd + hf + J T Jf Cd
(23)
 
pBd
qd = J +J_ CFM q_
CFM fH C DB !_ Bd __+ (H C DB + H C D B ) p_!B g (24)f Cd = [I 6 O3]fuI +G~ (M B !p_ Bd + hB )g
(25)
Bd
 
 
 Bd = D 1 T 0
p
B0
!_ Bd
uB
 
1
+ (D_ B0 T + DB10 T_ ) _0
(26)
C 0

where uB and uI are new inputs.

Figure 4: Contact Stability CornBy applying the controller to the system (22), we get
W (xd x) = 0. If W is nonsingular, we can get xd = x. Then, we get  C 0 = uB from (26) and k = uI from(25).
Letting C 0d and kd be the desired trajectory of C 0and k respectively , uB and uI are given by the following
servo controller
 C 0d + K V (_ C 0d _ C 0 )uB = + K P (C 0d C 0)(27)
Zt
uI = kd + K I(kd k)dt(28)
00where K P , K V and K I are gain matrices. Then, if W
is nonsingular, the object orientation error eP = C 0d
C 0 and the force error ef = kd k satisfy the following
equationseP + K V e_ P + K P eP = 0(29)
e_ f + K I ef = 0(30)
Thus, with appropriate K P , K V and K I , the actual object orientation  C 0 and the component of the internal
forces k will converge to the desired trajectories asymptotically.

4. Nonlinear Programming

Introducing of internal force, the contact force andmoment applied by ngers formulated in section 3 isn't
as small as possible. Then, in this section, we derivea magnitude of the component of the internal force bywhich the contact force and moment applied by ngers
can become as small as possible.
We consider the following nonlinear programming.
minsubj: to:h(k) = f TC J T Jf Cl(k) = V (



fS
ff



(31)

+ f ext)  0Here, l  0 express the translational frictional constraints which approximately represent the friction coneat the contact point Ci as the polyhedral convex
cone with Li faces [10], where V =diag[V 1 V 2 V 0 ]T ，
V i =col[(aik )T ] (k = 1; 2;    ; Li ), and aik denotes an outward normal vector of the kth face of the polyhedral
convex cone. Note that f ext in the constraint in the
programming denotes the contact force and moment related with an unexpected perturbation of input and theuncontrollable contact force and moment. we determine

f ext by using the concept of Contact Stability Corn pro-

posed by Nakamura et al.[12](Fig.4).
The contact force and moment related with an unexpected perturbation of input f SI can be expressed by
 
O3
1
~
f SI = diag[J I 3 ]GM B DB0 T
 C0I= [G^ T1 G^ T2 G^ T0 ]T  C 0

3

(32)

where  C 0 denotes the perturbation. On the other
hand, the contact force and moment related with uncontrollable contact force and moment f UC can be expressed by
0^ 1


J1
J~

f UC =
f~ C = @ J^ 2 A f~ C (33)T
T+e
(D B 0 H C 0 ) AJ^
0Note that fUC 0 in (33) expresses the contact force between the object and the environment occurred by uncontrollable contact force and moment applied by thengers where f SIi and f UCi denotes the components off SI and f UC at Ci respectively.
Now, the desirable internal force makesf Si +f SIi +f UCi can be in the approximated friction corn as well as the contact force and moment
applied by the ngers can be as small as possible. Then,
we consider
By supposing a maximum magnitude of the perturbation  C 0 is  and a maximum magnitude of normof uncontrollable contact force and moment f~ C is fCM ,the sets of f SIi and f UCi are given by

ff SIi jf TSIi f SIi  2 G^ Ti G^ ig
2 J~ T J~ ig
ff UCi jf TUCi f UCi  fCM
i

(34)
(35)
From this equation, we can determine the component off ext, f exti as follows.
p2
si + 1f exti =(Gimax + fCM Jimax )nCi
si

where Gimax and Jimax denote the maximum singularvalue of G^ i and J^ i respectively.
This nonlinear programming (31) can be solved by using Kuhn-Tucker optimality conditions and linear complementarily problem [11].
The obtained k become a desirable component of theinternal force.

5. Simulation

We show simulation results in this section to verify
the validity of our approach. We did the following 3
kinds of simulations; 1) dynamic control with constantinternal force 2) quasi-static control with constant internal force 3) dynamic control with obtained internal
force by nonlinear programming. Quasi-static control isfor the comparison and its scheme is given by neglecting the term related with both the acceleration and thevelocity in the equation of motion of the ngers and byassuming the direction of the displacement of the objects

Figure 5: Transition of State
coincides with the direction of the resultant force andmoment applied to the origin of B in dynamic control
scheme.
Suppose that the object is a regular prism whose
bottom is a regular pentagon whose side is 0:05[m]length, whose height is 0:1[m] length, and whose weightis 0:5[kg]. Each nger is a 3 D.O.F. parallel link robot(Fig.1) which is the same robot used in the experience in
Yokokohji et al.[3], and each ngertip is a spherical soft
ngertip with 0:018[m] radius. The initial con gurationof the object with respect to the reference coordinate
frame is (0:[m] 0:[m] 0:05[m] 0:[rad] 0:[rad] 0:[rad])Tand the orientation of the object is expressed byroll, pitch, and yaw. The initial position of F 1 is
( 0:0162[m]0:0498[m] 0:080[m])T and the oneof F 2 is ( 0:0162[m] 0:0498[m] 0:080[m] )T .
We set each dynamic rotational frictional coeÆcient
0:002[m], each maximum static rotational frictional coeÆcient 0:0023[m], and each maximum static translational frictional coeÆcients 1:0. Servo gains are set to
K P = diag[ 10000: 10000: 10000:](1=sec2), K V =diag[ 200: 200: 200:](1=sec), and K I = diag[ 100:
100: 100:](1=sec) in dynamic control and K P = diag[
3:425 3:425 3:425](Nm=rad), K V = diag[ 0:003425
0:003425 0:003425](Nmsec=rad), and K I = diag[100:100: 100:](1=s) in quasi-static control , and samplingtime is set to 1:0[msec].
We consider the desired trajectory as follows;
rst, tilting (raising) the object around the vertex
P (Fig.5(a)) until (0 =8 =20)T [rad] for the interval from t0 (= 0:0[sec]) to t1(= 0:5[sec]), then, rotating the object around the vertex P (Fig.5(b)) until (=8 =8 =20)T [rad] for the interval from t1 (=0:5[sec]) to t2 (= 1:[sec]), nally, lowering the object
around the vertex P (Fig.5(c)) until (=8 0 0)T [rad] forthe interval from t2 (= 1:[sec]) to t3(= 1:5[sec]). And thengers are removed form the object at t3(= 1:5[sec]).
When we let the internal force constant, we
set kd =[1: 1: 1:](N ) in dynamic control andkd =[200: 200: 200:](N ) in quasi-static control. On theother hand, when we use the nonlinear programming insection 4, we set the parameter as follows; 
 C 0d k> 10:)
C 0d k 0:1 (k  = ==k1:[rad=sec
2] (k  C 0d k 10:)
, fUCM = 0:1[N ], and we approximate the friction coneas the polyhedral convex cone with 10 faces.
The result is shown in Fig.6 and Fig.7. Fig.6(a)shows the actual object orientation (dot dash line) andthe desired one (solid line) in dynamic control, andFig.6(b) shows the ones in quasi-static control. From

Figure 7: Simulation Result of Pivoting (norm of force)

References

Figure 6: Simulation Result of Pivoting (orientation)these gures, we can see that the actual object orientation converge to the desired orientation and that the deviation after t3 in quasi-static control occurs because thevelocity of the object doesn't converge to 0 at t3 . This
result shows that dynamic control is more eÆcient thanquasi-static control. Fig.7 shows the norm of contact
force in case of dynamic control with constant internal
force (solid line) and in case of dynamic control with theobtained internal force by the nonlinear programming(dot line). From these gures, we can see that the magnitude of the forces applied to the object by the ngers
is smaller than by the environment, namely, we can manipulate the object by small contact force applied by thengers by using the reaction force from the environment.
we can also see that the contact force in case of using theobtained internal force by the nonlinear programming issmaller than in case of using constant internal force. So,
the validity of the nonlinear programming can be shown.

6. Conclusions

In this paper, we have proposed a dynamic control
method of soft- nger hands for pivoting an object incontact with the environment. The characteristics ofthe pivoting is that we can use a reaction force from theenvironment. By using this reaction force, we can expect
the magnitude of the forces applied to the object by thengers is smaller than the conventional manipulation of
the object by only ngers. In this paper, taking thischaracteristics of the reaction force into consideration,
we propose a dynamic control method for pivoting operation. To verify our approach, we have also presented
the simulation results.

[1] Z. Li, P. Hsu and S. Sastry, \Grasping and Coordinated
Manipulation by a Multi ngered Robot Hand,"The International Journal of Robotics Research, Vol. 8, No.4,
pp. 33{50, 1989.
[2] A.B.A. Cole, J.E.Hauser and S.S.Sastry, \Kinematics
and Control of Multi ngered Hands with Rolling Contact," IEEE Trans. on Automatic Control, Vol.34, No.4,
pp. 398{404, 1989.
[3] Y. Yokokohji, M. Sakamoto and T. Yoshikawa, \VisionAided Object Manipulation by a Multi ngered Hand
with Soft Fingertips," Proc. IEEE International Conference on Robotics and Automation, pp. 3201{3208, 1999.
[4] A. Bicchi, \On the Closure Properties of Robotic Grasping," The International Journal of Robotics Research,
Vol.14, No.4, pp. 319{334, 1995.
[5] Y. Aiyama, M. Inaba and H. Inoue, \Pivoting: A New
Method of Graspless Manipulation of Object by Robot
Fingers," Proc. IEEE/RSJ International Conference onIntelligent Robots and Systems, pp. 136{143, 1993.
[6] T. Yoshikawa and X. Zheng, \Coordinated Dynamic
Hybrid Position/Force Control for Multiple Robot Manipulators Handling One Constrained Object," The International Journal of Robotics Research, Vol.12, No.3,
pp. 219{230, 1993.
[7] L. Han, Y. S. Guan, Z. X. Li, Q. Shi and J. C. Trinkle, \Dexterous Manipulation with Rolling Contacts,"
Proc. IEEE International Conference on Robotics andAutomation, pp. 992{997, 1997.

[8] J. C. Trinkle and R. P. Paul, \Planning for Dexterous
Manipulation with Sliding Contacts," The International
Journal of Robotics Research, Vol.9, No.3, pp. 24{48,1990.
[9] J. C. Trinkle, R. C. Ram, A. O. Farahat and P. F. Stiller,\Dexterous Manipulation Planning and Execution of
Enveloped Slippery Workpiece," Proc. IEEE International Conference on Robotics and Automation, pp. 442{
448, 1993.
[10] J. Kerr and B. Roth, \Analysis of Multi ngered Hands,"
International Journal of Robotics Research, Vol.4, No.4,
pp. 3{17, 1986.
[11] C. E. Lemke, \Bimatrix Equilibrium Points and Mathematical Programming,"Man. Sci., Vol.11, pp. 681{689,1965.
[12] Y. Nakamura, K. Nagai and Y. Yoshikawa, \Dynamics and Stability in Coordination of Multiple Robotic
Mechanisms," International Journal of Robotics Reserach, Vol.8, No.2, pp. 44{61, 1989.