The definition of manipulability ellipsoids for dual robot systems is given. A suitable kineto-static formulation for dual cooperative robots is adopted which allows for a global task space description of external and internal forces, and relative velocities. The well known concepts of force and velocity manipulability ellipsoids for a single robot are formally extended and the contributions of the two single robots to the cooperative system ellipsoids are illustrated. Duality properties are discussed. A practical case study is developed

Chiacchio, P.

Chiaverini, S.

Sciavicco, L.

Siciliano, B.

English

Chiaverini, Stefano

IRIS Unicas (Università degli Studi di Cassino e del Lazio Meridionale)

On the manipulability of dual cooperative robots

NASA Technical Reports Server

ON THE MANIPULABILITY OF DUAL COOPERATIVE ROBOTS
P. Chiacchio,S. Chiaverini,L. Sciavicco,B. Siciliano
Dipartimento di Informatica e Sistemistica
Universit/t di Napoli
Via Claudio 21, 80125 Napoli, Italy
Abstract
In this paper the definition of manipulability ellipsoids for dual robot systems is given. A suitable
k:'neto-static formulation for dual cooperative robots is adopted which allows for a global task space
description of ezternal and internal forces as well as absolute and relative velocities. The well-known
concepts of force and velocity manipulability ellipsoids for a single robot are formally eztended and
the contributions of the two single robots to the cooperative system ellipsoids are evidenced. Duality
properties arc discussed. A practical case study is developed.
1. Introduction
Cooperative robots have been recognized by the robotics research community as offering enhanced
capabilities over current single robot structures. Dual robot cooperation allows for performing tasks
such as handling large, heavy and non-rigid objects, assembly and mating mechanical parts, which
could not be executed by a single robot. Another advantage is the enlargement of the reachable
workspace. All the above features play a crucial role, for instance, in space robotics applications
where cooperative manipulation is often considered as an essential requirement.
In spite of the potential benefits achievable with dual robots, the control problem becomes more
complex due to the kinematic and dynamic interactions. A must for the solution of this kind of
problem is constituted by an effective description of the kineto-static and dynamic relationship for a
general dual robot system. To this purpose, the formulation proposed by Dauchez and Uchiyama [1]
has been shown to be suitable to coordinated control schemes with equal importance attributed to the
two robots performing a given task. Their approach is somewhat opposite to the master-slave strategy
suggested by Luh and Zheng [2] which has been argued by Uchiyama et al. [3] to be unsuitable for
practical position/force control of dual robots.
It is believed that an important issue is the definition of quantitative measures of the enhanced
performance offered by dual robot cooperation. It is well-known that the manipulability ellipsoids
introduced by Yoshikawa [4] represent one of such measures for a single robot. The contribution of
this work is to provide a systematic way of extending the above concept [4] to the dual robot case. In
order to accomplish this goal, tlke formulation dictated in [3] is followed here. The motivation behind
this choice is that it leads to a natural, straightforward derivation of manipulability measures which be
consistent with those proposed in [4] and susceptible of an immediate physical interpretation for the
closed-chain system created by the dual robots tightly handling an object. A similar, parallel research
effort has recently been produced by Lee and Bejczy [5], although the definition of manipulability
measures for a dual robot system is obtained according to different criteria related to the effect of
one robot on the other, instead of regarding the closed-chain as a whole.
A practical case study is worked out for two simple planar cooperative robots. Velocity and force
static ellipsoids are obtained which show the correctness and functionality of the proposed approach.
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2. Kineto-static formulation for two cooperative robots
In the following the formulation of task space coordinates required for describing cooperative
tasks is briefly summarized from [2]. For the purpose of the present work, the case when the two
robots are rigidly attached to the object is considered, i.e. a rigid grasp.
According to [2], the cooperative task is described in terms of a set of absolute coordinates and a
set of relative coordinates. The static relationship between the generalized forces exerted by the two
robots and the generalized forces acting on the object -- external and internal -- is presented first.
The kinematic relationship will be derived by using the duality relation between forces and velocities.
Fig. 1 illustrates two cooperative robots tightly grasping an object. Let m be the common dimension
ofthe taskspacesofthe two robotsand
denote the vectors of the generalized forces (forces fl, f2 and moments nl, n2) exerted by the two
end-effectors, respectively. Let then
denote the vectors of external and internal forces/moments acting on the object, respectively. Let be
Itcan be shown that
where
h,=Wf (4)
' 0 , 0)-R1, I-R2_ I (5)
with Rl_, R2_ defined by fl × rl_ = -Rl_fl, f2 × r2_ -- -R2_f2, respectively. I and 0 denote identity
and null matrices of appropriate dimensions. Also it is
f = Vh, (6)
where
V = RI_
-X " (7)
-R2a
Itcan be recognizedthat the mapping V in (7) spans the nullspace of the mapping W in (5).This
means that the externaland internalforce/moment vectorsbelong to orthogonalsubspaces [6].
Once thestaticrelationshiphas been established,the differentialkinematicrelationshipisderived
in a similarmanner. Let
denote thevectorsofthe velocities(translational_i,_2 and rotationalcoi,_o2)atthe two end-effectors,
respectively.Let then
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denote the vectorsofexternal(absolute)and internal(relative)velocities,respectively.Let be
In forceof the dualityrelationbetween forcesand velocitieswhich isderivedfrom the principleof
virtualwork inmechanics,itcan be shown that
t=wT_a (11)
and
_', = vri
with W and V defined in (5) and (7), respectively.
(12)
3. Definition of manipulability ellipsoids
The idea of measuring the manipulating ability of robotic mechanisms was first introduced in [4].
According to that concept, a force manipulability ellipsoid and a velocity manipulability ellipsoid can
be defined for a single robot. Assume that an n-DOF robot is given and an m-dimensional task space
is of interest, usually with m _< n. It is well-known that
r = Jr(0)qt (13)
represents the static relationship between the task force vector 7 and the joint torque vector r through
the transpose of the Jacobian matrix J(0), with 0 denoting the joint displacement vector. Dually,
v = J(0)0 (14)
represents the kinematic relationship between the joint velocity vector 0 and the task velocity vector
v through the Jacobian J(O).
The unit sphere in the joint torque space
rrr = 1 (15)
.Tr (jjT).y __ 1 (16)
maps intothe taskforcespace ellipsoid
which is called force manipulability ellipsoid [4]. Dually, the unit sphere in the joint velocity space
maps into the task velocity space ellipsoid
_T_ = 1 (17)
v r (JJr)-zv = 1 (18)
which is called velocity manipulability ellipsoid [4]. Note that the explicit dependence on 0 has
been dropped in J. A direct comparison of (16) with (18) indicates that the principal axes (related
to the eigenvectors) of the two ellipsoids coincide, whilst the lengths of the axes (related to the
eigenvalues) are in inverse proportion. This inverse velocity-force relation is consistent with regarding
the manipulator as a mechanical transformer [7]. Conservation of energy dictates that amplification
353
in velocitytransmissionmust invariablybe accompanied by reductionin forcetransmission,and
vice-versa.
In the following the concepts of force and velocity ellipsoids defined in (16) and (18) are formally
extended to a two robotsystem, based on the kineto-staticformulationgivenin the previoussection.
Let nl and n2 be the DOF's of the robots,respectively.The staticrelationship(13)can be written
fora two robot system as
with f definedin (3),where
t:J_f (19)
,I
denotesthe extended jointtorquevectorin an (nl+ n2)-dimensionalspace,and
(20)
0
denotes the extended Jacobian matrix. Solving eq. (19) for f yields
f-- JTtt (22)
where the simbol _ t" denotes a pseudo-inverse of proper dimensions; in this case it is a left pseudo-
inverse.
The external force manipulability ellipsoid and the absolute velocity manipulability ellipsoid are
derived first. Plugging (22) into (4) gives
ha = j_tt (23)
which expressesthe relationshipbetween the extended jointtorque vectorand the externalforce
vector,through the matrix
j_t _=W jr: (24)
which is analogous to the pseudo-inverse of the Jacobian jr in (13) for a single robot. At this point
the formal definition of the external force manipulability ellipsoid can be given. The unit sphere in
the extended joint torque space
t rt : 1 (25)
maps into
h_ (J_J_)ha = 1 (26)
which is defined here as external force manipulability ellipsoid. Dually, the formal definition of the
absolute velocity manipulability ellipsoid can be given. The unit sphere in the extended joint velocity
space
_IT_I = 1 (27)
maps into
(28)•T,j jT,-1.Ya( a _) ya:l
which is defined here as absolute velocity manipulability ellipsoid. Notice that in (27)
01q=(02)  29,
354
indicates the extended joint vector.
An attractive mathematical expression can be found for the matrix J=J_ constituting the core
of the two manipulability ellipsoids just defined in (26) and (28), which is directly related to the
Jacobians of the two robots defined in (21). As shown in Appendix A, one may obtain
jojf __((j_.j_)_ _.+ (j_j_ )...1)_,. (30)
with ( o )jS t = jSt + __.(_t b i= 1,2 (31)
where the subscriptf referstothe upper blockofthe matrix jTt which maps thejointtorquevector
riintothe soleforcecomponents ofthe taskforcevectorf_definedin (1)(i.e.excludingthe moment
components). Itisworth noticingthatinthe particularcasewhen R/a --0,the matrices_rt simplify
to jTt.
In the same formalmanner asdone above for the externalforcemanipulabilityellipsoidand the
absolutevelocitymanipulabilityellipsoid,the derivationofthe internalforcemanipulabilityellipsoid
and the relativevelocitymanipulabilityellipsoidispresentednow. Plugging (22)into(6)gives
t = JTh, (32)
with
jr _ j_V. (33)
It is to be remarked that, by virtue of the definitions (24) and (33) and of the structure of the matrices
W and V, it results
j rt jr = 0. (34)
The unitsphere (25)maps into
hy(J, JT)h, = 1 (35)
which is defined as internal force manipulability ellipsoid. Dually, the unit sphere (27) maps into
(36)
which is defined as relative velocity manipulability ellipsoid. In this case too, an attractive mathe-
matical expression can be found for the matrix J, Jr constituting the core of the two manipulability
ellipsoids just defined in (35) and (36), which is directly related to the Jacobians of the two robots
defined in (21). As shown in Appendix B, one may obtain
j, jT = jl j_ + J2J_ (37)
with
jr =(_l),-1jT÷((_l),-,(j_),,Rxa 0) i= 1,2 (38)
where the subscript n refers to the lower block of the matrix J_ which maps the sole moment compo-
nents of the task force vector fi defined in (1) into the joint torque vector ri.
From eq. (34), it directly follows that
_t jr _ j_t j_ =0. (39)
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It is worth noticing that in the particular case when R_a -- 0, the matrices jT simplify to Jr; in this
case, thus, eq. (39) trivially holds.
Eqs. (30) and (37) suggest a nice interpretation of the way the Jacobians of the two robots
combine to form the respective cores of the ellipsoids defined above. If each term of the type jjT is
regarded as a generalized impedance, eq. (30) resembles the mathematical expression of the parallel
of two impedances, whilst eq. (37) resembles that of the series of two impedances. Therefore, one
would naturally be driven to generalize these results to the multiple robot case; this topic is under
investigation.
4. Case study
Two 3-DOF planar robots are considered for the purpose of illustrating the application of the
concepts presented in this work to a practical two robot system. For the sake of simplicity, the end-
effectors of the two robots are supposed to be located in the same point (i.e. the physical object is
removed); this implies that _ = J_ = Ji. This assumption is not restrictive at all, as formally shown
above. Moreover, a two-dimensional global task space is assumed, i.e. only forces and linear velocities
are of interest; the system thus possesses two redundant DOF's.
A CAD tool has been developed which is articulated into the following steps. The contact
point of the two end-effectors is input, then the two redundant DOF's are exploited to assign the
orientation angles of the end-effectors. A software package for solving the inverse kinematics of
general robot structures [8] is utilized to find the joint configurations and then the complete kineto-
static characterization of the system. An option is provided to compute the ellipsoid of interest.
The outputs are plotted by means of a graphic package. They illustrate the closed kinematic chain
together with the principal axes of the ellipsoids of the two single robots and those of the dual robot
system, in order that the manipulability of the cooperative system can be evaluated with respect to
the single robot manipulabilities.
Two complete sets of results for two different configurations of the dual robot system are displayed
in Figs. 2 and 3 respectively. The ellipsoids of each robot are included for a better comprehension
of the effects of the cooperation. It is remarkable that, in the second configuration, the two robots
are both proximal to singular configurations (see the shape of their ellipsoids). It can be recognized
that the external force ellipsoids (Figs. 2a and 3a) are improved in that the ability of each robot to
exert forces along a given direction is enhanced by the other, while the absolute velocity ellipsoids
(Figs. 2b and 3b) show that the ability of each robot to perform motions along a given direction is
penalized by the presence of the other. This result well agrees with practice, since it is intuitive that
when two robots cooperate the static force is shared by them whereas the faster robot is slowed down
by the other. Conversely, the ability of the system to absorbe forces along a direction is limited by
the weaker robot of the chain (Figs. 2c and 3c), while the ability of the system to give rise to relative
motions along a direction is supplied by both robots (Figs. 2d and 3d). All these conclusions reflect
the concept of duality which is at the basis of the definition of the manipulability ellipsoids presented.
5. Conclusions
The concept of manipulabilityellipsoidshas formallybeen extended to the case of dual robot
systems. A globalkineto-staticformulationof the closedchain createdby two tightlycooperating
robotshas been exploitedto defineexternaland internalforcemanipulabilityellipsoids.The corre-
sponding absoluteand relativevelocitymanipulabilityellipsoidshave been derivedon the basisofthe
dualityprinciplein mechanics. Functionalexpressionsfortheseellipsoidshave been obtainedthrough
the Jacobiansof the two robotsand a practicalruleof compositionhas been provided.The results
achieved for a simple casestudy have validatedthe theoreticalconclusionsin view of the physical
interpretationof the kineto-staticsof a dual robot system. Ithas been conjecturedthatthe proposed
definitionscan be extended to the multi-robotcase,although the formulationof internalforcesis
356
not straightforward. This issue, along with the analysis of different types of cooperation (e.g. loose,
soft) and the formulation of dynamic manipulability ellipsoids, will constitute the subject of further
investigation.
References
[1] J. Y. S. Luh and Y. F. Zheng, UConstrained relations between two coordinated industrial robots
for motion control, _ Int. J. Robotics Research, Vol. 6, No. 3, pp. 60-70, 1987.
[2] P. Dauchez and M. Uchiyama, "Kinematic formulation for two force-controlled cooperating
robots," Srd Int. Conf. on Advanced Robotics, Versailles, France, Oct. 1987.
[3] M. Uchiyama, N. Iwssawa, and K. Hakomori, "Hybrid position/force control for coordination of a
two-arm robot, _ #th IEEE Int. Conf. on Robotics and Automation, Raleigh, NC, Mar.-Apr. 1987.
[4] T. Yoshikawa, "Analysis and control of robot manipulators with redundancy," 1st Int. Syrup. on
Robotics Research, Eds. M. Brady & R. P. Paul, MIT Press, Cambridge, MA, pp. 735-748, 1984.
[5] S. Lee and A. K. Bejczy, "Dual redundant arm system manipulability," NATO Advanced Research
Workshop on Robots with Redundancy: Design, Sensing and Control, Salb, Italy, June-July 1988.
[6] Y. Nakamura, K. Nagal, and T. Yoshikawa, "Mechanics of coordinative manipulation by multiple
robotic mechanisms," 4th IEEE Int. Conf. on Robotics and Automation, Raleigh, NC, Mar.-
Apr. 1987.
[7] S. Chiu, "Control of redundant manipulators for task compatibility," 4th IEEE Int. Conf. on
Robotics and Automation, Raleigh, NC, Mar.-Apr. 1987.
[8] L. Sciavicco and B. Siciliauo, "Solving the inverse kinematic problem for robotic manipulators,"
6th CISM-IFToMM Ro.Man.Sy, Cracow, Poland, Sept. 1986.
Appendix A
The matrix j_t defined in (24), by substituting the expressions of W in (5) and J:2 in (21),
becomes 0, 0/(,:, 0)
= -RI. I -R1. jTt "
The matrices Jirt can be partitioned by rows as
J_t=((J_t)') i=1,2 (A-2)(j?t).
where the subscripts f and n refer to forces and moments respectively. Plugging (A-2) in (A-I) gives
jTt = (jrt jTt) + ( 0 0_R,_(j_ft) I _/_,(j2rt) I ) (A-3)
which, by virtue of (31), can be compactly written as
j[t (j It j_t). (A-4)
The matrix j,.jr in (26) can then be computed. First, one obtains
(jt(jrt jt + j:t J:)--I I _
C : _,jt(jrt jt + j:t fl)...,/ (A-S)
357
where the "~''s have been dropped without loss of generality. Eq. (A-5) leads to
joj: = (jr j_+j:t j:l_TJ,tj;(j, tj_+j:t j:)_,
+(jrt jt +j;tj:)-Tj;tj:(jrt j: + ,;t j:)-,
that can be compacted into
(A-6)
j.jy : (j_t j_ + j_t j_)-T. (A-7)
By virtue of the property j[t j t = (j,j[)-l, eq. (A-7) directly leads to (30 I.
Appendix B
The matrix _ defined in (33), by substituting the expression of Ji2 in (21) and V in (71, becomes
('..o, <B,,
-R2a
The matrices jr can be partitioned by columns as
J[ = ((J[)s (J[),) ;= 1.2.
Plugging (B-21 in (B-1 / gives
(J,_ (oJ_: -J;]+ o
which, by virtue of (38), can be compactly written as
(B-2)
(B-3)
(B--4)
Computing the matrix JrJ f in (35) directly leads to (37).
_" robot 2 ?
ro o/ I
q _ hand 2 frame _>
\ _haodl / + _ . .i, /
_:-P frame" ] "_ ,_ /
\ ",,,/_,._/_--<,. \,," /
_, -_ object -""'-.-_, " O/
%... fram. //
base frame
Fig. 1 - A dual cooperative robot system
358
robot t robot I ÷ robot 2 robot 2
robot I
( )
"obot I ÷ robot 2
/1{J
robot 2
I_HTERNAL FORCE
LLIP$OID 1
I_BSOLUTESPEED
LLIPSOID 1
robo_
a
robot 1 robot I * robot 2 robot 2 robot I
i
Y_
robot I + robot 2
//
/
robot 2
_NTERNAL FORCE
LLIPS01O 1
I_ELATIUE SPEED
LLIPSOIO 1
c d
Fig.2 - Manipulabilityellipsoids fora firstconfiguration
359
robot I
1'
robot I ÷ robot 2
_EXTERNAL
I _ kE"tPS°ID
FOR£E 1
robot I robot I + robot 2
tAB SOLUTE sweeo
_ I,ELLIPSOID
I' robot_
a b
%-
robot I robot I ÷ robot 2 robot 2
r,_.N,.o.cE]
robot I ,obot I + robot 2
,S
F
robot 2
fRELATIVE SPEED
c d
Fig. 3 - Manipulability ellipsoids for a second configuration
36O


On the manipulability of dual cooperative robots

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