Optimization of Grasping an Object by Using Required
Acceleration and Equilibrium-Force Sets
Tetsuyou Watanabe and Tsuneo Yoshikawa
Department of Mechanical Engineering, Kyoto University
Kyoto, 606-8501 Japante-watanabe@ieee.org, yoshi@mech.kyoto-u.ac.jp
Abstract
In this paper, we search optimal grasp points and configurations of fingers for not only resisting an external force
applied to a grasped object but also generating a desirable acceleration of the object. Based on the concept ofrequired external force set, we define required accelerationand equilibrium-force sets. By using the sets, we formulate
an optimization problem from the viewpoint of decreasing
the magnitudes of the joint torques required to generate therequired acceleration and equilibrium-force. We also showthat we can solve the problem by using a branch-and-boundmethod. The validity of our approach is shown by numerical examples.
1 INTRODUCTION
When we grasp an object by a robotic hand, force closure is one of the important properties of grasping [1].
Force closure is the concept which we can interpret in thefollowing two ways; ” any arbitrary acceleration and angular acceleration of a grasped object can be generated byjoint actuating torques. ” or ” the motion of a grasped
object can be completely constrained by virtue of contact
forces, whatever external force and moment are applied tothe object. ” Yoshikawa [2] called the former concept active closure, and the latter concept passive closure when the
motion of the object can be constrained without changing
the pre-loaded joint torques. In the case where there exist
multiple contact points between a finger and an object (forexample, enveloping grasp), or the case where the numberof joints of a finger is smaller than the number of the dimension of contact force applied by the finger, the contact force
between the finger and the object can be generated not actively by the joint torques but passively by the mechanismof the geometric constraints. In this case, even if we can resist certain external force and moment in a certain direction
and the motion of the object can be completely constrained,
there is no guarantee that we can generate an accelerationof the object in the same direction. So far, there doesn’t
exist any researches about searching optimal grasp points
on a grasped object, in which the above two concepts are
distinguished [3]–[8].

In order to deal with this problem, we deal with searching optimal both configuration of fingers and contact positions for not only resisting an external force applied to a
grasped object but also generating a desirable accelerationof the object.
This paper is organized as follows. At first, we set the
problem to solve. Then, we formulate the problem, andshow an algorithm to solve the problem. Lastly, numerical examples are presented to show the effectiveness of ourapproach.
2

PROBLEM DEFINITIONIn this section, we define an optimization problem ofgrasping. At first, we describe the target system. Then,
we define required acceleration and equilibrium-force sets.
Lastly, we give the optimization problem of grasping.
2.1 Target System
The target system is shown in Fig.1. In this paper, weconsider the case where an arbitrary shaped rigid object isgrasped by N fingers of a robotic hand (Note that N = 2
in Fig.1). We make the following assumptions: 1) Eachfinger makes a frictional point contact with the object. 2)
The unique normal direction at each contact point can beobtained. 3) There exists at most one contact point on eachlink of each finger. 4) Base doesn’t make any contacts withthe object. 5) In the manipulation of the object, both thenumber of contact points and the contact positions on theobject/fingers don’t change (We don’t consider the manipulation in which a certain contact point removes from theobject nor in which a certain point on a certain finger, which
isn’t a contact point, makes a new contact with the object).
Under these assumptions, we consider the case where in agiven workspace, we generate a desirable acceleration of
the object, resisting external force and moment, such as
gravitational force.
2.2 Required Acceleration and EquilibriumForce Sets
In this section, we define required acceleration andequilibrium-force sets based on the concept of required external force set [9] [10].

the number of the joints of the i th finger, d= 2/3 in 2/3
dimensional space, and Li denotes the number of the contact points on the i th finger. Note that we represent the
combination of configuration of the N fingers and the L(= ΣN
i=1 Li ) contact positions as C.

Figure 1: Target System

3.1.1 Kinematic Constraints
The relation between the velocities of pCij and q i , andthe one between the velocities of pCij and r are given as
follows respectively.
ṗCij = J ij q̇ i ,Required Acceleration Set (RAS) We call a set, composed of desirable accelerations which the grasped object must be able to generate, required acceleration set
AR ⊂ RD (D= 3/6 in 2/3 dimensional space).
Required Equilibrium-Force Set (REFS) We call a set,
composed of necessary resultant forces and moments tocounteract external forces and moments and to maintainthe current position and orientation of the object, required
equilibrium-force set WR ⊂ RD .
2.3 Problem to Solve
In general, there exist an infinite number of both configurations of fingers and contact positions, where we cangenerate the required acceleration and equilibrium-force.
We think it is suitable to minimize the magnitude of thejoint torques required to generate the required accelerationand equilibrium-force. Then, we consider the following
problem.
Find the configuration of fingers and the contact positions, which minimize the magnitude of the necessary joint torques to generate both any arbitrary acceleration contained in the given RAS and any arbitrary equilibrium-force contained in the given REFS, ateach position and orientation of the object in the given
workspace.
3 FORMULATION OF THE PROBLEM
In this section, we formulate the problem describedabove. At first, we consider the case where the object is
at certain position and orientation, and the fingers are in acertain configuration and grasp the object at certain contact points. The conditions for the joint torques to generate
certain required acceleration and equilibrium-force in thiscase are described. Then, we formulate the optimization
problem of grasping.
3.1 Conditions for Joint Torques
Let q i ∈ RMi (i = 1, 2, · · ·, N ) be the joint angles of
the i th finger. Let r∈ RD be the position of the origin
and orientation of, ΣO , the object coordinate frame fixedat the center of gravity of the object. Let pCij ∈ Rd (j =1, 2, · · · , Li ) be the position of, Cij , the j th contact point
between the object and the i th finger. Here Mi denotesṗCij = GTij ṙ

(1)where J ij ∈ Rd×Mi denotes the Jacobian matrix and Gij= ( I [(pCij − po )×] )T in 3 dimensional space. Here,I represents an identify matrix, po represents the origin ofΣO , and [a×] represents a skew symmetric matrix equivalent to the cross product operation ([ a × ]b = a × b).
Let M be ΣN
i=1 Mi . By using the following vectors andmatrices,Tq̇ = q̇ T1 q̇ T2 · · · q̇ TN∈ RM
G = G11 G12 · · · GNLN ∈ RD×Ld





J 21
J N1
J 11





J = diag  ...   ...  · · ·  ...  ∈ RLd×M
J 1L1

J 2L2

J NLN

where diag means a block diagonal matrix, we get the following expression from (1)GT ṙ = J q̇(2)

Differentiating both sides of this equation with respect totime, we can get the following relation.
T

GT r̈ + Ġ ṙ = J q̈ + J̇ q̇

(3)

3.1.2 Statics
Let f ∈ RLd be the contact force vector which combines the contact forces at all contact points, τ c ∈ RM be
the joint torque equivalent to f , and w∈ RD be the resultant force and moment applied to the object at ΣO . From(2) and the principle of virtual work, we get the following
relations.
τ c = JT f,

Gf = w(4)

3.1.3 Frictional Constraint
The frictional constraint at the j th contact point on thei th finger is represented by
Ffij = {f ij | t2fij ,1 + t2fij ,2 ≤ µij nfij , nfij ≥ 0}

(5)

in 3 dimensional space. Here, nfij denotes the normalcomponent of the contact force f ij , tfij ,1 and tfij ,2 denotethe tangential components of the contact force f ij , and µij
denotes the frictional coefficient at the contact point.

3.1.4 Equation of Motion
From (4), the equation of motion of the object and thefingers can be represented as follows respectively.
M B r̈ + hB − w e = Gf
M r q̈ + hr + g r + J T f

=

τ

(6)
(7)

where τ denotes the joint torques, M B and M r are the inertia tensors of the object and the fingers respectively, hB
and hr are the terms representing centrifugal and Coriolis forces of the object and the fingers respectively, g r is
the term representing gravitational force of the fingers, andwe denotes external force and moment applied to the object, such as gravitational force. Note that the force, whichcounteracts w e , is the required equilibrium-force w d (wd+ w e = o).
3.1.5 Conditions for Joint Torques
Based on the above discussion, we consider the conditions for the joint torques to generate both certain required
acceleration r̈d (which corresponds to r̈ in (3) and (6) )and certain required equilibrium-force w d , when the position and orientation of the object are r and the combinationof configuration of fingers and contact positions is C. From(3), (5), (6), and (7), The condition is that there exist joint
torques, τ , which satisfy the following constraints.
A1 τ + A2 f = GT r̈d + a(8)
Gf = M B r̈ d + wd + hB
(9)f ij ∈ Ffij (j = 1, 2, · · ·, Li )(i = 1, 2, · · · , N ) (10)
where
A1
a

=
=

−1 T
JM −1
r , A2 = −J M r J
T
JM −1r (g r + hr ) − J̇ q̇ + Ġ ṙ

3.2 Formulation of the Problem
In this subsection, we formulate the optimization problem of grasping. Let B be the position and orientation of
the object in the given workspace. Let SB be the set of all
possible candidates of B. In the given workspace, if theposition and orientation of the object change, the configuration of fingers also changes. Then, in order to clarify C
to be optimized, let Br be the representative B in the given
workspace and Cr be the C at Br . Now, we assume that
if C at Br , namely Cr is given, we can get each C at eachB∈ SB . Meaning, each C at each B∈ SB can be determined by the relation between the Cr and the each B. Then,
Cr can represent each C at each B∈ SB . Hence, optimizing Cr means optimizing each C at each B∈ SB . Let SC
be the set of all possible candidates of Cr . Let WcB be the
combinational required set given at B(∈ SB ) :
T
WcB = {x = r̈ Td w Td |r̈d ∈ AR , wd ∈ WR }(11)

Let φ be the largest magnitude of joint torque among all
magnitudes of all joint torques:φ = max |τij |

(12)

i,j

where τij denotes the j th joint torque of the i th finger.
Then, the problem can be formulated as follows.
Optimization Problem of grasping Find the combination of configuration of fingers and contact positions, Cr∗and ρ such that,max

ρ = minmin

Cr ∈SC x∈WcB ,B∈SB τ ,f satisfy(8)∼(10)

φ

(13)

4

ALGORITHM
In this section, we describe an algorithm to solve the optimization problem of grasping (13). In order to solve theproblem, we use a branch-and-bound method [11]–[13], byrepresenting the candidate configurations of fingers as discrete configurations, the candidate contact positions as discrete positions, and the candidate positions and orientationsof object as discrete positions and orientations.
We make the following assumptions: 1) The number ofthe candidate Cr ’s is finite nc . Let Crk (k = 1, 2, · · · , nc) be
the Cr contained in SC . 2) The number of the candidate B’s
is finite nb . Let BI (I = 1, 2, · · · , nb ) be the B contained in
SB . Let CkI be the C at BI in the case of Cr = Crk . If certain
Crk is given, each C at each BI (CkI ’s (I = 1, 2, · · · , nb ))can be uniquely obtained. 3) We consider only the casewhere the system is stationary state at each BI . Namely,
the velocities of the all variables in the constraints (8) and(9) are assumed to be zero. 4) The RAS and REFS (ARIand WRI ) given at each BI can be expressed by convex
polyhedrons composed of la and le vertices respectively.
5) When the object is at a certain BI and C is a certain CkI(k and I are fixed), g r in (7) can be expressed by a convex
polyhedron composed of le vertices, whose element corresponds one to one to the element contained in WRI . Notethat here, we may suppose the case where the direction ofgravitational force can change with respect to ΣO , resulting
from the manipulation of the robotic arm equipped with therobotic hand.
We remark g r (and WRI ). We consider the case where
the object is at a certain BI and C is a certain CkI . Let WRIg
be the set composed of the forces and moments, which arecontained in WRI and counteract gravitational force. Let
WRIo be the set composed of the forces and moments,
which are contained in WRI and counteract the other external forces and moments. We assume that WRIν (ν = gor o) can be expressed by a convex polyhedron composedof leν (≤ le ) vertices:
l

l

eν
eν
λeν i w Iνi , Σi=1
λeν i = 1, λeν i ≥ 0}
WRIν = {w = Σi=1

where w Iνi denotes the i th vertex of WRIν . Then, WRI

can be expressed by
WRI

=

leg
leo
{w = Σi=1
Σj=1
λei,j (w Igi + w Ioj ),legleoΣi=1 Σj=1 λei,j = 1, λei,j ≥ 0}
(14)

Let g̃ be gravitational acceleration vector and w be the correspondent element of the WRIg to the g̃. The relation
between g̃ and w can be expressed by −mb g̃= w where
mb denotes the mass of the object. Therefore, by usingg̃ i (= −w Igi /mb ) which corresponds to the i th vertex of
the WRIg , g̃ can be expressed by the convex combination
le

le

g
g
g̃ =Σi=1λeg i g̃ i , Σi=1
λeg i = 1, λeg i ≥ 0. If C (CkI )doesn’t change, g r can be expressed by Ag̃ where A isa certain constant matrix [14]. Then, by using g ri whichcorresponds to g̃ i which corresponds to the i th vertex of
the WRIg , g r can be expressed by the convex combina-

le

le

l

le

g
g
g
eo
λeg i g ri = Σi=1
Σj=1
λei,j g ri , Σi=1
λeg i =
tion g r = Σi=1

le

l

g
eo
Σi=1
Σj=1
λei,j = 1, λeg i , λei,j ≥ 0. Therefore, g r cancorrespond one to one to the element contained in WRI
given in (14).
In this paper, Let Subproblem 1 be the subproblemof the problem (13). In Subproblem 1, candidate Cr is
fixed to a certain Crk (∈ SC ) (k is fixed). If we can solve
this Subproblem 1 for all Crk ’s (k = 1, 2, · · · , nc), we
can get the optimal solution of the problem (13). Notethat in Subproblem 1, each C at each BI (namely CkI ’s(I = 1, 2, · · ·, nb )) can be uniquely obtained. Let xIjξT(= r̈TdIj w TdIξ ) be the combination of the required acceleration and equilibrium-force which correspond to the jth vertex of ARI and the ξ th vertex of WRI respectively.
Let Subproblem 2 be the subproblem of Subproblem 1. In
Subproblem 2, candidate B is fixed to a certain BI (∈ SB )and the candidate required acceleration and equilibriumforce are also fixed to a certain xIjξ ( (k,) I, j, and ξ are
fixed). Note that when we solve Subproblem 2 for BI (andxIjξ ) which is a subproblem of Subproblem 1 for Crk , C is
fixed to CkI . Let ρCrk ,Ijξ be the solution of the Subproblem
2. Now, we consider the magnitude of the necessary joint
torque φ (see(12)) to generate both an arbitrary acceleration contained in ARI and an arbitrary equilibrium-force
contained in WRI , in the case where Cr is a certain Crk
and B is a certain BI , namely C is CkI . Meaning, we consider Subproblem 2 for the BI , at the Crk . The constraintsof Subproblem 2 is composed of linear equations, linear
inequalities, and frictional conditions, which are all convex. Then, in this case, the necessary joint torque φ canbe represented by the convex combination of ρCrk ,Ijξ ’s (kand I are fixed, j = 1, 2, · · · , la , ξ = 1, 2, · · · , le ). Therefore, among all nb la le solutions of Subproblem 2, ρCrk ,Ijξ ’s(I = 1, 2, · · · , nb , j = 1, 2, · · · , la, ξ = 1, 2, · · · , le ), thelargest value is the solution of Subproblem 1 (for Crk ).
Meaning, we can solve Subproblem 1 by solving Subprob-

lem 2 for all xIjξ ’s and all BI ’s. Note that Subproblem
2 can be solved by simplex method, by approximating thefriction corn by a polyhedral convex corn [15]. Based on
the above discussion, we can solve the problem (13) by abranch-and-bound method.
4.1 Procedure of the Algorithm
In this subsection, we describe the algorithm to search
the optimal both configuration of fingers and contact positions. Let ρ̂ be the tentative optimal solution, ρCrk be theoptimal solution of Subproblem 1 for Crk , and ρ̂Crk be its
tentative optimal solution. We represent a list of feasible
Crk ’s as the LIST.
step 1 We put all candidate Crk ’s, whose corresponding
each C at each BI (namely CkI ’s (I = 1, 2, · · ·, nb )) satisfies some conditions, into the LIST. The some conditions
are, for example, no-interference between the object andthe fingers, no-interference between the fingers, and limitation of the joint angles of the fingers. Let ρ̂ be an appropriate lower bound value. Let each ρCrk and ρ̂Crk beappropriate upper and lower bound values respectively.
step 2 We solve Subproblem 1 for certain Crk contained in
the LIST.
step 3 If we can get the solution of the Subproblem 1 instep2, let ρ̂ (= ρCrk = ρ̂Crk ) be the solution, and let B̂I
and x̂Ijξ , respectively, be BI and xIjξ , which give the solution. Otherwise, we eliminate the Crk form the LIST andgo back to step2.
step 4 We solve Subproblem 2 for B̂I and x̂Iˆjξ , at every
Crk contained in the LIST. If we cannot get the solution atsome Crk , we eliminate this Crk form the LIST. If we canget the solution ρC ,Î at some Crk , we compute ρ̂Crk =rk jξ
max{ρ̂Crk , ρC ,Î }. If ρ̂ <ρ̂Crk , we eliminate this Crk
rk jξform the LIST.
step 5 Let Crk be the Crk at which ρC ,Î is the leastrk jξ
among those at all Crk ’s contained in the LIST.
step 6 We solve Subproblem 1 for Crk . If we can get thesolution, let ρĈ (= ρ̂Ĉ ) be the solution, and let B̂I andrkrkx̂Ijξ , respectively, be BI and xIjξ , which give the solution. If ρ̂ >ρĈ , ρ̂ =ρĈ . If we cannot get the solution orrkrkρ̂ <ρĈ , we eliminate this Crk form the LIST and go backrk
to step5.
step 7 If we can get the relation |ρ̂−ρCrk | <  ( denotes anarbitrary small positive value) for all Crk ’s contained in the
LIST, we finish the loop. Otherwise, we go back to step4.
5

NUMERICAL EXAMPLES
In order to show the effectiveness of our approach, weshow some numerical examples in this section. The target
system is shown in Fig.2(a). We set ΣO is placed at the geometric center of the object. Note that the scale markings
shown in Fig.2(a) represent the ones of the reference coordinate frame (the hand coordinate frame). Note also that

Figure 2: Target System in Numerical Examples

Figure 3: Target Operation with the Optimal Configurationin Case I

Figure 4: Target Operation with the Optimal Configurationin Case II

when the object is at the position and orientation shown in
Fig.2(a), the position and orientation of the reference coordinate frame become the same as the ones of ΣO . The
robotic hand is composed of 2 fingers which are same form.
We call the links of the fingers link 1 and link 2 in the orderof closeness to the base side. The length of link i (i = 1, 2)
is set to 0.05[m]. The mass of each link is set to 0.025[kg].
The limitations of the joint angles of the fingers are set as
follows: π/2 ≤ q11 ≤ π, −π ≤ q12 ≤ 0, 0 ≤ q21 ≤ π/2,0 ≤ q22 ≤ π, where qij ([rad]) denotes the j th (j = 1, 2)
joint angle of the i th (i = 1, 2) finger. In this numericalexample, we represent the configuration of fingers as the
position of base and the fingertip positions. Letting bx ([m])
and by ([m]) be the x and y components of the position of

the geometric center of the base respectively, we set theworkspace of the base as follows (see Fig.2(b)): −0.05 ≤bx ≤ 0.05, −0.155 ≤ by ≤ −0.055. The number of thecandidate positions of the base is set to 441 by giving thecandidate positions with steps of 0.005 in the horizontal direction (x direction) and 0.005 in the vertical direction (ydirection). The object is a 0.1([m])× 0.05([m]) quadrangle. The weight of the object is set to 4[N]. In Fig.2(b), thepoints on the object indicate the candidate contact points.
The number of the candidate contact points is 40. The frictional coefficients at the contact points are set to 0.5. Notethat we assume that the contact between link 1 of each finger and a vertex of the object can be represented by point
contact if the intersection between the link and the vertex(object) is small.
At first, we consider Case I shown in Fig.3. This task
is to rotate the object around the ”X” mark, resisting gravitational force. Note that we suppose the case where thedirection of gravitational force can be applied in multiple
directions, resulting from the manipulation of the roboticarm equipped with the robotic hand. The ”X” mark represents the point (−0.005, −0.065) with respect to ΣO (Notethat it isn’t easy to see the ”X” mark. Then, we also show
it in Fig.2(a)). The position and orientation of the object
surrounded with a rectangle are Br (This is the same in
the following Case II). Br is set to (0, 0, 0)T . The candidate positions and orientations of the object are obtained
as follows: the object, initially located at the Br , is rotated around the ”X” mark over an interval of −0.2[rad]to 0.2[rad] with steps of 0.1[rad] (nb = 5). We set the RAS
is composed of angular accelerations around the X” mark(−0.005, −0.065, 0), whose magnitude is 0.5[rad/s2 ]. We
set the REFS is a 33-side convex polyhedron which approximates the upward half circle whose radius is 4[N]. Weshow the RAS and the REFS at Br as follows.
AR1 = {r̈d = λ1 r̈ d1 +λ2 r̈ d2 , λ1 + λ2 = 1, λi ≥ 0,r̈d1 = (−0.0325, 0.0025, 0.5)T , r̈d2 = −r̈ d1 }32
WR1 = {wd |w d = Σ32
i=0 λi w di , Σi=0 λi = 1, λi ≥ 0,
w di = (4 cos(iπ/32), 4 sin(iπ/32), 0)T }The obtained optimal configuration of fingers and contact positions are shown in Fig.3. The obtained ρ was
0.270[Nm]. For someone’s information, the necessary joint
torque φ, in the case where Cr is the one shown in Fig.2(a),was 0.439[Nm]. At the optimal configuration, we can takeadvantage of the contact forces which can be generated
without changing the magnitudes of the joint torques, in order to generate the required equilibrium-force. We think it
is a reason why the configuration shown in Fig.3 is goodcompared with the configurations, at which each finger
contacts with the object at only one point.
Next, we consider Case II shown in Fig.4. This task is
to move the object in some translational directions, resist-ing gravitational force whose direction and magnitude are
the same ones in Case I. Br is set to (0, 0, 0)T . The candidate positions and orientations of the object are obtained
as follows: the object, initially located at the Br , is translated over an interval of −0.01[m] to 0.01[m] with stepsof 0.005[m] in x direction and an interval of −0.01[m] to0.01[m] with steps of 0.005[m] in y direction (nb = 25).
We set the RAS is composed of accelerations whose magnitude is 0.5[rad/s2 ] in x and y directions. We show the
RAS at Br as follows (Note that the REFS is the same one
in Case I).
AR1= {r̈d |r̈d = Σ4i=1 λi r̈ di , Σ4i=1 λi = 1, λi ≥ 0,r̈ d1 = (0.5, −0.5, 0)T , r̈ d2 = (−0.5, −0.5, 0)T ,r̈ d3 = (0.5, 0.5, 0)T , r̈d4 = (−0.5, 0.5, 0)T }The obtained optimal configuration of fingers and contact positions are shown in Fig.4. The obtained ρ was
0.393[Nm]. In order to generate the required accelerationsin this case, the number of the contact points between theobject and each finger must be one. We think it is a reason
why we can get such a result shown in Fig.4.
In the above examples, we use SELECT PRO JP
3YEAR /OX KER(CPU : AMD ATHLON 1.2GHZ) madeby GATEWAY. The running time is about 15 seconds forCase I and about 3 minutes for Case II.
6 CONCLUSION
In this paper, we have dealt with an optimization ofgrasping for not only resisting an external force applied to a
grasped object but also generating a desirable accelerationof the object. In order to achieve an optimal grasp in which
the two interpretations of force closure are distinguished,
we have defined required acceleration and equilibriumforce sets. By using the sets, we have formulated an optimization problem from the viewpoint of decreasing themagnitudes of the joint torques required to generate the required acceleration and equilibrium-force, and showed that
we can solve the problem by using a branch-and-boundmethod. We have also presented numerical examples inorder to show the validity of our approach.
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