Optimization of Grasping by Using A Required External Force Set
Tetsuyou Watanabe and Tsuneo Yoshikawa
Department of Mechanical Engineering, Kyoto University
Kyoto, 606-8501 Japan{watanabe-t@t99.mbox.media, yoshi@mech}.kyoto-u.ac.jp

Abstract
In this paper, we consider an optimization ofgrasping by using a required external force set. By
using the set, we can not only deal with whatevera desired grasp is, such as force closure or equilibrium grasp, but also evaluate the magnitudes of
the resistible external forces and moments. Then,
we define an optimization problem from the viewpoint of decreasing the magnitudes of the contact
forces required to resist the required external force,and show that we can solve the problem by usinga branch-and-bound method. Lastly we present
some numerical simulations to show the validity of
our approach.

1

Introduction

Consider lifting up an object on a table by a robotic
hand. If we don’t grasp the object with appropriate
contact positions between the object and the robotic
hand, we may not be able to counteract the external force such as gravitational force and fail to lift up
it. We must grasp an object with appropriate contact
positions, in order to manipulate the object in a desired direction. Therefore, much attention has been
attracted to search optimal grasp points on a grasped
object [1]〜[8].
In the above researches, there are 2 kinds of researches; 1) one is aimed at searching optimal grasp
points or regions for counteracting gravitational force(namely, achieving equilibrium grasp) [1] [2], 2) theother is aimed at searching optimal grasp points orregions for achieving force closure [3]〜[8]. Force closure indicates that if external force and moment in any
arbitrary direction applied to a grasped object, we cancounteract the external force and moment and restrain
the motion of the object.
There exist an upper bound of the magnitude of theresistible external force and moment in force closure
grasps. Then, the magnitude can be used as a criterion
for the grasps. Li et al. [9] evaluated a volume of thelargest task ellipsoid. It can be embedded in the set
constructed by resultant forces and moments, which

we can apply to a grasped object. But the computation of the volume is very complex. Then, it is hard
to search optimal grasp points by using this criterion.
On the other hand, researches about searching optimal grasp points, which minimize the magnitudes of
the necessary contact forces to resist an external force
and moment, has been done [4]〜[7]. Also here, a problem arises: how to handle the diﬀerence between theunit of force and the one of moment. Then, some devices, to resolve the problem, have been done in theseresearches. Markenscoﬀ et al. [4] considered the casewhere no external moment is applied to the object.
Mirtich et al. [5] evaluated the external force and moment separately. Mangialardi et al. [6] considered thecase where the external force or moment applied tothe object individually. Wang et al. [7] regarded themagnitude of the resistible external force and moment
as the magnitude of the contact force of the clamp.
In our previous paper, we optimized power grasp byusing required external force set [10] [11]. This set consists of external forces and moments, which we must
resist by contact forces applied by a robotic hand. The
magnitude of the external force and moment containedin the set corresponds to the magnitude of the resistible external force and moment. When the origin is
in the interior of the set, we can achieve force closure.
When the dimension of the set is 1, we can achieve
equilibrium grasp. Therefore we can deal with generaldesirable grasp including force closure and equilibrium
grasp by using the set. Teichmann et al. [8] considered
to search grasp points which minimize the number ofthe contact points needed to resist all external forces
and moments contained in the set, not considering friction. But in their approach, we can deal with only
force closure grasps.
In this paper, we search optimal grasp points which
minimize the magnitudes of the contact forces required
to resist all external forces and moments contained in
the required external force set. We consider the casewhere an arbitrary shaped object is grasped by an arbitrary number of ﬁngers with friction.
This paper is organized as follows. At ﬁrst, we setthe problem to solve. Then, we show an algorithm
to solve the problem. Lastly, numerical examples arepresented to show the eﬀectiveness of our approach.

2

Problem Definition

In this section, we give a problem for optimal grasp.
At ﬁrst, we describe the target system, statics between
a grasped object and a robotic hand, and the frictional
constraints. Then, we deﬁne required external force
set. Lastly, we give the problem for optimal grasp.

2.1

Target System

In this paper, we consider the case where an arbitrary shaped rigid object is grasped by m ﬁngers of arobotic hand. We make the following assumptions: 1)
Each ﬁnger makes a frictional point contact with theobject at the ﬁngertip, 2) The unique normal direction
at each contact point can be obtained.

2.2

Constraints for Problem

At ﬁrst, we describe the relation between the contact forces and the external force and moment applied
to the object. We denote the set of candidate contact
points by C and denote the combination of m selectedcontact points from C by
N ={pCN i (i = 1, 2, · · · , m)|pCN i ∈ C}

(2)

where GTN i = (I −[(pCN i −po )×] ) ∈ Rd×D in 3 dimensional space. Here, I represents an identify matrix,[a×] represents a skew symmetric matrix equivalent
to the cross product operation([ a × ]b = a × b), po
represents the origin of the frame ﬁxed at the object.
Next, the frictional constraint at the ith(i =1, 2, · · · , m) contact point is represented by
FfN i = {f N i | t2fN i ,1 +t2fN i ,2 ≤ µN i nfN i , nfN i ≥ 0}(3)
in 3 dimensional space, where nfN i denotes the normal
force component of f N i , tfN i ,1 and tfN i ,2 denote thetangential force components of f N i , and µN i denotes
the coeﬃcient of maximum static friction at the ith
contact point. In 2 dimensional space, we have only
to set tfN i ,2 = 0.

Required External Force Set

We deﬁne required external force set [10] [11] as follows.
Required External Force Set We call a set,
composed of all resultant forces and moments, which
we can apply to the object, possible-generated force set
W ⊂ RD . Let W − be {−w|w ∈ W}. Required external force set WR ⊂ RD is a set which W − must
contain.

2.4

Problem for Optimal Grasp

Based on the above discussion, we deﬁne a problem
to obtain an optimal grasp. In general, there exist
an inﬁnite number of combinations of contact points,
whose W − contains the given WR . We think it issuitable to minimize the magnitudes of the contact
forces needed to resist the external force and moment.
Now, let φ be the largest norm of contact force among
all norms of all contact forces.
φ = max |f N i |i

We consider the following problem.
Problem for Optimal Grasp Find the combination of contact points N ∗ such that(1)

where pCN i ∈ Rd (d= 2/3 in 2/3 dimensional space)
represents the position of the ith contact point (i =1, 2, · · · , m). Let f N i ∈ Rd be the contact force applied to the object at the ith contact point by theith ﬁnger and let w∈ RD (D= 3/6 in 2/3 dimensional
space) be the external force and moment applied tothe object(the origin of a frame ﬁxed at the object).
Then, the object can be in equilibrium if the following
equation holds.
Σm
i=1 GN i f N i = −w

2.3

ρ = min max

min

N ∈S w∈WR f
N i satisfy (2),(3)φ(4)

where S denotes the set of the candidate N . This
problem is to search N which minimizes φ required tocounteract w (∈ WR ).

3

Algorithm

In this section, we describe an algorithm to solve theproblem (4). In order to solve the problem, we use abranch-and-bound method [12] [13] by representing thecandidate contact points as discrete points. Note that
we can deal with any object whose geometry is arbitrary complex, by using the discrete candidate contact
points. The branch-and-bound method ﬁnds the optimal solution by enumerating the solutions of feasible
subproblems into which we can partition the original
problem. In the process, we cut the subproblems that
we don’t have to solve, by using a relaxed problem
whose constraints relax the constraints of the subproblem. It makes the computational time reduced.
In order to use the branch-and-bound method, we
make the 2 following assumptions: 1) the number ofthe candidate contact points on the object is ﬁniten, 2) WR can be expressed by a convex polyhedroncomposed of l vertexes.
From this assumption 1, the number of thecandidate combinations of contact points becomesC(n, m)(C(n, m) means the number of the combinations where we select m from n). In this assumption

2, if WR is not a convex polyhedron, we will deﬁne new
WR which is a convex polyhedral set which contains
the original WR . Then, WR can be represented byWR{w = Σlj=1 λj w vj , Σlj=1 λj = 1,
λj ≥ 0(j = 1, 2, · · ·, l)}=(5)where wvj denotes the jth vertex of WR .
Now, we deﬁne some subproblems and a relaxed
problem of the problem (4) at ﬁrst, and then we describe the procedure of the algorithm.

3.1

Subproblems and Relaxed Problem

At ﬁrst, we consider the case where we select someNk from the C(n, m) candidate combinations of contact points. Then, we obtain the following subproblem.
Subproblem 1
min

max

w∈WR fNk i satisfy (2),(3)φ(6)

If we solve this Subproblem1 for each Nk (k =1,2,· · ·,C(n, m)), we can obtain the solution of the original problem (4).
In order to solve Subproblem1, we consider the casewhere we select not only some Nk but also some vertex
wvj of WR . Then, we obtain the following subproblem
of Subproblem1.
Subproblem 2
minsubject toφf TNk i f Nk i ≤ φ (i = 1, 2, · · ·, m)Σm
i=1 GNk i f Nk i = −w vj(7)f Nk i ∈ FfNk i (i = 1, 2, · · · , m)Now, we consider the case where we can obtain each
optimal solution of Subproblem2 for each wvj (j =1, 2, · · · , l) at Nk . Let φNk ,vj be the solution of theSubproblem2 for wvj at Nk . Let f ∗Nk ,j = (f ∗Nk 1,j )T · · ·T(f ∗Nk m,j )T be the contact forces which give φNk ,vj .
Then, the convex polyhedron of the contact forces,
whose vertex is f ∗Nk ,j , can be represented by
FNk = {f = Σlj=1 λj f ∗Nk ,j , Σlj=1 λj = 1,
λj ≥ 0(j = 1, 2, · · ·, l)}(8)

Since the constraints of Subproblem2 is convex, we
can resist any w contained in WR , by using somecontact forces contained in FNk . Then, the largest
φNk ,vj among φNk ,vj (j = 1, 2, · · · , l) is the optimal solution of Subproblem1 (for Nk ). Namely, we can solve
Subproblem1 by solving Subproblem2 for each w vj .
Based on the formulation by Buss et al. [14], the inequality constraints of Subproblem2 can be rewrittenby the following constrains with respect to the symmetric matrices F Nk i and P Nk iφI
f Nk iO
F Nk i =f TNk iφ0
tfNk i ,1µNk i nfNk i
0µNk i nfNk i tfNk i ,2   O
P Nk i = 
tfNk i ,1
tfNk i ,2 µNk i nfNk i
where F Nk i  O means F Nk i is a positive semideﬁnite
matrix. Then, we can solve Subproblem2 by using
positive semideﬁnite program [15] [16].
Next, we deﬁne Relaxed Problem whose constraints
are linear and relax the constraints of Subproblem2.
Relaxed Problem is a problem to eﬀectively ﬁnd Nk
for which we don’t have to solve Subproblem1. We
can use Subproblem2 to ﬁnd the combinations. But,
since simplex method requires less computational timethan positive semideﬁnite program, we use Relaxed
Problem. Note that the optimal solution of Relaxed
Problem gives a lower bound of the optimal solutionof the corresponding Subproblem2, namely of the corresponding Subproblem1, since the constraints of Relaxed Problem contain the constraints of the corresponding Subproblem2. It makes us know Nk forwhich we don’t have to solve Subproblem1(See next
subsection).
Among the constraints of Subproblem2, we approximate f TNk i f Nk i ≤ φ by a convex polyhedron circumscribed in the set(Fig.1(a)). We approximate thefriction corn(3) by a L-side convex polyhedral corn circumscribed in the friction corn (Fig.1(b)) [17]. Then,
we can deﬁne Relaxed Problem as follows.
Relaxed Problem
minsubject toφeTκ V Nk i uNk i ≤ φ(i = 1, 2, · · · , m)(κ = 1, 2, · · · , d)−eTκ V Nk i uNk i ≤ φ(i = 1, 2, · · · , m)(κ = 1, 2, · · · , d)Σmi=1 GNk i V Nk i uNk i = −w vj(9)where V Nk i ∈ Rd×L denotes the matrix whose column
is a unit edge vector of the frictional convex polyhedral
cone v Nk ij (V Nk i =(v Nk i1 · · · v Nk im )), uNk i (≥ o)∈ RL
denotes the vector whose jth element represents themagnitude of the contact force in v Nk ij direction, andeκ ∈ Rd denotes the vector whose κth element is 1 andwhose other elements are 0(for example, e1 =(1 0 0)T
in 3 dimensional space). Note that we can solve Relaxed Problem by simplex method.

3.2

Procedure of the Algorithm

In this subsection, we describe the algorithm tosearch the optimal Nk . Now, let ρ̂ be the temporary optimal solution, ρNk be the optimal solution of

7

10

5
3

5

1
0

-1
-3

-5

-5


(a)

-7
-7

f TNk i f Nk i ≤ φ(b) Friction Corn

-10

-5

-3

-1

1

3

(a) Case I

5

7

-10

0

10

(b) Case II

Figure 1: Convex Polyhedron and Convex Polyhedral
Corn Circumscribing the Constraints of Subproblem2
Subproblem1 for Nk , and ρ̂Nk be its temporary optimal solution. We represent a list of feasible Nk as
LIST.
step 1 We put all candidate Nk into LIST. Let ρ̂ be
an appropriate lower bound value. Let each ρNk andeach ρ̂Nk for each Nk be appropriate upper and lower
bound values respectively.
step 2 We solve Subproblem1 for some Nk containedin LIST.
step 3 If we can get the solution of the Subproblem1
in step2, let ρ̂ (=ρNk = ρ̂Nk ) be the solution and ŵ vj
be w vj which gives the solution. Otherwise, we eliminate the Nk from LIST and go back to step2.
step 4 We solve Relaxed Problem for ŵ vj , at each
Nk contained in LIST. If we cannot get the solution atsome Nk , we eliminate this Nk from LIST. If we canget the solution ρ̂Nk ,v̂j at some Nk , we compute ρ̂Nk
= max{ρ̂Nk , ρ̂Nk ,v̂j }. If ρ̂ <ρ̂Nk , we eliminate this Nkfrom LIST.
step 5 Let Nˆk be Nk at which ρ̂Nk ,v̂j is the least among
those at all Nk contained in LIST.
step 6 We solve Subproblem1 for Nˆk . If we can get
the solution, let ρNk (= ρ̂Nk ) be the solution and ŵ vj
be w vj which gives the solution. If ρ̂ >ρN̂ , ρ̂ =ρN̂ .
kk
If we cannot get the solution or ρ̂ <ρN̂ , we eliminatekthis Nk from LIST and go back to step5.
step 7 If we can get the relation |ρ̂ − ρNk | <  (denotes an arbitrary small positive value) for all Nk
contained in LIST, we ﬁnish the loop. Otherwise, wego back to step4.

4

Numerical Examples

In order to show the eﬀectiveness of our approach,
we show some numerical examples in this section. We
show the target objects and the candidate contact
points in Fig.2. The objects in Case I and II are in
2 dimensional space. The object in Case III is a triangular prism√ whose base is a right isosceles triangle(
4 × 4 × 4 2). In this ﬁgure, the points on each object indicate the candidate contact points. Note that(c) Case III
Figure 2: Target Objects and Candidate Contact
Points(a) Case I(b) Case II

Figure 3: Required External Force Sets in 2 dimensional space
we show only the candidate contact points on the top
face and on one side in Case III for easy to see. We
give the candidate contact points on the bottom faceand on the other side in the same way as those on the
top face and on the side respectively. The number ofthe candidate contact points is 56 in Case I, 86 in Case
II, and 217 in Case III (36 for top and bottom face,45, 45, and 55 for 3 sides). Let ΣOi (i =I,II,III) be theobject coordinate frame ﬁxed at each geometric center of the object in Case i. We use a frictional 16-side
convex polyhedral corn in Relaxed Problem. We set
WR as follows:{(f T , mT )T ||f | ≤ sg, −0.8γsg ≤ m1 ≤ 0.8γsg} (in Case I and II)
WR = {(f T , mT )T | − sg ≤ fi ≤ sg,
−0.8γsg ≤ mi ≤ 0.8γsg
(i = 1, 2, 3)}(in Case III)
where f and m denote the external force and moment
respectively, fi and mi denote the ith component of f
and m respectively, s denotes the area or volume ofeach object, and g=0.01 denotes the speciﬁc gravity of

7

7

5

5

3

3

1

1

-1

-1

-3

-3

-5

-5

-7
-7

-5

-3

-1

1

3

5

7

-7
-7

(a) 2 Fingers

Table 1: ρ in the Problem (4)(a) Case I
µ
0.1
0.3
0.5
2 ﬁngers 3.71 1.38 0.919
3 ﬁngers 3.13 0.978 0.724
-5

-3

-1

1

3

5

7

(b) 3 Fingersµ
2 ﬁngers
3 ﬁngers

Figure 4: Optimal Grasp Points in Case I
10

10

5

5

0

0

-5

-5

(b) Case II
0.1
0.3
35.1 6.44
6.21 2.77

0.5
3.92
2.01

µ=0.1
µ=0.3

-10

µ=0.5

-10
-10

0

(a) 2 Fingers

10

-10

0

10

(b) 3 Fingers

Figure 5: Optimal Grasp Points in Case II

Figure 6: Optimal Grasp Points in Case III

each object. γ in Case i (i =I,II) denotes the distance
between ΣOi and the vertex which is closer to ΣOithan any other vertexes. γ in Case III denotes the
distance between ΣOIII and the face which is closer
to ΣOIII than any other faces. Here we suppose thecase where gravitational force (external force) can beapplied to each object in arbitrary direction and where
the external moment can be applied to each object
resulting from that the position of the center of gravity
can be located in the circle (Case I and II) or the sphere(Case III), whose center is the origin of ΣOi and whose
radius is 0.8γ. WR in Case I and II are convex sets
but not convex polyhedrons. Then, we approximate
the sets by 10-side convex polyhedrons circumscribedin the sets (Fig.3).
In Case I and II, we compute in the cases where
the number of the ﬁngers is 2 and 3 and where thefrictional coeﬃcient is 0.1, 0.3, and 0.5. In Case
III, we set the number of the ﬁngers is 3 and thefrictional coeﬃcient is 0.3. Note that if we obtain some optimal combinations of contact points
excepting symmetric arrangements, we search the
optimal combination which minimizes the criterion

T
min
Σm
i=1 f N i f N i among thesesatisfy (2),(3)obtained combinations.
Results about the optimal combination of contact
points are shown in Fig.4, Fig.5, and Fig.6. Results
about ρ in the problem (4) in Case I and II are shownin Table.1. ρ in Case III was 0.669. In Case III,the obtained optimal grasp points were (2/9,−4/3,0)T ,(2/3,2/3,0)T , and (−4/3, 2/9, 0)T (the origin (X) is theorigin of ΣOIII ). In Fig.4 and Fig.5, (a) and (b) show
the results in the cases where the object is grasped by
2 and 3 ﬁngers respectively, and the arrows show theobtained optimal grasp points. Note that we show only
one optimal combination of contact points where thearrangements of the obtained 2 optimal combinations
are symmetric. Note also that we show only one optimal combination of contact points in Fig.4 and Fig.5,
excepting the case shown in Fig.5(b), since the optimal
grasp points didn’t depend on the magnitude of thefrictional coeﬃcient. In Fig.6, the arrows, whose tip isblack sphere, show the obtained optimal grasp points,
X shows the origin of the reference frame. The graytriangle shows the projection drawing of the triangle,
made by the 3 optimal grasp points, on the bottom
face of the object.
From Fig.6, we can see that the each optimal grasp
point is located at a position close to each foot of each
perpendicular from ΣOIII to each side. For someone’s
information, we compute ρ in the case where we grasp
the object at these feet. ρ was 0.718 (which is bigger
than 0.669). From Table.1, we can see that the necessary magnitudes of the contact forces to resist therequired external force depends both on the magnitude of the frictional coeﬃcient and on the number ofmaxw∈WR fNithe ﬁngers in Case I and II.
The above algorithm was implemented in C/C++
and the above calculations were done on a PC with
1.2GHZ ATHLON. In 2 dimensional space (the ﬁctional coeﬃcient is 0.3), the running times for Case I
and II, respectively, were about 5 and 7 seconds in thecase where we grasp by 2 ﬁngers, and about 8 and 3
minutes in the case where we grasp by 3 ﬁngers. In
3 dimensional space, the running time was about 25
minutes. For someone’s information, we computed in
the case where we use Subproblem2 in place of Relaxed
Problem to eﬀectively ﬁnd Nk for which we don’t haveto solve Subproblem1 (the ﬁctional coeﬃcient is 0.3).
In this case, the running times in the cases where we
grasp by 2 and 3 ﬁngers, respectively, were about 24
seconds and 1.2 hours in Case I, and about 41 seconds
and 8.7 hours in Case II. The running time was about
4.35 hours in Case III. We think this shows the eﬀectiveness of using Relaxed Problem. The numbers of
Nk at which we solved Subproblem2 for all vertexes of
WR in the cases where we grasp by 2 and 3 ﬁngers,
respectively, were 6 and 278 in Case I (the ﬁctionalcoeﬃcient is 0.3), and 9 and 53 in Case II (the ﬁctional coeﬃcient is 0.3). In Case III, the number was
21. Note that the other Nk were eliminated. We think
this shows the eﬀectiveness of the branch-and-boundmethod.
In our approach, the number of the candidate combinations of contact points is C(n, m) and then thenumber become large when the number of the ﬁngers
or the candidate contact points is large. If the running time is too large to apply the above algorithm, we
can get approximate optimal grasp points in a smaller
running time by the following way; at ﬁrst we reducethe candidate contact points and search optimal grasp
points, and then we search optimal grasp points fromoriginal candidate contact points which is close to theobtained optimal grasp points.

5

Conclusion

In this paper, we have searched optimal grasp points
on a grasped object by using the concept of required
external force set. By using the required external force
set, we can deal with general desired grasps including
force closure and equilibrium grasp, and can evaluate
the magnitudes of the resistible external forces andmoments. We have deﬁned an optimal grasp that minimizes the magnitude of the contact force required toresist any external force and moment contained in thegiven required external force set. We have searchedthe optimal grasp points by using a branch-and-boundmethod. We have also presented some numerical examples in order to show the validity of our approach.

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