Mechanics of Hybrid Active/Passive-Closure Grasps
Tetsuyou Watanabe and Zhongwei Jiang

Tsuneo Yoshikawa

Department of Mechanical Engineering
Yamaguchi University
Ube, 755-8611, Japante-watanabe@ieee.org & jiang@yamaguchi-u.ac.jp

Department of Mechanical Engineering
Graduate school of Engineering
Kyoto University, Kyoto, 606-8501, Japan
yoshi@mech.kyoto-u.ac.jp

Abstract - In this paper, we discuss the directions of active andpassive force closures in hybrid active/passive-closure grasps. We
show the directions are orthogonal to each other. We also discuss
the magnitudes of the internal forces in the manipulation of theobject. In hybrid active/passive-closure grasps, there exist two
kinds of magnitudes of internal forces. One is the magnitudeof internal forces, which changes if the object moves and thegeometry of the fingers changes. The other is the one which
don’t change even when the object moves. We derive these two
magnitudes.
Index Terms - Active/Passive Force Closure, Grasping, Internal
ForcesI. I NTRODUCTION
When we grasp an object by a robotic hand, force closure is
one of the important properties of grasping [1]. Force closureis the concept which we can interpret in the following twoways; ” any arbitrary force and moment can be exerted on a
grasped object. ” or ” the motion of a grasped object can becompletely constrained without changing the pre-loaded joint
torques, whatever external force and moment are applied tothe object. ” Yoshikawa [2] called the former concept active
force closure, and the latter concept passive force closure. The
former concept corresponds to that the fingers (limbs) canmove the grasped object in arbitrary directions. Note that in
the definition of active force closure [2], we can regard active
force closure as included in passive force closure (active force
closure is necessary but not sufficient for passive force closure[3]). But active force closure in itself corresponds to the motionof the grasped object. Then, in this paper, we put attention
on this property of active force closure. On the other hand,passive force closure doesn’t correspond to the motion of thegrasped object. Consider the case where there exist multiple
contact points between a finger and an object (for example,
enveloping grasp). In this case, some contact forces between
the finger and the object can be generated not actively by thejoint torques but passively by the mechanism of the geometricconstraints. Then, 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 the motion of the object in the same direction.
There are some cases where certain directions of the grasped
object correspond to active force closure and the other directions correspond to passive force closure. Yoshikawa [2] called
such closures hybrid active/passive closure. In our previous
paper [4], we show hybrid active/passive closure is optimal ina certain planning of grasping to manipulate a grasped object.
But the relation between the directions of active force closureand those of passive force closure is still not clear.
In this paper, we show the directions of active force closure
are orthogonal to those of passive force closure in hybrid
active/passive-closure grasps.
In hybrid active/passive-closure grasps, not only certain
external forces and moments can be counteracted withoutchanging the pre-loaded joint torques, but also the object
can move in certain directions. When the object moves, theconfigurations of the fingers and the objects change. Then,the constant pre-loaded joint torques may or may not needto change for resisting external forces in the directions ofpassive force closure. This is related to the change of theinternal forces. The internal forces are to satisfy the frictional
constraints and to grasp the object stably. In the manipulationof the object, we have to control this internal force forthe stable grasp. However, since hybrid active/passive-closure
grasps have properties of passive force closure, there can exist
some internal forces, which don’t change even if the object
moves and the configurations of the fingers change. Then, we
also discuss this problem.
This paper is organized as follows. At first, the target system
is shown and the directions of active and passive force closures
are defined. Then, we show the directions of active force
closure are orthogonal to those of passive force closure. We
also discuss the internal forces in the manipulation of theobject when the grasp is of hybrid active/passive closure.
A. Related Works
The simplest case of active force closure is fingertip grasp.
As for the manipulation of an object grasped by fingertips, Liet al. [5], Cole et al. [6], and Yokokohji et al. [7] presented
control algorithms for the case of point contact, rolling contact,and soft-finger contact respectively. Cole et al. [8] extended thealgorithm to the case of sliding contact. But the algorithm waslimited in the 2 dimensional case. Zheng et al. [9] presented
a control algorithm for the case of sliding contact in the 3
dimensional case. As for the manipulation of multiple objects
grasped by fingertips or end links of fingers, Harada et al. [10]presented a control algorithm for the case of rolling contact.
Harada et al. [11] also discuss active force closure for the caseof grasping multiple objects.
On the other hand, passive force closure corresponds topower grasps. Power grasp is a grasp that can hold objects
stably without changing the pre-loaded joint torques of thefingers. The researches about power grasps have been done
from the following viewpoint [12]–[18] : the condition for
power grasps, the formulation of the contact force distribution,

Space of Passive Force Closure (SPFC); We call the direction, in which external force or moment can be counteracted
without changing the joint torques, the direction of passive
force closure (DPFC). We call the space, spanned by a set ofthe all DPFC’s, the space of active force closure (SPFC).

Fig. 1.

Target System

the optimization of the pre-loaded joint torques, and so on.
It is hard to manipulate an object under hybrid active/passive
closure. Trinkle et al. [19] discussed a grasp planning for
manipulating an enveloped object with sliding contacts in 2 dimensional space. Bicchi et al. [20] analyzed the manipulabilityof the general grasping systems including enveloping grasp.
Harada et al. [21] called the working style, where multiple
contacts are allowed between chains (fingers or legs) and an
object (or an environment), Envelope Family and presented asufficient condition for the manipulation of Envelope Family.
But, these researches didn’t put attention on resisting external
forces, which is another property of hybrid active/passive
closure. Passive force closure in hybrid active/passive-closure
grasps is discussed in [3], [22], [23]. But, these researches
didn’t discuss the manipulation of a grasped object. In thispaper, we discuss not only active force closure (motion of agrasped object) but also passive force closure in the case where
the grasped object is in hybrid active/passive closure.
II. T ARGET S YSTEM AND D EFINITION
A. 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 we show
the case where N = 2 in Fig.1. We make the following
assumptions: 1) Each finger makes a frictional point contact
with the object, and the contacts are neither rolling nor sliding.
2) The unique normal direction at each contact point can beobtained. 3) There exists at most one contact point on each linkof the fingers. 4) When the object moves, both the numbersof contact points and contact positions on the object/fingers
don’t change (we don’t consider the manipulation in which a
certain contact point removes from the object or in which a
certain point on a certain finger, which isn’t a contact point,
makes a new contact with the object).
B. Definition
Space of Active Force Closure (SAFC); Suppose that thegrasped object is in force closure. Then, we call the direction,in which the grasped object can move by the corresponding
motion of the fingers, the direction of active force closure(DAFC). We call the space, spanned by a set of the all DAFC’s,
the space of active force closure (SAFC).

III. O RTHOGONALITY OF D IRECTIONS OF A CTIVE AND
P ASSIVE F ORCE C LOSURESIn this section, we describe the orthogonality of SAFC and
SPFC. At first, we describe the kinematics and statics of thesystem. Then, we show the orthogonality.
A. Kinematics
Let q i ∈ RMi (i = 1, 2, · · · , N ) be the joint angles of theith finger of the robotic hand. Let r∈ RD be the position of
the origin and the orientation of the object coordinate frame
fixed at the object. Let pCij ∈ Rd (j = 1, 2, · · · , Li ) be theposition of the jth contact point between the object and theith finger. Here Mi denotes the number of the joints of the ith
finger, D= 3, d= 2 in 2 dimensional space and D= 6, d= 3 in
3 dimensional space, and Li denotes the number of the contact
points on the ith finger. The relation between the displacementsof pCij and q i , and the one between the displacements of pCij
and r, respectively, are given as follows;∆pCij = J ij ∆q i ,∆pCij = GTij ∆r(1)where J ij ∈ Rd×Mi denotes the Jacobian matrix and Gij
denotesI
Gij =[(pCij − po )×]in 3 dimensional space. Here, I represents an identify matrix,
po represents the position of the origin of the object coordinate
frame, [a×] represents a skew symmetric matrix equivalent tothe cross product operation ([ a × ]b = a × b).
N
Let M = ΣN
i=1 Mi and L = Σi=1 Li . By using the following
vectors and matrices,T∆q = ∆q T1 ∆q T2 · · · ∆q TN∈ RM ,





J 11
J 21
J N1





J = diag  ...   ...  · · ·  ...  ∈ RLd×M ,
G=
A=




J 1L1
G11
J

G12

J 2L2



J N LN· · · GN LN ∈ RD×Ld ,−GT ∈ RLd×(M+D)

where diag denotes a block diagonal matrix, we get thefollowing expression from (1);
T

= o.
(2)
A ∆q T ∆r T
B. Statics
Let f ∈ RLd be the contact force vector which combines
the contact forces at all contact points, τ ∈ RM be the joint
torques equivalent to f , and w∈ RD be the resultant force
and moment applied to the object at the object coordinateframe. From (2) and the principle of virtual work, we get thefollowing relation;


τT

−wT

T

= AT f =



J

−GT

T

f.

(3)

C. Orthogonality
Now, we consider the case where the object is grasped and isin stationary state with non-zero contact force (internal force)at every contact point, which satisfies the frictional constraint.
In addition, we assume that G has full row rank. This means
the grasp is of force closure. Namely, we consider SAFC andSPFC in force-closure grasps. Let τ pre (= 0) be the pre-loaded
joint torques corresponding to the internal forces. Then, from(3), we getτ TpreoT

T= AT f .

(4)1) SAFC: In order to move the grasped object, the corresponding motion of the fingers is needed. The motions of
the object and the fingers are constrained by the kinematicconstraints (2). Then, the allowable motions of the object andthe fingers are expressed by
 T 


EP 1
∆q
(5)
∆k1
= E TP ∆k1 =∆r
E TP 2where E P ∈ Ra×(M+D) is an orthogonal matrix whose rows
form bases of the null space of A, ∆k1 ∈ Ra is an arbitrary
vector expressing the magnitude of the each column of E TP ,a is the dimension of the null space of A, and E P 1 and E P 2
are, respectivelly, M × a and D × a block matricies.
Since G has full row rank, there don’t exist the cases where
in a certain column of E TP , the corresponding component
vector of E TP 1 is o and the corresponding component vectorof E TP 2 isn’t o. If in a certain column of E TP , both thecorresponding component vectors of E TP 1 and E TP 2 aren’t 0,
the motion of the object, in the direction corresponding to thevector of E TP 2 , can be achieved by the motion of the finger
in the direction corresponding to the vector of E TP 1 . If in acertain column of E TP , the corresponding component vector of
E TP 1 isn’t 0 and the corresponding component vector of E TP 2
is 0, the motion of the finger, in the direction corresponding
to the vector of E TP 1 , makes no influence on the motion of theobject. Then, E TP 2 expresses SAFC.
2) SPFC: For the sake of simplicity, we consider the casewhere all fingers are not in a singular configuration and there
is no kinematical redundancy with respect to the task in every
finger. This means J has full column rank. Now, we consider
the contact forces which can generate without changing the
pre-loaded joint torques. From (4), the contact forces can be
expressed by
f = (J T )+ τ pre + (I − (J T )+ J T )k̃2 ,(6)where (J T )+ denotes the pseudo-inverse matrix of J T andk̃2 denotes an arbitrary vector. Then, by substituting (6) into

(3), we get



  
O
I
τ
+
τ
k̃2
=
−w
−G(I − (J T )+ J T )−G(J T )+ pre
 


I
O
=
τ
+
k2 (from (4)),
(7)
O pre
−Ξ
where Ξ ∈ RD×p is a full column rank matrix whose columns
form bases of the G(I − (J T )+ J T ), p is the rank of theG(I − (J T )+ J T ), and k2 ∈ Rp is an arbitrary vector
expressing the magnitude of the each column of Ξ. Note thatτ pre corresponds to the internal forces and then G(J T )+ τ pre= o.
DPFC is the direction where an external force can becounteracted without influence on the joint torques. Then, Ξin the second term of (7) expresses SPFC.
From the definition of E P , AE TP = O. Then, from (3), (6),and (7), we can get
E P AT f
T +E P A (I − (J ) J )k̃2


O
EP
k2
−Ξ
T

T

= o,
= o,
= o.

k2 is an arbitrary vector in the last equation. Then, we can get


O
EP
= O,
−Ξ
E P 2 Ξ = O.
(8)(8) represents the orthogonality of SAFC (DAFC) and SPFC
(DPFC) because E TP 2 represents SAFC and Ξ represents
SPFC.
 In Tthe above
Tdiscussion, the generalized velocity vectorT∆q∆r
corresponds to the generalized force vector
 T
T
τ
−wT
. Then, equations (3) and (5) can correspondto the artificial constraints used in the context of hybrid
force/position control [24]. Since the second term of (7) is
included in the artificial constraint, we can regard the above
deviation as based on the orthogonality of the artificial constraint, though the purpose and the result of the derivation are
different from those of the hybrid force/position control (thepurpose is to divide into the directions of position-control andthe directions in which any external force can be counteracted
without changing the joint torques (not the directions of forcecontrol).).
D. Examples
Consider the cases shown in Fig.2. In Fig.2 (a), A is givenby
1 0 −1 00
 0 0 0 −1 1 
.
A=
(9)
 0 1 −1 00 
0 0 0 −1 −1(a)where the position of the origin of the object coordinate frame
is located at the first joint (the proximal end of the first link)of the left finger. In this case, E p and Ξ are, respectively,
expressed by
0.5125 0 0.64 −0.255 0 0 0.5125 ,
Ep =

T
1 0 0
.
(14)
Ξ =
0 1 0

(b)

From (14), we can see that SAFC (DAFC) corresponds to therotational direction of the object coordinate frame and thatSPFC corresponds to the translational directions of the object
coordinate frame. From the result that the second elements of
E TP in (14) is zero, we can also see that the object cannot
move with the change of the angle of the second joint of theleft finger (note that the left finger is the first finger).

(c)Fig. 2.

Examples of Hybrid Active/Passive Closure

Then, E p and Ξ are, respectively, expressed by−0.5774 0.5774 −0.5774 0
Ep =

T
0 1 0
Ξ =
.
0 0 1

0



,
(10)

From (10), we can see that SAFC (DAFC) corresponds to thehorizontal direction of this paper (x direction of the object
coordinate frame) and that SPFC corresponds to y and therotational directions of the object coordinate frame.
Next, consider the case shown in Fig.2 (b). The object ismanipulated, contacting with the environment. Let the lengthof each link of the fingers be 1. Then, A is given by
0
0 −1 000
0 −1 0 
.
√0
A=
(11)
 − 3/2 0 −1 0
0 
−1.5
−1 0 −1 −2
where the position of the origin of the object coordinate frame
is located at the contact point between the left finger and theobject. In this case, E p and Ξ are, respectively, expressed by
0 −0.8944 0 0 0.4472 ,
Ep =

T
1 0 0
.
(12)
Ξ =
0 1 0
From (12), we can see that SAFC (DAFC) corresponds to therotational direction of the object coordinate frame and thatSPFC corresponds to the translational directions of the object
coordinate frame.
Finally, consider the case shown in Fig.2 (c). The object ismanipulated, contacting with the left finger at two points. In
this case, A is given by
−0.353600
0−1 0 0.3536
 −0.3536
0000 −1 0.3536 
 −1.7678 −0.7071 00−1
0 1.7678 

.
A=
0
0 −1 0.3536  −0.3536 0.7071 000−1.5 −0.75 −1 0
1.5 
0
0
0 −1.299 0 −1 −0.6464(13)

IV. I NTERNAL F ORCES IN THE M ANIPULATIONIn this section, we discuss the internal forces in the manipulation when the grasp is of hybrid active/passive closure.
For the sake of simplicity, suppose that the normal unit
vector at every contact point is constant with respect to theobject coordinate frame. Since the contact positions are all
constant with respect to the object coordinate frame for theassumption given in section 2, the directions of the internal
forces are also constant with respect to the object coordinate
frame even if the object moves. Hence, whether the frictional
constraints can be satisfied depends on only the magnitudes of
the internal forces. In order to satisfy the frictional constraints
in the manipulation of the object, we take the following simple
way: at first find the appropriate constant magnitudes of theinternal forces for the grasping and the manipulation of theobject, and then assign the obtained constant appropriate magnitudes to the magnitudes of the internal forces and keep themagnitudes of the internal forces constant. Since there exists
SPFC in hybrid active/passive closure-grasps, it is possiblethat there exist the magnitudes of the internal forces, which
don’t change despite the motion of the object. We don’t haveto change the joint torques corresponding to the magnitudes,
while we have to change the joint torques corresponding to
the magnitudes of the internal forces, which change due tothe motion of the object. Namely, we have only to control themagnitudes of the internal forces, which change due to the
motion of the object. In the following, we derive the relation
between the joint torques and the magnitudes of the internal
forces, which do or don’t change due to the motion of theobject.
If A doesn’t have any null space, the object cannot move dueto the geometric constraints. Then, in hybrid active/passiveclosure grasps, A has null space. Here, we assume that A has
full row rank. Then, f , which satisfies (4), can be uniquely
obtained. Let such f be f int . f int is expressed by


τ pre
.
(15)
f int = (AT )+oSince f int is the internal forces, f int satisfies Gf int =o. Then, the following another expression of f int can beobtained.
∆

f int = (I − G+ G)k̃3 = Φk3(16)denotes an arbitrary vector, Φ ∈where k̃3 ∈ R
R(Ld)×(Ld−D) denotes an orthogonal matrix whose columns
form bases of the null space of G, and k3 ∈ RLd−D denotes
an arbitrary vector. Φ expresses the directions of the internal
forces and k3 expresses their magnitudes.
From (15) and (16), we get


τ pre
.
(17)
k3 = ΦT (AT )+o
Ld

If the object and the fingers infinitesimally move, (17) becomes


τ pre
∆k3 = (∆(ΦT (AT )+ ))
o


∆τ
pre
T
T +
+Φ (A )
.
(18)
o

T
Letting x = q T r T(∈ RM+D ) and B = ΦT (AT )+ (∈(LD−D)×(M+D)R), (18) becomes





∂B
τ pre
∆τ pre
∆k3 =
∆x
+B
.
(19)
o
o
∂x
∂B
where ∂B
∂x is a third-order tensor and ∂x ∆x is a second-ordertensor (matrix). By using the similar formulation of Chen and
Kao [25], the first term of (19) becomes



 

M+D 
∂B
∂B
τ pre
τ pre
∆x
∆xi ]
= [
o
o
∂x
∂xi
i=1


M+D 
∂B
τ pre
=
∆xi
o
∂xi
i=1

 



τ pre
τ pre
∂B
∂B
= [ ∂x1
∂x2
o
o



τ pre
∂B
···
]∆x
∂xM +D
o
∆

=

Ψ∆x.

Then, (19) becomes∆k3 = Ψ∆x + B∆τ preo(20)

hand, we don’t have to react the magnitudes of the internal
forces, contained in the null space of E P ΨT , namely{k3 |k3 = (I − ΨE TP (ΨE TP )+ )k5 , k5 ∈ RLd−D }where k5 denotes an arbitrary vector. We call such magnitudesof the internal forces reaction-not-needed magnitudes of theinternal forces.
In (22), letting ∆k3 = o and multiplying AT Φ from theleft, we can get∆τ pre(24)= −AT ΦΨE TP ∆k1 .
oThen, if the joint torques change from τ pre to τ pre + ∆τ pre(given by (24)) according to the change of the geometry, we
can manipulate the object with the constant internal forces
which satisfy the frictional constraints.
A. Examples
We consider the same cases as the previous examples. In
the following examples, let the each magnitude of the internal
forces be 1 (Let the every element of k3 be 1). We computeτ pre form the magnitude of the internal forces.
At first, consider the case shown in Fig.2 (a). In this case, thedirection of the internal force Φ and Ψ in (20), respectively,
are
Φ =
Ψ =



.

(21)

We consider hybrid active/passive closure here. Then, arbitrary∆x cannot be obtained. The allowable ∆x is expressed by (5).
Then, (21) becomes∆τ pre∆k3 = ΨE TP ∆k1 + B.
(22)o
In order to keep the magnitudes of the internal forces
constant in the manipulation, we have only to add ∆τ pre ,given by (22) with ∆k3 = o, to the joint torques. Then,ΨE TP ∆k1 is the magnitudes of the internal forces which wehave to react. We call such magnitudes of the internal forces
reaction-needed magnitudes of the internal forces. On the other

0

√1
2

0

0

− √12

0 0 0 .
0

T

,

(25)
(26)

It is clear that the magnitude of the internal force is reactionnot-needed magnitudes of the internal forces. Namely, even if
the object moves in the direction of active closure (DAFC),
we don’t have to change the pre-loaded joint torques in orderto keep the magnitudes of the internal forces constant and tosatisfy the frictional constraints.
Next, consider the case shown in Fig.2 (b). In this case, thedirection of the internal force Φ and Ψ in (20), respectively,
are
Φ =
Ψ =



0

√1
2

0 − √121.3856 0.8083 0

Then, from (12), ΨE TP



(23)

T

,

0

−1.3856

(27). (28)isΨE TP = −1.3426.

(29)

It is clear that the magnitude of the internal force is reactionneeded magnitudes of the internal forces. Namely, when theobject moves in the direction of active closure (DAFC), we
have to change the pre-loaded joint torques in order to keep
the magnitudes of the internal forces constant and to satisfy
the frictional constraints.
Finally, consider the case shown in Fig.2 (c). In order to beeasy to differentiate Φ with respect to x, we use the following
another expression of Φ given by [26];o e13 e12(30)Φ =  e23 o e21 
e32 e31 owhere eij is the vector directing from the contact point Ci tothe contact point Cj . Then, Φ can be expressed byT
00
1 −0.2678 −1 0.2678
Φ =  1 1.1464 00−1 −1.1464  .
0 1.4142 0 −1.4142 0
0
In this case, we use Φ+ in place of ΦT in (17) ∼ (24). Then,Ψ in (20) is expressed by
1.2249 0.2129 −0.9741 −1.4809 0 0 −0.2509
Ψ =−0.4992 −0.0867 0.7782 0.6256 0 0 −0.2790 .
−0.6811 0.2801 0.8118 1.2092 0 0 −0.1307
Then, from (14), ΨE TP isTΨE TP = 0.2535 −0.0603 −0.2049.

(31)This is reaction-needed magnitudes of the internal forces. In
this case, there also exist reaction-not-needed magnitudes of
the internal forces. These magnitudes are expressed by
0.18200.6181{k3 |k3 =  0.9812 −0.0637  k5 , k5 ∈ R2 } (32)−0.0637 0.7835


where k5 denotes an arbitrary vector. From this result, we
can see that in this case there exist both reaction-needed
magnitudes of the internal forces and reaction-not-needed
magnitudes of the internal forces.
V. C ONCLUSION
In hybrid active/passive-closure grasps, there are two kindsof essential directions. One is the directions of active force
closure. The other is the directions of passive force closure. In
this paper, we showed the directions of active force closure
are orthogonal to those of passive force closure in hybrid
active/passive-closure grasps. When the object moves, thegeometries of the fingers and of the objects change. Then,
we have to control the internal forces for satisfying thefrictional constraints. However, hybrid active/passive-closure
grasps have property of passive force closure. Then, it ispossible that there exist the magnitudes of the internal forces,
which we don’t have to control even when the object moves.
We derived the magnitudes of the internal forces. If in theinitial state, the object is stably grasped with appropriate τ pre ,we can manipulate the object by using the commands of joint
velocities q̇ = E Tp1 (k1d − k1 ) (k1d is a desired value of k1 )given by (5) and the commands of joint torques τ pre +∆τ pregiven by (24). But the development of more sophisticatedcontrol algorithm including feedback or dynamics is our futurework.
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