Towards Whole Arm Manipulation
by Contact State Transition
Tetsuyou Watanabe

Kensuke Harada

Tsuneo Yoshikawa

Zhongwei Jiang

Dep. of Mech. Eng.
Yamaguchi University
Ube, 755-8611, Japante-watanabe@ieee.org

National Institute ofAdvanced Industrial
Science and Technology
Tsukuba, 305-8568, Japankensuke.harada@aist.go.jp

Dep. of Human andComputer Intelligence
Ritsumeikan University
Kusatsu, Shiga 525-8577, Japan
yoshikawa@ci.ritsumei.ac.jp

Dep. of Mech. Eng.
Yamaguchi University
Ube, 755-8611, Japanjiang@yamaguchi-u.ac.jp

Abstract— This paper discusses the whole arm manipulation
allowing the contact state transition. For manipulation of an
object under fully constrained, the contact state transition becomes necessary. In order to realize the object manipulation,
we first derive the feasible direction of the object manipulation
by analyzing the active/passive closure properties for every
combination of contact states. Second, we derive the set of joint
torque to move the object in the feasible direction. These analyses
also provide the joint torque to realize the manipulation at theplanned contact states. Effectiveness of the proposed method is
confirmed by some simulation results.

I. I NTRODUCTION
In the grasping and manipulation of an object, the twokind of important classes are introduced by Yoshikawa [1]to explicitly take into account the specific mechanism andcapability of applying forces of the robotic hands, and also
to remove the confusion in the definition of force closure[2]. One is active force closure where any arbitrary force andmoment can be applied to a grasped object by fingers. The
other is passive force closure where any arbitrary external
force and moment exerted on a grasped object can be balanced
without changing the pre-loaded joint torque and the motionof the object can be completely constrained. Under active
force closure (such as fingertip grasps), the object can be
manipulated in any directions, but the robustness is lost. Underpassive force closure (corresponding to power grasps [3]), theobject can not be manipulated, but can be robustly grasped.
Some grasp configurations have both the active and passive
closure properties. Such intermediate grasp configurations are
named the hybrid active/passive closure grasps [1].
Fig.1 shows a 2D example of the whole arm manipulation [4] we have in mind. In this figure, the robotic handfirst grasps an object firmly by utilizing the passive closure.
Let us consider manipulating the grasped object from thisfully constrained configuration. In order to manipulate an
object arbitrarily, the robotic hand tries to change the grasp
configuration to the active closure. However, to realize theactive closure, the contact state between the finger and theobject has to change since the object cannot move unlessthe passive closure is satisfied. The intermediate condition
between the active and the passive closure corresponds to thehybrid closure grasp, and we can assume several grasp styles

Passive Closure Hybrid Closure

A ctive Closure:Sliding Contact:R olling Contact

Fig. 1.

Example of grasp transition from passive to active closureof hybrid closure grasp depending on the state of every contact
point. During the object is grasped under the hybrid closure,the object may slip or roll on the surface of the finger. Also,
the finger may detatch from the object. We can see from thisexample that, to realize this style of manipulation, the studyof contact state transition under hybrid closure is necessary.
More concretely, we have the following three research topics;
1) the feasible motion of the grasped object under given grasp
configuration, 2) how to apply the joint torque to realize
the change of contact conditions, and 3) calculation of thefeasible path of contact state transition to finally realize thedesired grasp configuration such as active closure. Amongthese problems, this paper aims to solve the first and second
ones.
In this paper, in order to realize such manipulation, we firstclassify the contact state and analyze the active and passive
closure properties for every combination of contact states. In
the analysis, we derive the feasible direction of the object
manipulation for every combination of contact states. Second,
we derive the set of joint torque enabling the object to movein the feasible direction and to achieve the change of contact
conditions. The obtained results enable us to not only ”plan
the contact state transitions for desired object manipulation”
but also ”provide the joint torque to realize the manipulation
at the planned contact states”.

ΣR
pO

pC21
ΣC21

ΣO
Object

2nd finger

1st finger
Base

Fig. 2.

Also, we will consider the transition of contact state. We
define that the contact state changes among the following fourstates:1) F-point : the contact point with static friction,2) N-point : the contact point without friction,3) S-point : the contact point with kinetic friction,4) D-point : the point about to detach.
We note that the contact force is zero at D-points.

Target System (N = 2)

III. BASIC F ORMULATION OF THE S YSTEM
A. Kinematics of the system
A. Related Works
As described in the introduction, Trinkle et al. [5] pointed
out that the confusion was introduced in the conventional definition of force closure. Yoshikawa [1] solved this confusion byredefining the force closure in the the active and the passive
ones. As for the recent work of the active closure, Haradaet al. analyzed the active force closure of multiple objects[6]. Also there are several works of formulating the general
grasping system. Park et al. [7] derived contact forces andaccelerations consistent with dynamics and friction law for agiven torque-wrench pair. Bicchi et al. [8], [9], Wen et al. [10],and Park et al. [11] analyzed the manipulability of the general
grasping systems. Harada et al. [12] presented a set of joint
torque required to move the object in a desired direction. In
the previous report, we showed the orthogonality between theactive and the passive parts, and developed a control algorithm
for manipulating an object under the hybrid closure [13], [14].
However, we have never shown the general framework ofmanipulation taking the contact state transitions into account.
On the other hand, Yashima and his colleagues have studied
about the motion planning of an enveloped object grasped
by multifingered hands with contact mode switching [15]–[18]. However, both detachment of the fingers and the passive
property in a grasping system were not considered.
II. TARGET S YSTEM AND D EFINITION
A. Target System
The target system is shown in Fig.2. In this paper, weconsider a quasi-static object manipulation in the case wherean arbitrary shaped rigid object is grasped by N fingers of arobotic hand. The nomenclatures are listed at appendix.
B. Definition
We consider manipulating the object taking the hybrid
closure properties into account. Under hybrid closure, both
the direction of the generalized force applied to the object
and that of the object motion are constrained. We define thefollowing two directions.
Direction of Active Force Closure (DAFC); The directionof the object’s unit twist in which the fingers can do positive
work is named the direction of active force closure (DAFC).
Direction of Passive Force Closure (DPFC); The direction of the object’s unit wrench in which external wrench canbe balanced without changing the joint torque is named the
direction of passive force closure (DPFC).

The relation of infinitesimal displacement among the contact
points, the center of gravity of the object, and the joints of thefingers are expressed by the following two equations;
∆pCF

ij

= J ij ∆q i ,

∆pCO

ij

= GTij ∆r(1)

where J ij ∈ Rd×Mi denotes the Jacobian matrix andI
Gij = [(p− p )×] .
COijoHere, I represents an identity matrix, [a×] represents a skew
symmetric matrix equivalent to the cross product operation ([a × ]b = a × b). From (1), we obtain


∆q
∆
(2)
A
= ∆pCF − ∆pCO = −∆pC .
∆r
Next, to classify (2) according to the state of contact, we introduce the following selection matrices to derive the kinematic
relation at every contact state.
(3)
H k = diag H kij ∈ RLk ×Ld (k = c, s, d),I
for F-point
H cij =,
nTij for S-point and N-point
H sij = I
H dij = nTij

for S-point,for D-point.

Using the selection matrices, (2) becomesTAc ∆q T ∆r T
= o,


T
≤ o,
Ad ∆q T ∆r T

T
T
T
∆r
As ∆q
= −pCs ,

(4)
(5)
(6)

where superscripts c, s and d expresses the corresponding stateof contact (for example, Ad = H d A) and o denotes a zero
vector.
By solving (4), we obtain




∆q
Λq
∆ζ
(7)
= Λ∆ζ =
Λr∆r
where Λ ∈ R(M +D)×a denotes an orthogonal matrix whose
columns form bases of the null space of Ac , ∆ζ ∈ Ra isan arbitrary vector expressing the magnitude of each columnof Λ. Note that ∆ζ corresponds to the object and the finger
motion within the constraint applied by the fingers.

B. Statics of the system

IV. F EASIBLE DIRECTION AND JOINT TORQUE

From (4) and the principle of virtual work, the following
relation is obtained:




τ
J Tc
T
= Ac f c =
(8)
fc
−w
−Gc
From (8), the following relation is obtained;
f c = (J Tc )+ τ + Γξ̃ = W f x(9)where (J Tc )+ denotes the pseudo-inverse matrix of J Tc
Lc ×p̃

andΓ ∈ R
is an orthogonal matrix whose columns formbases of the null space of J Tc , ξ̃ ∈ Rp̃ denotes an arbitrary
vector. Note that Γξ̃ expresses an internal force which makes
no influence on the joint torque. By substituting (9) into (8),we get∆w = Gc (J Tc )+ τ + Gc Γξ̃ = Dτ + Ξξ(10)where Ξ ∈ RD×p is an orthogonal matrix whose columns
form bases of the Gc Γ, p is the rank of the Gc Γ, and ξ ∈ Rp
is an arbitrary vector expressing the magnitude of each columnof Ξ. Note that the second term in the right side of (10) can
express a generable resultant force without changing the joint
torque.
Here, we consider the tangential force component of f ij at
S-point, which isn’t considered at the above discussion. tfij
is a kinetic friction force and can be expressed by
tfij = −bij nfij ∆pCij .

(11)

where bij (≥ 0) is a scalar value determined by frictional
constraint. Then, the resultant force exerted on the object by
tfs is given by
w s = Gs tfs = Gs N (f c , bij )∆pCs .

(12)

Note that nTij ∆pCij = o.
C. Frictional constraints
The frictional constraint for F-point can be represented by
Ff ij = {f ij | |tfij | ≤ µij nfij , nfij ≥ 0}.

(13)

The frictional constraint for N-point and the normal component of S-point can be represented by
Fnij = {nfij |nfij ≥ 0}.

(14)

The frictional constraint for the tangential component of Spoint can be represented by
Fsij = {tfij | |tfij | = µij nfij }.

(15)

Aggregating (13), (14) and (15) for all F-points, S-points, andN-points, we obtain
F = {f c |f ij ∈ Ff ij , ∀ Cij ∈ F-point,∀

nfij ∈ Fnij , Cij ∈ N-point, S-point},
Fs = {tfs |tfij ∈ Fsij ,∀Cij ∈ S-point}.

FOR OBJECT MANIPULATIONIn this section, we derive feasible direction of the object
motion and the joint torques to realize the feasible object
motion, in a grasping system under a hybrid closure. Also,
we derive the feasible finger motion to realize a regrasping
motion. From this analysis, we can obtain the feasible finger
and object motion and the joint torques for the manipulation
at every point of C-Space, when planning a manipulation withcontact state transitions.
Since the feasible object motion corresponds to DAFC, we
derive a set of DAFC with respect to arbitrary combinationof contact states. Also, we derive DPFC. Next, we derive thejoint torques to realize the object manipulation in the desired
DAFC direction. Also, we derive the finger motion to regrasp
the object without moving the object.
Let us consider the case where the all contact points are
assigned into F-points, N-points, S-points, and D-points. Here,we consider the joint torque, τ st , to grasp the object stably
and make the object be in stationary state. From (10) and (16),τ st holds−w ex = Dτ st + Ξξ st = Gc W f xst , W f xst ∈ F

where xst and ξst , respectively, represent x and ξ in the casewhere the system is in a stationary state, w ex denotes theexternal force such as gravitational force. Note that if there is
no external forces exerted on the object, wex = o. Note alsothat we assume that ws is not included in wex .
A. DAFC
First, we consider the kinematic constraints for the object
motion. Let Λ̃r ∈ RD×ã be a full column rank matrix whose
columns form orthonormal bases of the space spanned bycolumn vectors of Λr in (7).






Λ̃q
Λq
Λ̃qr
(19)
∆ζ =
∆ζ̃ +
∆ζ̃ r
Λr
O
Λ̃r
Here, the second term represents the redundancy of the joint
variables which do not affect the object motion (Λ̃qr ∈RM ×(a−ã) , ∆ζ̃ r ∈ Ra−ã ). From (4), (5), (7), and (19), theallowable object and finger motions within the constraints are
given by
Ar = {∆r|∆r = Λ̃r ∆ζ̃ = Φ∆ζ c , ∆ζ c ∈ A},
Aq = {∆q|∆q = Λ̃q Λ̃qr ∆ζ c , ∆ζ c ∈ A},Λ̃q Λ̃qr
A = {∆ζ c |Ad (∆ζ c ) ≤ o}.
Λ̃r O

(17)

(20)

The robot hand can do work on the object in the directions
contained in the set Ar .
Using (19), (6) becomes∆pCs = P s ∆ζ c , ∆ζ c ∈ A.

(16)

(18)

(21)

Here, we introduce the following theorem which provides
the generalized force set for DAFC.

Theorem 1: Suppose the object is grasped by τ st stasfying(18). Let τ ad be the joint torque added to τ st for doing workon the object. Let xad be x when τ ad is added to the system.
DAFC is included in the following set:
DDAF C = {∆r|∆r = Φ∆ζ c , ∆ζ c ∈ A,
W f (xad + xst ) ∈ F, N t (y)∆ζ c ∈ Fs ,

(23)
(24)

Second, the allowable object motion is given by (20).
Third, when moving the object in the allowable direction,
the work done by τ ad is given by∆r T w = ∆r T (Dxad + Ξξ ad + w s )TT= ∆ζ Tc (W 1 τ ad + N 1 (y)∆ζ c ) + ∆ζ̃ Λ̃r Ξξad , ∆ζ c ∈ A.
(25)
Here, Ξξ ad is resultant force and moment which work tocounteract external force and moment in the direction contained in Im(Ξ), only when the external force and moment
are exerted on the object. Then, the applied force and momentby τ ad in the direction contained in (Im(D)∩Im(Ξ)) willT
be counteracted. Here, we can easily show Λ̃r Ξ = O andrank(Ξ Λ̃r )= D by the similar way as our previous paper[13], [14], which corresponds to the orthogonality between
DAFC and DPFC. Therefore, the counteracted force and moT
ment is given by (I − Λ̃r Λ̃r )(Dτ ad +ws ) and the following
relation holds
T(I − Λ̃r Λ̃r )(Dτ ad + ws ) = −Ξξad .

In this subsection, we derive the joint torques to move theobject in a desired DAFC with the contact state transitions.
Using Λ̃qr in (19), τ can be represented by
τ = J Tc f c + Λ̃qr τ̃ .

∆ζ Tc (W 1 τ ad + N 1 (y)∆ζ c ) > 0,
W 2 xad + N 2 (y)∆ζ c = o}(22)
Proof: First, from (9), (12), (16), (17) and (21), thefrictional constraints are given by
f c = W f (xad + xst ) ∈ F,
tfs = N (f c , bij )P s ∆ζ c = N t (y)∆ζ c ∈ Fs .

C. Joint torques to realize the object motion in DAFC

(26)

T

From (20), (23), (24), (25), (26), and Λ̃r Ξ = O, DAFC isgiven by (22).

(29)

The second term represents the redundancy of the joint torques
which do not affect the contact forces applied to the object.
Here, we consider deriving the joint torques corresponding to
the first term.
In order to derive the joint torques for the desired object
motion, the kinetic frictional forces at S-point is needed to be
determined. But, when the fingers have the redundancy, thefinger motion, namely, the kinetic frictional forces at S-point
can not always be determined uniquely. Here, we determine
the finger motion by optimizing a criterion function φ withrespect to the redundancy component.
Let ∆r d = Λ̃r ∆ζ̃ d be the unit vector and the desired ∆r∈ DDAF C given by (22). We set the following set:T

W f (xad + xst ) ∈ F, N t (y)∆ζ c ∈ Fs
∆ζ Tc (W 1 τ ad + N 1 (y)∆ζ c ) > 0,TΛ̃qr τ ad = o, W 3 xad + N 3 (y)∆ζ c = o}.

−wp − w ex = Gc (f cst − f cp ) = Dτ st + Ξ(ξst − ξp ) (27)Here, f cp , ξ̃p , and ξp , respectively, are f c , ξ̃, and ξ corresponding to w p . If f cst − f cp ∈ F, Ξξp represents DPFC.
Then, DPFC is given by
DDP F C = {wp |w p = Ξξ p = Gc Γξ̃ p , f cst− Γξ̃p ∈ Ff }. (28)(30)

Here, to specify the direction in which the fingers do
work, we add the constraint in the direction Λ̃r (I −T

T(∆ζ̃ d ∆ζ̃ d )/( ζ̃ d ζ̃ d )).
Using the set S, we consider the following problem.
φmin∆ζ̃ r
subject to S

(31)

Solving this problem, the finger and the object motions (∆ζ cd )can be completely obtained. Then, the joint torques to realize
the desired object and finger motions are given by
T = {τ |τ = τ ad + τ st , S|∆ζ c =∆ζ c }.
dB. DPFC
The tangential component of the contact forces at S-points
works not for counteracting external forces but for avoiding
the object motion. Therefore, we do not regrad the directions
as DPFC. Consequently, we only have to consider the contact
forces at F-points and N-points, and the normal component of
the contact forces at S-points.
DPFC is the directions of external force and moment which
can be balanced without changing τ st in the state given by(18). Let be w p be such external forces and moments. w p
holds the following relation.

T

S = {∆r̂ d = Λ̃r ∆ζ̃ d ∈ DDAF C , ∆ζ c = [ ∆ζ̃ d ∆ζ̃ r ]T(32)

Note that τ st is added for keeping the grasping during themanipulation. If grasping is not needed during the manipulation (for example, dynamic manipulation), τ st can be set tozero in (32).
The frictional coefficient µ is labile. Therefore, we should
deal with various µ in the derivation of the joint torques
for the manipulation. At F-point, we only have to use theestimated smaller µ. However, at S-point, µ directly affects the
kinetic frictional forces. Here, we consider the case where µ
becomes νµ (1 ≥ ν ≥ 0) while from (32), the joint torques forthe desired finger and object motions, τ µ , have been already
obtained with respect to µ. Let Ls be the number of S-points
where the normal components of the contact forces induced
by τ µ are not zero. Let nsµi (i = 1, · · · , Ls ) be the normalcomponent of the contact force at each S-point induced byτ µ . If Ls = 0, the joint torques for the manipulation can beeasily obtained by the linear combination of τ µ and τ st . If
Ls > 0, the joint toruques can not be easily obtained for the

o

0.5

120

x

0.5

y
C1

60o

L

C2

C3 C4
0.1

Fig. 3.

Example1

rotation center

0.5
0

θ

s
λi τ sti where τ sti denotes
linear combination λ0 τ µ + Σi=1

the Ls sets of the joint torques at the steady state and nsstij
denotes the normal component of the contact force at the S
point corresponding to nsµi , induced by τ stj (j = 1, · · · , Ls ).
⎡
⎤ ⎡ ⎤
⎤⎡
λ0
(ν −1)nsµ1
νnsst11 . . . νnsst1L
0s⎢⎥ ⎢ .. ⎥ ⎢ . ⎥
..
..
..
..
⎢
⎢
⎥
⎥
.. ⎥
.
⎢
.
.
.
.
⎢
⎥
⎥⎢
⎥
⎢
⎥ ⎢ .. ⎥ = ⎢⎣ (ν −1)nsµL νnsstL 1 . . . νnsstL L ⎦ ⎣ . ⎦ ⎣ 0 ⎦
s
s
s s
1
λ 
1
1
...
1

0.5

0.25

0.5

nonliear constraints for the non-zero kinetic frictional forces.
However, if the following equation can be solved subject to
can be obtained by the
λ0 > 0 and λi ≥ 0, the joint toruques


−0.5
−1
−1
−0.5

1
0.5
0
0.5

Ls

0
−0.5
1 −1
y

: sliding contact: removing contactx

D. Finger motion for regrasping(a) DAFC

Here we consider regrasping without moving the object.
Letting ∆r d = o in (30) and (32), the realizable finger motion
can be obtained.

Fig. 4.
statesDAFC and the schema of DAFC for FSDS combination of contact

A = {∆q|∆q = Λ̃qr ∆ζ̃ r , J d Λ̃qr ∆ζ̃ r ≤ o,
D Gc Γ x − Gs N (x, bij )As Λ̃qr ∆ζ̃ r = −wex ,W f x ∈ F, −N (x, bij )As Λ̃qr ∆ζ̃ r ∈ Fs }.

C1

V. N UMERICAL E XAMPLES

T = {Σ5j=1 λj τ j , Σ5j=1 λj = 1, λj ≥ 0},
τ 1 = [−0.38 − 0.17 0.0093 0]T ,τ 2 = [−0.31 0.25 0.43 0]T , τ 3 = [−0.38 − 0.17 0 0]T ,τ 4 = [−0.15 0.12 0 0]T , τ 5 = [−0.31 0.25 0.43 0]T .
Next, we consider the case shown in Fig.5. Here, we
consider transiting the system from (1) to (4). In the transition
from (1) to (2) and from (2) to (3), (33) is used. In thetransition from (3) to (4), (30), (31) and (32) are used. We

C2
C4
C3
z
x
y

S-points

C5(33)

In order to verify our approach, we show some numericalexamples. Consider the case shown in Fig.3. The frictionalcoefficient is set to 0.5 for all contact points. We set thatw ex = [0 −1 0]T . The every contact state is denoted by ”F”(Fpoint), ”S”(S-point), and ”D”(D-point). The contact states are
expressed in the order of the number of contact point. For
example, SFSD means C1 is S-point, C2 is F-point, C3 is Spoint, and C4 is D-point. Using (22), we can know the feasible
direction of the object manipulation for every combinationof contact states. From the calculation, the following feasiblecombinations are obtained: FFDD, FFDS, FFSS, FSDS, FSSD,
FSSS, SSSS. Note that the symmetric combinations of contact
states are also feasible in the above list.
Here, we focus on FSDS combination of contact states.
From (22), the feasible direction of the object manipulation(DAFC) is calculated as shown in Fig.4 (a). The schema of
the obtained feasible object motion is shown in Fig.4 (b). From
Fig.4, it can be seen that x direction and clockwise rotation
around the rotation center shown in Fig.4 (b) are realizable.
Next, we consider the joint torques to move the object in[1 0.15 − 1]T , as an example. From (32), the set of the joint
torques is given by(b) schema of DAFC

(1)

(3)

(2)

(4)

D-points

Fig. 5.

Whole arm manipulation with several contact state transitions

set φ= ∆q T q, µ = 0.5, and w ex = [0 0 − 1 0 0 0]T .
The every link is set to 1. The frictional cone is approximatedby a 16-sided convex polyhedral cone.
At first, we consider transiting the system from (1) to (2).
The combination of the contact states is DDDDF. Then, therealizable finger motion is given by

A = {∆q| J d ∆q ≤ o}, J d = diag[J di ] (i = 1 ∼ 4),
J d1 = [−0.17 − 0.88 1.1 0.39],
J d2 = [0.083 0.51 1.3 0.47], J d3 = [0 0 1 0.5],
J d4 = [−0.083 − 0.51 1.3 0.47].
Note that the finger motion for the transition from (2) to (3)is similarly obtained.
Next, we consider transiting the system from (3) to (4). The
combination of the contact states is FSFSD. When we movethe object in the direction [0 0 1 0 0 0]T , the joint torques forthe manipulation is given by
T = {Σ2j=1 λj τ vj +Σ2i=1 βi τ wi , Σ2j=1 λj = 1, λj , βi ≥ 0},τ vj = col τ vjk (k = 1 ∼ 4), τ vj2 = τ vj4 = o, ∀ j,
τ wj = col τ wjk (k = 1 ∼ 4), τ wj2 = τ wj4 = o, ∀ j,
τ v11 = [−0.33 − 0.48 − 0.22 − 0.50]T ,τ v21 = [−0.28 − 0.41 − 0.22 − 0.50]T ,τ v13 = [0 0 0.72 0.29]T , τ v23 = [0 0 0.64 0.21]T ,τ w11 = [−0.28 − 0.41 − 0.22 − 0.50]T ,τ w21 = [−0.68 − 1.00 0.068 − 0.012]T ,τ w13 = [0 0 0.64 0.21]T , τ w23 = [0 0 1.00 1.15]T .
From the above analysis, it can be seen that the motions of
the fingers and the object shown in Fig.5 can be realized.
VI. C ONCLUSION
In this paper we have discussed whole arm manipulation
by taking the contact state transition into account. In order torealize the object manipulation, we have analyzed the active
and passive closure properties in a hybrid closure grasp, forevery combination of contact states. In the analysis, we havederived the feasible direction of the object manipulation forevery combination of contact states. We further derived the setof joint torque to move the object in the feasible directions.
Through some numerical examples, we showed that the grasp
transition from the passive to active closure can be realized
by using the proposed approach. In future, we plan to solve
the path planning for object manipulation by considering thecontact state transition.
A PPENDIXN OMENCLATURES
col
diag
N
Mi
Li
M
L
Lk
D
d
ΣR
ΣO
Cij
CFij
COij
ΣCI
r:
qi
∆q
p
I
∆p
J
nij
GT
J
A
w
f ij
f
τ
nfij
tfij
tf

a column vector or matrix formed by the following elements.
a block diagonal matrix.
Number of fingers.
Number of joints of the ith finger (i = 1, 2, · · · , N ).
Number of contact points on the ith finger.
Number of total joints (= ΣN
i=1 Mi ).
Number of total contact points (= ΣN
i=1 Li ) .
Number of columns of selection matrix (k= c,s,d).
3/6 in 2/3 dimensional space.
2/3 in 2/3 dimensional space.
Reference coordinate frame.
Object coordinate frame fixed at the object.
The jth contact point of the ith finger (j = 1, 2, · · · , Li ).
The contact point on the finger corresponding to Cij .
The contact point on the finger corresponding to Cij .
Coordinate frame fixed at CI (I = CFij , COij ).
Position and orientation of ΣO (∈ RD ).
Joint angle vector of the ith finger (∈ RMi ).
=col[∆q i ] ∈ RM .
Position of the origin of ΣI (I = O, CFij , COij , Cij ) (∈ Rd ).
=col[∆p ] ∈ RLd (I = CFij , COij , Cij , J = CF , CO , C).
I
Unit normal vector (directing to the inward of the object) at Cij .
=col[GT] ∈ RD×Ld
ij
2
32 .
3
33
22
J N1
J 21
J 11
6 . 7766 . 7 6 . 7
Ld×M
6
76
7
77
6
.
= diag 6
44 .. 5 4 .. 5 · · · 4 .. 55 ∈ R
J N LN
J 1L1
J 2L2
=[ J
−GT ] ∈ RLc ×(M +D) .
Resultant force and moment applied to the object (∈ RD ).
Contact force vector (∈ Rd ).
=col[f ij ] ∈ RLd . Note that f k = H k f (k= c,s,d).
Joint torque vector (∈ RM ).
Normal force component of f ij .
= [tfij ,1 tfij ,2 ]T where tfij ,k denotes the tangential force componentof f ij (k = 1, 2).
=col[tfij ].

µij
x
Wf
N
Φ
∆ζ c

Frictional coefficient at Cij .
T= [τ T ξ̃ ]T .
+= [(J T)Γ].
c
= N (f c , bij )=diag [−bij nfij ], ∀ Cij ∈ S-point.
= [Λ̃r O].
TT= [∆ζ̃ » ∆ζ̃ r ]T .
–
Λ̃q
Λ̃qr
Ps
.
= −As
O
Λ̃r
T
T
y
= [xT
ad xst b11 · · · ] .
N t(y)= N (xad , xst , bij )P s .
W 1 = ΦT D.
T[(I − Λ̃r Λ̃r )D Gc Γ].
W2 = "#TGc Γ(I − Λ̃r Λ̃
r )D
q
.
W3 =
T
T
T
(I − (∆ζ̃ d ∆ζ̃ d )/( ζ̃ d ζ̃ d ))Λ̃r D
OT
N 1(y)= Φ Gs N t (y).
T
N 2(y)= (I − Λ̃r Λ̃r )Gs N tq(y).
T

N 3(y)= (I − (∆ζ̃ d ∆ζ̃ d )/(TTζ̃ d ζ̃ d ))Λ̃r Gs N t (y).

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