Joint Torque-velocity Pair Based Manipulability for Grasping System

Abstract— This paper provides a new approach of manipulability for general grasping system. While conventional
manipulability is analysis in velocity domain and can notinclude force effect such as gravitational force, the proposing
approach can include the force effect to keep grasping. For the
purpose, an operation range is introduced. The operation range
is for actuator attached with every joint of robot and provides
generable joint torque and velocity and their relation (between
generating torque/velocity and addable velocity/torque). Using
the operation range, we derive manipulability set and measure
in velocity domain, including force effect. The proposing method
can evaluate not only the performance in velocity domain but
also effects of friction, contact state, and external forces, whichwere not obtained in conventional studies.

I. I NTRODUCTION
Manipulability is a well known concept to evaluate theperformance of robotic manipulator [1]. For a single-arm
manipulator, it is defined as the set of generable endeffector
velocity in the task space when the set of generable joint
velocity is given. When the given set of joint velocity is a unit
ball, the set of endeffector velocity becomes an ellipsoid. The
ellipsoid is called manipulability ellipsoid. The volume of theellipsoid is a quality measure to evaluate the performancein velocity domain. It is called manipulability measure.
Based on the manipulability, many quality measures suchas condition number are proposed [1].
This concept can be extended to the general constraining system such as a robotic hand [2]–[6]. In a general
constraining system, object velocity is evaluated instead ofendeffector velocity. For a dual-arm system, Chiacchio etal. [2] discussed manipulability. Bicchi et al. [3] analyzed
manipulability for general grasping system including wholearm manipulation system. After that, Bicchi et al. [4], Wenet al. [5], and Park et al. [6] analyzed manipulability for
general constraining system with underactuated joints.
In a grasping system, grasping itself is a key issue. But, theanalysis for grasping is in force domain, while manipulability
analysis is in velocity domain. In the above researches, force
closure grasp was assumed, and force analysis was evaded(Force closure is defined that any force and moment inany direction can be applied to the object). However, forthe assumption, the applicable classes of the analyses in
the above researches are limited. It can be said that closed
chained system was only considered. Therefore, exactly
saying, a manipulability analysis in velocity domain forgrasping system has not been done so far.
This work was not supported by any organization
T. Watanabe is with Graduate School of Natural Science &Technology, Kanazawa University, Kanazawa, 920-1192, Japante-watanabe@ieee.org

Rotational velocity [rad/s]

Tetsuyou Watanabe

1200
1000

Operation range

800
600
400
200
00

0.005 0.01 0.015 0.02 0.025 0.03

Torque [Nm]Fig. 1. Operation range for torque and velocity (maxon DC motor RE25
(20W))In this paper, we propose a new approach to analyze
manipulability in velocity domain, simultaneously including
the effect of force/torque needed to keep grasping. For the
purpose, we use an operation range (shown in Fig.1) ofactuator attached with every joint of robot (operation range
is originally used for motor selection). The operation range
provides not only information about how magnitudes of
torque and velocity the actuator can stably generate, butalso the relation between generating velocity/torque andaddable torque/velocity. Namely, we discuss manipulability
from the view point of power, which have not been done inconventional studies and is first try.
First, we derive joint torques needed to keep grasping.
We introduce required external force set (REFS) [7], [8]which is defined as a set of external wrenches (for example,
gravitational force and corresponding forces to acceleration)required to be balanced. REFS can handle any desired grasp
including force closure and equilibrium grasps. Using operation range and REFS, we compute necessary joint torques.
Next, using the necessary joint torques and the operation
range, we derive the set of generable object velocity. The
set corresponds to conventional manipulability ellipsoid. The
volume of the set corresponds to conventional manipulability measure. We call the set and its volume joint torquevelocity pair based manipulability set (TVMS) and measure(TVMM), respectively. The directions required to evaluate
object velocity are limited in some tasks. For example, it isenough if considering unconstrained translational directions
when lifting up an object on a table. To limit the directions
to evaluate, we introduce required velocity direction set(RVDS). The set is defined as a set of object’s twist directionin which it is required to evaluate how large velocity can begenerated.
However, the computational load to derive the TVMS
and TVMM is large since the set (polyhedron) used in
the computation is high dimensional and constructed bymany hyper planes and many vertices. The object can notalways move in any arbitrary directions in grasping. In which
direction the object can move depends on how to grasp theobject. Therefore, motion planning is needed, and TVMS canbe a criterion for the planning. For example, we can plan
how to grasp to keep the object movable along a desired
trajectory. Therefore, computational load should be lower.
Then, we propose another evaluation method letting RVDS
be a convex polyhedron (which is a set of scaled direction).
We compute maximum α such that RVDS multiplied by α
can be contained in the set of generable object velocity.
For the linearity, only vertices of RVDS have to be considered, and then the computational load is reduced. Since
the volume of RVDS is constant, the gauge parameter α cancorresponds to manipulability measure. Then, we call the αjoint torque-velocity pair based manipulability measure parameter (TVMMP). This approach has the following merits;
1) Friction effect can be included. 2) Contact state effect canbe included. 3) Effect of external forces such as gravitational
force can be included. These merits could not be obtainedin the conventional studies.
As for dynamical analysis, Zheng et al. [9] and Fujiwara
et al. [10] discussed a dynamic manipulability [1]. However,these analyses are not in velocity domain but in acceleration
domain. Since power is not included/considered, the obtained
generable accelerations are not associated with the generable
velocities (obtained in this paper). They discussed only
fingertip grasp. For the complexity, the computation load isvery large. The other information such as inertia tensor isneeded. In addition, in [10], internal force is fixed and thenapplicable class is limited.
II. P ROBLEM D EFINITION
A. Target System
The target system is shown in Fig.2. In this paper, weconsider a general grasping system where an arbitrary shaped
rigid object is grasped by N fingers of a robotic hand. The
nomenclatures are listed at appendix. We define that thecontact state is any of the following four states: 1) F-point :the contact point with static friction, 2) N-point : the contact
point without friction, 3) S-point : the contact point withkinetic friction, 4) D-point : the point about to detach.
We note that the contact force is zero at D-points. Also, we
assume that every contact position, every contact state, every
frictional coefficient are all given. For the convenient, thedifference between static and kinetic frictional coefficientsis not distinguished.
B. Problem Definition
After some definitions, we define the problem handled inthis paper.
Joint Torque-velocity Pair Set (TVS): The set of generable
joint torque and velocity at each joint, given by the operation
range of the corresponding actuator, is named joint torquevelocity pair set (TVS).
TVS is usually given for absolute value of joint torque andvelocity. Supposing that TVS is given as a convex polyheron

ΣR
pO

pC21
ΣC21

ΣO
Object

2nd finger

1st finger
Base

Fig. 2.

Target System (N = 2).

qijOperation range

τij

Fig. 3. Case when Stvij does not constitute a convex polyhedron withrespect to τij and q̇ij

with respect to absolute value of joint torque and velocity,
TVS is expressed by

 



ntvij
|τij ||τij |
τij vkStvij = {|= Σk=1 νk, νk ≥ 0,|q̇ij |
|q̇ij |
q̇ij vkntv

Σk=1ij νk = 1}(j = 1, 2, · · · , Mi , i = 1, 2, · · · , N ).

(1)

When using the both plus and minus values of joint torqueand velocity, the joint torque and velocity contained in Stvij
do not always constitute a convex polyhedron as shown in
Fig.3. To resolve it, we introduce the variables τij max , τij min ,q̇ij max , and q̇ij min , which, respectively, mean maximum andminimum values of joint torque, and maximum and minimum
values of joint velocity. We consider to construct convex
polyhedrons with respect to these four variables, instead of
τij and qij . Then, (1) becomes
Stvij = {y ij |τij min ≤ τij ≤ τij max , q̇ij min ≤ q̇ij ≤ q̇ij max ,τij max
τij max+++−∈ Stvij ,∈ Stv
ijq̇ij maxq̇ij min
τij minτij min
−+
−−
,
},
(2)
∈ Stv
∈ Stv
ij
ij
q̇ij maxq̇ij min
where


ntvij
ntv
s1 τij vk
s1 s2
Stv
=
{u
|u
=
Σ
ν
, Σk=1ij νk = 1,
ij
ij
k
k=1
ij
s2 q̇ij vk
νk ≥ 0}(s1 , s2 ∈ {+, −}).
Aggregating (2) for all joints, we obtain
Stv = {y|y ij ∈ Stvij ∀ i, j}.

(3)

In some tasks, the directions required to evaluate how large
object velocity can be generated are limited. For example,
when lifting up an object on a table, the target directions
required to evaluate are upward from the table and rotational

directions do not have to be considered. Here, we consider
only directions required to evaluate. For the purpose, we
introduce RVDS.
Required Velocity Direction Set (RVDS): The set ofobject’s twist direction in which it is required to evaluate howlarge velocity can be generated is named required velocity
direction set (RVDS).
RVDS is assumed to be given by a convex polyhedral
cone:
Srvd = {ṙ|Arvd ṙ ≤ o},(4)

where Arvd ∈ Rnrvd ×D .
In order to include the forces needed to be balanced tokeep grasping, we introduce REFS.
Required External Force Set (REFS): The set of object’s
external wrench required to be balanced is named required
external force set (REFS).
REFS is assumed to be given by convex polyhedron:
nref
nref
κi w vi , Σi=1
κi = 1, κi ≥ 0},
Sref = {w ex |wex = Σi=1(5)Note the REFS indicates desired grasp. For example, if
origin is in its interior, the desired grasp is force closure.
Then, we define the following problem:
Problem 1: Suppose that Stv , Srvd and Sref are all given.
In this case, find the set of generable object velocity, Sov
such that Sov ⊂ Srvd and ṙ ∈ Sov is the velocity which
can be generated even if any w ex ∈ Sref is exerted on theobject.
The obtained Sov corresponds to conventional manipulability ellipsoid, and its volume corresponds to conventional
manipulability measure. Therefore, we call Sov and its
volume joint torque-velocity pair based manipulability set(TVMS) and measure (TVMM), respectively.
III. BASIC F ORMULATION OF THE S YSTEM
The relation between velocities of pCF and q i , and theijrelation between velocities of pCO and r, respectively, areijgiven as follows:ij= J ij q̇ i ,

ṗCO

ij

= GTij ṙ,(6)where J ij ∈ Rd×Mi denotes Jacobian matrix andI
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).
To derive the relation between ṗCF and ṗCO accordingij
ij
to contact state, we introduce selection matrices. As for F andN-points and normal direction of S-point, the relation is
H cij (ṗCF − ṗCO ) = o,ijij
I for F-point
H cij =,
nTij for S-point and N-point

H dij (ṗCF − ṗCO ) ≤ o,ij

H dij = nTij

ij

(7)

(8)

for D-point .

As for tangential directions of S-point, the relation is
H sij (ṗCF − ṗCO ) = −ṗCsij ,ij

H sij = T ij

ij

(9)

for S-point ,aggregating (7), (9) and (8), we obtainTAc q̇ T ṙ T
= o,
 T
T
T
Ad q̇
≤ o,
ṙ
 T

T
= −ṗCs ,
As q̇
ṙ T

(10)
(11)
(12)

B. Statics of the system
First, we consider F and N-points and normal directionsof S-points. From (10) and the principle of virtual work, thefollowing relation is obtained:




τc
J Tc
T
(13)
= Ac f c =
f c.
−w c
−Gc
Next, we consider tangential directions of S-points. From(12) and the principle of virtual work, the following relation
is obtained:




τs
J Ts
(14)
= ATs f s =
f s.
−w s
−Gs
Here, f s is considered. When Cij is a S-point, assuming
friction obeys Coulomb’s law, kinetic friction force f sij canbe expressed byf sij = H sij f ij = −μij nfij ˆṗCsij = −μij ˆṗCsij f cij ,A. Kinematics of the systemṗCFwhere o denotes a zero vector. As for D-point, the relation
is

(15)

ˆ C = ṗC /|ṗC |. Aggregating for all S-points,
where ṗ
sij
sij
sij
(15) becomesˆ C f c = W sf c,
f s = col −μij ṗ
(16)
sij
ij


W s = diag W sij ,ˆ C for f c corresponding to S-point−μij ṗij
sij
W sij =.
nothing for f cij corresponding to F and N-points
From (14) and (16), (14) becomes

 

τs
J Ts
=
W sf c.
−w s
−Gs

(17)

Letting gravity term of fingers be g f , from (13) and (17),the total joint torque τ is expressed byτ = (J Tc + J Ts W s )f c + g f .

(18)

Also, total resultant force w is expressed byw = (Gc + Gs W s )f c .

(19)

C. Frictional constraints
Since kinetic friction is described above, static friction is
only described here. The frictional constraint for F-point canbe represented by
Ff ij = {f cij ||T ij f ij | ≤ μij nfij , nfij ≥ 0}.

(20)

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

(21)

Aggregating (20) and (21) for all F-points, S-points, and Npoints, we obtain
F = {f c |f cij ∈ Ff ij , ∀ Cij which is F-point,f cij ∈ Fnij , ∀ Cij which is N or S-point}.

(22)

IV. J OINT TORQUE - VELOCITY PAIR BASEDMANIPULABILITYFirstly, we consider the case when there is no S-point.
After that, we consider the case when S-points exist.
A. When there is no S-pointFirstly, we consider necessary joint torques to balancew vi ∈ Sref and keep gasping.
We approximate the frictional constraint (20) by a nf ric side convex polyhedral cone circumscribed in the friction
cone [11]. Then, (22) becomes∀

Flin = {f c |V ij f cij ≤ o, Cij which is F-point,f cij ≥ 0, ∀ Cij which is N or S-point}(23)

where V ij ∈ Rnf ric ×d . Then, from (19) and (23), in order tobalance wvi , the contact force has to satisfy the following
constraints: Gc f cvi + w vi = o, f cvi ∈ Flin where f cvi
is f c which balances w vi .From (18), the joint torque (τ vi )corresponding to f cvi is expressed by τ vi = J Tc f cvi + g f .
The τ vi has to be contained in TVS: y vi ∈ Stv where y vi
= [τ Tvi τ Tmax τ Tmin q̇ T q̇ Tmax q̇ Tmin ]T .
Then, the joint velocity has to satisfy the following constraints to balance w vi ∈ Sref :y vi ∈ Stv , τ vi = J Tc f cvi + g f , Gc f cvi + wvi = o,f cvi ∈ Flin .

(24)

Since Sref and the constraints (24) are all linear, if satisfying
the constraints (24) w.r.t. all vertices of REFS, the constraints
becomes the set of joint velocity which can be generated even
if any wex ∈ Sref is exerted on the object. Then, from (10),(11) and (24), Sov is expressed by

T

T
Sov = {ṙ|Ac q̇ T ṙ T = o, Ad q̇ T ṙ T ≤ o,y vi ∈ Stv , τ vi = J Tc f cvi + g f , Gc f cvi + w vi = o,f cvi ∈ Flin , (i = 1, 2, · · · , nref )}.

(25)

Since both equations and inequalities constituting Sov arelinear and τ and q̇ are bounded by Stv , Sov is a convex
polyhedron. Therefore, we call Sov joint torque-velocity
pair based manipulability polyhedron (TVMP). Sov can be

expressed using its vertices by a programming method suchas a pivoting algorithm [13]:
n

n

sov
sov
λi ṙ ovi , Σi=1
λi = 1, λi ≥ 0}, (26)
Sov = {ṙ|ṙ = Σi=1

where ṙ ovi denotes the vertex of Sov and nsov denotes thenumber of the vertices.
Here, we decompose Sov into simplices by, for example,
triangulation method using delaunay triangulation [12], [14]since the volume of simplex can be calculated. Letting ∫i(i = 1, · · · , nssov ) be the decomposed simplex of Sov andits volume V (∫i ), the volume of Sov (TVMM), Vsov , isnssov
V (∫i ).
calculated by Vsov = Σi=1
The derivation of both (26) and TVMM is sometimes timeconsuming. The computational complexity of the pivoting
algorithm (the derivation of (26)) is O(mnsov D) where mdenotes the number of inequalities constituting the convex
polyhedron (note that equations in the constraints of (25)can be easily transformed into inequalities). As for delaunay
triangulation, if the dimension of the polyhedron, l, is fixed,lthere is an incremental O(ns2ov ) algorithm [12]. Therefore,
whole calculation is time-consuming, especially in the caseof high dimensional space. Also, the calculation of TVMM
is available only when TVMP is full-dimensional.
Here, we consider to reduce the computational load. For
the purpose, we consider using RVDS which is given by
convex polyhedron:nv
nv
Prvd = {ˆṙ|ˆṙ = Σi=1rvd λi ˆṙ vi , Σi=1rvd λi = 1, λi ≥ 0}. (27)where ˆṙ denotes the direction of object velocity. Note thatsince a direction is determined by origin and a point, origin
should be included in Prvd . This RVDS can be regarded
as a set of scaled directions of interest object velocity. In
consistent with this RVDS, we modify Problem 1 as follows:
Problem 2: Suppose that Stv , Prvd and Sref are all given.
Consider the set of generable object velocity, Sov , such thatṙ ∈ Sov is the velocity which can be generated even if any
w ex ∈ Sref is exerted on the object. In this case, find the
maximum α(> 0) such that αPrvd ⊂ Sov .
Here, since the volume of Prvd multiplied by α corresponds to manipulability measure and the volume of Prvd
is constant, α can be regarded as a criterion to evaluate
TVMM. Therefore, we name α joint torque-velocity pairbased manipulability measure parameter (TVMMP). Notethat αPrvd can correspond to TVMP.
Here, we consider to generate object velocity in the
direction of ˆṙ vι ∈ Prvd , balancing wvk ∈ Sref . From (25),the maximum magnitude of generable object velocity in ˆṙ vι
direction is derived by solving the following problem:
αkι = max α subject to αˆṙ vι = ṙ ∈ S̄ov ,
x

T

T
S̄ov = {Ac q̇ T ṙ T = o, Ad q̇ T ṙ T ≤ o,(28)y ∈ Stv , τ = J Tc f c + g f , Gc f c + w vk = o, f c ∈ Flin }.
Since the constraints of the above problem (28) are all
linear and both RVDS and REFS are convex polyhedra, weonly have to consider the problem (28) with respect to thevertices of RVDS and REFS. The minimum value among theαkι ’s for all w vk ’s and ˆṙ vι ’s is the TVMMP. Therefore, theProblem 2 can be expressed as follows:minwvk ∈Sref ,ˆṙ vι ∈Prvd

max α
x

subject to αˆṙ vι = ṙ ∈ S̄ov
(29)

This problem can be solved by the following way:
1) Solve the problem (28) for every wvk and ˆṙ vι by a linear
programming (k = 1, 2, · · · , nref , ι = 1, 2, · · · , nvrvd ).
2) If obtaining the solution (αkι ) for every combination of
w vk and ˆṙ vι , mink,ι αkι is the overall solution. Otherwise,
there is no solution.
Remarks:
1. When obtaining the solution αkι for a certain combinationof w vk and ˆṙ vι in the problem (28), it means that theobject velocity with the magnitude of from zero to αkι canbe generated in the direction ˆṙ vι (balancing w vk ). Let f c ,τ max , τ min , q̇ max and q̇ min giving αkι be fixed among
the elements of x, while let the other elements q̇, ṙ and α(= αkι ) giving αkι be multiplied by  (0 ≤  ≤ 1):x → x = [f Tc τ Tmax τ Tmin q̇ Tmax q̇ Tmin q̇ T ṙ T αkι ]T .
Obviously, the constraints in the problem (28) can be satisfied
with respect to x .
2. Let α∗ be the overall solution. Then, object velocity withmagnitude of α∗ can be generated in any arbitrary direction
contained in Srvd .
3. We consider the case when one of the vertices of Prvd ,ˆṙ vi , is o. From the above remark, we can omit the calculation
with respect to ˆṙ vi = o direction if Prvd is not constructedby only o.
4. We can make the distribution of generable object velocity,
whose vertex is mink αkι ˆṙ vι (ι = 1, 2, · · · , nvrvd ). The
obtained distribution corresponds to the conventional manipulability ellipsoid.
B. When there are some S-points
In this case, by actuating joint velocities, not only object
velocity but also sliding velocity are generated. Then, if
focusing only on deriving the set of generable object velocity,in which direction sliding should be generated to generate thederived object velocity becomes unclear. Here, we focus onin which direction object velocity can be generated and how
large object velocity can be generated, with the finger sliding
direction (ˆṗCs = ṗCs /|ṗCs |) fixed.
Due to entity of kinetic friction, the terms and the constraints related with sliding and kinetic friction should be
added. From (12), (18) and (19), Sov (25) becomes

T

T
Sov = {ṙ|Ac q̇ T ṙ T = o, Ad q̇ T ṙ T ≤ o,TṗCs , (Gc + Gs W s )f cvi + w vi = o,As q̇ T ṙ T = −ρˆy vi ∈ Stv , τ vi = (J Tc + J Ts W s )f cvi +g f ,f cvi ∈ Flin (i = 1, 2, · · · , nref )}.

(30)

Note that Sov is also a convex polyhedron. Therefore, thevolume of Sov can be derived by the same way as the casedescribed in section IV-A. Similarly, Problem 2 (29) becomes
min

max

wvk ∈Sref ,ˆṙ vι ∈Prvd xs

αsubject toαˆṙ vι = ṙ ∈ S̄ov


T

T
S̄ov = {ṙ|Ac q̇ T ṙ T = o, Ad q̇ T ṙ T ≤ o,TAs q̇ T ṙ T = −ρˆṗCs , (Gc + Gs W s )f c + wvk = o,y ∈ Stv , τ = (J Tc + J Ts W s )f c +g f , f c ∈ Flin }.
Using the same way as the case described in section IV-A,
this problem can be solved.
V. N UMERICAL EXAMPLES
In order to verify our approach, we show some numericalexamples. Fig.4 shows the target system. ΣR is placed at
the contact point between the object and the base in theconfiguration shown in Fig.4. ΣO is placed at geometriccenter of the object. The object is a ball with radius of
0.1[m]. The robotic hand is composed of 4 fingers which
are same forms and have 4 joints. The length of every link
is set to 0.1[m] and the gravity center of every link is set
to geometric center of the link. Mass of every link is set to0.051[kg]. The actuator attached with every joint is all sameand attaches a gear with 1/40 of reduction ratio. The basepositions of fingers are set to [−0.1 −0.05 0]T , [0.05 0.1 0]T ,[0.05 0 0]T and [0.05 − 0.1 0]. We use the operation range
shown in Fig.1. We set nf ric = 16.
First, we consider Problem 1 for the case shown in Fig.5.
Here, we consider to grasp an object by only Finger 1 and3. ΣO is placed at [0.075 0 0.1]T . The contact position ofFinger 1 is [−0.022 0 0.074]T . The contact positions ofFinger 3 are [0.075 0 0]T and [0.171 0 0.072]T . μij is set
to 0.3. RVDS is not used and REFS is set as follows (Here,
we set mg = 1 [N]):Sref = {wex |w ex = [0 0 − mg 0 0 0]T }.

(31)

First, we consider the case when all contact points are Fpoints. The obtained TVMP is shown in Fig.6 (a). Denoting
the components of ṙ by [ẋ ẏ ż φ̇ ψ̇ θ̇]T and letting
β = [0.697 0 − 0.174 0 0.697 0]T , the TVMP is containedin the space expressed by [β ẏ φ̇ θ̇]T . The TVMP is notfull dimensional. Since the overall set can not be shown, theTVMP shown in Fig.6 (a) is the set mapped to θ̇ = 0. From
Fig.6 (a), it can be seen that the directions of generable velocities are limited for the geometrical constraints and large
object velocity can be generated in the specific directions.
Note that TVMM is zero but if calculating TVMM in the 4
D space, TVMM = 84.2.
In the first case, the translational velocity can be generated
only in y direction. Then, changing one of F-points to Spoint, we try to generate translational velocity in anotherdirection. We consider the case when the contact point on
Finger 3 closer to the base is S-point. H Ts ˆṗCs is set to[1 0 0 ]T (note that for easy to understand the directionof ˆṗCs , H Ts is added). Here, we only consider translational
directions (we set RVDS is φ̇ = ψ̇ = θ̇ = 0). The obtained
TVMP is shown in Fig.6 (b). From Fig.6 (b), it can beseen that velocity in another translational direction can be1.6

Finger2
Finger3

1.6
1.4

1.4

1.2

0.1
0.1

Finger4

Finger1

−0.1

−0.1

0
x [m]

Fig. 4. Target system in numerical examples

1

0

0
0.1

Fig. 5.

0.2

y [m]

0.8
1

α

1.2
α

z [m]

0.2

μ=0.1
μ=0.2
μ=0.3
μ=0.4
μ=0.5
1.25

1
0.8

1.5

z [m]

1.75

2

0.6
1

mg=0
mg=1
mg=2
mg=3
1.25

(a)

Case 1

1.5

z [m]

1.75

2

(b)

Fig. 9. TVMMP (α) for (a) various frictional coefficients and (b) various
magnitudes of the external force.
1
z [m/s]φ [rad/s]

5

T T T
is set such that ṗCs is in parallel with As [(J +c Gc ) I]T[0 0 1] according to the object motion. We use the same
REFS (31) as the previous case. RVDS is set as follows:

0

.

0

−1

.

−5

0

−5

0

5

−5

β

2

5

0
.
y [m/s].
x [m/s]

(a)

1

0
−1 .

4

y [m/s]ˆṙ v3 = [1 − 1 1 0 0 0]T , ˆṙ v4 = [−1 − 1 1 0 0 0]T , ˆṙ v5 = o.

(b)Fig. 6. TVMP: (a) when all contact points are F-points, (b) when contact
point on Finger 3 closer to the base is S-point

0.4

0.4

z [m]

z [m]

0.2

0.2

0.2

0.2
0
−0.2

0
−0.2

0 y [m]00 y [m]00.2 −0.2

x [m]

x [m]

(a)
Fig. 7.

0.2 −0.2(b)Case 2 : (a) initial state, (b) final state
2

α

1.5

1

0.5
1

no S−point
S−points exist
1.25

1.5

z [m]

1.75

Prvd = {ˆṙ|ˆṙ = Σ5i=1 λi ˆṙ vi , Σ5i=1 λi = 1, λi ≥ 0},
ˆṙ v1 = [1 1 1 0 0 0]T , ˆṙ v2 = [1 − 1 1 0 0 0]T ,2

Fig. 8. TVMMP (α). The solid line denotes the results when the contact
points on Finger 2 and 4 are N-points, and the dash line denotes the results
when the contact points on Finger 2 and 4 are S-points.

generated. The object velocity can be generated only inẋ > 0 direction due to the direction of ˆṗCs .
Next, in order to see the effects of changes of configuration, friction and external force, we consider Problem
2 for the case shown in Fig.7 where the object moves
in positive z direction. The contact points on Finger 1
and 3 are set to be F-point. The contact points on Finger
2 and 4 are set to be S or N-point. The contact pointon the base is set to be D-point if it exists. The contact
positions at the fingers and the base in the initial state are
set to [−0.077 − 0.038 0.086]T , [0.033 0.083 0.14]T ,[0.097 0 0.074]T , [0.033 − 0.083 0.14]T , and [0 0 0]T . ˆṗCs

Note that ṙ v5 can be omitted in the computation.
We compute TVMMP when the object moves from theinitial state to the final state shown in Fig.7. The results
are shown in Fig.8 whose horizontal axis denotes the z
coordinate of ΣO , and whose vertical axis denotes TVMMP(α). From Fig.8, it can be seen that as the object moves
in positive z direction, TVMMP becomes large at first andbecomes small after that. It can be also seen that there is little
difference between the cases when there is no S-point andwhen S-points exist. TVMMP is large since around the initialstate, every finger (especially finger 1 and 3) is in the state
where large contact velocity and force in any direction canbe generated. Around the final state, the elbow (third joint)of every finger is extended, and then TVMMP is small.
In order to investigate the effects of friction and external
forces, we compute TVMMP for various frictional coefficients and various magnitudes of external force. We computein the case when the contact points on Finger 2 and 4 are
N-points. We consider the following two cases: (1) μij ischanged to 0.1, 0.2, 0.3, 0.4, and 0.5 while mg = 2, (2)mg is changed to 0, 1, 2 and 3 while μij = 0.3. Fig.9 (a)shows the results when frictional coefficient changes, and(b) shows the results when mg changes. From Fig.9 (a), it
can been seen that TVMMP becomes large with the increaseof the frictional coefficient. When the frictional coefficientincreases, the range of applicable contact forces becomeslarge. Therefore, we can balance external force by smaller
joint torques, and then generable object velocity becomeslarge. From Fig.9 (b), it can been seen that TVMMP becomes
small with the increase of magnitude of external force. When
external force becomes large, necessary joint torques to keep
grasping becomes large. Therefore, generable object velocity
becomes small.
VI. C ONCLUSION
In this paper, we have proposed a new approach to analyze
manipulability for general grasping system. Using an opera-

tion range of every actuator attached with joint, we define a
set of generable joint torque and velocity, TVS (joint torquevelocity pair set). Also, we define a set of external force
required to be balanced (REFS (required external force set))
and a set of direction of object velocity required to evaluate((RVDS) required velocity direction set). Using these sets,
we have derived new manipulability set (joint torque-velocity
pair based manipulability set (TVMS)) and measure (joint
torque-velocity pair based manipulability measure (TVMM))
which can evaluate generable object velocities, including theeffect of force/torque needed to keep grasping.
As shown in the numerical examples, this approach canevaluate the effects of friction, contact state and external
force, which could not be included in the conventional
approaches.
A PPENDIXN OMENCLATUREScol
A column vector or matrix formed by the following elements.
diag A block diagonal matrix.
N
Number of fingers.
Number of joints of the ith finger (i = 1, 2, · · · , N ).
Mi
Number of contact points on the ith finger.
Li
M
Number of total joints (= ΣN
i=1 Mi ).
L
Number of total contact points (= ΣN
i=1 Li ) .
D
3/6 in 2/3 dimensional space.
d
2/3 in 2/3 dimensional space.
Reference coordinate frame.
ΣR
Object coordinate frame fixed at the object.
ΣO
Stvij TVS at every joint (j = 1, 2, · · · , Mi , i = 1, 2, · · · , N ).
τij
The jth joint torque of the ith finger (j = 1, · · · , Mi , i = 1, · · · , N ).
The jth joint angle of the ith finger (j = 1, · · · , Mi , i = 1, · · · , N ).
qij
uij = [τij q̇ij ]T .
τij vk (≥ 0)Torque component of the kth vertex of Stvij .
q̇ij vk (≥ 0)Velocity component of the kth vertex of Stvij .
ntvij Number of the vertices of Stvij .
y ij= [τij τij max τij min q̇ij q̇ij max q̇ij min ]T .
Stv TVS aggregating TVS at every joint.
y=col[yij ].
Srvd RVDS.
r
Position and orientation of ΣO (∈ RD ).
Arvd Matrix constituting inequality constraints representing Srvd .
nrvd Number of the rows of Arvd .
Sref REFS.
w
Resultant force and moment applied to the object (∈ RD ).
wex External wrench (∈ RD ).
wvi The ith vertex of Sref .
nref Number of the vertices of Sref .
Cij The jth contact point of the ith finger (j = 1, · · · , Li , i = 1, · · · , N ).
ΣCF Coordinate frame fixed at the contact point on finger corresponding to Cij .
ijΣCO Coordinate frame fixed at the contact point on object corresponding to Cij .
qi
p
O
p

Iij

nij
tkij
T ij
ṗCij

ij

= [qi1 qi2 · · · qiMi ]T (∈ RMi ).
Position of the origin of ΣO (∈ Rd ).
Position of the origin of ΣIij (I ∈ {CF , CO }) (∈ Rd ).
Unit normal vector (directing to the inward of the object) at Cij .
Unit tangential vector at Cij (k ∈ {1, 2}).
= [t1ij t2ij ]T (∈ Rd−1×d ). [t1ij ]T in 2 dimensional space.
= ṗC
− ṗC
.
Oij

ṗCsij = H sij ṗCij .

J
G
A
LK
Lc
Ls
Ld
Hk

H ij
q
p

I

Ak

Fij

=diag[col[J F1j ] · · · col[J FN j ]] ∈ RLd×M .
= [G11 G12 · · · GN LN ] ∈ RD×Ld .
= [J − GT ] ∈ RLd×(M +D) .
Number of K-points (K ∈ {F, N, S, D}).
= LF d + LN + LS .
= LS (d − 1).
= LD .
L ×Ld=diag[H(k ∈ {c, s, d}).
ij ] ∈ R kjwhen H kij exists
H kij
=
.
nothing when H kij doesn’t exist= col [q i ] (∈ RM ).
=col[p ] (I ∈ {CF , CO }) (∈ RLd ).
Iij= H k A (k ∈ {c, s, d}).

ṗCs

= col[ṗCsij ].

f ijContact force vector (∈ Rd ).
f kij = H kij f ij (k ∈ {c, s}).
f
fk

τ
τk
wk
nfij
μij
ˆ
ṗ

=col[f ij ] ∈ RLd .
= H k f = col[f kij ] (k ∈ {c, s}).

= col [τij ] (∈ RM ).
τ corresponding to f k (k ∈ {c, s}).
Resultant force and moment by f k (k ∈ {c, s}).
Normal force component of f ij (= nT
ij f ij ).
Frictional coefficient at Cij .
Csij = ṗCsij /|ṗCsij |.
Flin Linearized frictional constraints.
τ max = col[τij max ].
τ min = col[τij min ].
q̇ max = col[q̇ij max ].
q̇ min = col[q̇ij min ].
Prvd RVDS expressed by convex polyhedron.
ˆṙ
The direction of object velocity.
ˆ
The ith vertex of Prvd .
ṙ vi
nvrvd Number of the vertices of Prvd .
T
T
T
T
T
x
= [f Tṙ T α]T .
c τ max τ min q̇ max q̇ min q̇
ˆ
=
ṗ
/|
ṗ
|.
ṗ
Cs
Cs
Cs
ˆ
ρ
The magnitude of sliding velocity in ṗ
Cs direction.
TTxs= [x ρ] .

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