Effect of torque-velocity relationship on
manipulability for robot manipulators
Tetsuyou Watanabe
Assistant Professor
School of Mechanical Engineering
College of Science and Engineering
Kanazawa University
Kakuma-machi, Kanazawa, 920-1192
Email: te-watanabe@ieee.org

ABSTRACTThis paper presents novel manipulability analysis for robotic manipulators, taking the effect of generating joint
torques on generable joint velocities and vice versa into consideration. The conventional manipulability is analysis
in velocity domain and cannot concern force effect such as gravity of payload and external forces exerted on theendeffector. Gravitational force has been regarded that it just changes the origin of the manipulability ellipsoid
expressing the set of generable tip velocities, and its evaluation (its volume) does not change. However, if robot
grasps a heavy object, the robot cannot move with the same speed as the case of grasping a light object, because thepower of the robot is limited. It indicates that the robot performance evaluation by conventional manipulability has
serious problem that the force effect cannot be included. The power of the robot is determined by the operation rangeof every actuator, which tells us the relationship between generating torque/velocity and addable velocity/torque.
Then, this paper presents novel manipulability analysis which can take the force effect into consideration, basedon the torque-velocity relationship. This analysis shows that manipulability is influenced by payload, gravitational
force and external forces.

1

INTRODUCTION
Manipulability is a well known concept to evaluate the performance of robotic manipulator [1]. It is defined as theset of generable velocities of endeffector in the task space when the set of generable joint velocities is given. When thegiven set of joint velocities is a unit ball, the set of the endeffector velocities becomes an ellipsoid. The ellipsoid is called
manipulability ellipsoid. Its volume is a quality measure to evaluate the performance of manipulators in velocity domain,and called manipulability measure. Based on the manipulability, many quality measures such as condition number areproposed [1].
However, manipulability can not include the effects of gravity of link, payload and external forces. In actual systems,
manipulators are sometimes used under influence of gravity. If we do not concern the effect of gravity in such a case, we will
get the result that the performance when moving a heavy object is the same as the performance when moving a light object.
Then the performance evaluation becomes invalid. On the other hand, in some tasks such as tracing, manipulator is required
to keep contact with environment. In such a case, we should evaluate the performance, concerning the effect of the contact
forces. However, the conventional manipulability analysis can not include such a force effect. For example, gravitational
force has been regarded that it just changes the origin of the manipulability ellipsoid expressing the set of generable tip
velocities, and its evaluation (its volume) does not change.
Concerning described the above, this paper presents a new manipulability based approach to evaluate performance ofmanipulator in velocity domain, which can include force effect. For the purpose, we use an operation range (see Fig.1)of actuator attached with every joint of robot (operation range is originally used for motor selection). The operation range
provides not only the information about how much of magnitudes of torque and velocity the actuator can stably generate, butalso the relationship between generating velocity/torque and addable torque/velocity. If using this relationship, we can takethe magnitude of required joint torque into consideration when evaluating the generable tip/endeffector velocities.
The contributions of this paper are as follows.
1

Tetsuyou Watanabe:Paper JMR-10-1163

2 MANIPULABILITY TAKING FORCE EFFECTS INTO CONSIDERATION
1) We derive a manipulability convex polyhedron corresponding to the conventional manipulability ellipsoid, taking force
effects into consideration.
2) We derive 3 kinds of corresponding manipulability measures: the volume of the manipulability convex polyhedron, themagnitude of generable tip velocity which is generable in any direction, and the maximum magnitude of generable tip
velocity.
3) We also evaluate how much magnitude of tip forces the robot can generate, considering the effect of generating/required
tip velocities.
First, we derive required joint torques to compensate gravity effect and external forces, and keep configuration/pose ofmanipulator. Using the required joint torques and the operation range, we derive usable (actually generable) joint velocities.
Using the usable joint velocities, we derive a set of generable tip/endeffector velocities. The set is described by a convex
polyhedron. Based on this set, we derive the new manipulability measures described the above.
In some tasks such as assembling, drilling and deburring, the tip of the manipulator is required to contact with environment such as a table, and to generate tip force. To include the effect of the tip force, we introduce required external force set(REFS) [2,3] which is defined as a set of external forces required to be compensated. Based on the REFS, we derive required
joint torque to compensate any force contained in REFS, and derive a set of generable endeffector/tip velocities and thecorresponding manipulability measures. In this paper, we present an approach when REFS is given as a convex polyhedronor ellipsoid.
We also present a way to evaluate the generable tip forces. For example, if we want to grind a cup with a certain constant
speed, large contact force for keeping contact would be preferable. Then, to include the effect of endeffector/tip velocities,
we introduce a required velocity set (RVS) defined as a set of required endeffector/tip velocities to be generated in a giventask. Based on the RVS, we derive required joint velocity to generate any endeffector/tip velocity contained in RVS, andderive a set of generable tip forces and the corresponding manipulability measures.
Manipulability analysis was extended to various fields such as robotic hand [4–6] and musculo-skeletal analysis [7].
However, there is no manipulability research for manipulator from the viewpoint of power to include both force and velocity
effects.

2

Manipulability taking force effects into consideration
In this paper, we consider a serial n link manipulator with n joints. A payload is attached to the tip of the manipulator.
In order to evaluate the effect of the payload, external forces and gravity of manipulator, we introduce the following set:
Joint Torque-velocity Pair Set (TVS): The set of generable joint torques and velocities at each joint, given by the corresponding actuator and gear specifications, is named joint torque-velocity pair set (TVS).
The specification for actuators (operation range) is usually given with respect to the absolute values of torque andvelocity. We express generable maximum absolute values of joint torque and velocity with |τimax | and |q̇imax |. Let |q̇iUmax |(≥ 0)
be the usable maximum absolute value of joint velocity, determined by currently generating joint torque. Similarly, let |τiUmax |(≥ 0) be the usable maximum absolute value of joint torque, determined by currently generating joint velocity. We describe
this relationship with the following functions.
τ→q̇(|τi |),q̇→τ

(|q̇i |) = (ξi|q̇iUmax | = ξi|τiUmax | = ξi

(1)
τ→q̇ −1) (|q̇i |).

(2)

τ→q̇

This function ξican be derived from the actuator and gear specifications. For example, if using DC motor under the
τ→q̇
constant nominal voltage, ξican be derived by utilizing maximum speed under the voltage, maximum torque, and torqueτ→q̇
speed constant. An example of TVS is the area surrounded by the rigid lines in Fig.1, and ξigives the rigid lines. In thiscase, the TVS is convex. If we can not get the information about torque-speed constant, we use power. Let ψi be the constant
τ→q̇
can be expressed bypower for the evaluation. Then, ξi
τ→q̇
ξi
=



ψi /|τi | ψi ≤ |τi ||q̇imax |.
otherwise|q̇imax |(3)Then, we consider the following problem:
Problem 1: Suppose that TVS is given. In this case, derive the set of generable endeffector/tip velocities, Sev , andcorresponding (manipulability) measures.
First, we consider the relationship between joint velocity and endeffector/tip velocity. Let joint angle vector be q
= [q1 q2 · · · qn ]T , and the tip position of the manipulator r∈ R D where D = 3 in two dimensional space and D = 6 in three2

Tetsuyou Watanabe:Paper JMR-10-1163

2 MANIPULABILITY TAKING FORCE EFFECTS INTO CONSIDERATION
dimensional space. Then, the relationship is expressed by:ṙ = J q̇,(4)where J ∈ R D×n denotes Jacobian matrix. From (4), we getq̇ = J + ṙ + Ekq ,(5)where J + denotes the psedo-inverse matrix of J, E denotes an orthogonal matrix whose columns form bases of the null spaceof J, and kq denotes an arbitrary vector.
From (4) and the principle of virtual work, the following relation is obtained:τq = J T f ,(6)where τq ∈ R n denotes joint torque vector corresponding to q̇, and f ∈ R D denotes tip force corresponding to ṙ.
Let the gravity term of the manipulator be gm = ∂ ∑i Ui T /∂q where Ui is the potential energy of the ith link of themanipulator due to the gravitational force. Let also the mass of the payload be m p , gravitational acceleration vector be g, andthe other external force exerted on the tip be f ex . Then, overall joint torque, τ = [τ1 τ2 · · · τn ]T ∈ R n , is expressed byτ = gm − J T m p g − J T f ex .

(7)

where −J T f ex is the term for compensating f ex .
If f ex = o, from (1), the usable maximum joint velocity vector, q̇Um = [q̇1Umax q̇2Umax · · · q̇nUmax ]T ∈ R n is expressed by⎤q̇1Umax⎢ q̇2Umax ⎥⎥⎢τ→q̇q̇Um = {⎢ . ⎥ , |q̇iUmax | = ξi (|τi |), τ = gm − J T m p g}.
⎣ .. ⎦⎡(8)q̇nUmaxThen, from (4), and (8), Sev is expressed by

Sev = {ṙw |ṙw = MJ q̇, −|q̇iUmax | ≤ q̇i ≤ |q̇iUmax |,τ→q̇|q̇iUmax | = ξi(|τi |), τ = gm − J T m p g}.

(9)Since q̇Um is uniquely determined and the other equations and inequalities constituting Sev are linear, Sev is a convex
polyhedron. Therefore, we call Sev joint torque-velocity pair based manipulability polyhedron (TVMP). Here we will consider the following 3 manipulability measures: 1) αv : the volume of Sev , 2) αallmax : the maximum tip velocity magnitude
which is generable in any arbitrary direction, and 3) αmax : the maximum magnitude of generable tip velocity. The first one
is typical measure. The second one corresponds to the radius of the hypersphere inscribed in Sev . The third one corresponds
to the radius of the hypersphere circumscribed in Sev . Note that in order to treat translational and rotational velocities atthe same time, we use the weight matrix, M, such that ṙw = Mṙ can appropriately evaluate the (weighted) endeffector/tip
velocity in every direction: for example, every direction can have the same unit. Note also that the choice of M affects theevaluation, and then we should carefully choose M such that the evaluation can be valid and reasonable.
Sev can be expressed using its extremes:
nsev
sev
Sev = {ṙw |ṙw = Σni=1
λi ṙvi , Σi=1
λi = 1, λi ≥ 0},(10)

where ṙvi denotes the vertex of Sev and nsev denotes the number of the vertices. This expression is called V-representation [8].
Generally, this transformation can be done by a programming method such as double description method [9]. However, since3

Tetsuyou Watanabe:Paper JMR-10-1163

2 MANIPULABILITY TAKING FORCE EFFECTS INTO CONSIDERATION
every |q̇iUmax | is unique, every V-representation for q̇i can be easily derived as follows. The generable joint velocity set forevery joint is expressed by{q̇i = −η1 |q̇iUmax | + η2 |q̇iUmax |, Σ2j=1 η j = 1, η j ≥ 0}

(11)

Since the generable joint velocity vector set is given by the direct sum of the generable joint velocity sets for all joints, it canbe expressed bynn{q̇ = Σ2i=1 λi q̇vi , Σ2i=1 λi = 1, λi ≥ 0}
where q̇vi denotes the vertices of the set. Then, the V-representation for Sev is expressed bysev
sev
Sev = {ṙw |ṙw = Σi=1
λi ṙvi , Σi=1
λi = 1, λi ≥ 0},

n

n

(12)

where
ṙvi = MJ q̇vi ,nsev = 2n .

Let ∫ be a simplex in d dimensional space, and vsi (i = 0, · · · , d) be its vertex. In this case, the volume of the simplex isgiven byV (∫ ) =| [vs1 − vs0 vs2 − vs0 · · · vsd − vs0 ] |
.
d!

(13)

Therefore, if polyhedron Sev can be decomposed into simplices, we can calculate the volume of Sev . One of the famous
methods for the decomposition is triangulation method using delaunay triangulation [8, 10]. Then, letting ∫i (i = 1, · · · , nssev )be the decomposed simplices of Sev , the volume of Sev (TVMM), αv , is calculated bynssev
V (∫i ).
αv = Σi=1(14)

Note that qhull [11] can also compute the volume of the convex polyhedron (αv ).
Next, we consider the maximum tip velocity magnitude which is generable in any arbitrary direction, αallmax . From (12),we will derive H-representation (expression by half planes) of Sev .

Sev = {ṙw |Aw ṙw ≤ bw }.

(15)

where Aw and bw are constant matrix and vector resulted from the transformation. Cdd librarly [8] or qhull [11] can beused to derive this H-representation. If letting Aw = col[aTwi ] and bw = col[bwi ] where col denotes a column vector or matrix
formed by the following elements, every aTwi ṙw ≤ bwi expresses every face of Sev . The distance between the face aTwi ṙw ≤ bwi
and the origin can be computed as follows.
dwi = bwi /|awi |.
Therefore, αall
max can be obtained by
αallmax = min dwi

(16)

i

Lastly, we consider the maximum magnitude of generable tip velocity αmax . From (12), it can be obtained by
αmax = max |ṙvi |.
i

4

(17)
Tetsuyou Watanabe:Paper JMR-10-1163

2.1 When tip is constrained

2 MANIPULABILITY TAKING FORCE EFFECTS INTO CONSIDERATION

From (5), (9) and (15), its alternative way is to solve the following convex quadratic programming problem.
αmax = maxṙ,kqṙT M T Mṙ = ṙTw ṙw− q̇Um ≤ J + ṙ + Ekq ≤ q̇Um

subject toorαmax = maxṙ,kqṙT M T Mṙ = ṙTw ṙw

subject to Aw ṙw ≤ bw

(18)

Remark that the required joint torques to compensate payload, link gravity, and external forces determine the range offeasible joint velocities. If the required joint torques are large (close to their maximum), the range of feasible joint velocities
and the generable tip/endeffector velocities would be small. If the required joint torques exceed their maximum, the robot
cannot keep its configuration and then the robot cannot generate tip/endeffector velocities.
2.1

When tip is constrained
Consider the case when a task for manipulator is given, and its tip is constrained. In such a case, We need to generate tip
forces, for example, to keep the contact. In order to consider the effect of the tip forces, we introduce the following Required
External Force Set (REFS):
Required External Force Set (REFS): The set of the endeffector’s external force required to be compensated is named
required external force set (REFS). We suppose that the we can cope with any external force if any arbitrary external force
contained in REFS can be compensated.
REFS is assumed to be given by a convex polyhedron or an ellipsoid:
n

n

re f
re f
Srepolf ={ f ex | f ex = Σi=1
κi f vi , Σi=1
κi = 1, κi ≥ 0},

(19)

T
T
Sreelipf = { f ex | f ex M f M f f ex ≤ 1}.

(20)

Here, f vi denotes the vertex of the convex polyhedron, nre f denotes the number of the vertices, M f denotes a weight matrix.
Note that f ex can have moment component. M f is added similarly to M in (9) such that every component of f ex can beequally treated: for example, every direction can have the same unit, or the magnitude of external forces can be normalized.
Note also that M f has to be selected such that the choice of M f can be consistent with the choice of M. One of the ways
might be M f = M T . Another way will be described in the section 3 for numerical examples.
Using this REFS, we consider the following modified problem:
Problem 2: Suppose that TVS and REFS are given. In this case, find the set of generable endeffector velocities, Sev ,such that the velocity contained in Sev can be generated even if any f ex contained in REFS is exerted on the manipulator. In
addition, find the corresponding (manipulability) measures: αv , αall
max and αmax .
First, we consider the case when REFS is given as a convex polyhedron Srepolf (19). We derive minimum required joint
torque to compensate any f ex contained in REFS. If we can compensate every f vi , we can compensate any f ex containedin REFS. Therefore, from (7), the minimum required joint torque τr = [|τr1 | · · · |τrn |]T can be obtained by the following
problem.
minτr|τrcι | ≤ τr
τrcι = gm − J T m p g − J T f vι(ι = 1, · · · , nre f ).

(21)

Since τrcι is constant for ι, we calculate τrcι for every ι and just compare its magnitude at every joint. Then, we can get
the the minimum required joint torque τr .
5

Tetsuyou Watanabe:Paper JMR-10-1163

2.2 Force evaluation

2 MANIPULABILITY TAKING FORCE EFFECTS INTO CONSIDERATION

Next, we consider the case when REFS is given as an ellipsoid Sre f (20). Let f̂ ex = M f f ex . Then, from (7), the minimum
required joint torque τr has to satisfy the following condition to compensate the external force f̂ ex
elip

A f 1 f̂ ex + A f 2 τr ≤ b f

(22)

where
Af1 =

−J T M −1−I−gm + J T m p g
f, Af2 =
, bf =.
−1T−I
gm − J T m p g
J Mf

Let A f 1 = col[aTf1i ]. If we had already obtained τr , (22) could express a convex polyhedron for f̂ ex , and a f 1i could express
the normal of the face of the convex polyhedron. Generally, the distance between the origin and a hyperplane aT x = b isgiven by b/|a|. Therefore, if holding the following conditions, we can compensate any arbitrary f ex ∈ Sreelipf .
col[|a f 1i |] + A f 2 τr ≤ b f .

(23)Here, taking into consideration that A f 2 is constructed by identity matrices, from (23), we can getτr ≥ b̂ f(24)

where b̂ f is the vector resulted from the transformation of the inequality, and indicates the minimum required joint torque.
Now, we have minimum required joint torque. From (1), we can get the usable maximum joint velocity vector, q̇Um .
Then, we can use the same way as the discussion from (10) to (18), in order to get Sev , and its corresponding manipulability
measures: αv , αall
max and αmax .
2.2

Force evaluation
In tasks such as deburring and drilling, manipulator moves with its endeffector contacting with environment. In thiscase, it is also important to evaluate how magnitude of tip/endeffector force can be generated and in which directions theforce can be generated. It is also important to consider manipulator motion, namely, manipulator speed. To take the effect ofmanipulator motion into consideration, we define the following set:
Required Velocity Set (RVS): The set of endeffector’s velocities required to be generated is named required velocity
set (RVS). We suppose that any endeffector velocity required in a given task can be generated if any endeffector velocity
contained in RVS can be generated.
Using RVS, we will evaluate tip/endeffector forces in the case when manipulator moves with a velocity contained in
RVS. RVS is assumed to be given by a convex polyhedron or an ellipsoid:
pol
rv
rv
Srvs
= {ṙ|ṙ = Σni=1
λi ṙvi , Σni=1
λi = 1, λi ≥ 0},elip
T T
Srvs = {ṙ|ṙ M Mṙ ≤ 1}.

(25)
(26)

where ṙvi denotes the vertex of RVS and nrv denotes the number of vertices of RVS. M denotes a weight matrix, for example,
to consider the difference between the units of translational and rotational velocities, or to normalize the magnitude of thevelocity.
Using this RVS, we consider the following problem:
Problem 3: Suppose that TVS and RVS are given. In this case, derive the set of generable tip/endeffector forces Se fsuch that the force contained in Se f can be generated even if any tip/endeffector velocity ṙ contained in RVS is generating.
In addition, find the corresponding (force manipulability) measures: βv (the volume of Se f ), βallmax (the maximum magnitudeof tip/endeffector force which is generable in any arbitrary direction) and βmax (the maximum magnitude of generable
tip/endeffector force).
pol(25). We derive required joint velocity
First, we consider the case when RVS is given as a convex polyhedron Srvs
to generate any tip velocity ṙ contained in RVS. If we can generate every ṙvi , we can generate any ṙ contained in RVS.
Therefore, from (5), the required joint velocity q̇r = [|q̇r1 | · · · |q̇rn |]T holds the following constraints.
−q̇r ≤ J + ṙvι + Ekq ≤ q̇r6(ι = 1, · · · , nrv ).

(27)
Tetsuyou Watanabe:Paper JMR-10-1163

2.2 Force evaluation

2 MANIPULABILITY TAKING FORCE EFFECTS INTO CONSIDERATION

Next, we consider the case when RVS is given as an ellipsoid Srvs (26). From (5), the required joint velocity q̇r has tosatisfy the following condition to generate ṙ (or ṙw = Mṙ):elip

Ar1 ṙw + Ar2 kq + Ar3 q̇r ≤ o

(28)

where

Ar1 =

J + M −1,
−J + M −1

Ar2 =

E
,
−E

Ar3 =

−I
.
−I

Now, we remove kq from (28). First, we transform (28) to its V-representation.
ArAr
xr = Σni=1
λi xrvi , Σni=1
λi = 1, λi ≥ 0

where
⎤
ṙw
xr = ⎣ k q ⎦ ,q̇r⎡

and xrvi denotes the vertex and nAr denotes the number of the vertices. We eliminate the terms of kq by multiplying thefollowing matrix from the left side
I OO
.
OO I
After that, we transform it to its H-representation.
Âr1 ṙw + Âr3 q̇r ≤ b̂r(29)This two transformations can be done by cdd library [8].
Let Âr1 = col[âTr1i ]. Now, if q̇r had been given, (29) could express a convex polyhedron for ṙw , and âr1i could express
the normal of the face of the convex polyhedron. Therefore, similarly to the previous subsection, if holding the following
elip
conditions, we can generate any arbitrary ṙ ∈ Srv .
col[|âr1i |] + Âr3 q̇r ≤ b̂r .

(30)

Now, we have H-representation for q̇r , similarly to (27).
Here, we will assume that the relationship between joint torque and velocity (2) can be represented by
Atq1i |τiUmax | + Atq2i |q̇ri | ≤ btqi

(31)

where Atq1i Atq2i and btqi are the matrices and vector expressing the boundary of the region. If aggregating this relationshipfor all joints, we get

Atq

τUm
≤ btq
q̇r

(32)


Atq = diag[Atq1i ] diag[Atq2i ] , btq = col[btqi ].
7

Tetsuyou Watanabe:Paper JMR-10-1163

3 NUMERICAL EXAMPLES
Then, with the same way as the way of form (28) to (29), from (32) and (27) if using Srv or (30) if using Srv , we get
the usable range of joint torques
pol

AUtau τUm ≤ bUtau

elip

(33)

Then, from (7) and (33), Se f is expressed by

Se f = { f̂ ex | − τUm ≤ τc ≤ τUm , AUtau τUm ≤ bUtau
τc = gm − J T m p g − J T M −1
f f̂ ex }.

(34)

Here, Sev is expressed by H-representation. It can be transformed to V-representation with cdd librarly [8].
nsns

Se f = { f̂ ex | f̂ ex = Σi=1e f λi f̂ exvi , Σi=1e f λi = 1, λi ≥ 0}.

(35)

where f̂ exvi denotes the vertex and nse f denotes the number of the vertices.
If J has row full rank, we get+Tf̂ ex = (J T M −1
f ) (−τc + gm − J m p g).

(36)

In this case, we have alternative way for the transformation for the V-representation of Se f . First, we will get the Vrepresentation of the range for τc using (33) and −τUm ≤ τc ≤ τUm . Then, from (36), we can get V-representation for
Se f .
all
The manipulability measures, βv , βmax and βallmax can be obtained by the same way as the deviation of αv , αmax and αmax
(see from (12) to (18)).

3

Numerical examples
In order to verify our approach, we show some numerical examples. For the convenient, we only considered translational
directions in task space. First, we considered Problem 1 for 2-link planar manipulator shown in Fig.2. Base position was
located at origin. The length of every link was set to 0.1[m] and the gravity center of every link was set to the geometriccenter of the link. Mass of every link was set to 0.051[kg]. Gravitational acceleration vector was set to [0 − 9.8]T [m/s2 ].
We used the operation range shown in Fig.1. The actuators are all the same.
Here, we considered the case when the tip position changes from origin to [0.2 0]T as shown in Fig.2. We calculated
the proposed new manipulability polyhedron (TVMP) and measures, αv , αall
max , and αmax when mass of the payload attachedto the tip of the manipulator, m p , is 0.0[kg], 0.1[kg], 0.2[kg], 0.3[kg] and 0.4[kg]. For the comparison, we also calculated
original manipulability ellipsoid and measure.
Fig.2 (a)∼(f) and Fig.3 (a)∼(c) show the results. Note that as it can be seen from Fig.3, when m p = 0.4 and x coordinateof the tip position is around 0.2, αv and αall
max are 0 (and αmax becomes around 0.2). It means that manipulator can not generate
endeffector velocity in arbitrary directions due to the large mass of the payload. In other words, endeffector velocity can begenerated in only the specified direction as shown in Fig. 2 (f). Then, αv , αall
max and αmax decreases largely. Comparing Fig.2(a) with Fig.2 (b)∼(f), it can be seen that the direction of maximum (and minimum) generable velocity is different. Also, it
can be seen that with the increase of mass of payload, maximum generable velocity becomes smaller. Especially, the velocity
in y direction in which gravitational force is applied decreases. From Fig.3, it can be seen that the tip position whose αv ismaximum becomes closer to that for original manipulability measure with decrease of the payload mass. Note that we will
not compare the magnitudes of new and original manipulability measures since the given set of generable joint velocities isdifferent although maximum generable joint velocity is the same. The tip positions for maximum αv , αall
max , and αmax are,andα
all different. With the increase of mass of the payload, all manipulability measures, αv , αall
max become smaller.
max
αmax increase monotonously with the incase of x coordinate of the tip position while the other measures increases firstly but
decrease after reaching their maximums. It is considered to be related with the shape of the TVMP. These results indicatethat we can not evaluate the effect of the payload mass by the original manipulability while the proposed manipulability andits measures can evaluate the effect.
Next, we considered√ Problem 2 for the manipulator shown in Fig.4. Its tip moved on the spherical environment whose
center is [0 0 − 0.1/ 2] and whose radius is 0.1. Note that in Fig.4, the part (z ≥ 0) of the environment is only shown. We
8Tetsuyou Watanabe:Paper JMR-10-1163

3 NUMERICAL EXAMPLES
call the joints and the links, joint 1, 2, 3, 4, and link 1, 2, 3, 4, respectively, in the order of closeness to the base side. The
lengths of link 1 and 2 were set to be 0. The lengths of link 3 and 4 were set to be 0.1[m] and the gravity center of eachlink was set to the geometric center of the link. Masses of link 3 and 4 were set to 0.051[kg]. Mass of the payload was set
to 0.1[kg]. Base position was located at [−0.1 0 0]T . We used the operation range shown in Fig.1. The actuators are all thesame.
Here, we considered to move the tip in parallel with x axis, contacting with the environment. Let θ be the angle between z
axis and outward normal vector of the environment at the contact point between the environment and the tip. The manipulator
moved the tip such that θ changes from 0 to π/4.
The maximum contact force was supposed to be 1[N], and the maximum frictional coefficient was supposed to be 0.3.
Under these suppositions, REFS was set as follows:

Srepolf ={ f ex | f ex = RE n fn , fn = κ1 + κ2 ·0,Σ2i=1 κi = 1, κi ≥ 0},T
2 2
Sreelipf ={ f ex | f ex = RE t f t , f t f t ≤ 0.3 f n },

where
⎤
⎡
⎤
⎡ ⎤
1
0
0
10
0
R = ⎣ 0 cos(θ) sin(θ) ⎦ , E t = ⎣ 0 1 ⎦ , E n = ⎣ 0 ⎦ .
0 − sin(θ) cos(θ)
001⎡

We used Sre f and Sre f together. We derived the constraints (24) for every vertex of Sre f . Then, we can get the combined
pol

elip

pol

problem for Sre f and Sre f . By solving the combined problem, we derived minimum required joint torques. Since the tip
moves contacting with the environment, we considered the velocities in the tangential directions at the contact point. We set
M as follows.
pol

elip

M = RE t .
The target manipulator has redundancy. Therefore, the configuration of manipulator was calculated, letting q2 be constant. Here, we considered the case when q2 = 0, −π/16, −π/8, −3π/16, −π/4.
Fig.5 and Fig.6 show the results. It can be seen that when θ is small (around 0), the smaller q2 is, the larger the volumeof TVMP (αv ) is. On the other hand, if θ is large, then, the larger q2 is, the larger the volume of TVMP (αv ) is. αall
max also
shows similar tendency. If focusing on the shape of the TVMP when θ is small (around 0), it shows larger anisotropy with thedecrease of q2 . On the other hand, if θ is large, then, the larger q2 is, the larger the anisotropy is. This tendency corresponds
to αmax . Summarizing, in order to conduct the given task efficiently, the larger θ is, the larger q2 we should take.
Lastly, we considered Problem 3 for the same manipulator shown in Fig.4. Here, we considered the same task. But, in
this case, we suppose that we need to generate a velocity in any arbitrary tangential direction with the maximum magnitudeof 0.15[m/s]. Under this condition, we derived generable tip forces. We set RVS as follows
elip
Srvs
= {ṙ|ṙ = 0.15RE t ṙt , ṙtT ṙt ≤ 1}.

We set M f as follows.

M f = R 0.3E t E n .
Here we took the friction effect into consideration.
Fig.7 and Fig.8 show the results. It can be seen that the volume of force manipulability polyhedron (βv ) becomes larger
with the decrease of q2 while becomes smaller with the increase of θ. If focusing on the shape of the force manipulability
polyhedron, the smaller q2 is, the larger anisotropy we get. This tendency is also true of βmax . Around θ = 0.2π ∼ 0.25π,
βmax for small q2 becomes smaller with the increase of θ while that for large q2 becomes larger with the increase of θ. Oneof the reasons would be that with the decrease of q2 , the contact tangential surface becomes closer to the plane constructed9

Tetsuyou Watanabe:Paper JMR-10-1163

REFERENCES
by links 3 and 4 (the plane where links 3 and 4 move). βall
max becomes smaller with the increase of θ. The ratio of the decrease
becomes smaller with the increase of q2 . Therefore, when θ is small, the smaller q2 is, the larger βall
max is, while aroundθ = 0.2π ∼ 0.25π, the larger q2 is, the larger βall
max is. Summarizing, if we want to generate large contact forces in this task,
we should keep q2 small. If we want to uniformly generate large contact forces in this task, we should keep q2 small if θ issmall and keep q2 large if θ is large, especially θ is around 0.2π ∼ 0.25π.
Remark that here we showed that our proposed approach can evaluate generable tip/endeffector velocities and forceseven when the tip is constrained by an environment and applying tip forces are required, and the directions of generable tip
velocities are limited. The performance evaluation in such situations could not be conducted by the conventional approaches.

4

Conclusion
In this paper, we proposed a novel approach to analyze manipulability of manipulators including the effect of force.
First, we derived required joint torques to compensate link gravity, payload, tip forces for keeping contact, and so on. Using
operation range attached to every actuator, we formulated the relationship between the required joint torques and actually
generable joint velocities. Using the relationship, we derived generable endeffector velocities from the actually generable
joint velocities. The derived generable endeffector velocities can concern the effect of force exerted on the manipulator.
We derived the corresponding manipulability measures: the volume of the generable endeffector velocity set, the maximum
magnitude of endeffector velocity which is generable in any direction, and the maximum magnitude of generable endeffector
velocity. In addition, we considered the evaluation when the forces exerted on the manipulators are not specified but thecandidates (the set of possible exerted forces) are given. For example, it is true of the situation when a grinding task is given.
To deal with this situation, we introduced required external force set (REFS), and presented how to derive manipulability.
We also presented how to evaluate end effector force, taking required endeffector velocities into consideration. The validityof our approach was shown by numerical examples. This analysis can evaluate the cases which could not be evaluated byconventional manipulability approaches: for example, the case when gravity effect such as a payload has to be concerned, orthe case when the manipulator contacts with an environment.

References[1] Yoshikawa, T., 1990. Foundations of Robotics. MIT Press, Cambridge.
[2] Watanabe, T., and Yoshikawa, T., 2007. “Grasping optimization using a required external force set”. IEEE Transactions
on Automation Science and Engineering, 4(1), pp. 52–66.
[3] Watanabe, T., and Yoshikawa, T., 2003. “Optimization of grasping an object by using required acceleration andequilibrium-force sets”. Proc. of IEEE/ASME International Conference on Advanced Intelligent Mechatronics,
pp. 338–343.
[4] Chiacchio, P., Chiaverini, S., Sciavicco, L., and Siciliano, B., 1991. “Global task space manipulability ellipsoids for
multiple-arm systems”. IEEE Transactions on Robotics and Automation, 7(5), October, pp. 678–685. correspondence.
[5] Bicchi, A., Melchiorri, C., and Balluchi, D., 1995. “On the mobility and manipulability of general multiple limb
robots”. IEEE Transactions on Robotics and Automation, 11(2), April, pp. 215–228.
[6] Bicchi, A., and Prattichizzo, D., 2000. “Manipulability of cooperating robots with unactuated joints and closed-chain
mechanisms”. IEEE Transactions on Robotics and Automation, 16(4), August, pp. 336–345.
[7] Hara, K., Yokogawa, R., and Yokogawa, A., 1998. “A graphical method for evaluating static characteristics of thehuman finger by force manipulability”. Proc. of the IEEE International Conference on Robotics and Automation,
pp. 1623–1628.
[8] Fukuda, K., 2004. Frequently Asked Questions in Polyhedral Computation. [Online]. Available: http://www.
ifor.math.ethz.ch/∼fukuda/polyfaq/polyfaq.html.
[9] Fukuda, K., and Prodon, A., 1996. Double description method revisited In M. Deza, R. Euler, and I. Manoussakis,
editors, Combinatorics and Computer Science, volume 1120 of Lecture Notes in Computer Science, pages 91–111.
Springer-Verlag. [Online]. Available: http://www.ifor.math.ethz.ch/∼fukuda/cdd home/index.
html.
[10] Fukuda, K., 1999. cdd/cdd+ Reference Manual. [Online]. Available: http://www.ifor.math.ethz.ch/∼fukuda/cdd home/index.html.
[11] Barber, C. B., Dobkin, D. P., and Huhdanpaa, H., 1996. “The quickhull algorithm for convex hulls”. ACM Transactions
on Mathematical Software, 22(4), December, pp. 469–483.

10

Tetsuyou Watanabe:Paper JMR-10-1163

LIST OF TABLES

LIST OF TABLES

List of Tables

11

Tetsuyou Watanabe:Paper JMR-10-1163

LIST OF FIGURESLIST OF FIGURES

List of Figures
1
Operation range of torque and velocity (maxon DC motor Amax16 (2W) with gear (ratio: 371:1)); supposing
to control with nominal voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Manipulability for 2-link planar manipulator. (a) original manipulability ellipsoid. (b)∼(f) new manipulability polyhedron (TVMP): (b) m p = 0.0, (c) m p = 0.1, (d) m p = 0.2, (e) m p = 0.3, and (f) m p = 0.4. . . . . .
3
Manipulability measures for 2-link planar manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4Target system in 3 dimensional space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5New manipulability polyhedra (TVMP) for the manipulator shown in Fig.4: (a) q2 = 0, (b) q2 = −π/16, (c)q2 = −π/8, (d) q2 = −3π/16, and (e) q2 = −π/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Manipulability measures for the manipulator shown in Fig.4: (a) αv , (b) αall
max and (c) αmax . . . . . . . . .
7
New force manipulability polyhedra for the manipulator shown in Fig.4: (a) q2 = 0, (b) q2 = −π/16, (c)q2 = −π/8, (d) q2 = −3π/16, and (e) q2 = −π/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8Force manipulability measures for the manipulator shown in Fig.4: (a) βv , (b) βall
max and (c) βmax . . . . . .

12

13
14
15
16
17
18
19
20

Tetsuyou Watanabe:Paper JMR-10-1163

LIST OF FIGURES

Rotational velocity [rad/s]

LIST OF FIGURES

3

Operation range

2.5
2
1.5
1
0.5
0

0

0.2

0.4

0.6

0.8

1

Torque [Nm]Fig. 1. Operation range of torque and velocity (maxon DC motor Amax16 (2W) with gear (ratio: 371:1)); supposing to control with nominal
voltage

13

Tetsuyou Watanabe:Paper JMR-10-1163

LIST OF FIGURESLIST OF FIGURES

0.15

0.15

0.15

0.1

0.1

0.1

y 0.05y 0.05y 0.05

0

0

0

0

0.1

x

0.2

0

(a)

0.1

x

0.2

0

(b)
0.15

0.15

0.1

0.1

0.1

y 0.05y 0.05y 0.05

0

0

0

0.1

x
(d)

0.2

0

0.1

x
(e)

0.2

0.1

0.2

(c)

0.15

0

0.1

x

0.2

0

x
(f)

Fig. 2. Manipulability for 2-link planar manipulator. (a) original manipulability ellipsoid. (b)∼(f) new manipulability polyhedron (TVMP): (b)m p = 0.0, (c) m p = 0.1, (d) m p = 0.2, (e) m p = 0.3, and (f) m p = 0.4.

14

Tetsuyou Watanabe:Paper JMR-10-1163

LIST OF FIGURESLIST OF FIGURES

0.35

αv

mp=0.00.3

mp=0.10.25

mp=0.2
mp=0.2
mp=0.30.2

mp=0.4

Conventional

0.15
0.1
0.05
0
0

0.05

0.15

0.1
x [m]

0.2

(a) αv
0.25

0.2
mp=0.0

0.15
all
αmax

mp=0.1
mp=0.20.1

mp=0.3
mp=0.4

0.05

0
0

0.05

0.1
x [m]

0.15

0.2

(b) αall
max
0.9
0.8
0.7
0.6
αmax

mp=0.00.5

mp=0.10.4

mp=0.20.3

mp=0.30.2

mp=0.4

0.1
0
0

0.05

0.1
x [m]

0.15

0.2

(c) αmax
Fig. 3.

Manipulability measures for 2-link planar manipulator

15

Tetsuyou Watanabe:Paper JMR-10-1163

LIST OF FIGURESLIST OF FIGURES

0.1

q4

z 0.05

q3

0
y

0
0.05
0.1 0.1

0

0.05

−0.1

−0.05
x

z(=z1)
x q1
x1

z2
y1
y

y2
q2

x1(=x2) y1

Fig. 4. Target system in 3 dimensional space

16

Tetsuyou Watanabe:Paper JMR-10-1163

LIST OF FIGURESLIST OF FIGURES

0.1

0.1

0.1

z 0.05z 0.05z 0.05

0

0

−0.1
−0.05

0
y 0.05

0

x

y

0.1

0

0.05

−0.1
−0.05

0

x

y

0.1

(a)

0.1

z 0.05z 0.05
0

−0.1
−0.05
y

0

0.05

−0.1
−0.05

0

x

y

0.1

(d)

x

0.1

(c)

0.1

0

0

0.05

(b)

0

Fig. 5.

0

−0.1
−0.05

0

0

0.05

x

0.1

(e)

New manipulability polyhedra (TVMP) for the manipulator shown in Fig.4: (a) q2 = 0, (b) q2 = −π/16, (c) q2 = −π/8, (d)q2 = −3π/16, and (e) q2 = −π/4.

17Tetsuyou Watanabe:Paper JMR-10-1163

LIST OF FIGURESLIST OF FIGURES

0.275

0.6

0.27
0.5

0.265
0.4

q2=0

αv 0.3all

αmax

q2=-π/16

q2=0

0.26

q2=-π/16

0.255

q2=-π/8

q2=-3π/16

0.25

q2=-3π/16q2=-π/4

0.245

q2=-π/80.2
0.1

q2=-π/4

0.24

0
0

0.05

0.1

0.15

0.2

0

0.25

0.05

0.1

0.15

0.2

0.25

θ [π rad]θ [π rad](a) αv(b) αall
max
0.75
0.7
0.65
0.6

αmax

q2=0

0.55

q2=-π/16

0.5

q2=-π/8

0.45

q2=-3π/16

0.4q2=-π/4

0.35
0.3
0

0.05

0.1

0.15

0.2

0.25

θ [π rad](c) αmax
Fig. 6. Manipulability measures for the manipulator shown in Fig.4: (a) αv , (b) αall
max and (c) αmax

18

Tetsuyou Watanabe:Paper JMR-10-1163

LIST OF FIGURESLIST OF FIGURES

0.1

0.1

0.1

z 0.05z 0.05z 0.05

0

0

−0.1
−0.05

0
y

0

0.05

0

−0.1

x

−0.05

0
y

0.1

0

0.05

−0.1

x

y

0.1

(a)

−0.05

0

(c)

0.1

0.1

z 0.05z 0.05
0

−0.1
−0.05

0
y

0

0.05

x

−0.1
−0.05

0
y

0.1

(d)

x

0.1

(b)

0

0

0.05

0

0.05

x

0.1

(e)

Fig. 7. New force manipulability polyhedra for the manipulator shown in Fig.4: (a) q2 = 0, (b) q2 = −π/16, (c) q2 = −π/8, (d) q2 =−3π/16, and (e) q2 = −π/4.

19Tetsuyou Watanabe:Paper JMR-10-1163

LIST OF FIGURESLIST OF FIGURES

80000
70000
q2=0

60000

q2=-π/16βv

q2=-π/850000

q2=-3π/1640000

q2=-π/4

30000
0

0.05

0.1

0.15

0.2

0.25

θ [π rad](a) βv
11
10
9

q2=0

all

q2=-π/16

βmax 8q2=-π/87

q2=-3π/166q2=-π/4

5
0

0.05

0.1

0.15

0.2

0.25

θ [π rad](b) βall
max
80
75
70

q2=0
q2=-π/16

βmax 65q2=-π/860

q2=-3π/16q2=-π/4

55
50
0

0.05

0.1

0.15

0.2

0.25

θ [π rad](c) βmax
Fig. 8. Force manipulability measures for the manipulator shown in Fig.4: (a) βv , (b) βall
max and (c) βmax

20

Tetsuyou Watanabe:Paper JMR-10-1163