Manipulability Analysis

Abstract

The ability of computing a quality index for given configurations can be useful for several applications in the context of robotic manipulation. E.g. it can be used for monitoring the current state of the system or it can support decision processes, such as grasp selection in humanoid robotics. Here, a large set of precomputed grasps for a given object have to be quickly filtered in order to select the reachable sub set, for which the inverse kinematics (IK) problem has to be solved. In this work, we present an approach for analyzing the workspace capabilities of a manipulator in order to store a representation for efficient online processing. Compared to existing work, where usually reachability information is used to represent the robot's capabilities, we show how an extended manipulability measure can be used to build a quality distribu-tion in workspace. The proposed approach for manipulability measurement is suitable for redundant manipulators while considering joint limits and self-distance. Further, task specific distributions are introduced and several applications for the humanoid robots ARMAR-III [1] and iCub [2] are presented.

Full text

Manipulability Analysis
Nikolaus Vahrenkamp
∗†
, Tamim Asfour
∗
, Giorgio Metta
†
, Giulio Sandini
†
and R
¨
udiger Dillmann
∗
∗
Institute for Anthropomatics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
†
Robotics, Brain and Cognitive Sciences Departement, Istituto Italiano di Tecnologia (IIT), Genova, Italy
Abstract—The ability of computing a quality index for
given configurations can be useful for several applications in
the context of robotic manipulation. E.g. it can be used for
monitoring the current state of the system or it can support
decision processes, such as grasp selection in humanoid robotics.
Here, a large set of precomputed grasps for a given object
have to be quickly filtered in order to select the reachable sub
set, for which the inverse kinematics (IK) problem has to be
solved. In this work, we present an approach for analyzing
the workspace capabilities of a manipulator in order to store
a representation for efficient online processing. Compared to
existing work, where usually reachability information is used
to represent the robot’s capabilities, we show how an extended
manipulability measure can be used to build a quality distribu-
tion in workspace. The proposed approach for manipulability
measurement is suitable for redundant manipulators while
considering joint limits and self-distance. Further, task specific
distributions are introduced and several applications for the
humanoid robots ARMAR-III [1] and iCub [2] are presented.
I. INTRODUCTION
The ability of supervising own movements is essential for
robots operating in such different domains as industrial ap-
plications, service or humanoid robotics. Hence, the systems
must be equipped with methods to measure their perfor-
mance, accuracy or success rate. In this context, measuring
the manipulability is a well-known technique for determining
the ability to maneuver in workspace.
The manipulability index of given robot poses, introduced
by Yoshikawa in 1985 [3], can be calculated as a quality
value that gives information about how good an adjustment
in workspace is possible. With such information, redundancy
of the robot can be exploited to favor poses which result in
higher manipulability and which hence allow better adaption
during execution. In case the manipulability is used to
support a decision process, e.g. when a grasp should be
selected out of a set of potential grasps, a precomputed
representation can be useful in order to serve manipulability
queries efficiently.
Several approaches of determining the reachability of
manipulators are known in literature, which are based on
the idea that a voxelized 3D or 6D grid is filled with reach-
ability information (see [4], [5], [6]). Due to the discretized
representation the reachability entries can just give a hint,
whether a pose in workspace will lead to a valid IK solution
or not. In this work we will show, how a similar data structure
can be used to capture the manipulability distribution of
a manipulator in workspace. The resulting data therefore
extends the approximated reachability information with a
quality index. This additional information can be used to
Fig. 1. The humanoid robots iCub and Armar-III performing grasping tasks
in cluttered environments.
support grasp selection in manipulation tasks (see Fig. 1) in
order to favor grasps that result in a high manipulability. High
manipulability allows to adapt movements during execution,
e.g. when visual servoing approaches are used to improve
the accuracy [7]. The proposed algorithms are published
within the open-source robotics toolbox Simox, which can
be accessed online
1
.
A. Manipulability Measure
With Yoshikawa’s manipulability index [3] a quality mea-
sure for redundant manipulators was introduced, which de-
scribes the distance to singular configurations. The approach
is based on analyzing the manipulability ellipsoid that is
spanned by the singular vectors of the Jacobian. In [3] the
manipulability measure is proposed as
w =
q
det(JJ
T
) = s
1
s
2
···s
n
. (1)
Since w can be rewritten by multiplying the singular
values s
i
, the measure is proportional to the volume of
the manipulability ellipsoid. The inverse condition number
measurement was introduced in [8] as the relation of the
smallest to the largest singular value
c =
1
cond(J)
=
s
n
s
1
. (2)
In [9], the following penalization was introduced to con-
sider the distance to the lower (l
−
j
) and the upper (l
+
j
) joint
limits:
P (θ
j
) = 1 − exp(−k
n
Y
j=1
(θ
j
− l
−
j
)(l
+
j
− θ
j
)
(l
+
j
− l
−
j
)
2
). (3)
In this penalty function k is a scaling factor that can be used
to adjust the behavior near joint limits. By multiplying the
manipulability with P , configurations which are near joint
1
http://simox.sourceforge.net

limits are penalized, but redundancy is not considered, which
can lead to an erroneous quality information (e.g. one joint is
close to a boundary while the end effector’s maneuverability
is not affected due to redundancy).
A task specific quality function for a redundant manipula-
tor is considered in [10] in order to compute optimal poses.
A dexterous performance measure is presented in [11]. By
analyzing the so-called augmented Jacobian matrix, which
combines information about the position, orientation, and
joint limits of the end-effector, the manipulability of joint
configurations can be measured. It is shown that this ap-
proach is more accurate than Yoshikawa’s manipulability
measure and exemplary applications for optimizing base
placements are given. However, the approach relies on the
generation of surface patches to represent the reachable
workspace, which can lead to challenging computational
problems. Joint velocity limits can be considered [12], re-
sulting in an analysis of a scaled Jacobian.
As shown in [12] or [13], the use of manipulability
ellipsoids can lead to issues that are caused by mixing
translational and rotational sub spaces for manipulability
measurement. By using manipulability polytopes the de-
scribed problems can be avoided, but higher computational
costs have to be taken into account. In this work, we use
a weighting matrix W
x
in order to set the translational
components of the Jacobian matrix in relation with the
rotational components (see [13]).
B. Precomputed Reachability Information
In literature the reachability of a manipulator is loosely
defined as the volume of the workspace that can be reached
by the end effector. Usually the three- or six-dimensional
Cartesian space around the robot is investigated, whereas
only the position (3D) or position and orientation (6D) is
considered. The reachability data is created by filling a
voxelized data structure in an offline step. The grid entries
represent either a probability that a given pose that lies within
the corresponding workspace voxel is reachable or binary
information is used to indicate that a voxel lies (partly) within
the reachable workspace (see [4], [5], [6]).
In this work we pick up the idea of having a pre-
computed workspace representation of a manipulator. Instead
of holding information about the reachability, we store a
quality index that serves information about the manipulability
in workspace (see Sec. III).
II. EXTENDED MANIPULABILITY MEASURE
In this work we use an extended manipulability measure-
ment in order to consider constraints that limit the maneuver-
ability in workspace. Such constraints are introduced by joint
boundaries and workspace (self-)collisions, but any other
constraints can be incorporated as long as the derivation with
respect to joint movements can be built.
In the context of optimal control, several approaches are
known to avoid joint limits and obstacles [14], [15]. Com-
pared to these works, we are not interested to find an optimal
control strategy for a requested end effector movement, but
we want to investigate the limitations that are given for any
possible movements in workspace.
A. Joint Limit Penalization Function
The influence of joint limits on the manipulability measure
can be considered by analyzing a modified Jacobian matrix
J. The entries of J are penalized according to the quality
measure that arises from the current distance to joint limits
(l
−
j
and l
+
j
). Since the distance to lower or upper joint
limit usually differs, the corresponding penalization terms
also differ and hence, the quality measure depends on the
investigated workspace movement.
In [14] a joint limit potential function h(θ) is proposed
and the joint limit gradient function 5h(θ) is derived. For
each joint θ
j
the corresponding entry of 5h(θ) can be with
5h(θ)
j
=
∂h(θ)
∂θ
j
=
(θ
j
− l
−
j
)
2
(2θ
j
− l
+
j
− l
−
j
)
4(l
+
j
− θ
j
)
2
(θ
j
− l
−
j
)
2
. (4)
This gradient is equal to zero, if the joint is at the middle
of its range and goes to infinity at either limit. It can be
used to avoid joint limits during online control as shown in
[14] or [15]. In order to construct an according penalization
term we have to distinguish the different potential movement
directions in workspace (see Sec. II-C). Depending of the
actual position of the joint (either located in the lower or the
upper half of its range), the following two penalization terms
are built:
p
−
j
=
(
1, |θ
j
− l
−
j
| > |l
+
j
− θ
j
|
1
√
1+|5h(θ)
j
|
, otherwise
p
+
j
=
(
1
√
1+|5h(θ)
j
|
, |θ
j
− l
−
j
| > |l
+
j
− θ
j
|
1, otherwise
(5)
In Eq. 5, the penalization term p
−
j
stands for the pe-
nalization that has to be applied when investigating the
joint’s movement in negative direction. If the current joint
value is located in the upper half of its range, the value
is 1, which means that the movement does not underlie
any penalizations. Otherwise, the movement is penalized
according to Eq. 4. The second part of Eq. 5, describes the
construction of p
+
j
, the penalization term that is applied when
a movement in positive joint direction is investigated. In this
case, the penalization is only considered when the current
joint position is located in the upper half of its range.
B. Obstacle Penalization Function
Obstacles, such as environmental objects or parts of the
robot, limit the possibility of a manipulator to maneuver
in workspace. Hence, the manipulability measure should
consider a penalization that is derived from the distance and
the position of any limiting obstacles. By determining the
nearest points p
o
and p
m
on the surface of the obstacle
and the manipulator, one can determine the corresponding
obstacle vector v
0
= p
o
− p
m
and the obstacle distance

d = |v
0
|. The vector v ∈ R
3
is extended to R
6
by setting the
rotational components to zero: v = [v
0
, v
1
, v
2
, 0, 0, 0]
T
.
A collision function P (θ, d) should go to infinity for
d → 0 and decays exponentially to zero as d increases. As
proposed in [15], we choose
P (θ, d) = e
−αd
d
−β
. (6)
The parameters α and β can be used to adjust the obstacle
influence (see [15] for details). The gradient of P is the
collision gradient function, which gives information about
how each joint influences the obstacle distance:
5P (θ, d) =
∂P
∂θ
=

∂P
∂θ
1
, . . . ,
∂P
∂θ
n

=
∂P
∂d
∂d
∂θ
. (7)
As shown in [15] 5P can be computed with
∂P
∂d
= −e
−αd
d
−β
(βd
−1
+ α),
∂d
∂θ
=
1
d

J
T
v

T
.
(8)
Finally, two penalization terms can be constructed similar to
Eq. 5. Again, we are interested in all potential movements of
the end effector and therefore a distinction is made, whether
a workspace movement in negative (o
−
i,j
) or positive (o
+
i,j
)
direction is investigated:
o
−
i,j
=
(
1, v
i
> 0, i > 3
1
√
1+|5P
j
|
, otherwise
o
+
i,j
=
(
1
√
1+|5P
j
|
, v
i
> 0, i ≤ 3
1, otherwise
(9)
In Eq. 9 the terms o
−
i,j
and o
+
i,j
represent the penalizations
that have to be applied when investigating movements in neg-
ative respectively positive Cartesian directions. Penalizations
are only applied for translational dimensions (i ≤ 3) and if
the obstacle is located in the corresponding direction (v
i
).
C. Computing the Augmented Jacobians
Compared to [14] or [15], we cannot use the gradients
directly since an explicit movement x ∈ R
6
is needed in
order to optimize the control law. In the context of ma-
nipulability, we are interested in a measurement that serves
information about the possibility to maneuver in workspace
and hence all potential movements must be covered. Since
the manipulability is affected by joint limit and obstacle
constraints and, as shown above, these constraints directly
depend on the investigated movement, we have to build a
representation that covers all potential directions. Therefore,
the space of workspace movements is partitioned by 2
6
hyperoctants, which are identified by Γ ∈ {−1, +1}
6
.
For each Γ an augmented Jabobian matrix
˜
J is built by
considering the corresponding penalization terms. Finally,
the manipulability of
˜
J is computed, which gives information
about the manipulability of all potential movement directions
that are represented by the corresponding hyperoctant Γ.
1. Joint Limit Penalization Matrix
Depending on the investigated movement of θ
j
, either p
−
j
(move towards lower joint limit) or p
+
j
(move towards upper
Fig. 2. (a) The classical manipulability ellipsoid without joint limit penal-
ization (c = 0.170). (b) For each quadrant different penalization terms are
applied according to the corresponding joint limit influence (˜c
ext
= 0.047).
(c) The obstacle influences the penalization terms (˜c
ext
= 0.067).
joint limit) can be applied in order to penalize the computed
maneuverability. Therefore, the following penalization matrix
is computed:
L
i,j
(Γ, θ) =

p
−
j
, sgn(J
i,j
(θ)) sgn(Γ
i
) < 0
p
+
j
, otherwise
(10)
The selection which penalization has to be applied depends
on two facts. Firstly, the sign of the corresponding Jacobian
entry J
i,j
indicates the direction a positive Cartesian move-
ment causes in joint space. Further, the sign of i-th entry of
Γ indicates which hyperoctant is considered. E.g. a positive
sign of J
i,j
and a hyperoctant in a negative coordinate axis
results in a joint space movement towards the negative joint
limit and hence, p
−
j
has to be considered for penalization.
2. Collision Penalization Matrix
In addition, collision penalizations can be computed. Here,
either o
−
i
(movement in negative coordinate axis) or o
+
i
(movement in positive coordinate axis) are considered:
O
i,j
(Γ, θ) =

o
−
i,j
, sgn(Γ
i
) < 0
o
+
i,j
, otherwise
(11)
3. The Augmented Jacobian
Finally, the augmented Jacobian
˜
J is constructed as fol-
lows:
˜
J
i,j
(Γ, θ) = L
i,j
(Γ, θ)O
i,j
(Γ, θ)J
i,j
(θ). (12)
By applying Eq. 12 to all possible permutations of Γ,
64 Jacobians are generated, each of which describes the
maneuverability in the corresponding hyperoctant of the
workspace. These matrices can be analyzed for manipu-
lability in order to retrieve a quality index for a given
configuration. Therefore all Jacobians are decomposed via
Singular Value Decomposition (SVD) in order to retrieve
the corresponding sets of singular values ˜s
Γ
. To compute
the Extended Inverted Condition Number the minimum and
maximum singular value is determined:
˜c
ext
=
min{ ˜s
Γ
}
max{ ˜s
Γ
}
.
(13)
The quality measure ˜c
ext
implicitly considers redundancy,
since a penalization due to joint limits can be compensated
by redundant joints in the kinematic structure. The results
of this measure applied to a 4 degrees of freedom (DoF)
planar manipulator can be seen in Fig. 2. The figure on the

Fig. 3. Top row: (a) The distribution of the classical manipulability measure
is analyzed for a planar 4 DoF robot. (b) By applying a global penalization
term, joint limits can be considered, but redundancy is not reflected by the
measure. Bottom row:(c) The distribution of the extended manipulability
measure ˜c
ext
in workspace. (d) The obstacle is considered additionally.
left shows a 2D visualization of the manipulability ellipsoid
without considering joint limits, whereas the second visual-
ization shows the scaled manipulability ellipsoids in each of
the four quadrants. On the right, an obstacle is considered and
the minimum distance is visualized. The according joint limit
and obstacle penalization terms are applied as described in
Eq. 12. Note, that only parts of the manipulability ellipsoids
are visualized, so that one can see the penalization effects in
the different quadrants. Due to this visualization technique,
discontinuous transitions arise between the quadrants. Such
artifacts only affect the visualization and have no influence
on the computation, since for each quadrant (respectively
hyperoctant in the general case) the whole manipulability
ellipsoid is considered.
III. MANIPULABILITY ANALYSIS
In order to build up a representation of the manipulability
distribution in workspace, we propose a 6D voxelized data
structure holding manipulability reference values in each
voxel. Since the mapping between C-space and workspace
is not unique for redundant manipulators, either the average
manipulability or the lower respectively the upper bound of
the achievable manipulability within a voxel can be stored.
Hence, the manipulability data cannot be seen as an exact
representation of the manipulators capabilities, but it can be
used to efficiently serve an approximation of the expected
manipulability of a given pose in workspace. This can be
useful in several situations, e.g. when the selection of a grasp
should be made and a large number of potential grasping
poses are available. Further, the expected manipulability can
help in choosing a suitable location for a hand-over process.
A workspace representation of the robot’s reachability or
manipulability capabilities can be built by two ways. Either
work or joint space is discretized and all configurations are
processed in order to update the corresponding 6D voxel, or
randomized approaches are used. In the following, we build
the manipulability representation by sampling a large set of
configurations randomly. For each sample, the corresponding
˜c
ext
value is computed and the 6D voxel v is determined
Fig. 4. (a) The task specific manipulability distribution for an upright
prototype direction without considering joint limits (top). The distribution
of the task specific manipulability ˜c
p
(bottom). (b) The task-specific
manipulability distribution for the left arm of ARMAR-III considering an
upright lifting movement.
by computing the forward kinematics of the manipulator in
order to determine the location of the end effector. If the
actual entry of v is lower than ˜c
ext
, the value is updated.
This leads to an upper-bound representation of the achievable
manipulability which can be seen as an optimistic view on
the robot’s possibility to maneuver in workspace.
Fig. 3 shows a visualization of the manipulability distri-
bution of a 2D planar robot with 4 DoF. For each position
that is indicated, the maximum manipulability is shown,
whereas the color indicates the magnitude of the achiev-
able manipulability (blue:low, red:high). On the top left
the classical manipulability index c is used, whereas the
top right figure shows the distribution when a penalization
term according to Eq. 3 is applied when measuring the
manipulability. Since redundancy is not considered in Eq. 3,
the resulting distribution gives only limited information about
the achievable manipulability of a redundant manipulator.
When applying the proposed penalizations, the corre-
sponding distribution can be seen in Fig. 3. The bottom
left figure shows the result when joint limits are penalized
according to Eq. 10. On the bottom right figure, the obstacle
penalization (Eq. 11) is additionally applied and the com-
bined Jacobians are analyzed as described in Section II-C.
A. Task-Specific Manipulability Distributions
In case the robot is supposed to frequently execute a spe-
cific task, as it is the case e.g. for pick-and place operations,
task-specific manipulability distributions can be precomputed
in order to support the execution. Such distributions give
information about the possibility to move in a specific
direction v
p
∈ R
6
.
The penalization terms are computed according to Eq. 12,
and since a hyperoctant 
p
is defined by the direction v
p
,
only one augmented Jacobian
˜
J(
p
, θ) has to be assembled.
Further, the magnitude of the Jacobian’s gradient with respect
to the given prototype direction v
p
is used as a quality index:
q
p
(θ) = |
˜
J(
p
, θ)
T
v
p
|. (14)

Fig. 5. (a) A 3D visualization of the 6D manipulability distribution
of iCub’s 10 DoF kinematic chain covering torso and right arm. (b)
A cut through the manipulability distribution of the ARMAR-III’s right
manipulator (torso and arm). Joint limits and self-distance are considered
by the underlying manipulability measure.
Together with the inverted condition number ˜c

p
for the
corresponding hyperoctant, the resulting manipulability can
be computed with:
˜c
p
= q
p
(θ)˜c

p
. (15)
The distribution of the corresponding manipulability val-
ues when considering an upright prototype direction v
p
can
be seen in Fig. 4(a). The distribution of the manipulability
w.r.t. an upright movement for the left arm of ARMAR-III
is shown in Fig. 4(b).
IV. APPLICATIONS
A. Manipulability Distribution of Humanoid Robots
In Fig. 5 a 3D visualization of the manipulability dis-
tributions of iCub and ARMAR-III can be seen. In both
cases the kinematic chain covering three torso and seven arm
joints has been considered. The underlying manipulability
measure incorporates penalizations due to joint limits and
self distance as described in Sec. II. The left figure shows a
3D visualization of the 6D distribution, which was generated
by showing the orientation with maximum manipulability
at each 3D position. On the right, a cut through the 6D
distribution is shown.
B. Performance Evaluation
The main advantage of an offline generated manipulability
distribution is given by the possibility to structure the data
in order to speed up information retrieval during online
processing. In the following experiment we evaluate the
effect of using precomputed manipulability representations
in the context of grasp selection.
In Fig. 6 the model of a water bottle for which 1000
precomputed grasps are available can be seen on the left. An
exemplary subset of reachable grasps is depicted, whereas
the manipulability is encoded by color (blue:low, red:high).
To measure the performance, the bottle is placed randomly
in front of ARMAR-III and different strategies for grasp
selection are evaluated. Firstly, no precomputed data is used
and the IK-problem is solved for all grasps for the 10 DoF
kinematic chain covering torso and arm. In case an IK-
solution could be retrieved, the manipulability is computed
Fig. 6. (a) The manipulability of a set of reachable grasps is encoded by
color. (b) A grasp with ˜c
ext
= 0.02. (c) The optimal grasp in terms of
manipulability (˜c
ext
= 0.12).
for the corresponding joint configuration in order to deter-
mine the grasp with maximum ˜c
ext
. In the second approach
all grasps are filtered according to their reachability. Here,
reachability is given when the corresponding entry of the
manipulability distribution is greater zero. This leads to a
binary representation of the 6D reachability similar to the
approaches of [4], [5], [6]. For the remaining sub-set of
reachable grasps, the IK-solutions are computed in order
to determine the corresponding manipulability. The third
approach takes full advantage of the precomputed manip-
ulability data. Due to the efficient data structures an ordered
set of reachable grasps with corresponding manipulability
values can be retrieved in less than one millisecond and the
IK-problem has only to be solved for the first entry of this
list (see Table I).
TABLE I
COMPARISON OF DIFFERENT APPROACHES FOR GRASP SELECTION
(AVERAGE OF 100 TESTS).
Overall
Reach. / IK Manip.
Manip. access Solving Comp.
No precomputed
9.97s - 9.07s 0.90s
data
Reachability
2.24s <0.001s 1.75s 0.49s
data
Manipulability
0.01s <0.001s 0.01s -
data
C. Grasp Selection based on Manipulability
In this setup, manipulability distributions are used to
support the grasp selection process, which is part of a
grasping pipeline implemented on the humanoid robot iCub.
The grasping pipeline takes advantage of offline generated
data, such as object specific sets of grasps and manipulability
data as proposed in this work. During online processing
a subset of reachable grasps is determined according to
the object’s pose. Therefore, all grasps are transformed to
Cartesian grasping poses by combining the object’s pose in
workspace with the object-grasp relation. These poses can
then be quickly filtered according to their manipulability
and the resulting ordered subset gives information about
reachability and manipulability of the remaining grasps. In
addition, the inverse kinematics (IK) problem is solved and
a collision-free motion is generated with algorithms related

Fig. 7. iCub’s grasping pipeline is supported by precomputed manipulability distributions. (a) The virtual representation of the scene. (b) All reachable
grasps are colorized according to their manipulability. (c) The IK-solution of the selected grasp. (d) A collision-free grasping motion.
Fig. 8. The grasp selection of iCub’s grasping pipeline is supported by precomputed manipulability distributions.
to Rapidly-exploring Random Trees (RRT) [16]. Fig. 7
depicts the results of the different processing stages and the
execution on iCub is shown in Fig. 8.
V. CONCLUSION
In this work, we presented an integrated approach for
determining the manipulability for redundant manipulators.
We showed how classical measurements, which are mostly
based on analyzing the manipulability ellipsoid, can be
extended in order to consider constraints coming from joint
limits, workspace obstacles or self-distance while explicitly
considering redundancy. This extended manipulability mea-
surement is used to build the distribution of a manipulator’s
manipulability in workspace, allowing to capture the capabil-
ities of a robot in terms of operational freedom. In addition
we showed how task specific data can be generated and how
the manipulability data can be used for grasp selection on
humanoid robots.
ACKNOWLEDGMENT
The research leading to these results has received funding
from the European Union Seventh Framework Programme
under grant agreement 270273 (Xperience).
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