Jacobian, manipulability, condition number and accuracy of
parallel robots
J-P. Merlet
INRIA, BP 93, 06902 Sophia-Antipolis, France
Abstract: Although the concepts of jacobian matrix, manipulability
and condition number have been floating around since the early beginning
of robotics their real significance is not always well understood, although
these conditioning indices play an important role e.g. for optimal robot
design. In this paper we re-visit these concepts for parallel robots and
exhibit some surprising results (at least for the author!) that show that
these concepts have to be manipulated with care for a proper understanding
of the kinematics behavior of a robot.
1 Introduction
Parallel robots are nowadays leaving academic laboratories and are finding
their way in an increasingly larger number of application fields such as
telescopes, fine positioning devices, fast packaging, machine-tool, medical
application. A key issue for such use is optimal design as performances
of parallel robots are very sensitive to their dimensioning. Optimal design
methodologies have to rely on kinetostatic performance indices and accuracy
is clearly a key-issue for many applications. It has also be a key-issue for
serial robots and consequently this problem has been extensively studied
and various accuracy indices have been defined. The results have been in
general directly transposed to parallel robots. We will now review how well
these indices are appropriate for parallel robots.
2 Jacobian and inverse Jacobian matrix
Let X
a
denotes the generalized coordinates of the end-effector composed
of parameters describing the available n d.o.f. of the end-effector while X
denotes all the generalized coordinates of the end-effector. We will impose
no constraints on the choice of X (e.g. for a Gough robot with a planar
platform the pose may be represented by the 9 coordinates of 3 particular
points on the end-effector).
The geometry of the robot is described by its joints variables vector Θ.
The twist W of the end effector is composed of its translational and angular
velocities and the restricted twist W
a
is defined as the restriction of W to
the available d.o.f. of the robot. It is well known that for robot having
at least 2 rotational d.o.f. W is not the time-derivative of X as there is
no representation of the orientation whose derivatives corresponds to the
angular velocities. However there exists usually a matrix H such that
W = H
˙
X (1)
1
In the usual approach the jacobian matrix J
k
linearly relates the actuated
joint velocities
˙
Θ
a
to W
a
:
W
a
= J
k
˙
Θ
a
(2)
In this paper we consider only non-redundant robots so that matrix J
k
is
square and we will call it the kinematic jacobian. A feature of parallel robots
is that it is usually easy to establish an analytical form for J
−1
k
while it is
often impossible to obtain J
k
.
But we may also define other jacobian matrices by first changing the
parameters in Θ. Indeed parallel robots differ from their serial counterpart
by a larger number of passive joints and it may thus be interesting to include
the m passive joints variables Θ
p
. If Θ is defined as (Θ
a
, Θ
p
) we may then
define write the l inverse kinematics equations as F(Θ, X
a
) = 0 from which
we derive
∂F
∂Θ
˙
Θ +
∂F
∂X
a
˙
X
a
= U
˙
Θ + V
a
˙
X
a
= 0 (3)
where U is (l ×(n + m)) and V
a
is (l ×n). This relation allows to quantify
the influence of the measurement errors on the passive and actuated joints
variables on the positioning errors ∆X
a
on the n d.o.f. of the end-effector
by using (1).
Although we say that some robot have n < 6 d.o.f., still the end-effector
is a 6 d.o.f. rigid body and positioning errors on all d.o.f. should be exam-
ined. It is thus interesting to determine an inverse jacobian that involves
the full twist W of the end-effector. In that case we write the kinematics
equations as G(Θ, X) = 0. If we fix X we know that these kinematics
equations have a finite number of solutions, which implies that the number
of equations in G should be n + m. By differentiation we get:
∂G
∂Θ
˙
Θ +
∂G
∂X
˙
X = A
˙
Θ + B
˙
X = 0 (4)
where A is a square n + m × n + m matrix while B is n + m × 6. Provided
that H is square and not singular we may now derive an inverse jacobian
such that
˙
Θ = −A
−1
BH
−1
W = J
−1
W (5)
where J
−1
is n + m ×6 In most cases however a velocity analysis allows one
to obtain a simpler inverse jacobian matrix through a relation that involves
only
˙
Θ
a
:
˙
Θ
a
O
= J
−1
fk
W (6)
where J
−1
fk
is n+m×6 and will be called the full inverse kinematics jacobian.
We may further extend this approach to take into account the design
parameters P of the robot (e.g. the location of the anchor points of the legs
in a Gough platform). For that purpose the kinematics equations will be
written as G(P, Θ, X) = 0 and the matrix of the partial derivatives of G
with respect to P will allow one to quantify the influence of the errors on
P on the positioning error of the end-effector.
2
As may be seen there is not a single inverse jacobian matrix but a multi-
plicity of them. Note however an important property of the inverse jacobian
J
−1
of (5) with respect to J
−1
fk
: the rank of J
−1
fk
is the same than the rank
of J
−1
.
It is also important to note that any inverse jacobian involving the full
twist of the end-effector W will not be homogeneous in terms of units. This
will be true also for the inverse kinematic jacobian for robot involving both
translation and rotational d.o.f. for the end-effector. Consequently many
matrix properties (such as the trace, determinant) will not be invariant
under a change of units.
In this paper we will focus on the influence of ∆Θ
a
on the positioning
errors of the end-effector through J
−1
fk
. The necessity of using the full inverse
kinematic jacobian will be emphasized on an example.
2.1 Example: the 3 − UP
U robot
Tsai [10] has proposed this robot as a 3 d.o.f. translation robot (figure 1).
Each leg of this robot is constituted, starting from the base, by a U joint
followed by an extensible leg terminated by another U joint whose axis are
the same than the U joint on the base. This constraint allows theoretically
to obtain only translation for the end-effector. This example will allow
Figure 1: The 3 − UP U robot
us to establish a methodology for determining the full inverse kinematic
jacobian. But it will also enable to show the importance of this matrix.
The story is that such a robot was designed at Seoul National University
(SNU) and that is was exhibiting a strange behavior: although the prismatic
actuators were locked, the end-effector was exhibiting significant orientation
motion. This phenomena was explained by Bonev and Zlatanov [1] and later
in [2, 11]. Furthermore motion sensitivity to manufacturing tolerances has
been studied [5, 8] and has shown that this robot was very sensitive.
3
We will denote by B
1
, B
2
, B
3
the center of the U joints on the platform
and will now calculate the full inverse kinematic jacobian matrix. The
velocity V
B
of the B points is V
B
= V + BC × Ω. Let us define n as the
unit vector of the leg and compute the dot product of the right and left
terms of the previous equation:
V
B
.n = ˙ρn = V.n + (BC × Ω).n = V.n + (CB × n).Ω (7)
Now let us define u
i
, v
i
the unit vectors of the two joint axis of the U joint
at B
i
. These vectors are the same for the base and platform. The angular
velocity of the leg ω
l
with respect to the base and the angular velocity of
the platform ω
p
with respect to the leg are
ω
l
=
˙
θ
i
A
u
i
+ ˙α
i
A
v
i
ω
p
=
˙
θ
i
B
u
i
+ ˙α
i
B
v
i
The angular velocity of the platform is
Ω = ω
l
+ ω
p
= K
i
1
u
i
+ K
i
2
v
i
where K
i
1
, K
i
2
can be obtained from the previous equations. Now define
s
i
= u
i
×v
i
and compute the dot product of the right and left terms of the
previous equation by s
i
:
s
i
.Ω = 0 (8)
Combining equations (7, 8) we get the full velocities equations involving the
twist W as
˙ρ
i
0
= J
−1
fk
W =
n
i
(CB
i
× n
i
)
0 s
i
W (9)
which establish the full inverse kinematic jacobian. The inverse kinematic
jacobian may be extracted from J
−1
fk
as the 3 ×3 matrix whose rows are the
n
i
vectors. But an important point for accuracy analysis is to consider the
lower part of J
−1
fk
which shows that if s
1
.(s
2
× s
3
) = 0 the platform may
exhibit orientation motion that may be infinitesimal or finite according to
the geometry of the U joint. It happens that the design of the SNU robot
was in the later category.
3 Manipulability
It is realistic to assume that the joint errors are bounded and consequently
so will be the positioning errors. The norm of the bound may be chosen
arbitrary as (6) is linear so that a simple scaling will allow to determine the
positioning error from the errors obtained for a given bound. A value of 1
for the bound is usually chosen:
||∆Θ|| ≤ 1 (10)
4
∆θ
1
∆θ
2
∆x
∆y
J
−T
J
−1
σ
min
σ
max
1
1
-1
-1
∆θ
1
∆θ
2
∆x
∆y
λ
1
λ
2
1
-1
1
-1
J
Figure 2: The mapping between the joints errors space and the generalized
coordinates error space induced by J
−T
J according to the norm: on top the
Euclidean norm and on bottom the infinity norm.
which leads to
∆X
T
J
−T
J
−1
∆X ≤ 1 (11)
A classical geometrical interpretation of this relation is presented for the
2D case in figure 2. If the Euclidean norm is used (10) represents a circle
in the joints errors space. This circle is mapped through matrix J
−T
J
−1
into an ellipse in the generalized coordinates error space. More generally
the mapping transform the hyper-sphere of the joints errors space into an
ellipsoid, usually called the manipulability ellipsoid.
In fact the use of the Euclidean norm is not realistic: it implies for exam-
ple that if one of the joint error is 1, then by some mysterious influence all
the other joint errors are 0. The appropriate norm is the infinity norm that
states that the absolute value of the joint errors are independently bounded
by 1. With this norm (10) represents a n-dimensional square in the joints
errors space that is mapped into the kinematics polyhedron, that includes
the manipulability ellipsoid, in the generalized coordinates errors space.
Figure 2 illustrates this mapping in the 2D case. It must be noted that,
apart of being more realistic, the previous mapping leads to geometrical
object that can be more easily manipulated than the ellipsoid. For example
assume that one want to determine what are all the possible end-effector
velocities that can be obtained in 2 different poses of the end-effector. For
that purpose we will have to calculate the intersection of the 2 polyhedra
obtained for the 2 poses, a well known problem of computational geometry,
that can be much more easily solved than computing the intersection of 2
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