HAL Id: hal-00401752
https://hal.archives-ouvertes.fr/hal-00401752
Submitted on 6 Jul 2009
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Identiable parameters for parallel robots kinematic
calibration
Sébastian Besnard, Wisama Khalil
To cite this version:
Sébastian Besnard, Wisama Khalil. Identiable parameters for parallel robots kinematic calibration.
ICRA-IEEE Robotics and Automation Conference, 2001, Seoul, South Korea. pp.2859-2866. �hal-
00401752�
1/6
Identifiable Parameters for Parallel Robots Kinematic Calibration
S. BESNARD, W. KHALIL
Institut de Recherche en Communications et Cybernétique de Nantes, U.M.R. CNRS 6597
1, rue de la Noë, B.P. 92101 F-44321 Nantes Cedex 3, France
Wisama.Khalil@irccyn.ec-nantes.fr
Abstract
This paper presents a numerical method for the
determination of the identifiable parameters of parallel
robots. The special case of Stewart-Gough 6 degrees-of-
freedom parallel robots is studied for classical and self
calibration methods, but this method can be generalized
to any kind of parallel robot. The method is based on QR
decomposition of the observation matrix of the calibration
system. Numerical relations between the identifiable and
non identifiable parameters can be obtained.
1.
Introduction
The classical methods for parallel robot calibration need
external sensors to measure the position and orientation of
the mobile platform [1] [2] [3] [4] [5]. The calibration
problem is formulated in terms of minimizing the
difference between the measured and computed motorized
joint variables, it uses the inverse kinematic model which
is easy to calculate for parallel robots. Self calibration
methods using extra sensors on the passive joints have
been also proposed for parallel robots [6] [7] [8] [9].
These methods are based on the use of redundant sensors
on the passive joints and to adjust the values of the
kinematic parameters in order to minimize a residual
between the measured and the calculated values of the
angles of these joints. As many parallel robots don’t have
redundant sensors on the passive joint, mechanical
constraints on the leg can also be used [10] [11].
For some calibration methods, all the geometric
parameters cannot be identified. In previous work, the
identifiable parameters of parallel robots are derived by
intuition. In the case of serial robots, the identifiable
parameters are computed from a
QR
decomposition of the
analytical observation matrix [12]. We propose to extend
this method for parallel robots even in the case where the
Jacobian matrix cannot be obtained analytically.
2.
Description of the robot
The parallel robot studied here is the Stewart-Gough 6
degrees of freedom robot (Figure 1). The base
connections are composed of Universal joints (U-joints)
and the platform connections are composed of Spherical
joints (S-joints). The centers of the U-joints and S-joints
are denoted by A
i
and B
i
(
i
= 1 to 6) respectively. The
configuration of the parallel robot is given by the (6x1)
vector
L
representing the leg lengths A
i
B
i
for
i
=1,…,6:
L
= [
l
1
l
2
l
3
l
4
l
5
l
6
]
T
(1-a)
Typically each variable is given as:
ioffii
qql
,
+=
(1-b)
where
q
i
is the prismatic position sensor reading and
q
off,i
is a fixed offset value.
A
1
A
2
A
3
A
4
A
5
A
6
B
2
B
3
B
4
B
5
mobile
platform
mobile
fixed
base
prismatic joint
(motorized)
S-joint
(passive)
U-joint
(passive)
Figure 1: Stewart-Gough parallel robot
Let the frame F
0
be fixed with respect to the base and
frame F
m
fixed to the movable platform, such [7]:
- A
1
is the origin of frame F
0
, while the x
0
axis is
determined by (A
1
A
2
) and x
0
y
0
plane is determined by the
points A
1
, A
2
and A
6
.
- similarly, B
1
is the origin of frame F
m
, while B
1
B
2
represents its x
m
axis and B
1
B
2
B
6
its x
m
y
m
plane.
With this definition of F
0
and F
m
we have:
0
Px
A1
=
0
Py
A1
=
0
Pz
A1
=
0
Py
A2
=
0
Pz
A2
=
0
Pz
A6
= 0
2/6
m
Px
B1
=
m
Py
B1
=
m
Pz
B1
=
m
Py
B2
=
m
Pz
B2
=
m
Pz
B6
= 0
Where
j
P
Pi
denotes the coordinates of the point P
i
with
respect to coordinate system F
j
and:
j
P
Pi
= [
j
Px
Pi
j
Py
Pi
j
Pz
Pi
]
T
Thus, the robot is described by 24 constant parameters
which may be not equal to zero.
A
1
A
6
A
5
A
4
A
3
A
2
B
1
B
6
B
5
B
4
B
3
B
2
F
m
F
0
F
E
F
-1
-1
T
0
m
T
E
O
E
O
-1
Figure 2: Definition of the frames
The (4x4) transformation matrix between frames F
0
and
F
m
giving the location (position and orientation) of the
platform with respect to the base is denoted by:
0
T
m
=
1000
PA
m
0
m
0
(2)
The location of frame F
0
with respect to the world
reference frame F
-1
of the environment is given by a
transformation matrix Z. In addition, the matrix E denotes
the location of the end-effector frame F
E
in frame F
m
(cf.
Figure 2). The location of the end- effector frame relative
to the world reference frame is:
ETZT
m
0
E
1
⋅⋅=
−
(3)
Thus, the coordinates of A
i
relative to frame F
-1
are:
⋅=
⋅=
−
−
1
P
Z
1
P
T
1
P
iii
A
0
A
0
0
1
A
1
(4)
The coordinates of point B
i
relative to frame F
E
are:
⋅=
⋅=
1
P
E
1
P
T
1
P
iii
B
m
B
m
m
E
B
E
(5)
The matrices Z and E can be defined using 6 independent
parameters. Thus, we can describe the geometry of the
robot using 36 constant parameters: either by
–1
P
Ai
and
E
P
Bi
, or by
0
P
Ai
,
m
P
Bi
and the matrices E and Z. The total
number of parameters is thus equal to 42, after taking into
account the 6 joint variables.
For the calibration, we propose to use the coordinates of
points A
i
and B
i
in frames F
-1
and F
E
respectively in order
to have homogeneous parameters to identify (only
lengths). From these coordinates, it is easy to find the
transformations Z and E, and the coordinates of the points
of the base and the platform in frames F
0
and F
m
.
2.1
Kinematic modeling
The inverse kinematic model (IKM) which computes the
leg lengths vector for a desired
-1
T
E
is unique and easy to
obtain [13]. While, the direct kinematic model (DKM),
which gives the matrix
-1
T
E
as a function of a given leg
lengths vector, is difficult to obtain analytically and up to
40 solutions may exist [14]. A numerical iterative method
based on the inverse Jacobian matrix is used to find a
local solution for the DKM.
3.
General calibration models
The aim of the kinematic calibration is to estimate
accurately the geometric parameters. All the calibration
methods are based on calculating a function, for sufficient
number of configurations, in terms of the robot parameters
and some external variables. The model parameters are
estimated by minimizing this function by solving a
nonlinear system of equations. The general form of the
calibration equation is:
11
1
ee
e
f( , , )
F( , , ) 0
f( , , )
qx
QX
qx
η
η
η
==
!
(6)
where
η
denotes the geometric parameters,
Q
={
q
1
,…,
q
e
}
T
contains the prismatic positions of the robot for
e
different configurations, and
X
= {
x
1
,…,
x
e
}
T
are the
corresponding external measured variables such as the
Cartesian coordinates. This nonlinear optimization
problem can be solved by the
leastsq
function of Matlab
based on the Levenberg-Marquardt method.
Supposing that the U- and S-joints are perfect, we have to
identify the error
∆
-1
P
Ai
,
∆
E
P
Bi
,
∆q
off
,
i
(with
i
= 1,…,6).
They will be collected in the vector
∆
η
. Before solving
the calibration equation, it is important to define the
identifiable parameters, because only these parameters can
3/6
be identified without ambiguity. We propose to determine
these parameters using
QR
decomposition of the
observation matrix of the linearized model of randomly
e
configurations satisfying the constraints of the calibration
procedure. The outlines of this algorithm is given
references[12,15]. The linearized equations corresponding
to the nonlinear equation (6) can be written as:
)W(Q,),(Q,
ρ
ηηη
+∆⋅=∆ XY
(7)
where
∆Y
is the difference between the model and the real
robot, W is the (
r
,
np
) observation matrix of the system,
with
np
the number of geometric parameters and
r
>>
np
,
The vector
ρ
indicates the residual errors owing to noise
or modeling errors.
For parallel robots, the observation matrix W can be
obtained analytically for the calibration method which is
based on the IKM. For all the other methods we have to
calculate W numerically by supposing small variations
ε
on each geometric parameter and calculating the
corresponding
∆Y
i. The
j
th
column of W corresponding to
that parameter will be computed as
∆Y
j
/
ε
. Good results
are obtained with
ε
=
10
-6
meter for each parameter.
The number of the identifiable parameters denoted by
b
.
The
QR
decomposition will provide as a set of identifiable
parameters those corresponding to the first
b
independent
columns of W. We assign a priority number to each
parameter, the parameters with higher priority will be
placed at first in
η
. We place at first the offsets
q
off,i
(priority 3), and we place at the end the 12 coordinates of
the points defining frames F
0
and F
m
(
-1
Px
A1
,
-1
Py
A1
,
-1
Py
A2
,
-1
Pz
A1
,
-1
Pz
A2
,
-1
Pz
A6
,
E
Px
B1
,
E
Py
B1
,
E
Py
B2
,
E
Pz
B1
,
E
Pz
B2
,
E
Pz
B6
) (priority 1), the other parameters will get
priority 2 and will be placed after the offset parameters in
the following order:
-1
Px
A2
,…,
-1
Px
A6
,
-1
Py
A3
,…,
-1
Py
A6
,
-1
Pz
A3
,…,
-1
Pz
A6
, then
E
Px
B2
,…,
E
Px
B6
,
E
Py
B3
,…,
E
Py
B6
,
E
Pz
B3
,…,
E
Pz
B6
.
4.
Application to calibration methods
We compute the identifiable parameters for several
calibration methods for the parallel robot whose nominal
parameters are given in Table 1. The obtained identifiable
parameters are valid for any robot of the Stewart-Gough
type. The grouping relations of the non identifiable
parameters are functions of the numerical values of the
geometric parameters.
leg123456
-1
Px
Ai
0 0,8426 0,9382 0,5168 0,3258 -0,0955
-1
Py
Ai
0 0 0,1654 0,8952 0,8952 0,1654
-1
Pz
Ai
000000
E
Px
Bi
0 0,1042 0,3340 0,2819 -0,1777 -0,2298
E
Py
Bi
0 0 0,3980 0,4883 0,4883 0,3980
E
Pz
Bi
000000
q
off,
i
,85 0,85 0,85 0,85 0,85 0,85
Table 1: Nominal values of the geometric parameters
4.1
Calibration using the IKM
Measuring the location of the platform, the inverse
kinematic model (IKM) can be used to compute the 6 leg
lengths of the robot. The calibration method consists in
minimizing the residual between the computed and the
measured prismatic variables [2].
The equation for each leg and each configuration
is:
()
()
∆
∆
∆
−
−−−
−−
=∆
−
−−−−
−−−
i
i
i
i
q
L
L
q
,off
B
E
A
1
T
E
1
T
E
1
B
E
E
1
A
1
T
E
1
B
E
E
1
A
1
i
i
ii
ii
P
P
.A.PP.AP
PP.AP
1
(8)
Applying this equation for the 6 legs of the robot and
e
configurations, we have the relation:
ηη
∆=∆
.),W(
XQ
(9)
where
∆Q
is the difference between the measured
prismatic joint values and those computed by the IKM.
Note that the observation matrix W can be computed
analytically. Using (9) for
e
random configurations, with
e
>> 7 such that the number of rows of W is greater than
the number of the parameters. The rank of the matrix W is
42. Thus, all the parameters can be identified. The
condition number of W can be used as a measure of the
excitation of the parameters by the calibration method.
Using e configurations such that the number of equations
is 4 times the number of parameters we find that The
condition number of W is about 350.
4.2
Calibration with measurement of the position of
the platform
Measuring only the position of the platform, we cannot
use the IKM of the robot since we have only 3 equations
to solve a system of 6 unknowns (the 6 leg lengths of the
4/6
robot). Nevertheless, using the direct kinematic model
(DKM), if we consider a configuration
q
of the robot and
p
E
the measured position of the effector in frame F
-1
, we
can write the nonlinear model of calibration as:
1
EE
P(, ) 0
qp
η
−
−=
(10)
The corresponding linear differential model is:
ηη
∆Ψ=∆
.),(p
E
q
(11)
The Jacobian matrix
Ψ
is obtained numerically by
supposing small variation on each parameter and
calculating the corresponding variation on
E
p
∆
.
Measuring the position of the end-effector for a sufficient
number
e
of random configurations (minimum 14
configurations), we have:
ηηη
η
η
∆=∆
Ψ
Ψ
=
∆
∆
=∆
.),W(.
),(
),(
11
1
E
E
Q
q
q
p
p
P
ee
e
!!
Ε
ΕΕ
Ε
(12)
The rank of the matrix W is obtained as 39. The
identifiable parameters of the system are obtained by the
QR
decomposition of this matrix. Applying the rules of
priority described in section 3, the errors
∆
E
Py
B2
,
∆
E
Pz
B2
and
∆
E
Pz
B6
are not identifiable and their effect are
grouped on the other parameters which defines the
positions of the S-joints on the mobile platform. We
propose to fixe these parameters such that:
0
622
B
E
B
E
B
E
=== PzPzPy
(13)
This makes that the orientation of frame E, which cannot
be determined, is such that the x axis is along the
measured point and the point B
2
while the xy plane is
along the measured point and the points B
2
and B
6
. The
condition number of the observation matrix W for this
calibration method using a number of equations which is
equal to 4 times the number of parameters is about 2000.
4.3
Calibration using two inclinometers
In this calibration method the rotation angles of the
platform of the robot about x
m
and y
m
axis are measured
by two inclinometers fixed on the platform [5]. For a
given configuration q, the theoretical values
α
1
and
α
2
of
the inclinometers can be computed using the DKM. These
values are functions of some elements of the orientation
matrix
–1
A
E
and of the angle
γ
between the inclinometers
axes. The linear differential model can be written as:
η
γ
ηΦ
∆Ψ=∆
.),,(
q
(14)
where
∆
Φ
is the difference between the inclinometers
measured values
Φ
m
and those computed by the model
Φ
,
and
Ψ
is the numerical Jacobian matrix (cf. section 3).
Using a sufficient number
e
of configurations:
ηγη
Φ
Φ
∆⋅=
∆
∆
),,W(
1
Q
e
(15)
The rank of W is 36, there are 4 non identifiable
parameters concerning the U-joints (
∆
-1
Px
A1
,
∆
-1
Py
A1
,
∆
-1
Pz
A1
and
∆
-1
Py
A2
) and 3 on the position of the S-joints
(
∆
E
Px
B1
,
∆
E
Py
B1
and
∆
E
Pz
B1
). The effect of these
parameters are grouped on the other parameters of the
base (U-joints) and the platform (S-joints) respectively.
These results are confirmed by the study of the geometry
of the system. The position coordinates of the
inclinometers on the platform have no effect.
Consequently, we can consider that the origin O
E
, which
cannot be determined by this method, is aligned with the
origin of frame F
m
. Then we have by convention:
0
111
B
E
B
E
B
E
=== PzPyPx
(16)
Similarly, the position of the base of the robot with
respect to F
-1
has no influence on the inclinometers
measurement, as well as its orientation around the vertical
axis. We can define arbitrarily the origin of frame F
-1
as
A
1
and the axis x
-1
and z
-1
such that A
2
is in the plane
(A
1
x
-1
z
-1
). Then we have by definition:
0
2111
A
-1
A
-1
A
-1
A
-1
==== PyPzPyPx
(17)
The condition number of the linear observation matrix W
for this method using a number of equations which is
equal to 4 times the number of parameters is about 2000.
4.4
Calibration with mechanical constraints on the
orientation of the legs
This method uses the variables of the motorized prismatic
joints corresponding to configurations where either one U-
joint or one S-joint is fixed by mechanical lock, thus the
leg direction is constant with respect to the base or with
the movable platform [11].
Each U-joint
i
is described by 2 angles
θ
1,i
and
θ
2,i
, while
each S-joint
i
is defined using three angles
θ
3,i
,
θ
4,i
and
θ
5,i
.
