Reduced Error Model for Learning-based Calibration of Serial
Manipulators
Nadia Schillreff and Frank Ortmeier
Chair of Software Engineering, Otto-von-Guericke-University Magdeburg, Germany
Keywords:
Modeling, Parameter Identification, Calibration.
Abstract:
In this work a reduced error model for a learning-based robot kinematic calibration of a serial manipula-
tor is compared with a complete error model. To ensure high accuracy this approach combines the geo-
metrical (structural inaccuracies) and non-geometrical influences like for e.g. elastic deformations that are
configuration-dependent without explicitly defining all underlying physical processes that contribute to po-
sitioning inaccuracies by using a polynomial regression method. The proposed approach is evaluated on a
dataset obtained using a 7-DOF manipulator KUKA LBR iiwa 7. The experimental results show the reduction
of the mean Cartesian error up to 0.16 mm even for a reduced error model.
1 INTRODUCTION
For a robot manipulator that is mainly used in repet-
itive applications (e.g. pick-and-place operations)
where the desired poses (position and orientation) of
the manipulator’s end-effector (EE) can be manually
taught, high repeatability is important to successfully
perform defined tasks. This ability to repeat a known
pose has submilimeter values for modern manipula-
tors. However if a task is unique, the robot is mostly
given a target pose defined in some relative or abso-
lute coordinate system. Such situations arise often
when robot’s poses are obtained through a simulation
during which the layout of the working environment
and a model of the robot are used. This requires spe-
cial attention to the accuracy of the simulated robot
model, and whether it corresponds to the actual kine-
matics of the robot.
The process of robot calibration that consists of
developing a mathematical model and identification
of parameters that are able to reflect the actual be-
havior of the investigated robot can be divided into
three categories (Elatta et al., 2004). The first is joint
calibration, which is also called first level calibra-
tion, where the difference between the actual joints
displacements and the encoder signals is considered.
Level two involves kinematic calibration, where the
robots kinematic parameters are determined. Level
three takes into account non-kinematic error sources
like elasticity of the links or the backlash of the joints.
The main sources for positioning inaccuracies
can be divided into geometric and non-geometric
errors.nGeometric errors are present when nominal
kinematic parameters of the robot do not correspond
to actual parameters due to for ex. manufacturing er-
rors.
Non-geometric errors include among others the
link and joint compliance, elastic deformations, trans-
mission nonlinearities, and thermal expansion. To in-
clude these effects into the robotic model the under-
lying processes can be expressed with gear or elastic
models as in (Klimchik et al., 2015) or (Marie et al.,
2013). Considering all sources that can potentially
contribute to the end-effector positioning errors, it is
difficult to model all relevant parameters explicitly.
Instead, learning-based approaches are proposed
to model the behaviour of a robot. This allows us
to solve the calibration problem as a function esti-
mation task based on the measured data. In general
errors of the robot’s EE may originate from five fac-
tors (Liou et al., 1993): environmental (e.g. tem-
perature or the warm-up process), parametric (e.g.
Kinematic parameter variation due to manufacturing
and assembly errors, influence of dynamic parame-
ters, friction and other nonlinearities), measurement
(resolution and nonlinearity of joint position sensors),
computational (computer round-off and steady-state
control errors) and application (e.g. installation er-
rors). To be able to reflect different influences in the
model, experiments that include variations of the rel-
evant factors would have to be conducted. This work
considers the influence of only a limited amount of
478
Schillreff, N. and Ortmeier, F.
Reduced Error Model for Learning-based Calibration of Serial Manipulators.
DOI: 10.5220/0009835804780483
In Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2020), pages 478-483
ISBN: 978-989-758-442-8
Copyright
c
2020 by SCITEPRESS Science and Technology Publications, Lda. All rights reser ved
described factors, mainly concentrating on the inter-
nal parameters of the robot. There is a number of cal-
ibration approaches that use joint-dependant errors,
but they often rely on complicated models with a big
number of parameters (Ma et al., 2018).
In Section 2, a full and reduced error model is pre-
sented. Regression analysis is than used to find the
values of the introduced error parameters. In Section
3, we applied this method to the performance evalu-
ation of the KUKA LBR iiwa 7 manipulator to com-
pare and analyze the two error models. We conclude
our paper in Section 4.
2 JOINT-DEPENDENT ERROR
MODELING
In this section, we describe a method used to include
relevant processes influencing position and orienta-
tion of the end-effector in the modeling of a robot.
First, D-H representation and generalized error mod-
eling is used to establish the transformation from the
base frame of the robot to the flange in presence of
errors. Then, redundant error parameters are elimi-
nated and regression analysis is used to include the
joint-dependent influences.
2.1 Robot Kinematic Model
D-H parameters are used to describe the kinematics
of a manipulator. The position and orientation of ref-
erence frame A
A
A
i
with respect to the previous frame
A
A
A
ii
is defined by a homogeneous transformation that
depends on the geometric parameters of the manipu-
lator: skew angles α
i
, link lengths a
i
, joint offsets d
i
and joint angle offsets θ
i
(c
α
:= cos(α), s
α
:= sin(α)
for compact notation):
A
A
A
i
=
c
θ
i
s
θ
i
c
α
i
s
θ
i
s
α
i
a
i
c
θ
i
s
θ
i
c
θ
i
c
α
i
c
θ
i
s
α
i
a
i
s
θ
i
0 s
α
i
c
α
i
d
i
0 0 0 1
(1)
The transformation A
A
A
T
from the base to the end-
effector of a robot with n joints is then obtained by:
A
A
A
T
=
n
i
A
A
A
i
(2)
2.2 Generalized Error Model
To include differences between the ideal robot and the
actual one, the kinematic model represented by (2)
can be extended by translational and rotational error
parameters. The introduction of these errors leads to
Figure 1: Frame transformations in presence of errors.
displacement of joint frames from their nominal loca-
tions as shown in Figure 1. For joint i this difference
in frames can be represented by a homogeneous ma-
trix E
E
E
i
with 6 error parameters: e = (e
i1
, . . . , e
i6
). The
rotational part of matrix E
E
E
i
consists of e
i4
, e
i5
, e
i6
,
which denote rotation about X,Y and Z axes with re-
spect to A
A
A
i
. The e
i1
, e
i2
, e
i3
represent translation in
X,Y and Z direction respectively.
These errors for modern manipulators are ex-
pected to be small, so that the generalized error for
a frame A
A
A
i
can be expressed as:
E
E
E
i
=
1 e
i6
e
i5
e
i1
e
i6
1 e
i4
e
i2
e
i5
e
i4
1 e
i3
0 0 0 1
(3)
Considering the introduced error parameters, the
general transformation model A
A
A
E
in presence of trans-
lational and rotational errors for a robot with n joints,
is given by:
A
A
A
E
=
n
i
A
A
A
i
E
E
E
i
(4)
When analyzing the effect rotational and transla-
tional components of the error parameters have on the
resulting error of the EE, it is clear that e
i4
, e
i5
and e
i6
have much greater influence. Taking this into account
we can also further investigate the extend of these in-
fluences with the following error models:
1. Only rotational errors are present, resulting in 3 ×
n error parameters (RotXYZ-error model):
E
E
E
i
=
1 e
i6
e
i5
0
e
i6
1 e
i4
0
e
i5
e
i4
1 0
0 0 0 1
(5)
2. Errors in translational as well as in rotational
parts, corresponding to the most general case as
in (3), leading to 6 × n error parameters (RotXYZ-
TransXYZ-error model).
Reduced Error Model for Learning-based Calibration of Serial Manipulators
479
2.3 Error Parameters Calculation
Now that we defined error models, the introduced er-
ror parameters should be estimated based on the ex-
perimental data. The nominal pose of the end-effector
defined by (4) should be as close as possible to the
measured reference pose p
iT
P
T
. It can be written
in general form as:
p
iT
= f
f
f
e
(e
e
e), (6)
where f
f
f
e
is a non-linear function of error param-
eter vector e
e
e. Since these errors are small, this func-
tion can be linearized at 0
0
0. If the difference between
nominal position and measured position is p
i
, for l
measurements:
p =
p
1
.
.
.
p
l
=
J
e1
.
.
.
J
el
e
e
e = J
J
J
e
e
e
e (7)
where J
J
J
e
is a Jacobian function of f
f
f
e
with respect
to the elements of error vector e
e
e, evaluated at 0
0
0.
Under assumption that introduced error parame-
ters are constant the above equation can be solved
with for example a least squares technique, in which
case the solution would be of the form:
e = (J
J
J
T
e
J
J
J
e
)
1
J
J
J
T
e
p (8)
where (J
J
J
T
e
J
J
J
e
)
1
J
J
J
T
e
is a left pseudo-inverse matrix
of J
J
J
e
. But considering the need to include the non-
geometric influences (which are configuration depen-
dent and as result can not be constant) into developed
model, this approach can no longer be used. In this
case we can consider each measurement individually:
p
i
= J
J
J
ei
e
e
e (9)
After solving (9) for every measured-nominal
point pair, all introduced error parameters can be cal-
culated.
2.4 Indistinguishable Error Parameters
Some generalized errors from link i 1 contribute to
the same EE pose errors as errors from link i and their
individual influence can not be distinguished. To re-
move this redundancy from the model an analytical
approach described in (Meggiolaro and Dubowsky,
2000) can be used. For each link i following com-
binations have the same effect on the position or ori-
entation of the end-effector:
e
i2
= e
i13
s
α
i
= e
i16
a
i
c
α
i
e
i3
= e
i13
c
α
i
= e
i16
a
i
s
α
i
e
i5
= e
i16
s
α
i
e
i6
= e
i16
c
α
i
. (10)
If joint i is prismatic additional combinations are
present:
e
i1
= e
i11
e
i2
= e
i12
c
α
i
= e
i13
s
α
i
= e
i16
a
i
c
α
i
e
i3
= e
i12
s
α
i
= e
i13
c
α
i
= e
i16
a
i
s
α
i
e
i5
= e
i16
s
α
i
e
i6
= e
i16
c
α
i
. (11)
When only positional part of the end-effector pose is
considered and the last joint n is revolute and a
n
= 0:
e
(n1)1
= e
(n1)5
d
n
e
(n1)2
= e
(n1)4
d
n
. (12)
The above equations can now be used to elimi-
nate the redundunt error parameters depending on the
structure of the manipulator. It should be noted that
apart from this approach that studies the structural
properties of the manipulator a numerical approach
can also be used. But considering that it needs to
be carried out for every data point pair, an analytical
method is preferred.
2.5 Error Parameters Modeling
After calculating the values of the error parameters for
every measurement and eliminating the redundant pa-
rameters, an estimator based on the obtained data can
be constructed and error parameters can be modeled
as functions of joint variables. For this, the available
experimental measurements of the position of EE are
first divided into training and testing sets. The train-
ing set is used to determine the regression coefficients
of the model and for tuning model parameters, and
performance evaluation is done on the test set.
2.5.1 Regression Analysis
As input parameters, the joint configurations of the
manipulator were chosen. In order to model nonlin-
ear relationships, they were also extended with com-
binations of different joint configurations with a poly-
nomial degree of up to 4. Even if higher polynomial
degrees can better model some functions we should
avoid overfitting. To prevent features that have big
variance from dominating the objective function, all
input variable were centered and scaled. Ridge re-
gression was chosen for modeling error parameters
of the manipulator, because it uses regularization by
minimizing a penalized residual sum of squares. This
approach is effective in case of highly correlated input
variables.
ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics
480
To determine the regularization parameters of the
models m-fold cross-validation (CV), during which
the training set is split into m smaller sets, is used.
Then for each of these sets a model is trained using
m 1 of the remaining sets, and the resulting model
is validated on the remaining part of the set. The re-
sulting performance measure of the m-fold CV is cal-
culated as the average of the values computed in the
loop. Those parameters that result in the smallest CV
error are chosen for model training. To evaluate the
performance of regression models, R
2
metric (the co-
efficient of determination) was used.
It should also be noted that for the resulting model
the rotational part of the estimated resulting matrix
can no longer be a rotation matrix. Therefore, the ob-
tained values should go through an additional orthog-
onalization procedure to project the rotational part to
the closest rotation matrix. If the unprojected rota-
tional part of the matrix is denoted as
˜
R
R
R it can be
singular-value-decomposed as:
˜
R
R
R = U
U
US
S
SV
V
V
T
(13)
Then the closest rotation matrix to
˜
R
R
R would be
given by (Horn et al., 1988):
R
R
R = sign(det(S
S
S))U
U
UV
V
V
T
(14)
3 EVALUATION
To show the validity of the proposed approach the
modeled values were compared with values measured
from a real robot. A 7DOF LBR iiwa, a lightweight
robot with non-linear influences in the structure was
used in the experiments.
3.1 Experimental Setup
In order to compare considered input parameters, re-
sulting models are evaluated on an example of KUKA
LBR iiwa 7 robot. The position of the EE was mea-
sured using a FARO laser tracking system. The accu-
racy of the laser tracker is up to 0.015 mm. Because
only positional data of the manipulators end-effector
could be recorded no comparison of the rotational re-
sults would be made. The measured dataset includes
position data from 800 uniformly distributed points,
measured in the working volume of the robot. Con-
sidering the redundancy of the 7-DOF manipulator re-
garding the task space, we need additional parame-
ter to uniquely specify a configuration for a given EE
pose. For a S R S (spherical-rotational-spherical)
manipulator structure, the first and last three joints can
Table 1: LBR IIWA R800 nominal D-H parameters.
Frame d (mm) a (mm) α (rad) θ (rad)
A
1
340 0
π
2
θ
1
A
2
0 0
π
2
θ
2
A
3
400 0
π
2
θ
3
A
4
0 0
π
2
θ
4
A
5
400 0
π
2
θ
5
A
6
0 0
π
2
θ
6
A
7
126 0 0 θ
7
be represented as a shoulder (S) and wrist (W ) spheri-
cal joints while θ
4
is then called the elbow (E). Using
this notation, one of the additional parameters can be
an arm angle, which corresponds to the angle between
the plane spanned by S, E, and W , and a reference
plane of a virtual non-redundant manipulator. This
reference plane can be chosen for the case when joint
angle θ
3
is fixed to zero. All of the experimental mea-
surements were taken for the same value of the arm
angle.
The D-H parameters of LBR IIWA R800 are ob-
tained from nominal data according to the official
manufacturer specification and are listed in Table 1.
Considering that LBR IIWA has all revolute joints,
(10) and (12) result in the following combinations of
error parameters:
e
22
= e
13
, e
25
= e
16
, e
32
= e
23
, e
35
= e
26
e
42
= e
33
, e
45
= e
36
, e
52
= e
43
, e
55
= e
46
e
56
= e
65
= d
7
e
61
, e
62
= e
53
, e
64
= e
62
d
7
e
73
= e
63
, e
66
= e
76
. (15)
This reduces the number of error parameter func-
tions to 12 for the RotXY Z-error model, and 25 for
the RotXYZTransXY Z-error model.
The available dataset was split into training (100
points) and testing (600 points) sets. First the two
models were compared with different polynomial de-
grees. The models were trained using joint angles
and 5-fold CV to choose the hyper-parameters. Re-
sults presented in Table 2 show that there is no sub-
stantial difference in resulting accuracy between the
RotXYZ and RotXY ZTransXY Z-error models. In the
following the RotXY Z-error model with 3rd polyno-
mial degree was chosen for the further analysis. To
illustrate the results of the modeling, the frequency
plot of the resulting positional errors, measured as the
Cartesian distance between the nominal and measured
position (red) alongside with the distance from mod-
eled to measured position (blue) is presented in Figure
2.
To investigate how the size of the training set in-
fluences the performance of the modeling approaches,
Reduced Error Model for Learning-based Calibration of Serial Manipulators
481
Table 2: Distance error (mm) comparison for different error models.
Error model degree 2 degree 3 degree 4
mean std max mean std max mean std max
RotXYZTransXY Z 0.1658 0.0972 0.8147 0.1583 0.0889 0.5612 0.1570 0.0878 0.5242
RotXYZ 0.1759 0.1134 0.8953 0.1648 0.0975 0.6241 0.1563 0.0885 0.5102
Figure 2: Frequency histogram of the distance error, mm,
for RotXYZ model of 3rd degree.
error models were trained using [25, 35, . . . , 125]
points from the available dataset. To ensure that each
point was used for training and testing at least once
5-fold CV was used. To account for the variance in
the algorithm itself the cross validation procedure was
run 25 times, giving an estimate of the performance of
the algorithm on the dataset and an estimation of how
robust its performance is.
The box plots of the resulting positional errors for
RotXYZ (Figure 3) show that the mean value stays al-
most the same and only the maximum values change.
From the Table 3, that presents the statistical infor-
mation of the regression models comparison, we can
see the steady decline of mean, standard deviation and
maximum values as the number of training points in-
creases.
Table 3: Distance error comparison for different number of
training points.
T
raining points
mean,
mm std, mm max, mm
25 0.2979
0.2965 8.5011
35 0.2469
0.2047 6.5084
45 0.2316
0.2047 4.4081
55 0.2163
0.1634 4.8526
65 0.2039 0.1553 5.2877
75 0.2013
0.1456 5.3608
85 0.1933
0.1579 3.4861
95 0.1911
0.1279 3.7388
105 0.1879
0.1265 3.8411
115 0.1841 0.1162 3.5205
125 0.1799
0.1044 2.4463
Figure 3: Distance error for the test set for different number
of training points.
4 CONCLUSIONS
In this paper, a full RotXY ZTransXY Z and only ro-
tational RotXY Z-error model for learning-based cal-
ibration were compared. Both models showed sig-
nificant improvement in positional error. And it was
shown that reduced model performed comparably
with the full model, but with half of the needed error
functions. The RotXY Z model was then used to ana-
lyze the performance of modeling based on the num-
ber of training points. Of course as with any model-
based approach the bigger number of training set re-
sulted in improved performance, but starting from 85
points the rate of improvement declined. Because
the measurements were uniformly distributed in the
working volume of the robot and the training points
were chosen randomly, the results contain some out-
liers. In the future work we will consider finding opti-
mal training points distribution to prevent such cases.
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