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

 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

with respect to the previous frame

i−i

is deﬁned by a homogeneous transformation that

depends on the geometric parameters of the manipu-

lator: skew angles α

, link lengths a

, joint offsets d

and joint angle offsets θ

:= cos(α), s

:= sin(α)

for compact notation):







−s

−c

0 s

0 0 0 1







(1)

The transformation A

from the base to the end-

effector of a robot with n joints is then obtained by:

∏

(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

with 6 error parameters: e = (e

, . . . , e

). The

rotational part of matrix E

consists of e

, e

which denote rotation about X,Y and Z axes with re-

spect to A

. The e

, e

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

can be expressed as:







1 −e

−e

1 e

0 0 0 1







(3)

Considering the introduced error parameters, the

general transformation model A

in presence of trans-

lational and rotational errors for a robot with n joints,

is given by:

∏

(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

, e

and e

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):







1 −e

−e

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

∈ P

. It can be written

in general form as:

= f

e), (6)

where f

is a non-linear function of error param-

eter vector e

e. Since these errors are small, this func-

tion can be linearized at 0

0. If the difference between

nominal position and measured position is ∆p

, for l

measurements:

∆p =







∆p



















e = J

e (7)

where J

is a Jacobian function of f

with respect

to the elements of error vector e

e, evaluated at 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

)

−1

∆p (8)

where (J

)

−1

is a left pseudo-inverse matrix

of J

. 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

= J

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

i−13

= e

i−16

= e

i−13

= −e

i−16

= e

i−16

= e

i−16

. (10)

If joint i is prismatic additional combinations are

present:

= e

i−11

= e

i−12

= e

i−13

= e

i−16

= −e

i−12

= e

i−13

= −e

i−16

= e

i−16

= e

i−16

. (11)

When only positional part of the end-effector pose is

considered and the last joint n is revolute and a

= 0:

(n−1)1

= e

(n−1)5

(n−1)2

= −e

(n−1)4

. (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

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 it can be

singular-value-decomposed as:

R = U

(13)

Then the closest rotation matrix to

R would be

given by (Horn et al., 1988):

R = sign(det(S

S))U

(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)

340 0 −

0 0

400 0

0 0 −

400 0 −

0 0

126 0 0 θ

be represented as a shoulder (S) and wrist (W ) spheri-

cal joints while θ

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

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

, e

= e

, e

= e

, e

= e

= −e

, e

= −e

, e

= −e

, e

= −e

= e

= d

, e

= e

, e

= −e

= e

, e

= e

. (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

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.

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.

REFERENCES

Elatta, A., Gen, L. P., Zhi, F. L., Daoyuan, Y., and Fei, L.

(2004). An overview of robot calibration. Information

Technology Journal, 3(1):74–78.

Horn, B. K., Hilden, H. M., and Negahdaripour, S. (1988).

Closed-form solution of absolute orientation using or-

thonormal matrices. JOSA A, 5(7):1127–1135.

Klimchik, A., Furet, B., Caro, S., and Pashkevich, A.

(2015). Identiﬁcation of the manipulator stiffness

model parameters in industrial environment. Mech-

anism and Machine Theory, 90:1–22.

Liou, Y. A., Lin, P. P., Lindeke, R. R., and Chiang, H.-D.

(1993). Tolerance speciﬁcation of robot kinematic pa-

rameters using an experimental design technique—the

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

482

taguchi method. Robotics and computer-integrated

manufacturing, 10(3):199–207.

Ma, L., Bazzoli, P., Sammons, P. M., Landers, R. G., and

Bristow, D. A. (2018). Modeling and calibration of

high-order joint-dependent kinematic errors for indus-

trial robots. Robotics and Computer-Integrated Man-

ufacturing, 50:153–167.

Marie, S., Courteille, E., and Maurine, P. (2013). Elasto-

geometrical modeling and calibration of robot manip-

ulators: Application to machining and forming appli-

cations. Mechanism and Machine Theory, 69:13–43.

Meggiolaro, M. A. and Dubowsky, S. (2000). An analytical

method to eliminate the redundant parameters in robot

calibration. In Robotics and Automation, 2000. Pro-

ceedings. ICRA’00. IEEE International Conference

on, volume 4, pages 3609–3615. IEEE.

Reduced Error Model for Learning-based Calibration of Serial Manipulators

483