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Production Engineering
https://doi.org/10.1007/s11740-020-00975-8
MACHINE TOOL
Probabilistic information fusion tomodel thepose‑dependent
dynamics ofmilling robots
MaximilianBusch1 · FlorianSchnoes1· ThomasSemm1· MichaelF.Zaeh1· BirgitObst2· DirkHartmann2
Received: 22 May 2020 / Accepted: 27 July 2020
© The Author(s) 2020
Abstract
Conventional industrial robots are increasingly used for milling applications of large workpieces due to their workspace and
their low investment costs in comparison to conventional machine tools. However, static deflections and dynamic instabilities
during the milling process limit the efficiency and productivity of such robot-based milling systems. Since the pose-dependent
dynamic properties of the industrial robot structures are notoriously difficult to model analytically, machine learning meth-
ods are recently gaining more and more popularity to derive system models from experimental data. In this publication, a
modeling concept based on a modern information fusion scheme, fusing simulation and experimental data, is proposed. This
approach provides a precise model of the robot’s pose-dependent structural dynamics and is validated for a one-dimensional
variation of the robot pose. The results of two information fusion algorithms are compared with a conventional, data-driven
approach and indicate a superior model accuracy regarding interpolation and extrapolation of the pose-dependent dynam-
ics. The proposed approach enables decreasing the necessary amount of experimental data needed to assess the vibrational
properties of the robot for a desired pose. Additionally, the concept is able to predict the robot dynamics at poses where
experimental data is very costly to gather.
Keywords Robotic milling· Structural dynamics· Machine learning· Multi-fidelity information fusion
1 Introduction
In order to increase the economic efficiency of milling pro-
cesses in terms of investment, operating and maintenance
costs, conventional industrial robots are an attractive alter-
native for machining of large workpieces [9, 26]. However,
the low static and dynamic stiffness of industrial robots
often lead to static displacements of the tool or to dynamic
instabilities, also called chatter [1, 10, 21, 25]. The static
displacements of the tool during the process result in devia-
tions from the target workpiece geometry, whereas dynamic
instabilities result in an insufficient surface quality or might
even lead to increased tool wear as well as failure of spindle
components.
Thus, current research projects address the precise mod-
eling and the system identification of the static and dynamic
structural behavior of milling robots. This allows to compen-
sate the estimated errors by choosing a compensated tool
path or by choosing stable process parameters [22].
The dynamics of the robot structure are usually formu-
lated in the following, analytic form, depending on the gen-
eralized coordinates
q
, the velocities
q
and accelerations
q
[14]:
M
represents the mass matrix,
C
the Coriolis matrix,
N
con-
tains gravitational terms and other generalized forces such
as joint forces due to the internal joint stiffness and damp-
ing properties.
𝜏
are the resulting generalized forces and
moments. The efficient simulation of the dynamic behavior
of such a rigid body model is provided by software pack-
ages, such as the Dynamics Animation and Robotics Toolkit
(DART) [12] or the Rigid Body Dynamics Libary (RBDL)
[4]. Nonetheless, the identification of the model param-
eters, such as the inertia, stiffness and damping terms of
the joints, remain open issues. Consequently, an incorrect
(1)
M
(
q
)
q
+C(
q,
q
)
q
+N(
q,
q
)=𝜏
.
* Maximilian Busch
maximilian.busch@iwb.tum.de
1 Institute forMachine Tools andIndustrial Management,
Technical University ofMunich (TUM), Boltzmannstr. 15,
85748Garchingb.Muenchen, Germany
2 Siemens AG, Otto-Hahn-Ring 6, 81739Muenchen, Germany
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parameterization can lead to a potentially inaccurate struc-
tural model. In comparison to conventional machine tools,
where flexible multi-body simulations are used to model the
position dependent dynamics [23], this is especially crucial
for robotic milling processes of large workpieces, since the
dynamic properties may vary significantly during the pro-
cess due to the highly nonlinear kinematic structure. Thus,
modeling the pose-dependent properties of the robot’s struc-
ture is still an unsolved problem [2].
Nguyen etal. [15] propose a data-driven approach to
model the position dependency of the modal parameters.
This approach circumvents the issue of modeling the
dynamic properties of a physics-based model accurately, but
a large data set is needed to incorporate the full workspace
into the data-driven model.
However, a rigid body simulation can still provide a rough
estimate of the robot dynamics. Hence, taking knowledge
from both information sources into account can alleviate the
burden of gathering experimental data in the whole workspace
and reduce the necessary amount of experiments. This paper
introduces an approach to model and predict the pose-depend-
ent robot dynamics precisely by fusing simulation results with
experimental data. The applicability of two information fusion
algorithms is demonstrated in comparison to a conventional
machine learning approach by inter- and extrapolating the
dynamics at the tool center point (TCP) for a variation of the
robot’s third axis. The concept serves as a feasibility study on
how advanced algorithms from the field of machine learning
can be fused with physical knowledge to increase the reli-
ability of robot based machining processes.
The paper is structured as follows: the concept is moti-
vated in Sect.2 on the basis of a comparative study of the
vibrational properties determined by a rigid body simulation
and the measured dynamic properties. Section3 provides an
overview on methods for information fusion and introduces
the main concept of the proposed approach. The model setup
and training processes of two information fusion algorithms
and a conventional data-driven approach are presented in
Sect.4, followed by a discussion of the results in Sect.5.
Section6 concludes this work by discussing the present lim-
its and gives an outlook on future research.
2 Model analysis
In order to visualize the strength of the proposed
approach, the dynamics of a KUKA Quantec Prime 240
robot with a high-speed milling spindle were simulated
and experimentally measured in dependence of the fre-
quency f (
5 Hz ≤f≤30 Hz
) and the robot’s third axis
𝜙z,3
(
70◦
≤
𝜙z,3
≤
120◦
). In both cases, the vibrational behavior
is represented by the frequency response function (FRF)
H(f,𝜙z,3)
at the spindle close to the TCP, resulting from
an excitation in horizontal direction at the same point (see
Fig.1).
2.1 Rigid body simulation
The rigid body simulation of the robot dynamics was car-
ried out using Matlab Simscape Multibody with estimated
stiffness and damping properties of the joints and estimated
mass and inertia properties of the bodies. The model pro-
vides three rotational degrees of freedom (DOFs)
𝜙x
,
𝜙y
and
𝜙z
at each joint i with corresponding rotational stiffness
and damping properties. Figure2 illustrates the rigid body
model. The model parameters are provided in the appendix.
The frequency response function
H(f,𝜙z,3)
is calculated
from a linear state space model at a given axis angle of
𝜙z,3
.
Figure3 illustrates the frequency response function in the
range
70◦
≤
𝜙z,3
≤
120◦
. The simulation was carried out in
discrete steps of
𝛥sim𝜙z,3 =0.1◦
, resulting in 501 simulation
samples with a frequency resolution of
𝛥simf=0.01 Hz
.
As illustrated in Fig.3, it is clearly visible that the rigid
body simulation captured two resonance frequencies:
– The first eigenfrequency starts at
7 Hz
and increases with
increasing axis angle
𝜙z,3
to
11 Hz
.
Fig. 1 Pose variation of the third axis
𝜙z,3
for the data set generation.
The excitation’s driving point is marked with an arrow
Fig. 2 Rigid body model with associated, rotational DOFs at each
joint
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– The second eigenfrequency starts at
28 Hz
and
decreases with increasing axis angle
𝜙z,3
to
20 Hz
.
The corresponding two mode shapes are illustrated in
Fig.4.
2.2 Experimental data
Similar to the simulation, the FRF
H(f,𝜙z,3)
was meas-
ured experimentally at discrete axis configurations in the
same range
70◦
≤
𝜙z,3
≤
120◦
via impact testing at the
driving point illustrated in Fig.1. The experiment was
conducted in discrete steps
𝛥exp 𝜙z,3 =2◦
, resulting in
26 measurement samples with a frequency resolution of
𝛥exp f=0.0455 Hz
(visualized in Fig.5).
In contrast to the rigid body simulation, the experi-
mentally captured dynamics include three resonance
frequencies:
– The first eigenfrequency starts at
8 Hz
and increases
with increasing axis angle
𝜙z,3
to
10 Hz
.
– The second eigenfrequency starts at
22 Hz
and
decreases with increasing axis angle
𝜙z,3
to
17 Hz
.
– The third eigenfrequency starts at
25 Hz
and decreases
with increasing axis angle
𝜙z,3
to
19 Hz
.
The shapes of the first modes of the used milling robot had
been experimentally identified in previous works [21, 27].
The mode shapes which are assumed to correspond to the
three measured modes in this publication are illustrated
in Fig.6.
The general pose-dependent behavior of the vibration
modes is roughly captured in the rigid body simulation, as
the general shape of the pose-dependencies are adequately
captured: the pose-dependent behavior of the first mode
is well captured in the simulation and the pose-dependent
behavior of the second and third measured mode is roughly
comparable to the behavior of the second simulated mode.
φz,3in ◦
→
fin Hz →
Hin m
N→
80 100 120
10
20
30
10−8
10−6
10−4
Fig. 3 Simulated pose-dependent frequency response at the driving
point; the amplitude
H(f,𝜙z,3)
is illustrated in color code
Fig. 4 Mode shapes of the two captured modes in the simulation for
an axis angle of
𝜙z,3 =90◦
φz,3in ◦
→
fin Hz →
Hin m
N→
80 100 120
10
20
30
10−8
10−6
10−4
Fig. 5 Measured pose-dependent frequency response at the driving
point; the amplitude
H(f,𝜙z
,3
)
is illustrated in color code
Fig. 6 Mode shapes of three experimentally captured modes for the
depicted axis configuration (illustrations taken from [21])
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Nonetheless, the simulation does not capture a third
mode, but only predicts a second mode with a comparable
pose-dependent behavior. It is assumed, that the simulation
does not account for the mode which corresponds to the
shape in Fig.6b). The reasons for this issue are twofold:
– Inaccurate model parameterization Since the rigid body
model is based on a large number of parameters, the
identification procedure might have failed to estimate all
physical parameters accurately. An inaccurate identifi-
cation of the mass, stiffness or damping parameters can
lead to significantly different dynamic properties. For
example, if damping parameters are assumed too high, a
mode can be erroneously damped in the simulation.
– Unmodeled physical effects Although it is assumed that
the measured mode shapes are representing rigid body
modes, the simulation might not have taken all physical
degrees of freedom into account. Additionally, the simu-
lation does not account for mode coupling, which can be
a significant vibrational effect of industrial robots [25].
3 Probabilistic information fusion
In order to cope with such issues where approximate models
can be cheaply evaluated, but precise data is rare or costly to
gather, modern machine learning algorithms are capable of
fusing information from different data sets and information
sources with different fidelity levels.
Hence, such algorithms are also referred to as multi-fidel-
ity information fusion algorithms [5, 16].
Meng etal. [13] proposed a deep learning approach,
where two artificial neural networks are coupled hierarchi-
cally to fuse information from two data sets with a different
fidelity level. An uncertainty estimation is included based on
a Dropout approach [7]. Additionally, since the implemen-
tation of this approach makes use of automatic differentia-
tion methods, the training objective can also take physically
motivated bounds in an analytic form into account, as previ-
ously published in [19, 20].
Similarly, the information fusion can be based on proba-
bilistic, Bayesian inference [3, 6, 11, 17, 18]. The approaches
mainly make use of Gaussian and deep Gaussian process
regression techniques and differ in the ability to fuse data
with linear or nonlinear correlations. In contrast to deep
learning-based approaches, Bayesian inference algorithms
rely on a statistically well-founded theory. Thus, they pro-
vide a reliable uncertainty estimation.
Such a probabilistic information fusion scheme can
be set up to infer the mapping between a simulation of
the robot’s structural dynamics and the experimental
data: The pose-dependent dynamic behavior can be eas-
ily approximated by a rigid body simulation. In contrast,
experimental data, for example gathered via impact testing
or automated shaker experiments, is costly to generate, but
provides a precise representation of the pose-dependent
dynamics. The approach presented in this paper addresses
this issue by fusing the information of both sources using
a multi-fidelity information fusion algorithm.
The setup of such a multi-fidelity information fusion
scheme is described more formally in the following:
As previously introduced, the general objective is to find
an unknown mapping
𝜓
between an input space
X
and an
output space
Y
. The latter is assumed to be one-dimensional:
To find such a mapping, a training data set
D
is given, which
consists of m input data samples
x∈X
with corresponding
output data
y∈Y
:
In the present case, the objective is to find a mapping
between the input data
x
=[f
𝜙z,3
]
T
and the output data
space
y=H
. For the depicted scenario, there are two infor-
mation sources, which provide information on this unknown
relationship. The goal of the information fusion algorithm
is to find a suitable mapping between the approximative but
cheap data
DLF
(also called low fidelity data) and the precise
but costly data
DHF
(also called high fidelity data).
If there is a linear relationship between the two fidelity
levels, the relationship between low fidelity
𝜓LF(x)
and
high fidelity
𝜓HF
(
x)
can be described with a constant scal-
ing factor
𝜌c
and a bias term
𝛿(x)
[3]:
In case of a nonlinear relationship between the two fidelity
levels, a constant scaling factor is not sufficient. Instead, the
functional relationship between low and high fidelity data
must incorporate a nonlinear transformation
𝜌nl(𝜓LF (x),x)
[3]:
For the given scenario, the two nonlinear information fusion
algorithms NARGP [17] and MFDGP [3] are used to infer
the possibly nonlinear mapping between a simulation of the
robot’s structural dynamics and the experimental data. Both
approaches have a hierarchical, layer wise (deep) architec-
ture of Gaussian processes in common, where a separate
Gaussian process is conditioned for each fidelity level. The
approaches differ in terms of which data set is used to con-
dition each Gaussian process and in their specific training
procedure.
In the following, the setup and the training processes of
both information fusion algorithms NARGP and MFDGP
(2)
𝜓∶x∈X
↦
y∈Y.
(3)
D
= {(
x
1
,y
1
)…(
x
m
,y
m
)}
.
(4)
𝜓HF
(
x)=
𝜌c
⋅
𝜓LF
(
x)+
𝛿
(
x)
.
(5)
𝜓HF
(
x
)=
𝜌nl
(
𝜓LF
(
x
)
,x
)+
𝛿
(
x
)
.
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are described. For comparison, the setup and the training
process of a conventional, Gaussian process model are also
described. In Sect.5, the performance of the information
fusion approaches is assessed. The conventional Gaussian
process regression model serves as a benchmark to illus-
trate the superior performance of the information fusion
techniques.
4 Model setup andtraining
To illustrate the performance of such a data-driven modeling
approach, the data set
DHF
is divided into a training data
set
DHF,train
and a test data set
DHF,test
. The training data
set
DHF,train
consists of the first 70 % of the data regard-
ing the axis angle
𝜙z,3
, which corresponds approximately
to the range of
70◦
≤
𝜙z,3
≤
105◦
, whereas the test data set
DHF,test
consists of all remaining data samples in the range
105◦<𝜙
z,3
≤
120◦
(illustrated in Fig.7).
DLF,train
consists of
12801 data points, while
DHF,train
only consists of 2622 data
points. Both data sets are taken equally distributed from the
data shown in Sect.2.1.
In the following, it will be shown, that both information
fusion algorithms are able to combine the information from
both data sets
DLF
and
DHF,train
. The model setup and train-
ing process was conducted with the Python library Emukit
[24].
As proposed by Perdikaris etal.[17], the most general
NARGP model is based on two consecutive Gaussian pro-
cesses with a radial basis function kernel
k(x,x�)
:
where
k(x,x�)
is the kernel to model the covariance between
two data points
x
and
x′
,
𝜎2
is the variance and l the length
scale of the kernel.
The MFDGP model incorporates a more complex kernel
design. This layer-wise kernel design is described in [3]. The
(6)
k
(x,x�)=𝜎2⋅exp
−
x−
x
�
2
l2
,
radial basis function kernel was extended in such a way, that
linear relationships between the low and high fidelity data
are better addressed.
Although different kernel designs are possible, the model
setup in this work follows the proposed kernel design of
Perdikaris etal. [17] and Cutajar etal.[3], since physical
expert knowledge is deliberately not included by manual
kernel shaping.
To speed up the training process, MFDGP relies on a
sparse variational approximation method with 800 induc-
ing points and a mini-batch size of 50. Comparable to the
approach of Cutajar etal.[3], the model training was con-
ducted consecutively: in the first step, the model was trained
with a fixed variance, followed by a second training step
which includes the model variance in the training process.
Table1 summarizes the model and training parameters
for the MFDGP model.
As previously mentioned, a conventional Gaussian pro-
cess regression model can serve as a reference benchmark.
The model training was conducted using the Python library
Gpy [8]. The Gaussian process regression model was set up
with an additive standard radial basis function kernel and a
bias kernel to shift the mean as follows:
In the following section, the training results are compared
and the model performance of each algorithm is assessed.
5 Results
The proposed information fusion approach can improve the
model accuracy significantly in comparison to conventional
machine learning approaches. Figure8 shows the predic-
tion results for the training and the test data set. Since all
three algorithms provide probabilistic models, the frequency
response function
H(f,𝜙z,3)
is not only represented by the
expected value
𝔼(f,𝜙z,3)
, but also by an uncertainty estima-
tion based on the standard deviation
𝜎(f,𝜙z,3)
.
The model performance can be evaluated by consider-
ing the model’s capabilities to interpolate and extrapolate
the dynamic properties of the robot: It can be observed,
that the conventional Gaussian process model is capable to
represent the robot dynamics for axis angles where exper-
imental data is nearby. As seen in Fig.8a), the model
is able to interpolate within the training data set (axis
angles between
70◦
and
105◦
), but the model’s accuracy
quickly deteriorates with increasing axis angle in the test
data set (from axis angle
105◦
onwards). Thus, a conven-
tional data-driven approach is unable to extrapolate the
robot dynamics, which results in poor predictions of the
(7)
k
(x,x�)=𝜎2⋅exp
−
x−x�
2
l
2
+
1.
Fig. 7 Division into training and test data sets
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robot dynamics for workspace regions, where no experi-
mental data has been gathered by an end user. The model
takes this issue into account by increasing the uncertainty
𝜎GP(f,𝜙z,3 )
, as the expected value
𝔼GP(f,𝜙z,3 )
tends to a
mean value (see Fig.8d).
The MFDGP model can extrapolate the dynamics better
that the conventional Gaussian process regression model
(see Fig.8b). Nonetheless, similar issues arise: the MFDGP
model lacks a precise extrapolation of the frequency
response, while the uncertainty increases with increasing
axis angle in the test data set from axis angle
105◦
onwards
(see Fig.8e).
φz,3in ◦→
fin Hz →
(a)
EGP(x)
Hin m
N→
φz,3in ◦→
fin Hz →
(b)
EMfdgp(x)
Hin m
N→
φz,3in ◦
→
fin Hz →
(c)
ENargp(x)
Hin m
N→
φz,3in ◦→
fin Hz →
(d)
σGP(x)
Hin m
N→
φz,3in ◦→
fin Hz →
(e)
σMfdgp(x)
Hin m
N→
φz,3in ◦
→
fin Hz →
(f)
σNargp(x)
Hin m
N→
70 80 90 100 110 120
10
20
30
10−7
10−6
10−5
70 80 90 100 110 120
10
20
30
10−7
10−6
10−5
70 80 90 100 110 120
10
20
30
10−7
10−6
10−5
70 80 90 100110 120
10
20
30
10−7
10−6
10−5
70 80 90 100110 120
10
20
30
10−7
10−6
10−5
70 80 90 100110 120
10
20
30
10−7
10−6
10−5
Fig. 8 The model prediction using the expected value
𝔼(x)
and the uncertainty estimation using the standard deviation
𝜎(x)
for all three
algorithms: the conventional Gaussian process (GP), the MFDGP algorithm and the NARGP algorithm; the start of the test data set at
𝜙z,3 =105◦
is marked with a white, dashed line
Table 1 MFDGP model and training parameters
Parameter Value
Training step 1: fixed variance
Algorithm AdAm
Learning rate 0.01
Iterations 20,000
Training step 2: variable variance
Algorithm AdAm
Learning rate 0.001
Iterations 15,000
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In contrast, the trained NARGP model is able to extrap-
olate the pose-dependent dynamic behavior (see Fig.8c).
Since the amplitudes of the measured frequency response
functions vary especially at the resonance frequencies due to
a nonlinear structural dynamic behavior, the NARGP model
is able to incorporate this knowledge implicitly in its uncer-
tainty
𝜎NARGP(f,𝜙z,3 )
(see Fig.8f).
In the following, the model accuracy is assessed in detail.
Figure9 illustrates the accuracy of a NARGP model for the
training as well as the test data. Additionally, the simulation
accuracy is illustrated as a benchmark. As indicated by the
black dots, the NARGP model performs very well on the
training data set. Additionally, it is visible that the model
also performs well on the test data set (indicated by the blue
dots), whereas the rigid body simulation accuracy is sig-
nificantly worse, since it does not incorporate all vibrational
modes. Furthermore, the NARGP model’s standard devia-
tion provides a comprehensible uncertainty quantification.
Table2 summarizes the model performance by quantify-
ing the prediction accuracy of the rigid body simulation and
the three models, pointing out the significantly more accu-
rate prediction of the NARGP information fusion scheme in
regard to the coefficient of determination
R2
and the root-
mean-square error RSME.
In order to assess the interpolation capabilities and the
efficiency of the depicted approach using the NARGP
algorithm, the model performance is evaluated by gradu-
ally reducing the number of experiments used in the training
data set. For each assessment
i∈{1, 2, 3, 4}
, only every
ith
experiment of the whole data set was used for training as
high fidelity data. On the contrary, the complete simulation
data was used as the low fidelity data set for each training.
The results, illustrated in Fig.10, indicate that the pre-
diction accuracy decreases with fewer experiments used for
training, especially regarding the second and third measured
eigenfrequency. Nonetheless, the prediction remains highly
accurate, even when the number of considered experiments
is reduced from 26 to 9 measured frequency response func-
tions (i.e. a reduction of over 65%).
6 Conclusion
In this paper, an approach to improve the modeling accu-
racy of structural dynamics models of milling robots was
presented. This approach is based on an information fusion
algorithm to combine physics-based rigid body simulation
models with experimental data. The results show, that it out-
performs both, the rigid body simulation and conventional
machine learning approaches in estimating and extrapolating
the pose-dependent robot dynamics. It is worth mentioning
that the presented approach is not only limited to model the
pose-dependent of dynamic properties of industrial robots
such as milling robots, but could also be used to model the
position-dependent dynamics of conventional machine tools.
Nonetheless, there are open issues which need to be
addressed in future research:
– Economic utilization of the prediction uncertainty An
essential advantage of Bayesian approaches, such as the
used deep Gaussian processes, is the well-founded incor-
poration of a statistically motivated model uncertainty. It
remains an open issue how the prediction uncertainty can
10−710−610−510−4
10−7
10−6
10−5
10
−
4
measured Hin m
N→
predicted Hin m
N→
HNargp(f, φz,3≤105◦)
HNargp(f, φz,3>105◦)
HSim(f, φz,3>105◦)
σNargp(f, φz,3>105◦)
Fig. 9 Prediction accuracy assessment for the NARGP model in com-
parison to the rigid body simulation; the prediction of the NARGP
model is marked in black and blue for the training and test data sets.
The uncertainty of the NARGP model for the test data set is high-
lighted in blue
Table 2 Model performance evaluation
Approach
R2
RMSE
Training and test data
Simulation 0.34462 0.040286
GP 0.94341 0.004157
NARGP 0.99249 0.000938
MFDGP 0.93175 0.029387
Only test data
Simulation 0.35282 0.062790
GP 0.78473 0.01432
NARGP 0.98197 0.000497
MFDGP 0.75790 0.106640
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be incorporated into the robust design and parametriza-
tion of robotic milling processes.
– Computational complexity As of now, the power of
this approach has only been illustrated regarding a one-
dimensional robot work space (as a variation of the third
axis). In order to make use of such models in industrial
applications, the depicted approach needs to be extended
to more degrees of freedom, which results in significantly
larger data sets. Nonetheless, the computational com-
plexity of the given algorithm scales nonlinearly with
φz,3in ◦→
fin Hz →
E
Nargp,1
(
x
)
Hin m
N→
φz,3in ◦→
fin Hz →
σNargp,1
(
x
)
Hin m
N→
φz,3in ◦→
fin Hz →
ENargp,2(x)
Hin m
N→
φz,3in ◦→
fin Hz →
σNargp,2(x)
Hin m
N→
φz,3in ◦→
fin Hz →
ENargp,3(x)
Hin m
N→
φz,3in ◦→
fin Hz →
σNargp,3(x)
Hin m
N→
φz,3in ◦
→
fin Hz →
ENargp,4(x)
Hin m
N→
φz,3in ◦
→
fin Hz →
σNargp,4(x)
Hin m
N→
70 80 90 100 110 120
10
20
30
10−7
10−6
10−5
70 80 90 100110 120
10
20
30
10−7
10−6
10−5
70 80 90 100 110 120
10
20
30
10−7
10−6
10−5
70 80 90 100110 120
10
20
30
10−7
10−6
10−5
70 80 90 100 110 120
10
20
30
10−7
10−6
10−5
70 80 90 100110 120
10
20
30
10−7
10−6
10−5
70 80 90 100 110 120
10
20
30
10−7
10−6
10−5
70 80 90 100110 120
10
20
30
10−7
10−6
10−5
Fig. 10 Assessment of the NARGP model performance by reducing the number of experiments in the high fidelity data set. The used experi-
ments are marked with white, dashed lines
Production Engineering
1 3
the number of training samples (see [3]). Thus, it is nec-
essary to examine other feature space representations.
In addition, reducing the computational complexity also
enables more sophisticated validation schemes, such as
cross validation, which is favorable when the approach
is extensively used in larger workspace areas.
– Nonlinear simulation model The depicted approach has
been illustrated using a linear, pose-dependent state
space model of the robot’s structural dynamics. Since
the information fusion heavily relies on an accurate esti-
mation of the most important vibrational properties, the
physics-based model should also incorporate nonlinear
effects.
Acknowledgements Open Access funding provided by Projekt DEAL.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article’s Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.
Appendix
Mass and inertia properties in [x, y, z]
Parameter Value Unit
Mass of base 82.3950000000 kg
Center of base
[−0.004506000, 0.000002000, 0.092993000]
m
Moments of inertia of base [3.7240884840,3.9490065970,6.8761129230] kg · m2
Mass of shoulder 231.8910000000 kg
Center of shoulder
[−0.0143440000, 0.0188780000, 0.2622950000]
m
Moments of inertia of shoulder [7.6416913860,17.2508730530,14.4043881140] kg · m2
Mass of bicep 239.748 kg
Center of bicep
[0.4324000000, −0.0018890000, 0.2481170000]
m
Moments of inertia of bicep [3.6204536900,38.3609963770,38.3764146450] kg · m2
Mass of elbow 163.9880000000 kg
Center of elbow
[0.0248650000, 0.1558280000, −0.1796220000]
m
Moments of inertia of elbow [16.8260426100,2.3710703450,16.5413372330] kg · m2
Mass of forearm 39.7430000000 kg
Center of forearm
[−0.0013410000, 0.0000000000, 0.1220490000]
m
Moments of inertia of forearm [0.4256140500,0.4198806530,0.0971138130] kg · m2
Mass of wrist 31.0220000000 kg
Center of wrist
[0.000000000, 0.0540810000, −0.0375520000]
m
Moments of inertia of wrist [0.4758308100,0.3952674750,0.2490081860] kg · m2
Mass of palm 78.0020000000 kg
Center of palm
[−0.0659490000, 0.0000000000, 0.1573850000]
m
Moments of inertia of palm [0.7658796760,1.4016328790,1.2958580130] kg · m2
Production Engineering
1 3
Rotational joint stiffness for [x, y, z]
Parameter Value Unit
Joint base—shoulder [16028000,16028000,5823000]
N
⋅
m
rad
Joint shoulder—bicep [9566000,4255000,11361000]
N
⋅
m
rad
Joint bicep—elbow [3766000,2717000,13247000]
N
⋅
m
rad
Joint elbow—forearm [3246000,3246000,1056000]
N
⋅
m
rad
Joint forearm—wrist [2528000,2528000,1203000]
N
⋅
m
rad
Joint wrist—palm [2191000,2191000,454000]
N
⋅
m
rad
Rotational joint damping for [x, y, z]
Parameter Value Unit
Joint base—shoulder [16028000,16028000,5823000]
N
⋅
m
rad
Joint shoulder—bicep [9566000,4255000,11361000]
N
⋅
m
rad
Joint bicep—elbow [3766000,2717000,13247000]
N
⋅
m
rad
Joint elbow—forearm [3246000,3246000,1056000]
N
⋅
m
rad
Joint forearm—wrist [2528000,2528000,1203000]
N
⋅
m
rad
Joint wrist—palm [2191000,2191000,454000]
N
⋅
m
rad
References
1. Lejun C, Melkote Shreyes N (2017) Effect of robot dynamics on the
machining forces in robotic milling. Procedia Manuf 10:486–496
2. Marcel C, Wolfgang H, Yusuf A (2019) Chatter stability in robotic
milling. Robot Comput Integr Manuf 55:11–18
3. Cutajar K etal (2019) Deep Gaussian processes for multi-fidelity
modeling (2019). arXiv :1903.07320
4. Felis Martin L (2017) RBDL: an effcient rigid-body dynamics
library using recursive algorithms. Auton Robots 41(2):495–511
5. Fernáandez-Godino MG etal (2016) Review of multi-fidelity
models (2016). arXiv :1609.07196
6. Forrester AIJ (2010) Black-box calibration for complex-sys-
tem simulation. Philos Trans R Soc A Math Phys Eng Sci
368(1924):3567–3579
7. Yarin G (2016) Uncertainty in Deep Learning. University of Cam-
bridge, Dissertation
8. GPy (2012) A Gaussian process framework in python. http://githu
b.com/Sheffi eldM L/GPy
9. Wei J, Lihui W (2019) Industrial robotic machining: a review. Int
J Adv Manuf Technol 103:1239–1255
10. Karim A, Verl A (2013) Challenges and obstacles in robot-
machining. IEEE ISR 2013, Seoul, 2013, pp 1–4. https://doi.
org/10.1109/ISR.2013.6695731
11. Kennedy MC, O’Hagan A, Higgins N (2011) Bayesian analysis
of computer code outputs. In: Quantitative methods for current
environmental issues, pp 227–243 (2011)
12. Lee J, Grey MX, Ha S, Kunz T, Jain S, Ye Y, Srinivasa SS, Stilman
M, Liu CK (2018) DART: dynamic animation and robotics toolkit. J
Open Source Softw 3(22):500. https ://doi.org/10.21105 /joss.00500
13. Meng X, Karniadakis GE (2019) A composite neural network that
learns from multi-fidelity data: application to function approxima-
tion and inverse PDE problems. arXiv :1903.00104
14. Murray RM, Li Z, Sastry SS (1994) Grasp statics. In: A math-
ematical introduction to robotic manipulation, Vol 29. CRC, Boca
Raton, pp 214–222
15. Vinh N, Toni C, Shreyes M (2019) Data-driven modeling of the
modal properties of a six-degrees-of-freedom industrial robot and
its application to robotic milling. J Manuf Sci Eng 141(12):1–12
16. Peherstorfer B, Willcox K, Gunzburger M (2018) Survey of mul-
tifidelity methods in uncertainty propagation, inference, and opti-
mization. arXiv :1806.10761
17. Perdikaris P, Raissi M, Damianou A, Lawrence N, Karniadakis G
(2017) Nonlinear information fusion algorithms for dataefficient
multi-fidelity modelling. Proceedings of the Royal Society A:
Mathematical, Physical and Engineering Science. 473:20160751.
https ://doi.org/10.1098/rspa.2016.0751
18. Raissi M, Karniadakis G (2016) Deep Multi-fidelity Gaussian
processes. arXiv :1604.07484
19. Raissi M, Perdikaris P, Karniadakis GE (2017) Physics informed
deep learning (part I): data-driven solutions of nonlinear partial
differential equations. arXiv :1711.10561 v1
20. Raissi M, Perdikaris P, Karniadakis GE (2017) Physics informed
deep learning (part II): data-driven solutions of nonlinear partial
differential equations. arXiv :1711.10561 v1
21. Oliver R (2014) Steigerung der Arbeitsgenauigkeit bei der Fräs-
bearbeitung metallischer Werkstoffe mit Industrierobotern. Ph.D.
thesis, pp 658–662
22. Schnoes F, Zaeh MF (2019) Model-based planning of machining
operations for Industrial Robots. Procedia CIRP 82:497–502
23. Semm T, Rebelein C, Zaeh MF (2019) Prediction of the position
dependent dynamic behavior of a machine tool considering local
damping effects. CIRP J Manuf Sci Technol 27:68–77
24. Paleyes A, Pullin M, Mahsereci M, Lawrence N, González J (2019)
Emulation of physical processes with Emukit. In: Second Work-
shop on Machine Learning and the Physical Sciences, NeurIPS
25. Lei Y etal (2018) A review on chatter in robotic machining pro-
cess regarding both regenerative and mode coupling mechanism.
IEEE/ASME Trans Mechatron 23(5):2240–2251
26. Zaeh MF, Schnoes F, Obst B, Hartmann D (2020) Combined
offline simulation and online adaptation approach for the accu-
racy improvement of milling robots. CIRP Annal 69. https ://doi.
org/10.1016/j.cirp.2020.04.045
27. Zaeh Michael F, Roesch O (2014) Improvement of the machining
accuracy of milling robots. Prod Eng Res Dev 8(6):737–744
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