Automated Identification of Linear Machine Tool Model Parameters Using Global Sensitivity Analysis

Abstract

High-fidelity machine tool models are needed for condition monitoring, machine tool development, and process simulation. To accurately predict the dynamic behavior of their real counterparts, these models have to be identified, meaning that the values for the involved physical model parameters have to be found by comparing the model with measured data from its real counterpart. As of now, this can only be performed automatically for comparably simple models, which are only valid under limiting assumptions. In contrast, parameter identification for predictive high-fidelity models requires cumbersome manual effort in many intermediate steps. The present work addresses this problem by showing how to automatically identify the parameters of a complex structural dynamic machine tool model using global sensitivity analysis. The capability of the proposed approach is demonstrated in two steps for simulated reference data: first, with a model being able to perfectly replicate the reference data, and second, with a disturbed model, which can only approximate the reference because modeling is present. It is shown that, in both cases, globally valid model parameters, which lead to high conformity with the reference data, can be found, paving the way for calibrating models based on experimental reference data in future work.

Full text

Citation: Ellinger, J.; Zaeh, M.F.
Automated Identification of Linear
Machine Tool Model Parameters
Using Global Sensitivity Analysis.
Machines 2022,10, 535. https://
doi.org/10.3390/machines10070535
Academic Editor: Gianni Campatelli
Received: 2 June 2022
Accepted: 30 June 2022
Published: 1 July 2022
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4.0/).
machines
Article
Automated Identification of Linear Machine Tool Model
Parameters Using Global Sensitivity Analysis
Johannes Ellinger * and Michael F. Zaeh
Institute for Machine Tools and Industrial Management (iwb), TUM School of Engineering and Design,
Technical University of Munich (TUM), Boltzmannstr. 15, 85748 Garching, Germany; michael.zaeh@iwb.tum.de
*Correspondence: johannes.ellinger@iwb.tum.de
Abstract:
High-fidelity machine tool models are needed for condition monitoring, machine tool
development, and process simulation. To accurately predict the dynamic behavior of their real
counterparts, these models have to be identified, meaning that the values for the involved physical
model parameters have to be found by comparing the model with measured data from its real
counterpart. As of now, this can only be performed automatically for comparably simple models,
which are only valid under limiting assumptions. In contrast, parameter identification for predictive
high-fidelity models requires cumbersome manual effort in many intermediate steps. The present
work addresses this problem by showing how to automatically identify the parameters of a complex
structural dynamic machine tool model using global sensitivity analysis. The capability of the
proposed approach is demonstrated in two steps for simulated reference data: first, with a model
being able to perfectly replicate the reference data, and second, with a disturbed model, which can
only approximate the reference because modeling is present. It is shown that, in both cases, globally
valid model parameters, which lead to high conformity with the reference data, can be found, paving
the way for calibrating models based on experimental reference data in future work.
Keywords: machine tools; local damping; parameter identification; optimization
1. Introduction
The dynamic structural behavior of machine tools greatly influences the success of
cutting processes, making it the objective of numerous modeling efforts. Fundamentally,
two modeling strategies can be distinguished [
1
]: On the one hand, there is the so-called
experimental approach, which is based on measurements of the real system’s behavior and
results in a gray-box or black-box model replicating the input–output behavior (e.g., in the
form of transfer functions [
2
,
3
]). Their identification, that is the process of determining
values for the involved model parameters, is often referred to as “system identification” or,
in case in situ machining measurements are used, as “rapid identification”. Even though
the experimental approach is generally fast and accurate, it offers only limited physical
insights due to its high abstraction level. Furthermore, it is only valid in close vicinity to
the experimental data it was built with [1].
On the other hand, the theoretical approach is based on physical laws and leads
to a bottom-up white-box model of the system. This type of model directly contains
physical parameters describing the system. If formulated correctly, the model has a broad
range of validity. For machine tools, simple models can be derived analytically [
4
]. More
complex theoretical models can be built by performing finite element analyses (FEAs) [
5
,
6
].
With the help of linear and nonlinear local damping models, very accurate models can be
achieved [6].
Independent of their origin, theoretical models of machine tools contain physically
interpretable parameters [
1
] such as stiffness or damping values. Since the final simula-
tion outcome crucially depends on these parameters, they must be identified accurately.
Machines 2022,10, 535. https://doi.org/10.3390/machines10070535 https://www.mdpi.com/journal/machines

Machines 2022,10, 535 2 of 18
Determining their values is often referred to as “(parameter) identification”, “parametriza-
tion”, or “model updating”. For this, reference (i.e., training) data are required, which,
for example, can be measured on the model’s real counterpart. A comprehensive overview
of conventional and modern experimental reference data identification methods and pos-
sible sources of inaccuracies can be found in Iglesias et al. 2022 [
7
]. In the literature,
many approaches for parameter identification exist, which can be classified as either direct
or indirect.
Direct methods aim at identifying the parameters by solving the system of differential
equations of motion for the unknown stiffness and damping matrices in one step. However,
parameters identified this way may not be physically meaningful anymore. As for the
experimental models, this leads to simulation results being only valid in close vicinity to
the machine configuration in which the reference data have been recorded [8,9].
In contrast, indirect methods try to identify the model parameters by adapting them
iteratively, considering their sensitivity with respect to a chosen simulation outcome [
10
].
Another way to group parameter identification methods is by the complexity of the under-
lying system: model updating can either be performed on full-scale machine tool systems
or smaller subsystems. The latter can mean, for example, that the parameters of theoretical
machine tool simulation models are identified separately on distinct test benches [
11
,
12
].
Research with simplified feed drive models [
13
,
14
] also belongs to this group, as does the
work of Mehrpouya et al. 2015, who identified the stiffness and damping properties of four
joints between two plates using receptance coupling [15].
A completely different parameter identification approach on a subsystem is the method
of sequential assembly [
11
,
12
]. Here, the system is assembled step by step. In parallel,
the simulation model is also built incrementally so that it can be compared with the real
system in every step. As for the identification using test benches, the basic idea of this
method is that, in each step, only a few parameters, corresponding to the component that
has been added last, have to be identified. This drastically reduces the complexity of the
parametrization. However, identifying the overall model parameters on subsystems may
not always be applicable since either appropriate test benches are not available or the
disassembly of the machine tool is not possible for economic reasons. Furthermore, it is not
guaranteed that the identified parameters are also valid on the completely assembled ma-
chine tool. For example, the leveling mount parameters exhibit large deviations depending
on the assembly history of the machine tool [10].
Global methods aim to directly identify the parameters of theoretical models on the
machine tool in its final assembly state. In Witt 2007, data from experimental modal analyses
(EMA) were used as the input for an indirect stochastic optimization of
41
the stiffness
and damping parameters of a machine tool [
16
]. Here, a large FEA model representing
the machine tool in a single-axis position was combined with a genetic algorithm (GA).
The optimization problem was constrained by expert knowledge, meaning that, for each
parameter, a different mode was selected as the optimization target. Garitaonandia et al.
2008 also manually selected three eigenmodes as the target of their optimization, but
used a Bayesian parameter-updating technique to identify stiffness parameters for the
leveling mounts and the axial ball screw drive (BSD) stiffness [
17
]. Again, the reference
data were acquired through an EMA. In contrast to the approaches presented so far,
Hernandez-Vazquez et al. 2014 used two different machine tool configurations, that is
two different axis positions, to update the Young modulus, joint stiffnesses, and lumped
masses of the considered model [
18
]. By comparing the parameters with the ones obtained
from using only one axis position, it was shown that globally valid parameters could be
determined. The approaches presented so far used the computationally expensive FEA or
flexible multibody simulation (MBS) models in their optimization routines. In Hernandez-
Vazquez et al. 2018, the computational workload was drastically reduced by using quadratic
response surfaces as surrogate models [
19
]. This way, optimal parameters for nine joint
stiffnesses were found by matching the surrogate models with data from an EMA. However,
no globally valid stiffness parameter for the axial replacement stiffness of the unconstrained

Machines 2022,10, 535 3 of 18
degree of freedom (DOF) between the machine bed and the ram could be found. In contrast
to the linear approaches presented so far, Semm et al. 2020 used a nonlinear flexible MBS
model in their work [
12
]. However, before they identified the parameters of the leveling
mounts and the profile rail system, the model was linearized. Again, the reference data
were acquired by an EMA. Furthermore, the performance of two stochastic methods,
namely a GA and a particle swarm optimization (PSO), and a deterministic sequential
quadratic programming algorithm were compared. All these methods either require expert
knowledge and manual effort [
12
,
16
,
17
] or can only be applied to moderately complex
machine tool models with a limited number of unknown parameters [15,18,19].
In contrast, the present paper targets a fully automated identification of all unknown
parameters for highly complex machine tool models. For this, global sensitivity analysis
(GSA) is used to partition the overall identification problem into smaller, solvable subprob-
lems, in a way similar to the involvement of expert knowledge in [
12
,
16
,
17
]. Since this has
only been demonstrated exemplarily and only with limited success [
20
], this is the first
such implementation in the field of structural dynamic machine tool models.
The remainder of this paper is structured as follows: In Section 2, requirements for
the considered complex machine tool models and the available reference (i.e., training)
data are defined, and the theoretical basics of GSA are given. After that, the proposed
method for automatically identifying machine tool model parameters is explained generally.
Section 3starts with a detailed description of the used machine tool model and the involved
model parameters. Following this, the effectiveness of the proposed approach is shown
in two steps: First, reference data are simulated deploying a well-parameterized machine
tool model and used to identify another instance of the model while pretending that its
parameters are unknown. Here, the term “instance” refers to a copy of the model that may
have a different set of machine tool parameters and, thus, may behave differently. Second,
the very same reference data are used to parameterize an artificially disturbed model,
hinting at the proposed approach’s robustness in terms of modeling errors. The paper is
concluded with a summary and an outlook for future research (see Section 4).
2. Proposed Method
Rather than applying expert knowledge and manually splitting up the overall identifi-
cation problem [
12
,
16
,
17
], an automated procedure is to be followed here. The involved
machine tool model and its prerequisites are presented in Section 2.1. Section 2.2 gives an
overview of the application of GSA for machine tool model identification. Last, a detailed
description of the proposed approach is given in Section 2.3.
2.1. Starting Point
For this work, a position-flexible, dynamic, structural machine tool simulation model
needs to exist, which can be evaluated for the modal parameters:
(fr,Φr,ξr) = g(p,r)(1)
Here,
fr
is a vector of eigenfrequencies at the machine tool’s axis position
r
,
Φr
is the
corresponding matrix of eigenvectors,
ξr
is the corresponding vector of modal damping
ratios,
g
is a function representing the machine tool model, and
p
is a vector containing all
model parameters. Furthermore, four assumptions need to be made:
A1
A set of reference (i.e., training) modal parameters
(fr,re f
,
Φr,re f
,
ξr,re f )
, which govern
the machine tool’s vibrational behavior, exists. Since there is no significant coupling
between modes, the eigenvectors and eigenfrequencies are independent of the struc-
ture’s damping. This holds true for weakly damped structures [9].
A2
There is a set of parameters po pt such that
(fr,re f ,Φr,re f ,ξr,re f ) = g(popt ,r), (2)
meaning that the model is capable of representing the reference data.

Machines 2022,10, 535 4 of 18
A3
Parameter bounds need to be known such that
popt ∈[pmin
,
pmax ]
, that is the search
space must contain the true and globally valid machine tool parameters
popt
, which
are to be identified.
A4
The machine tool model contains linear damping sources only. Thus, the overall
modal damping of a mode ican be expressed as
ξi,r=
ND
∑
n=1
∆ξi,n,r(3)
with
ND
being the number of damping sources in the model and
∆ξi,n,r
being the
contribution of damping source ito the overall damping [9].
Note that these assumptions have little effect on the universality of the proposed
approach: Reference modal parameters can be easily acquired by an EMA, and machine
tool structures are generally considered lightly damped [
9
,
11
,
21
], such that modal coupling
can be neglected (A1). Assumption A2 only needs to be approximately true for a predictive
model. Furthermore, usually, at least some knowledge exists from data sheets, experience,
or comparable machines to estimate sufficient parameter bounds (A3), and even if non-
linear damping (A4) is present, it can be easily linearized using the approach presented
in [
21
]. The linearization also remedies the need for extensive time-domain simulations and,
thus, speeds up the model evaluations. This is beneficial for both GSA and optimization
algorithms that require many model evaluations (see Equation (1)). As examples of suitable
models for the approach presented in this work, the position-flexible MBS models of [
22
]
or models created by the simulation environment MORe [23] can be named.
2.2. Global Sensitivity Analysis for Machine Tool Models
For partitioning the overall parameter identification problem into smaller, solvable
subproblems, variance-based GSA is used here. Eventually, this leads to a set of parameters
such that the model predicts the reference modal parameters well (see Section 2.2). In this
publication, the match between the predicted and the reference modal parameters is
quantified by the extended modal assurance criterion (MACXP) [24]:
MACXP(ϕ1,ϕ2) = |ϕ∗
1ϕ2|
|λ∗
1+λ2|+|ϕT
1ϕ2|
|λ1+λ2|2
|ϕ∗
1ϕ1|
2|Re(λ1)|+|ϕT
1ϕ1|
2|λ1| |ϕ∗
2ϕ2|
2|Re(λ2)|+|ϕT
2ϕ2|
2|λ2|(4)
and the squared natural damping difference (NDD):
NDD2(ξ1,ξ2) = (N DD(ξ1,ξ2))2=ξ1−ξ2
ξ22
(5)
analogous to the natural frequency difference (NFD) [
25
]. Here,
ϕ1
and
ϕ2=ϕre f
are the
calculated and reference eigenvectors,
λ1
and
λ2=λre f
are the calculated and reference
eigenvalues, and
ξ1
and
ξ2=ξre f
are the calculated and reference modal damping values.
Compared to the standard modal assurance criterion (MAC), the MACXP significantly
improves the distinction of modes when only a limited number of measurements is avail-
able [
24
]. Because of this, the MACXP is used not only to measure the model’s accuracy,
but also to find the right mode to compare in the first place.
Both the MACXP and the squared natural damping difference (NDD
2
) can be repre-
sented by the so-called high-dimensional model representation (HDMR):
f(p) = f0+∑
i
fi(pi) + ∑
i
∑
j>i
fi,j(pi,pj) + . . . +f1,2,...,k(p)(6)

Machines 2022,10, 535 5 of 18
defined for square-integrable functions
f
(e.g., MACXP or NDD
2
) whose argument
p
(e.g.,
the machine tool model parameters) is defined in an n-dimensional unit hypercube [
26
].
In Equation (6),
fi(pi)
represents the (first-order) effect of parameter
pi
on the model’s
conformity,
fi,j(pi
,
pj)
the joint contribution of the parameters
pi
and
pj
exceeding their
first-order effects, and so on. Equation (6) can be reformulated to the analysis of variance
(ANOVA) HDMR using the orthogonality properties between any pairs of terms and square
integration [26]:
V=V0+∑
i
Vi+∑
i
∑
j>i
Vi,j+. . . +V1,2,...,k, (7)
with
V=V(f)
being the total variance,
Vi=V(fi)
being the variance caused by parameter
pi
, and
Vi,j=V(fi,j)
being the variance caused by the interaction of parameters
pi
and
pj
[
26
]. In other words,
V
represents, for example, the total variance of the model’s NDD
2
value and
Vi
the contribution of parameter
pi
to the total variance. The so-called Sobol
indices are calculated by dividing both sides by the total variance V:
1=∑
i
Si+∑
i
∑
j>i
Si,j+. . . +S1,2,...,k, (8)
indicating shares of variance [26]. The total effect of a parameter pi:
STi =Si+∑
j6=i
Si,j+. . . +S1,2...,k, (9)
is simply the sum of all its shares. Here,
STi =
0 is a necessary and sufficient condition for
parameter pibeing non-influential [26].
The GSAs in this work were implemented using the Python package SALib [27].
2.3. Identification Procedure
The parameters of the machine tool model (see Section 2.1) can be identified in three
steps, which are visualized in Figure 1: Starting from a preliminary (i.e., only roughly
parameterized) machine tool model, the overall identification problem with a high number
of unknown parameters is divided into many smaller identification problems with just a
few unknown parameters each. This is performed by GSA. Second, all stiffness parameters
can be identified by using classical optimization algorithms. Third, the model’s damping
parameters are estimated using the least-squares (LS) method, leading to the final identified
machine tool model.
Start:
Preliminary model
Result:
Identified model
1) Global sensitivity analysis (GSA) for each mode
at each axis position
3) Least squares (LS) damping
identification
2) Brute-force
stiffness identification
Figure 1. Overview of the proposed identification approach.
2.3.1. Partitioning of the Overall Identification Problem via GSAs
For each reference mode at each machine tool axis position, two GSAs are conducted.
One uses the MACXP between the reference and the simulated mode, and the other uses
the NDD
2
between the reference and simulated damping as scalar model output. This
results in

Machines 2022,10, 535 6 of 18
NGSA =2Nmodes Npos (10)
separate GSAs with
Nmodes
being the number of reference modes and
Npos
being the
number of axis positions, emphasizing the need for a computationally efficient model.
As a result, the model parameters
p
can now be ranked by their degree of influence on
each mode’s shape and damping. In other words, modes can be found that are only
affected by a handful of model parameters, leading to the desired partitioning of the overall
identification problem. More information can be found in [
20
], where this was already
shown exemplarily.
2.3.2. Stiffness Parameter Identification
As the model’s damping parameters do not influence the simulated mode shapes and
eigenfrequencies (see Assumption A1 in Section 2.1), the overall identification problem can
be further simplified by determining the stiffness parameters first. The partitioning via
GSAs leads to
NmodesNpos
(see Equation (10)) identification problems with only a handful
of search parameters each [
20
]. Even though the partitioning eliminates most of the local
minima of the overall optimization problem, which would lead to only locally valid and
non-physical machine tool model parameters, some are still present [
20
]. To overcome this
problem, a brute-force approach, which is illustrated by Figure 2, is deployed here: Each of
the
NmodesNpos
smaller identification problems is repeated
Nopt
times. It is assumed that if
all repetitions have ended up at (nearly) the same result (i.e., the same values for the search
parameters), a global optimum is found. This is assessed by the standard deviation of each
parameter for each identification subproblem. In the end, the mean value of all repetitions
from the subproblem with the least standard deviation for each parameter is chosen as the
final result.
Note that, in contrast to the depiction in Figure 2, not all, but just a few parameters are
searched for each position–mode combination as a result of the preceding GSA.
2.3.3. Damping Parameter Identification
For lightly damped machine tool models with linear damping only (see Assumptions A1
and A4 in Section 2.1), the overall modal damping of mode
i
at machine tool axis position
r
can be split up as
ξi,r=ξi,p,r+ξi,pr(11)
with
ξi,p,r
being the share depending on the unknown model parameters
p
and
ξi,pr
already-
identified damping sources in the model, such as known damping parameters or material
damping. Making use of their linearity [9], the former can be expressed by
ξi,p,r=∑
n∈ND∗
qi,n,rpn=ξi,r−ξi,pr=:ξi,res,r. (12)
Here,
ND∗
is the set of all yet-unidentified damping sources with
ND∗
members,
ξi,res,r
is the residual damping of mode
i
at position
r
, and
qi,n,r
is a factor quantifying the
contribution of model parameter
pn
to the overall modal damping, which depends on the
system’s eigenfrequencies and mode shapes [9]. In matrix form, Equation (12) becomes
Qp =ξres,r, (13)
with the dimensions of Qbeing NmodesNpos ×ND∗, as Equation (12) can be set up for each
reference mode at each machine tool axis position. If
NmodesNpos ≥ND∗
, the application of
the LS on Equation (13) yields a unique solution, resulting in estimates for all unknown
damping model parameters.

Machines 2022,10, 535 7 of 18
p1
Mode position
combination 1
p2pND∗
· · ·
...
.
.
.
.
.
..
.
.
Mode position
combination 2
Mode position
combination NmodesNpos
.
.
.
· · ·
· · ·
· · ·
Figure 2.
Exemplary depiction of the brute-force identification results for overcoming local minima;
for each position–mode combination, an optimization problem searching for the unknown stiffness
model parameters is set up and repeated, leading to different values due to local minima. The
position–mode combination with the least standard deviation across all repetitions yields the final
identified value for each parameter. This is indicated by red solid lines instead of black dashed lines.
3. Sensitivity-Guided Parameter Identification
The capability of the proposed approach (see Section 2) is demonstrated in two steps:
First, a machine tool model is parameterized using simulated reference (i.e., training) data
from another instance of the model with known parameters in Section 3.2. In a second step,
a similar, but not identical, model is used for generating reference data for the parameter
identification. This is shown in Section 3.3. This procedure is also illustrated in Figure 3. All
steps refer to a four-axis machine tool model in uniaxial configuration, which is described
in detail in Section 3.1
3.1. Machine Tool Structure and Model Description
Here, the proposed approach was applied to a four-axis machine tool model in uni-
axial configuration, which is depicted in Figure 4. For this system, a high-fidelity and
well-parameterized model with a high conformity to the real machine tool structure ex-
ists [
6
], which can be used to simulate reference data and validate the method presented
here without measurement and modeling errors. Due to a beneficial combination of sub-
structuring, MOR [
28
], and the linearization of the involved nonlinear friction models [
21
],
the model is computationally efficient, parametric, and, at the same time, position-flexible.
Note that some parameters of the original unreduced model, as, for example, the compo-
nents’ Young’s moduli, are fixed after the MOR and cannot be identified here. However,
this is not a drawback of the presented approach, but the examined model and could be
overcome with more advanced MOR approaches. The model has already been subjected to
a dimensionality reduction step [
20
], resulting in the following
27
unknown machine tool
parameters representing the lumped stiffness and damping properties of:
• The three mounting elements (MEs) (kx,ky,kz,bx,bz,dyeach)

Machines 2022,10, 535 8 of 18
• The fixed bearing (FB) supporting the BSD (kz)
• The coupling (CPL) between the motor shaft and the BSD (krz ,drz)
• The BSD (kz,drz );
• The linear guiding system (LGS) (kx,ky,krx ,kry used for all four shoes)
Here, indices indicate the direction of the stiffness (
k
) and viscous (
b
) and hysteretic
(d) damping parameters.
Reference
data
Model
"Ideal model vs.
simulated data"
(Section 3.2)
Ideal
Simulated
Disturbed
"Disturbed model vs.
simulated data"
(Section 3.3)
Figure 3.
Overview of the different setups considered in Section 3; the blurred illustration indicates a
similar, but non-matching model with modeling inaccuracies.
Note that the considered machine tool system fulfills all assumptions stated in
Section 2.1
:
Modal parameters can be simulated with high accordance [
6
], and the machine tool is lightly
damped [
11
,
21
] (A1 and A2). Suitable bounds can, in this case, be easily chosen since the
true model parameters are already known (A3). Here, the intervals:
pi∈([0.7pi,o pt , 1.3pi,o pt]for stiffness and
[1
3pi,opt, 3 pi,opt]for damping parameters (14)
are used. In real-world scenarios, where the true model parameters are unknown, these
intervals are believed to be large enough to ensure the validity of assumption A3 even
when only data sheet values provided by the machine tool component manufacturers or
values from similar machine tools are known. Lastly, all nonlinear damping sources were
linearized and replaced by spring–damper systems (A4) [21].
For evaluating the machine tool’s modal parameters,
32
nodes (see Table 1and Figure 4)
were used in the simulations. Both the computational modal analysis and the EMA were
conducted at four axis positions covering the full motion range of the
z
-axis (
z1=−294 mm
,
z2=−60 mm
,
z3=135 mm
, and
z4=330 mm
). Note that there is no unconstrained DOF
in the model since the motor brake of the
x
-axis is applied and the
z
-axis is constrained by
the linear replacement stiffness resulting from the friction model linearization [12].

Machines 2022,10, 535 9 of 18
x
y
z
Machine bed
Workpiece table (WPT)
Figure 4.
Illustration of a four-axis machine tool model in uniaxial configuration, consisting of a
machine bed and a workpiece table (WPT); the non-hidden nodes (see Table 1) considered in the
identification are indicated by circular markers.
Table 1. Considered model nodes; overlapping nodes are indicated by alphabetic subscripts.
Node Description
N1a, N1bShoe and rail nodes of the first LGS shoe
N2a, N2bShoe and rail nodes of the second LGS shoe
N3a, N3bShoe and rail nodes of the third LGS shoe
N4a, N4bShoe and rail nodes of the fourth LGS shoe
N5a, N5bNut and shaft nodes of the BSD
N6, N7Workpiece table (WPT) nodes
N8a, N8bLinear encoder bed and WPT nodes
N9a, N9bBed and shaft nodes of the x-axis brake
N10a, N10bBed and shaft nodes of the z-axis brake
N11, N12, N13, N14, N15 Bed nodes
N16, N17, N18 ME nodes
N19a, N19bFB bed and shaft nodes
N20a, N20bLoose bearing (LB) bed and shaft nodes
N21a, N21bCPL motor and BSD shaft nodes
3.2. Parameter Identification of an Ideal Machine Tool Model
To demonstrate the effectiveness of the proposed approach, an ideal test case was set
up first (i.e., “ideal model vs. simulated data”). Reference data were simulated using the
model described in Section 3.1. Thus, in this case, the parameters to be identified are already
known in advance, enabling a direct evaluation of the final identification results in addition
to an indirect evaluation in terms of model evaluation criteria (e.g., frequency response
assurance criterion (FRAC) [
29
], cross-signature scale factor (CSF) [
30
], MACXP, and NDD).
The data comprise the first
20
modes covering the frequency range up to
311 Hz
and four
axis positions, of which three are used in the identification (
z2
,
z3
, and
z4
) and one is held
back for validation purposes (
z1
). Modes with higher eigenfrequencies are not related to the
machine tool structure, but the workpiece or the tool [
31
,
32
] and are excluded from further
consideration. Afterward, random values (within their bounds defined in Equation (14))
are assigned to all
27
model parameters, providing the starting point for the identification

Machines 2022,10, 535 10 of 18
procedure. All computations were performed on a state-of-the-art workstation with
40
Intel®Xeon®Gold 6148 CPU cores.
Following the approach in Section 2.3,
120
GSAs (see Equation (10)) were performed
for each of the
20
modes at three axis positions for both the MACXP and NDD
2
. On the
available workstation, this resulted in a computation time of
1 h 4 min
. Figure 5exemplarily
shows two resulting parameter rankings for different position–mode combinations. It can
be seen that instead of all
27
model parameters, in both cases, only a few parameters
significantly affect the specific model outcomes (i.e., the fit criteria for the displayed modes
at the chosen position). The GSA results can also be confirmed by looking at the mode
shapes: For example, mode
8
at position
z3=135 mm
(see Figure 5a) at
75.5 Hz
is the first
bending mode of the machine tool bed in the
yz
-plane. This mainly involves large vertical
(
y
-axis) movements of the back MEs (i.e., ME2 and ME3), making them the most significant
parameters. Other ME parameters are significant as well due to smaller movements because
of asymmetries in the machine tool model. More importantly, feed drive parameters (e.g.,
BSD, CPL, and FB) are missing here since there is no deformation of the feed drive, but
only movement in the unconstrained screw DOF of the BSD. Note that there is, per the
definition (see assumption A1 in Section 2.1), no influence of the damping parameters on
the MACXP.
0 20 40 60
Sobol’ indices in % →
BSz k z
ME1 k x
ME1 k z
ME2 k z
ME1 k y
ME3 k x
ME2 k x
ME3 k z
ME3 k y
ME2 k y
Model parameters
Total
First-order
(a) MACXP for mode 8 at position z3=135 mm
0 20 40 60
Sobol’ indices in % →
ME3 k x
ME3 k z
ME1 k y
ME2 k x
ME3 k y
ME2 k z
ME3 d y
ME1 k x
ME2 d y
ME2 k y
Model parameters
Total
First-order
(b) NDD2for mode 3 at position z2=−60 mm
Figure 5.
Sobol indices and corresponding 95% confidence intervals indicated by whiskers for two
different position–mode combinations; for better readability, only the ten most significant model
parameters are shown. The notation is explained in Section 3.1.
This knowledge can be exploited by setting up optimization problems searching
for only the significant parameters while using fixed arbitrary values for the rest. Here,
significant parameters were distinguished from non-significant ones by a threshold of
1% regarding their total effect. To circumvent still-existing local minima [
20
],
60
(i.e.,
NmodesNpos
) optimization problems were set up for all position–mode combinations with
the MACXP and repeated
Nopt =100
times, resulting in total in
6000
independent opti-
mization problems. As the number of unknown parameters is small for each individual
run, gradient-based algorithms can be efficiently used. The sequential least-squares pro-
gramming (SLSQP) approach [
33
] was implemented in this work, resulting in an overall
computation time of
1 h 8 min
. Each optimization run ended up at different final values for
the significant (search) parameters because local minima are present, and the non-significant
parameters and the initial values for the search parameters were chosen randomly. The final
identified stiffness parameter values were determined as the mean value of all repetitions

Machines 2022,10, 535 11 of 18
from the position–mode combination with the smallest standard deviations (see Section 2.3).
Figure 6a shows the relative deviation of the selected stiffness parameter values from their,
in this case, true known values. It can be seen that all parameters but two were iden-
tified almost perfectly with deviations of less than
5 %
. Only the stiffnesses of the first
and second MEs in the
z
-direction (ME1 k z and ME2 k z) were identified poorly with
deviations of
−21.2 %
and
−8.3 %
, respectively. However, the high conformity with the
reference frequency response function (FRF) shown in Figure 7indicates that the influence
of these model parameters is small. Minimal deviations toward higher stiffnesses of the
other model parameters seem to be able to compensate the underestimated MEs’ stiffnesses
in the z-direction.
012345678
Counts →
−20
−15
−10
−5
0
Relative deviation in % →
Median
90 % percentile
(a) Stiffness parameters
012
Counts →
−50
0
50
100
150
200
Relative deviation in % →
Median
90 % percentile
(b) Damping parameters
Figure 6.
Histogram plots showing the relative deviation of the identified stiffness and damping
parameters from their true known value for the case “ideal model vs. simulated data”; additionally,
their median value and 90% percentiles are shown.
In the next step, the model’s damping parameters can be identified using an LS
approach, which was described in Section 2.3. In principle, an LS problem could be
set up with all position–mode combinations with NDD
2
. To avoid the propagation of
errors, only the
20
position–mode combinations with the lowest influence of all non-
search parameters on the NDD
2
were selected. This information can be directly taken
from the GSAs conducted before. To reduce numerical errors, the number of equations
was further reduced by removing all rows from Equation (13) with a residual damping
ξi,res,r
smaller than 0.5%, resulting in
18
equations for
11
unknown damping parameters.
The relative deviations of the identified damping parameters from their, in this case,
known values can be found in Figure 6b. It can be seen that approximately half of the
parameters can be identified with a still reasonably good accuracy of 30%. The rest is
spread all over the allowed interval (see Equation (14)) with some parameters even at
their limits as, for example, the rotary hysteretic damping of the coupling (CPL d rz) and
the viscous damping of the second ME in the
x
-direction (ME2 b x) with deviations of
200% and
−
67.7%, respectively. It is important to note that this is not a shortcoming of the
damping parameter identification approach. The targeted parameters were found almost
perfectly with maximum deviations of 0.01% when it was assumed that the perfect stiffness
parameters were found before. This can be explained by again looking at Figure 5b, which
shows that the model’s stiffness parameters influence the NDD
2
value and, with it, the
modal damping. From the latter, the right-hand side of the LS problem for the damping
estimation was constructed (see Section 2.3.3), leading to the displayed deviations.

Machines 2022,10, 535 12 of 18
Table 2demonstrates the accuracy of the identified model in terms of common model
evaluation criteria (i.e., FRAC, CSF, MACXP, and NDD) for the evaluated modes at the
three axis positions considered in the identification (
z2
,
z3
, and
z4
). In general, FRAC and
CSF values above 80% [
11
] or even above 70% [
21
] are considered a good match. Thus, it
can be stated that the model’s conformity is very high, with worst-case FRAC, CSF and
MACXP values of 84.9% and mean values close to 100%. Note, that the worst-case NDD
indicates a relative and not absolute deviation of 4.1%, which is also very low. Table 3
shows the same data considering only the validation position
z1
. The FRAC and CSF values
are a bit lower, but in the same range. However, the modal fitness values are very similar
with worst-case MACXP and NDD values of 99.9% and 5.5%, respectively.
This indicates very high conformity of the model, which is also confirmed by the WPT
FRF in the
x
-direction shown in Figure 7. Here, hardly any deviations from the reference
FRF can be seen at all. Note that this holds for the whole frequency range up to
500 Hz
,
even though only modes up to
311 Hz
were considered in the identification, indicating
that the global optimum of the model parameters has been found. Additionally, Figure 7
stresses the importance of the damping parameter identification by also showing an FRF of
a model with the same damping parameter deviations, but randomly reassigned to other
(damping) parameters. For example, the deviation of the rotary hysteretic damping of the
coupling (CPL d rz) from its true reference value of
−
67.7% could be used to calculate a
value for the viscous damping of the first ME in the
z
-direction (ME3 b z), and so on. This
leads to a poor qualitative match with the reference FRF, especially in comparison to the
actual identification results.
Table 2.
Statistics of model performance indicators using the identified stiffness and damping
parameters for the case “model vs. ideal model” for three considered axis positions z2,z3, and z4.
FRAC in % CSF in % MACXP in % NDD in %
Worst 84.93 91.72 99.86 4.10
5 % percentile 86.21 92.78 99.88 1.99
Mean 94.92 97.17 99.97 0.52
Median 96.12 98.04 100.00 0.27
Table 3.
Statistics of model performance indicators using the identified stiffness and damping
parameters for the case “model vs. ideal model” for the validation position z1.
FRAC in % CSF in % MACXP in % NDD in %
Worst 79.17 88.43 99.86 5.48
5 % percentile 79.41 88.78 99.89 4.73
Mean 90.21 94.19 99.98 1.07
Median 91.63 95.58 100.00 0.24
Summarizing the above, it can be said that the shown approach provided stiffness
parameters very close to their global optimum (see Figure 6a). Note that this can only
be stated since, in this case, simulated reference data from a model whose parameters
are known were used as a starting point for the identification. In general, one would
validate the model against reference data not used in the identification, as shown in Table 3
and Figure 7. However, this might lead to a misleading conclusion in the case of the
damping parameters: the conformity of the model is very high (see Table 3), as is the
overall deviation of the parameters from their true known reference (see Figure 6b). It is
assumed that this conflict originates from the high influence of the stiffness parameters
on the damping parameters (see Figure 5b), which would mean that the found damping
parameters still represent the global optimum of the model. This is supported by the fact
that the damping parameters’ deviations almost vanish when it is assumed that the true
stiffness parameters have been found. However, it cannot be completely ruled out that
this conflict originates from a validation position too close to the identification positions

Machines 2022,10, 535 13 of 18
or very different significance values of the damping parameters, resulting in only those
with high sensitivity being calculated correctly. This would indicate local and, thus, non-
transferable solutions for the damping parameters. More insights into this will be targeted
in further research. For the machine tool model considered here (see Section 3.1), the final
parameter identification results were found to be comparably insensitive to the sensitivity
threshold, the number of repetitions of each optimization problem
Nopt
, and the number of
involved position–mode combinations in the damping parameter identification. However,
the importance of these hyperparameters will be reinvestigated in further research.
10−6
10−5
10−4
Receptance amplitude
in mm/N →
0100 200 300 400 500
Frequency in Hz →
−150
−100
−50
0
Phase in deg →
Model (CSF = 98.8 %; FRAC = 99.4%)
Model with permuted damping
Reference
Figure 7.
Identified and reference FRFs in the
x
-direction from force to displacement at the WPT at the
validation position
z1=−294 mm
for the case “ideal model vs. simulated data”; for better readability,
the highlighted section has a linear amplitude scale in contrast to the main plot. Additionally, a model
FRF is shown where the identified damping deviations were permuted randomly, emphasizing the
need for properly identified damping parameters.
3.3. Parameter Identification of a Disturbed Machine Tool Model
In this section, the effectiveness of the proposed approach is shown by using reference
data from a similar, but not matching model for the identification. Here, the model to be
identified is disturbed by setting different values for
44
non-sensitive parameters [
20
]. Note
that, in contrast to Section 3.2, assumption A2 from Section 2.1 is now only approximately
true. As the same simulated reference data as in Section 3.2 were used, the identification
results, that is the modal parameters, can again be evaluated directly by comparing them
to the reference model’s parameters.
Based on the results of the GSAs for the disturbed model, the stiffness parameters
of the model were identified first (see Sections 2.3 and 3.2). The results can be found in
Figure 8a. It can be seen that most of the stiffness parameters were identified well with
deviations of even less than 3%. Similar to Section 3.2, the stiffness in the
x
-direction of
the first ME (ME1 k x) shows a larger deviation of 12.1%. The identified and the reference
FRFs in Figure 9match very well. Additionally, both the modal and the frequency-based
conformity measures for the position considered in the identification process in Table 4are
very high, suggesting that the global optimum was approximated well.
This is also supported by Table 5, which, apart from the worst-case and 5% percentile
MACXP (and NDD) values, also shows very high conformity for the validation position.
The reason for this exception is that there are three poorly identified modes in the range
265 Hz
to
311 Hz
, in which the FRF has a low amplitude (see Figure 9), leading to only a
minor influence on the FRAC (and the NDD).

Machines 2022,10, 535 14 of 18
012 3 45 6
Counts →
−12
−10
−8
−6
−4
−2
0
2
Relative deviation in % →
Median
90 % percentile
(a) Stiffness parameters
012
Counts →
−60
−40
−20
0
20
40
60
Relative deviation in % →
Median
90 % percentile
(b) Damping parameters
Figure 8.
Histogram plots showing the relative deviation of the identified stiffness and damping pa-
rameters from their true known value for the case “disturbed model vs. simulated data”; additionally,
their median value and 90% percentiles are shown.
10−6
10−5
10−4
Receptance amplitude
in mm/N →
0100 200 300 400 500
Frequency in Hz →
−150
−100
−50
0
Phase in deg →
Model (CSF = 99.9 %; FRAC = 100.0%)
Model with permuted damping
Reference
Figure 9.
Identified and reference FRFs in the
x
-direction from force to displacement at the WPT at
the validation position
z1=−294 mm
for the case “disturbed model vs. simulated data”; for better
readability, the highlighted section has a linear amplitude scale in contrast to the main plot. Addi-
tionally, a model FRF is shown where the identified damping deviations were permuted randomly,
emphasizing the need for properly identified damping parameters.
Table 4.
Statistics of model performance indicators using the identified stiffness and damping
parameters for the case “model vs. disturbed model” for three considered axis positions
z2
,
z3
, and
z4
.
FRAC in % CSF in % MACXP in % NDD in %
Worst 92.15 95.73 99.92 1.69
5 % percentile 96.70 97.15 99.96 1.51
Mean 99.26 99.49 99.99 0.40
Median 99.85 99.91 100.00 0.25

Machines 2022,10, 535 15 of 18
Table 5.
Statistics of model performance indicators using the identified stiffness and damping
parameters for the case “model vs. disturbed model” for the validation position z1.
FRAC in % CSF in % MACXP in % NDD in %
Worst 97.53 97.36 88.57 40.65
5 % percentile 98.41 98.36 89.78 39.83
Mean 99.60 99.64 98.65 6.05
Median 99.83 99.91 99.99 0.43
Similar to Section 3.2, an LS problem was set up and solved for the yet-unknown
damping parameters. Again, only the
20
position–mode combinations with the lowest
interaction of the non-search parameters on the NDD
2
were selected. The chosen residual
damping threshold of
0.5 %
did not lead to any further reduction of the number of equations
in this case. The final deviations depicted in Figure 8b are slightly higher than in the
“ideal model vs. simulated data” case (see Figure 6b). However, it is believed that the
identified damping parameters still represent the true and global optimum since the overall
conformity shown in Tables 4and 5is, except for three modes in the range
265 Hz
to
311 Hz, very high. Furthermore, the identified model’s FRF in Figure 9matches well with
the reference data. Again, the comparison with similar, but randomly assigned damping
deviations stresses the importance of the damping parameter identification.
4. Conclusions
In this paper, a method was presented to identify unknown parameters of machine tool
models based on the outcomes of GSAs. In contrast to related work, the shown approach
also works for highly complex models, needs only a minimum of manual effort, and does
not require subsystems for intermediate modeling and measurement steps.
The presented approach relies on breaking down the overall identification problem
by means of GSA into considerably smaller subproblems, which can be solved easily and
efficiently. This already reduces the occurrence of local minima, which past works have
suffered from (e.g., [
21
]). Additionally, a brute-force approach exploiting the computational
efficiency of the model was developed, which led to a globally valid solution for the model’s
stiffness parameters. Based on these, the model’s damping parameters can be identified by
utilizing their linear influence on the overall model’s energy dissipation. This was proven
in two cases: first for an ideal model, which is able to replicate the reference data perfectly,
and second, for the more realistic case of a disturbed model with slight imperfections.
For both, statistics of common model conformity measures were given, showing a high
agreement between the model’s predictions and the provided reference data. Furthermore,
it was demonstrated that the model can also replicate reference data that have not been
used as the input for the identification process. Additionally, exemplary FRF results were
provided, confirming the success of the presented identification approach. Over the course
of the parameter identification, it was found that the model’s damping parameters showed
strong interaction effects with the previously identified stiffness parameters. Thus, in this
paper, the original model’s damping parameters, which were used to simulate the reference
(i.e., training) data, were identified correctly only in some cases and with large deviations
otherwise. This may imply that the damping parameters are not transferable to other
models in general or only in combination with the identified stiffness distribution, which
will be examined in more detail in future research.
In order to be fully applicable to full-scale machine tools, the presented approach must
be extended to the multi-axis setup of state-of-the-art machine tools. On the one hand, it is
believed that the identification will become more difficult because of the increased number
of unknown model parameters and the corresponding increase of the number of possible
local minima. On the other hand, however, more axes offer more flexibility in finding axis
positions that are not prone to local solutions, which is beneficial for the proposed method.
Another transition that needs to be made in future research is the one from simulated to
measured reference (i.e., training) data. As the shown approach heavily relies on high-

Machines 2022,10, 535 16 of 18
quality modal parameters (i.e., eigenfrequencies, mode shapes, and modal damping), ways
must be found to determine these inputs with, in the spirit of the automated solution
proposed here, as little manual effort as possible.
Author Contributions:
Conceptualization, J.E.; methodology, J.E.; software, J.E.; validation, J.E.;
investigation, J.E.; resources, J.E. and M.F.Z.; writing—original draft preparation, J.E.; writing—review
and editing, J.E. and M.F.Z.; visualization, J.E.; supervision, J.E. and M.F.Z.; project administration, J.E.
and M.F.Z.; funding acquisition, J.E. and M.F.Z. All authors have read and agreed to the published
version of the manuscript.
Funding:
This work was supported by the Bavarian State Ministry for Economic Affairs, Energy and
Technology (StMWi) within the research project “Artificial Intelligence and Digital Twin for Predictive
Maintenance of Machine Tools (KIDZ)” under Grant Agreement Number DIK0140/01. We would
like to thank the StMWi for the funding and the VDI|VDE|IT for the pleasant cooperation.
Data Availability Statement:
The reference data presented in this study are available upon request
from the corresponding author.
Conflicts of Interest: The authors have no conflicts of interest to declare.
Abbreviations
The following abbreviations are used in this manuscript:
ANOVA analysis of variance
BSD ball screw drive
CPL coupling
CPU central processing unit
CSF cross-signature scale factor
DOF degree of freedom
EMA experimental modal analyses
FB fixed bearing
FEA finite element analysis
FRAC frequency response assurance criterion
FRF frequency response function
GA genetic algorithm
GSA global sensitivity analysis
HDMR high-dimensional model representation
KIDZ Artificial Intelligence and Digital Twin for Predictive Maintenance of Machine Tools
LB loose bearing
LGS linear guiding system
LS least squares
MAC modal assurance criterion
MACXP extended modal assurance criterion
MBS multibody simulation
ME mounting element
MOR model order reduction
NDD natural damping difference
NDD2squared natural damping difference
NFD natural frequency difference
PSO particle swarm optimization
SLSQP sequential least-squares programming
StMWi Bavarian State Ministry for Economic Affairs, Energy and Technology
WPT workpiece table
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