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Prediction of the Posture-Dependent Tool Tip Dynamics in Robotic
Milling Based on Multi-Task Gaussian Process Regressions
Yang Leia, Tengyu Houa and Ye Dinga,†
aState Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering,
Shanghai Jiao Tong University, Shanghai 200240, China
Abstract: Chatter vibration is one of the main factors that limit the productivity and
quality of the robotic milling process. To predict the robotic milling stability, it is
essential to obtain the tool tip frequency response function (FRF). The tool tip
dynamics of a robot heavily depend on its postures and used tools. A state-of-art
methodology of combining the regression model with the Receptance Coupling
Substructure Analysis (RCSA) method is proved to be effective in predicting tool tip
FRFs of machine tools for different positions and tools. However, for the milling
robot, the cross coupling FRFs have an obvious influence on the dynamic property of
the milling robot, thereby greatly affecting the milling stability boundary. It is of great
challenge to directly integrate the effect of the cross coupling FRFs into the
state-of-art approach to predict the tool tip dynamics. To tackle this challenge, in this
paper, we propose an approach to predict the posture-dependent tool tip dynamics for
different tools in robotic milling considering the cross coupling FRFs. First, a more
comprehensive RCSA procedure is adopted to include the cross coupling FRFs. Then,
the impact test is designed to measure the required FRF matrix. By fitting the
measured FRF matrix with the multiple-degree-of-freedom (MDOF) model, the
number of modal parameters is significantly reduced. Next, the Multi-Task Gaussian
Process (MTGP) regression model is employed to mine the physical correlations
between different modal parameters. Compared to the ordinary Gaussian Process
regression model, the number of required regression models in MTGP is reduced and
the prediction performance is improved in terms of accuracy and robustness.
Furthermore, the effectiveness of the proposed approach is validated by the impact
test and milling experiment on an industrial robot.
Keywords: Tool tip dynamics; Robotic milling; Cross coupling FRFs; Receptance
coupling substructure analysis; Multi-Task Gaussian Process regression
1. Introduction
There is a growing tendency to utilize industrial robots in machining applications
to achieve more flexibility, larger workspace and lower costs of production compared
to conventional machine tools [1,2]. However, as industrial robots are less rigid than
machine tools, they are more prone to chatter vibrations. Chatter vibrations would
increase tool wear, shorten tool life, reduce machining accuracy and surface quality,
and even cause tool fracture and workpiece scrapping in serious cases [3,4], thus
limiting the productivity and quality of the robotic machining process [5]. Selecting
the combination of stable machining parameters using the Stability Lobe Diagram
(SLD) [6–8] is a widely adopted strategy to prevent chatter in robotic milling [4,9,10].
The tool tip FRFs are essential for establishing the SLD.
Tool-tip FRFs can be directly obtained by the Experimental Modal Analysis
† Corresponding author. E-mail address: y.ding@sjtu.edu.cn (Dr. Ye DING).
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(EMA). However, the experimental studies [4,11–13] based on EMA show that the
tool tip dynamics change significantly for the milling robot due to its low structure
stiffness. The EMA procedure is time-consuming for the robotic system since it has to
be repeated for different postures. Hence, it is necessary to establish the predictive
models to obtain the tool tip FRF in arbitrary postures.
To predict the tool tip dynamics in robotic milling, many methods were raised in
previous research. To minimize experimentations, the methods based on the
multibody dynamics model were implemented to predict the posture-dependent tool
tip dynamics in robotic milling. Mousavi et al. [14,15] established a multibody
dynamic model of a serial robot using Euler–Bernoulli beam elements and the model
was further developed by applying the flexible joints and bodies approach coupled
with the Matrix Structural Analysis method [10]. Baglioni et al. [16] developed a
parametric multibody modeling procedure based on the Denavit-Hartenberg
convention. Huynh et al. [17] presented two multibody dynamics models of
articulated industrial robots suitable for machining applications and Chen and
Ahmadi [18] further use Gaussian Process (GP) regression model to predict the
posture-dependent joint elastic parameters in the multibody models. However,
multibody dynamics models usually include a large number of inertial and joint
elastic parameters that must be identified experimentally, which is pretty complicated.
Multibody dynamics models are subject to inaccuracies because of the simplified
assumptions in modeling [19].
The experimental results of the impact test are more accurate compared to the
results predicted by multibody dynamics since the practical dynamic responses at the
tool tip are obtained [18]. To reduce the number of impact tests and improve
prediction accuracy, some researchers worked to establish regression models to
predict the posture-dependent tool tip dynamics of an industrial robot within its
workspace based on experimental results. Chen et al. [20] used the inverse distance
weighted (IDW) model to interpolate the tool tip FRF at any posture based on the
sample information that contains the tested robot postures and corresponding modal
parameters. Nguyen et al. [21] used the GP regression model to predict the
posture-dependent tool tip FRF based on the FRF data measured from EMA and the
model is further combined with Operational Modal Analysis (OMA) to maximize
both testing efficiency and spatial resolution of EMA [22]. Busch et al. [23] improved
the performance of the GP regression model for predicting the pose-dependent robot
dynamics by fusing simulation results with experimental data. Wang et al. [24] trained
the random forest to predict the posture-dependent modal parameters considering the
cross coupling FRFs. However, the regression models mentioned above only aim at
the tested tool, so when the tool changes, the established model will no longer work
[18]. The procedure is time-consuming since it has to be repeated for different tools.
To achieve tool tip FRFs prediction for different tools, Schmitz et al. [25–27]
first introduced the RCSA method. The RCSA method has been studied and applied in
machining areas for many years [28–32]. The RCSA method is proven to be an
effective method to establish the relationship of tool tip dynamics between the tested
tool (named the reference tool) and a new tool (named the target tool) with finite
element (FE) analysis [33,34]. Recently, the RCSA method has been used in the
prediction of the posture-dependent tool tip FRF [35,36]. Deng et al. [35] first
introduced the RCSA method into the regression model to predict position-dependent
tool tip FRFs of the machine tools. They trained a GP (or Kriging) model to predict
the tip FRFs of the machine tool frame-holder base at any position and then calculate
the position-dependent tool tip FRFs using the RCSA method with the finite element
3
model (FEM) of the tool. Their approach combines the GP regression model with the
RCSA method to effectively predict position-dependent tool tip FRFs of the machine
tools for different tools without having to perform repetitive impact tests. In their
work, due to the high stiffness of machine tools, the contribution of vibration modes
generated in directions orthogonal to the excitation direction due to cross coupling
was neglected and the FRFs in different directions were treated separately. The
measured direct FRFs were fitted with the “peak-picking” method [37] separately to
identify the modal parameters. As the structural stiffness of the milling robot is much
lower than the machine tools, the cross coupling FRFs have a more obvious influence
on the dynamic property of the system [24,38] and then affect the milling stability
boundary greatly [39,40]. When cross coupling FRFs are considered, the RCSA
procedure will be much more complicated [41], which requires a new experimental
design of the impact test to collect more FRFs. Fitting these measured FRFs
separately following the method in Ref. [35] will result in too many modal parameters
for each mode. As the robot system usually exhibits multiple modes of vibration
[24,38,42], the total number of identified modal parameters will be very large. In
addition, in Ref. [35], the modal parameters were respectively trained and predicted
by the GP regression model which is a single-output predictive model. As a result, the
number of established models equals the extracted modal parameters and the intrinsic
correlations between different modal parameters are ignored. In summary, the
approach to predict the position-dependent tool tip FRFs of the machine tools with
different tools is hardly used for industrial robots considering the multi-mode and
cross coupling effects of robotic milling systems. It will encounter great difficulties in
the experimental design of the impact test, the extraction and prediction of the modal
parameters.
In this paper, an effective approach to predict the posture-dependent tool tip
dynamics for different tools in robotic milling considering multi-mode and cross
coupling FRFs is proposed. Through combining the RCSA method with the MTGP
regression model, the proposed approach achieves more comprehensive and general
prediction of the tool tip dynamics in robotic milling than the previous works. The
effects of multi-mode and cross coupling of the robotic milling system and the change
of the tool are simultaneously considered. A new experimental design of impact test is
proposed to collect the required FRFs for the RCSA procedure considering the cross
coupling FRFs. The measured FRFs can be summarized in the FRF matrix. Then, the
measured FRF matrix is used to identify the modal parameters with the MDOF model,
which means all measured FRFs are fitted together [38,40]. Through considering the
physical relationship between the measured FRFs, the number of modal parameters
required is significantly reduced. To train and predict the extracted modal parameters,
the MTGP regression model, which is an extension of the ordinary Single-Task
Gaussian Process (STGP, usually denoted as GP) regression model, is adopted. With
the MTGP regression model, the modal parameters with physical similarity are
trained and predicted together. The number of established regression models is
reduced and the prediction performance is improved in terms of accuracy and
robustness. The approach proposed in this paper considers the physical knowledge of
the system in modal parameter extraction and prediction. This makes it more feasible
to predict the posture-dependent tool tip FRFs of the robot for different tools
considering the cross coupling FRFs.
Henceforth, the remainder of this paper is organized as follows. The overview of
the proposed approach is described in Section 2. In Section 3, a more comprehensive
RCSA procedure is adopted to establish the relationship of tool tip dynamics between
4
different tools considering the non-negligible cross coupling FRFs. The improved
procedure to predict the posture-dependent reference tool tip dynamics is presented in
Section 4. The proposed approach is experimentally verified in Section 5 and the
paper is concluded in Section 6.
2. Overall Approach Description
This paper proposes an improved approach to achieve effective prediction of the
posture-dependent tool tip dynamics for different tools in robotic milling considering
the cross coupling FRFs, as shown in Fig. 1.
Tool tip dynamics of
the reference and target tools
at a certain posture
FEM of
the reference and
target tools
FE analysis
Posture-dependent target
tool tip dynamics
Designed impact test
Relationship of dynamics
between the reference and target tools
RCSA
Tool tip dynamics of
the reference tool
at the arranged postures
Posture-dependent reference
tool tip dynamics
MTGP
Modal parameters
MDOF model
Fig. 1. Overview of the proposed approach.
Firstly, the RCSA method is utilized to establish the relationship of tool tip
dynamics between the reference and target tools to make the prediction method
effective for different tools. To predict the cross coupling FRFs of the target tool tip
dynamics of the milling robot, the full receptance matrices are required in the RCSA
procedure. Then, the FRF matrix required to estimate the full receptance matrix of the
reference tool is determined and measured with the designed impact test at the
arranged postures. The MDOF model is applied to extract modal parameters from the
measured FRF matrix and the training data set is built. Next, the MTGP regression
model is established to predict the posture-dependent modal parameters of the
reference tool. The proposed MTGP regression model takes advantage of the intrinsic
correlation between similar modal parameters to improve the performance of the
modal parameter prediction in terms of accuracy and robustness. Combining the
established relationship of tool tip dynamics and the predicted posture-dependent
reference tool tip dynamics, the posture-dependent target tool tip dynamics containing
the cross coupling FRFs can be calculated.
5
3. The relationship of tool tip dynamics Establishment
The RCSA method is utilized in Ref. [35] to predict tool tip dynamics for
different tools, but only direct FRFs were considered due to the high stiffness of
machine tools. As the cross coupling FRFs have a more obvious influence on the
dynamic property of the robot system, the RCSA method in Ref. [35] needs to be
expanded. In this section, the RCSA method based on the full receptance matrices in
Ref. [41] is adopted. The adopted RCSA method is well-suited for establishing the
relationship of tool tip dynamics between different tools considering the cross
coupling FRFs.
As shown in Fig. 2(a), the milling robot (the assembly spindle-tool holder-tool,
S-TH-T) can be divided into two parts: one is the robot body assembled with the
spindle-tool holder (the substructure S-TH) and the other is the tool (the substructure
T). According to the RCSA method [28,37], the dynamics of the assembly can be
computed by the dynamics of its substructures. The two substructures of the milling
robot are coupled with a flexible connection
K
, as shown in Fig. 2(b).
Substructure
S-TH Substructure
T
[K]
1 2 1
Substructure S-TH
Substructure T
(a) (b)
Fig. 2. Two substructures of the milling robot.
Since the stiffness along the tool axis is much larger than that in other
orthogonal directions [24,36], the axial receptance (i.e., the receptance of the Z
direction) is ignored in this paper. To take the cross coupling FRFs into consideration,
the dynamics should be expressed by the full receptance matrix
ij
T
as follows [41]:
__
__
ix jx ix jy
ij
iy jx iy jy
RR
TRR
=
, (1)
where
_ix jx
R
and
_iy jy
R
are the direct FRF submatrices,
_ix jy
R
and
_iy jx
R
are the cross FRF submatrices. For the direct FRF submatrix
_ix jx
R
, the first
subscript “ix” refers to the position “i” and direction “x” of the measurement while the
second one “jx” refers to the position “j” and direction “x” of the excitation. The
symbols of other FRF submatrices are defined by analogy.
The direct FRF submatrix
_ix jx
R
has the following form:
__
___
ix jx ix jx
ix jx ix jx ix jx
HL
RNP
=
, (2)
where
_ix jx
H
is the displacement-to-force receptance,
_ix jx
L
is the
displacement-to-moment receptance,
_ix jx
N
is the rotation-to-force receptance,
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_ix jx
P
is the rotation-to-moment receptance. Their subscripts are defined in the same
way as the submatrix. The elements of other FRF submatrices are defined by analogy.
So all the receptances in the full receptance matrix
ij
T
are as follows [41]:
_ _ _ _
_ _ _ _
_ _ _ _
_ _ _ _
ix jx ix jx ix jy ix jy
ix jx ix jx ix jy ix jy
ij iy jx iy jx iy jy iy jy
iy jx iy jx iy jy iy jy
H L H L
N P N P
TH L H L
N P N P
=
. (3)
With the RCSA theory [28,41], the relationship of tool tip dynamics between the
reference tool T1 and the target tool T2 can be established:
( )
()
1
-1
-1 -1 1
11 11 12 22 12 11 11 21 22 21
2 2 2 2 1 1 1 1 1 2
C A A A A A C A A A
T T T T T T T T T K T
−
−
= − + − − +
, (4)
where A and C represent the substructure T and the assembly S-TH-T respectively.
The right subscript of the receptance matrix (“1” or “2”) represents the installed tool.
K
is the complex contact stiffness matrix representing the contact dynamics at the
holder-tool interface, which is determined by the stiffness and damping constants for
translation and rotation [41].
1 1 1
21
=K K K
− − −
−
reflects the difference in
interface contact dynamics between the reference tool T1 and the target tool T2. The
full receptance matrices of two tool substructures
1
A
ij
T
,
2
A
ij
T
can be obtained
through FE analysis [33,34,43–45] and the difference in interface contact dynamics
1
K−
can be identified through the impact test conducted at a certain posture.
With the established relationship, when the posture-dependent full receptance
matrix of the reference tool
11 1
C
T
is predicted by regression models, the tool tip
dynamics for different tools can be obtained directly without the need to retrain the
regression models.
4. Posture-dependent reference tool tip dynamics prediction
As shown in Section 3, the posture-dependent reference tool tip dynamics are
expressed by the full receptance matrix, which contains more FRFs than Ref. [35] due
to the consideration of the cross coupling FRFs. This brings great challenges to the
experimental design of the impact test, the extraction and prediction of the modal
parameters. In this section, the improved procedure to predict the posture-dependent
reference tool tip dynamics is presented. Firstly, the impact test is designed based on
the finite-difference technique to estimate all the elements in the full receptance
matrix of the reference tool. Then the modal parameters are extracted from the
measured FRFs with the MDOF model and used to build the training set. Finally, the
MTGP regression model is trained to map posture property vectors from the input
space to modal parameters in the output space.
4.1. The experimental design of the impact test
The full receptance matrix used to present reference tool tip dynamics includes
displacement-to-force, displacement-to-moment, rotation-to-force, and
rotation-to-moment receptances. While the displacement-to-force elements in the full
receptance matrix are straightforward to obtain by the standard impact test, it is
difficult to directly measure other elements due to the lack of moment exciter and
7
rotational sensors [46].
The finite-difference technique is one of the most practical methods to measure
rotation-related FRFs from displacement-to-moment FRFs [47,48]. The effectiveness
of the finite-difference technique has been widely verified in RCSA-based
applications to predict tool tip FRFs [33,34,44,49]. Based on the finite-difference
technique, a new experimental design of impact test is proposed to collect the required
FRFs for the full receptance matrix.
The experimental design of the impact test is illustrated in Fig. 3. As shown in
Fig. 3, two locations, which are located at the tool tip (1X,1Y) and S mm away from
the tool tip (1aX,1aY), are selected to excite and measure the FRFs. Note that (1X,1Y)
and (1aX,1aY) are the symbols to indicate the locations. During the impact test, the
impact locations are set at the tool tip (1X,1Y) and the location (1aX,1aY)
respectively. Four sensors simultaneously work to measure the displacements of the
points (1X,1Y) and (1aX,1aY). When the excitation is conducted at the tool tip
(1X,1Y), the impact directions are shown in Fig. 3 and similarly for the excitation at
the location (1aX,1aY). When the excitation is conducted at the tool tip (1X,1Y), four
displacement-to-force receptances,
1 _1xx
H
,
1 _1yx
H
,
1 _1xy
H
and
1 _1yy
H
, as
illustrated in Fig. 3, are measured by the sensors at the tool tip (1X,1Y). Another four
displacement-to-force receptances,
1 _1ax x
H
,
1 _1ay x
H
,
1 _1ax y
H
and
1 _1ay y
H
, are
measured by the sensors at the location (1aX,1aY). When the excitation is conducted
at the location (1aX,1aY), the other eight displacement-to-force receptances,
1 _1x ax
H
,
1 _1y ax
H
,
1 _1x ay
H
,
1 _1y ay
H
,
1 _1ax ax
H
,
1 _1ay ax
H
,
1 _1ax ay
H
and
1 _1ay ay
H
, are measured in
the same way. A total of 16 displacement-to-force receptances are measured from the
impact test and can be summarized in the measured FRF matrix
Η
:
1 _1 1 _1 1 _1 1 _1
1 _1 1 _1 1 _1 1 _1
1 _1 1 _1 1 _1 1 _1
1 _1 1 _1 1 _1 1 _1
x x x y x ax x ay
y x y y y ax y ay
ax x ax y ax ax ax ay
ay x ay y ay ax ay ay
H H H H
H H H H
H H H H
H H H H
=
Η
. (5)
A-A section view
H1x_1x H1x_1y
H1y_1y
H1y_1x
Impact direction
Sensor
1aX
s
A A
Tool
Sensors
z
x y
Tool holder
1aY
1X1Y
Fig. 3. The experimental design of the impact test to get the full receptance matrix.
As the cross coupling between different directions for the milling robot reflects
non-symmetric [38], the measured displacement-to-force receptances are considered
to be different from each other. All the elements in the full receptance matrix
11 1
C
T
of
the assembly for the reference tool shown in Eq. (3) can be obtained from the
measured FRF matrix
Η
, as described in Appendix A. The posture-dependent tool
8
tip dynamics of the reference tool can be determined by predicting all 16
displacement-to-force receptances in the FRF matrix
Η
. The above procedure to
measure the displacement-to-force receptances of the reference tool can be repeated at
a series of training postures for prediction.
4.2. Identification of modal parameters
In the previous work on modal parameter prediction, most studies [21,35,36]
focus on the direct FRFs, so each direct FRF is independent of each other, and the
modal parameters are extracted separately. Three modal parameters are identified for
each mode in each FRF. Recent research work [24] considers the cross coupling
FRFs, but since only two degrees of freedom of the tool tip are considered, only four
FRFs need to be processed, including the direct FRFs and the cross coupling FRFs.
Therefore, extracting the modal parameters for each FRF separately is still
manageable in Ref. [24]. In this paper, the measured FRF matrix
Η
considers
the degrees of freedom of two locations, i.e., (1X,1Y) and (1aX,1aY), and it contains
16 displacement-to-force FRFs. If the modal parameters are still extracted separately
for each FRF as in Refs. [21,24,35,36], the 16 FRFs will correspond to 48 modal
parameters for each mode. As the robot system usually exhibits multiple modes of
vibration [24,38,42], the total number of identified modal parameters will be very
large. This will bring great difficulties to the subsequent modal parameter prediction.
In Refs. [38,40], the MDOF model has been applied to the tool tip FRF matrix to
consider the relationship between the degrees of freedom at the tool tip. Through
considering the physical relationship between the measured tool tip FRFs, the effect
of the cross coupling FRFs on the milling stability was studied. The measured FRF
matrix
Η
reflects the relationship between the degrees of freedom of two
locations, so the MDOF model is still applicable. Through considering the physical
relationship between the FRFs in the measured FRF matrix
Η
, the number of
modal parameters to be identified is significantly reduced.
The robot system is a multiple-degrees-of-freedom system and its dynamics can
be described by the differential equation of motion as:
( ) ( ) ( ) ( )t t t t+ + =Mu Cu Ku f
, (6)
where
nn
M
,
nn
C
and
nn
K
denote the system mass, damping and
stiffness matrices respectively.
() n
tu
is the generalized coordinate vector and
() n
tf
is the excitation vector. n is the number of degrees of freedom.
By taking the Laplace transform of Eq. (6), the dynamics of the system are
given as:
( ) ( ) ( )s s s=G U F
, (7)
where
2
()s s s= + +G M C K
,
()sU
is the generalized coordinate vector. Then the
transfer function matrix
()sH
can be derived from Eq. (7):
1
( ) adj( ( ))
( ) ( ( ))
( ) ( )
ss
ss
ss
−
= = =
UG
HG
FG
, (8)
where
adj( )
is the adjoint matrix and
is the determinant of the matrix.
The FRF matrix
()
H
can be obtained by substituting s = jω [50]:
9
adj( ( ))
() ()
=G
HG
. (9)
From Eq. (9), it can be seen that all the measured FRFs of the MDOF system
have a common denominator
()
G
, which indicates that there is an internal
physical connection between the measured FRFs. The MDOF model captures the
physical connection between the measured FRFs, allowing all measured FRFs to be
fitted simultaneously with much fewer modal parameters.
By introducing the state vector
T
T
T
( ) ( ) ( )t t t
=
v u u
, the Eq. (6) can be
transformed into the state-space form:
ˆˆ
( ) ( ) ( )t t t+=
v
Av Bv f
, (10)
where
()
ˆˆ
, , ( ) t
t
= = =
−
v
C M K 0 f
A B f
M 0 0 M 0
.
Due to the coupling structure of the robot system, the system is regarded as
non-proportionally damped and the mass, damping and stiffness matrices are
non-symmetric [38]. The modal model consists of normalized left eigenvector
Ln
r
, normalized right eigenvector
Rn
r
and eigenvalue
r
. The
eigenvectors and eigenvalues together determine the normalized modal
transformation matrix
22
,1 ,1 , ,
=nn
s s s n s n
where
T
TT
,, 1, ,
s r r r r rn
= =
. The eigenvalues form the eigenvalue matrix
22
diag , , , nn
rr
=
. The normalized eigenvectors satisfy the properties:
TT
L R L R
ˆˆ
, = = −A I B
. (11)
Using the modal transformation
R
( ) ( )tt= v
vq
and combining the properties
in Eq. (11), Eq. (10) can be transformed into the modal space:
T
L
( ) ( ) ( )t t t− =
v v v
q q f
, (12)
where
2
() n
t
v
q
is the modal coordinate vector.
By taking the Laplace transform, the modal transfer function matrix is given as:
( ) ( )
TT
1T
RL
1
( ) [ ] ( )
R L R L
nr r r r
rrr
ss ss
−
=
= − = +
−−
HI
, (13)
where I is the identity matrix. The eigenvalue
r
is related to the natural frequency
in rad
,nr
and damping ratio
r
of the corresponding modes and can be
expressed as
2
,,
1
r r n r n r r
j
= − −
. (14)
By substituting
=si
in Eq. (13), the FRF matrix can be obtained:
( ) ( )
TT
1
( ) ( )
R L R L
nr r r r
rrr
ii
=
=+
−−
H
. (15)
It can be seen from Eq. (15) that for each mode,
L
r
,
R
r
and
r
can grasp
the physical relationship between the FRFs in the FRF matrix and determine all the
FRFs simultaneously. It should be noted that multiplying
R
r
and dividing
L
r
by
the same constant can get exactly the same result in Eq. (15). This indicates that the
10
properties of the normalized eigenvectors in Eq. (11) can not work when determining
the normalized eigenvectors
L
r
,
R
r
with Eq. (15). To address this problem,
L
r
is further normalized by taking its last element as 1 so that
L
r
and
R
r
can be
uniquely determined with Eq. (15).
The elements of the FRF matrix in Eq. (15) can be expressed as:
R L R L
1
( ) ( )
njr kr jr kr
jk rrr
ii
=
=+
−−
H
, (16)
where
R
jr
is the j-th element in
R
r
and
L
kr
is the k-th element in
L
r
.
For the measured FRF matrix
Η
in Eq. (5), both measurement coordinates
and excitation coordinates are {1X,1Y,1aX,1aY}, i.e., j,k∈{1X,1Y,1aX,1aY}. For
each mode, the modal parameters corresponding to the measured FRF matrix are the
eigenvalue
r
and
L
kr
,
R
jr
for j,k∈{1X,1Y,1aX,1aY}. As the last element of
L
r
is fixed to 1 after normalization and other elements in the normalized
eigenvectors are complex, 16 real parameters are required to fit the measured FRF
matrix for each mode and they are summarized in the parameter vector
r
p
:
L L L L L L
, 1 , , 1 , , 1 , , 1 , , 1 , , 1 , ,
R R R R R R R R
1 , , 1 , , 1 , , 1 , , 1 , , 1 , , 1 , , 1 , ,
[ , , , , , , , ,
, , , , , , , ]
r n r r X r real X r imag aX r real aX r imag Y r real Y r imag
X r real X r imag aX r real aX r imag Y r real Y r imag aY r real aY r imag
f
=p
, (17)
where
,nr
f
is the natural frequency in Hz corresponding to
,nr
. The subscript
“real” refers to the real part of the elements in the eigenvectors and the subscript
“imag” refers to the imaginary part of the elements in the eigenvectors.
As shown in Eq. (17), by considering the physical relationship between the
measured FRFs with the MDOF model, the number of modal parameters to be
identified for each mode is reduced to 16, which is one-third of the number of modal
parameters required by the method of extracting modal parameters for each FRF
separately [21,24,35,36]. This will make the subsequent modal parameter prediction
more feasible.
4.3. Training data set building
After determining the experimental design of the impact test and the method to
identify the modal parameters based on the MDOF model, it is necessary to plan the
collection postures reasonably to construct the data set for training. Considering that
the machining task is defined in Cartesian space, the Cartesian positions of the robot
tool tip in the base frame are used as inputs in this paper. Since the workpiece to be
machined is mounted on a fixed table with a certain height, the tool tip has a
significantly larger range of position in the X-Y plane than in the Z direction (height)
[9]. So the tool tip is limited to staying at a constant height of the Z-axis of the base
frame and the position changes in the X-Y plane in this work. To ensure the milling
tool is perpendicular to the workpiece, the tool axis is constrained to be perpendicular
to the X-Y plane of the robot’s base frame in all experiments. The redundant axis,
which has a significant impact on the joint angles under the same tool tip position
[10,21], is also taken into consideration. There are different rotational orientations
about the tool axis at each tool tip position to change the posture. The redundant axis
angle
is used to express the rotational orientation of the tool coordinate system
with respect to the tool axis, as shown in Fig. 4. Therefore, the posture is determined
by the tool tip position in the X-Y plane and the redundant axis angle. Denote X as the
11
input space, namely the posture properties of the robot. For
iXx
,
= , ,
i i i i
xy
x
.
,
ii
xy
are the coordinates in XY plane and
i
is the redundant axis angle.
z
x
y
z
x
y
Base
Tool
Fig. 4. The redundant axis angle for the milling robot.
The training postures are arranged to make the training set cover the milling
tasks on the fixed worktable as much as possible. The range of the XY plane is chosen
to cover the installable area of the workpiece on the fixed worktable and the range of
redundancy angles is as large as possible in the feasible range. The procedure to
measure the displacement-to-force FRF matrix of the reference tool and extract the
modal parameters is repeated at
train
n
arranged training postures in the input space to
collect the training data. For each mode, the training set
r
D
of
train
n
observations for
a corresponding modal parameter
(
, 1,...,
rrn
=p
) can be denoted as
T
i
{κ} {(x , ) | 1,..., } {( , , , ) | 1,..., }
r i train i i i i train
i n x y i n
= = = = =D X,
, (18)
where
i
denotes the order of the samples. The column vector inputs are aggregated
in the
3train
n
matrix
X
and the outputs are collected in the
1train
n
matrix
κ
.
The built training data set will be further used to establish the regression prediction
model to predict the posture-dependent reference tool tip dynamics.
4.4. Establishing the prediction model based on the MTGP regression
model
With the aid of the powerful prediction capability of unknown information based
on the known sample information and the advantage of estimating the confidence
bounds [51], the ordinary Single-Task Gaussian Process (STGP) regression model has
been successfully used to describe the relationship between the related modal
parameters and postures [21,22,35]. Nonetheless, the STGP regression model is
unable to utilize the correlation between different modal parameters. In addition, as
the robot system usually exhibits multiple modes of vibration [24,38,42], the total
number of identified modal parameters is very large. There are so many regression
models required to build if the single-output STGP predictive model is used.
The Multi-Task Gaussian Process (MTGP) regression model [52] has been
successfully applied to other fields, such as healthcare [53] and image reconstruction
[54]. The previous work reflected the advantages of the MTGP regression model in
12
prediction performance compared with the STGP regression model. To overcome
these shortcomings of the STGP regression model while retaining the advantages, the
MTGP model is introduced to predict the posture-dependent modal parameters for the
first time to solve the difficulties in modal parameter prediction.
This section establishes the regression prediction model to predict the
posture-dependent reference tool tip dynamics based on the MTGP regression model.
The STGP regression model is briefly introduced first and then extended to the
MTGP regression model which can mine the correlation between different modal
parameters and reduce the number of required regression models.
4.4.1. Single-Task Gaussian Process regression
The ordinary Gaussian Process [55] is also called Single-Task Gaussian Process
(STGP) because of the single output [53]. A Gaussian process used to describe the
distribution over function
f
is completely specified by its mean function
()mx
and
covariance function
( , ')kxx
:
( ) ~ ( ( ), ( , '))f GP m kx x x x
. (19)
As one of the most used kernel functions, the non-isotropic squared exponential
kernel function is chosen to specify the covariance between pairs of random variables:
2
1
1
cov[ ( ), ( ')] ( , ') exp[ ( ')]
2
D
f d d d
d
f f k w
=
= = − −
x x x x x x
, (20)
where
cov
denotes the covariance.
D
represents the dimension of the input vector
and its value is 3.
2, ( 1,..., )
fd
w d D
=
are the parameters of the kernel function.
Assume that the observed target value
differs from the function value
(x)f
by additive noise
which follows an independent, identically distributed Gaussian
distribution with zero mean and variance
2
n
:
2
(x) , ~ (0, )
n
fN
=+
, (21)
where
N
denotes Gaussian distribution. The variance of the noisy observations
becomes
2
cov( ) ( , ) n
K
=+κX X I
, (22)
where
( , )KXX
is the
nn
covariance matrix which can be calculated elementwise
based on the input vector of the training set with the kernel function mentioned in Eq.
(20).
I
is the identity matrix of size
n
.
Given
*
n
test inputs
*
X
, the joint distribution of the observed target values
κ
and the function values
*
f
at the test locations
*
X
can be written as:
2*
***
*
( )
()
~ (0, )
( )
()
nK
K
NK
K
+
X,X
κX,X I
fX ,X
X ,X
, (23)
where
*
()KX,X
denotes the
*
nn
covariance matrix evaluated at all pairs of
training and test points, and similarly for other matrix elements
**
( )KX ,X
,
*
()KX ,X
.
Through conditioning the joint Gaussian distribution, the key predictive
equations for GP regression can be derived as [55]:
* * * *
| , ~ ( ,cov( ))NfXκ,X f f
, (24)
where
21
* * * *
E[ | ] ( )[ ( ) ]
n
KK
−
=+f f X,κ, X X,X X,X I κ
(25)
13
21
* * * * *
cov( ) ( ) ( )[ ( ) ] ( )
n
K K K K
−
= − +f X , X X , X X,X I X,X
. (26)
*
f
and
*
cov( )f
are the posterior mean and covariance matrix for the function values
*
f
corresponding to test inputs
*
X
.
It can be seen from the above procedure that the hyperparameters of the model,
which can be collected in a vector
θ
, contain both the parameters of the kernel
function (i.e.,
2, ( 1,..., )
fd
w d D
=
) and the variance of the Gaussian noise
2
n
.
Observing that
2
~ (0, )
n
NK
+κI
, the negative logarithmic marginal likelihood
(NLML) can be written as
T1
11
log ( | , ) log | | log2π
2 2 2
n
p K K
−
− = + +κXθ κ κ
, (27)
where
2
()
n
KK
=+X,X I
is the covariance matrix for the noisy targets
κ
. The
hyperparameters in the vector
θ
are determined by minimizing the NLML [55].
4.4.2. Multi-Task Gaussian Process regression
When there are m posture-dependent modal parameters to be predicted, a
straightforward approach is to train the STGP regression model for each modal
parameter independently, as illustrated in Fig. 5(a). In contrast, the MTGP regression
model combines the m posture-dependent modal parameters within one model to
learn the correlation within the modal parameters concurrently to improve the
performance of the prediction, as shown in Fig. 5(b).
*
1
x
*
m
x
*
1
*
m
MTGP
*
1
x
*
m
x
*
1
*
m
( , , , , )
ct
Xκ θ θl
a) b)
( , , )
m m m
STGP Xκ θ
( , , )
1 1 1
STGP Xκ θ
Fig. 5. Graph models of STGP and MTGP for the modal parameter prediction. (a) Multiple
STGP regression models, (b) one MTGP regression model
For the m posture-dependent modal parameters, the training postures can be
denoted as
= | 1,..., ; 1,...,
jj
i train
j m i n==Xx
and the observations for the modal
parameter
(
, 1,...,
rrn
=p
) can be denoted as
= | 1,..., ; 1,...,
jj
i train
j m i n
==κ
,
where
j
train
n
is the number of the postures of the j-th modal parameter. As all the
modal parameters are measured and identified for the
train
n
arranged postures, their
training numbers are the same, i.e.,
j
train train
nn=
. To distinguish different modal
parameters, a label
j
l
has to be added as an additional input with
j
lj=
. The
covariance function of MTGP can be expressed as
( ) ( ) ( )
, , , , ,
MTGP c t
k l l k l l k
=x x x x
, (28)
where
c
k
represents the correlation between the modal parameters, which is a
function of the labels.
t
k
is similar to the covariance functions of STGP, which
represents the resemblance between different postures within the same modal
parameter. Then the covariance matrix
MTGP
K
for m modal parameters can be written
as
( , , , )
MTGP c t
KXθ θl
:
14
( ) ( )
( , , , ) , ,
MTGP c t c c t t
K K K=Xθ θ θ Xθll
, (29)
where
is the Kronecker product,
= | 1,...,
j
l j m=l
. The vectors
c
θ
and
t
θ
contain the hyperparameters for
c
K
and
t
K
, respectively. As
c
K
and
t
K
have the
size of
mm
and
train train
nn
, the covariance matrix
MTGP
K
has the size of
train train
mn mn
. The diagonal elements of
c
K
describe the self-correlation of the
modal parameters and the non-diagonal elements represent the cross-correlation
between different modal parameters.
To construct a positive semidefinite covariance function
c
K
that fulfills
Mercer’s theorem, the Cholesky decomposition is utilized and the elements of the
lower triangular matrix
L
can be parameterized:
1
,2 ,3
T
, 1 , 2 ,
00
0
==
c
cc
c
c k m c k m c k
θ
θ θ
K
θ θ θ
− + − +
，
， LL L
, (30)
where
1
m
i
ki
=
=
is the number of the hyperparameters of
c
K
. Similar to the STGP
model, the hyperparameters for the MTGP model can be optimized by minimizing the
NLML. Given a testing input
**
,lx
, the corresponding prediction can be obtained
by computing the conditional probability
( )
* * *
| , , , ,pl
Xκxl
.
The MTGP regression model takes advantage of the intrinsic correlation
between different modal parameters to improve the performance of prediction, but the
prediction quality depends strongly on the correlation between the modal parameters.
Additionally, one limitation of the MTGP model is its higher computational costs. The
computational cost for learning an MTGP model is higher than the computational cost
for learning m STGP models independently. Therefore, the modal parameters
gathered in the MTGP regression model should be chosen carefully to possess a high
correlation and the number of the modal parameters gathered in one MTGP regression
model cannot be too large. The way to determine the value of m based on the training
time and choose the modal parameters in one MTGP model based on the correlation
will be discussed in detail in Section 5.2.
5. Experimental validations
In this section, the MTGP regression models to predict the posture-dependent
reference tool tip dynamics are established and validated in the experiment. Then the
predicted reference tool tip dynamics are combined with the relationship of tool tip
dynamics established in the experiment to predict the posture-dependent target tool
tip dynamics containing the cross coupling FRFs. Finally, the stability lobe diagrams
calculated by the predicted target tool tip dynamics are validated with the milling
experiment.
5.1. Experimental Setup
The proposed tool tip dynamics prediction approach is applied to an industrial
ABB IRB6660 robot. The impulse excitation is applied by an impact hammer (PCB
086C01). The high-frequency modes are measured with four single-axis
accelerometers (PCB 352C23), as seen in Fig. 6. Because accelerometers cannot
15
effectively measure low-frequency modes [36,38], four laser displacement sensors
(KEYENCE LK-H025) are used to measure the low-frequency modes, as shown in
Fig. 7. The distance S between the two locations described in Section 4.1 is chosen to
be 20 mm for the accelerometers and is determined to be 33.4 mm due to the
installation dimension for the laser displacement sensors. The acquired FRFs are
processed with ModalVIEW software. The impact test is repeated three times to
assure the validity of experimental data.
The reference tool is chosen to be a smooth rod with a 20 mm diameter, also
called the dummy tool [56]. The tool length is 100 mm and the overhang length is 50
mm. The reference tool is made of cemented carbide whose material properties are
listed in Table 1. The adopted target tool is a two tooth PCD end mill with an 18 mm
diameter. The tool length and the overhang length are 95 mm and 45 mm respectively.
The cutting edges are made from polycrystalline diamond (PCD) material and brazed
on the tool body whose base is the same with the reference tool. The adopted target
tool and the FEM of its overhang portion are shown in Fig. 8.
S-TH-T
Tool
holder
Reference
tool
Accelerometer
Impact hammer
s
Fig. 6. The experimental setup to measure high-frequency modes.
16
S-TH-T
Tool
holder
Reference
tool
Laser displacement
sensor
Impact hammer
s
Fig. 7. The experimental setup to measure low-frequency modes.
(a)
Modeled Section Clamped Section
(b)
Fig. 8. (a)The adopted target tool and (b) the FEM of the overhang portion.
Table 1
Material properties of cemented carbide.
Modulus of elasticity
(GPa)E
Density
3
(kg/m )
Poisson's ratio
structural damping ratio
540
14500
0.25
0.001
The machining stability for the predicted target tool tip dynamics is validated by
a series of milling experiments, as shown in Fig. 9. The workpiece material is
aluminum alloy 6061. The cutting force coefficients have been identified as
2
880 /
t
K N mm=
and
2
52 /
r
K N mm=
with the mechanistic approach [50]. The
accelerometer (PCB 356A01) is used to measure the vibrations during the milling
operation.
17
Accelerometer
Workpiece
Fig. 9. The experimental setup for stability validation.
5.2. Prediction of the posture-dependent reference tool tip dynamics
In this section, the procedure to predict the posture-dependent reference tool tip
dynamics presented in Section 4 is implemented in the experiment. Firstly, the
training postures to conduct the impact test designed in Section 4.1 are determined.
Then the measured FRFs are fitted with the adopted MDOF-based method to identify
the posture-dependent modal parameters, thus the training data set is established.
Finally, the proposed MTGP regression models are established to predict the modal
parameters and their prediction performances are validated through 5-fold
cross-validation and external validation.
As described in Section 4.3, the posture is determined by the tool tip positions in
the XY plane and the redundant axis angle. To cover the installable area of the
workpiece on the fixed worktable, the tool tip positions of the training postures are
equally spaced at the intervals of 200 mm in the XY plane under the base coordinate
system as shown in Fig. 10. To cover the feasible range of redundant angles as much
as possible, seven rotational orientations about the tool axis
(
60 ,-40 ,-20 ,0 ,20 ,40 ,60-
=
) are arranged for each tool tip position. Therefore,
the total number of training postures is 112.
18
z
x
y
1300 1500 1700 1900
-300
-100
100
300
X(mm)
XOY Plane
Y(mm)
Fig. 10. The arrangement of tool tip positions in impact tests.
The measured FRFs at each posture are fitted simultaneously with the
MDOF-based method in a nonlinear least square approach [38] to identify the modal
parameters. The FRFs measured by the laser displacement sensors and accelerometers
are used to identify the low-frequency modes and high-frequency modes respectively.
Five dominant low-frequency modes (modes 1-5) and four dominant high-frequency
modes (modes 6-9) are identified. Taking one representative posture (1300 mm, 100
mm,
0
=
) as a sample, the identified modal parameters of all 9 modes are listed in
Appendix B. Note that the last element
L
r
is fixed to 1 due to the normalization
constraint in the identification. Taking the tool tip FRFs as the representative of the
16 displacement-to-force FRFs, the fitting results of the first five dominant modes
and the last four dominant modes are shown in Fig. 11 and Fig. 12 respectively. The
FRFs indicate that the robot shows multi-mode dynamics and the modes exhibit
strong non-symmetric cross coupling, which is consistent with the results in Refs.
[24,38]. By comparing the measured and fitted FRFs, the fitted curves agree well with
the measured curves. This shows that the method to fit the measured FRFs based on
the MDOF model is effective.
19
H1x_1x H1x_1y
H1y_1x H1y_1y
Fig. 11. The amplitude of the measured and fitted tool tip FRFs of the first five modes.
H1x_1x H1x_1y
H1y_1x H1y_1y
Fig. 12. The amplitude of the measured and fitted tool tip FRFs of the last four modes.
The number of the modal parameters gathered in one MTGP regression model,
i.e., the value of m, is determined based on the training time for the whole training
data. The STGP model and MTGP model with different values of m are trained 50
20
times on the same computer to calculate their average training time. All the programs
are run in Matlab 9.7 on the same personal PC with an Intel core i5-7300HQ CPU of
2.5 GHz and 8G RAM. The average training time for the STGP model is 0.46s. When
the value of m is taken from 1 to 5, the average training time for the MTGP model is
2.06s, 3.06s, 3.36s, 51.13s and 88.86s. It can be seen that when the value of m is not
greater than 3, the training time of MTGP is not much longer than that of STGP. When
the value of m is greater than 3, the training time of MTGP increases significantly.
Therefore, the value of m selected in the experiment is not greater than 3.
The modal parameters in one MTGP model are chosen based on their physical
similarity. For the same mode, the elements in the eigenvectors whose corresponding
coordinates are in the same direction (1X vs 1aX, 1Y vs 1aY) have a similar trend of
changing with the posture due to the similarity of the corresponding coordinates. This
can be illustrated with two representative modal parameters,
R
1 ,5,Y imag
and
R
1 ,5,aY imag
.
For two different rotational orientations about the tool axis (
12
-20 0
==，
), the
values of the two modal parameters at the training positions are shown in Fig. 13.
Because the corresponding coordinates are in the same direction, these two modal
parameters have a consistent trend of change with the pose. Therefore, in the same
eigenvector for the same mode, the modal parameters of the same type corresponding
to two different coordinates in the same direction can be grouped together in one
MTGP model. On the other hand, for the modal parameters that do not have the same
direction correspondence in the same mode, the modal parameters are combined with
the modal parameters of the same type in other modes. In order to ensure the
correlation between the modal parameters of different modes, the low and high
frequency modes are combined separately. Note that the number of modal parameters
in the same MTGP model, i.e., the value of m, is not greater than 3. Through the above
steps, the combination of the modal parameters for the right eigenvectors of the
identified nine modes is shown in Fig. 14 and the combination of the other modal
parameters is shown in Fig. 15. For a total of 144 modal parameters, 70 MTGP models
were built after the combination of modal parameters. Compared with the STGP
models, the number of required regression models has been reduced by 51.39%.
Fig. 13. The values of
R
1 ,5,Y imag
and
R
1 ,5,aY imag
in different postures.
21
R R R R R R R R
1 ,1, 1 ,1, 1 ,1, 1 ,1, 1 ,1, 1 ,1, 1 ,1, 1 ,1,
R R R R R R R R
1 ,2, 1 ,2, 1 ,2, 1 ,2, 1 ,2, 1 , 2, 1 ,2, 1 ,2,
R
1 ,3, 1 ,
X real aX real X imag aX imag Y real aY real Y imag aY imag
X real aX real X imag aX imag Y real aY real Y imag aY imag
X real aX
R R R R R R R
3, 1 ,3, 1 ,3, 1 ,3, 1 ,3, 1 ,3, 1 ,3,
R R R R R R R R
1 ,4, 1 ,4, 1 ,4, 1 ,4, 1 ,4, 1 , 4, 1 ,4, 1 ,4,
RR
1 ,5, 1 ,5, 1 ,5,
real X imag aX imag Y real aY real Y imag aY imag
X real aX real X imag aX imag Y real aY real Y imag aY imag
X real aX real X ima
R R R R R R
1 ,5, 1 ,5, 1 ,5, 1 ,5, 1 ,5,
R R R R R R R R
1 ,6, 1 ,6, 1 ,6, 1 ,6, 1 ,6, 1 ,6, 1 ,6, 1 , 6,
R R R R
1 ,7, 1 ,7, 1 ,7, 1 ,7, 1
g aX imag Y real aY real Y imag aY imag
X real aX real X imag aX imag Y real aY real Y imag aY imag
X real aX real X imag aX imag
R R R R
,7, 1 ,7, 1 ,7 , 1 ,7,
R R R R R R R R
1 ,8, 1 ,8, 1 ,8, 1 ,8, 1 ,8, 1 ,8, 1 ,8, 1 ,8,
R R R R R
1 ,9, 1 ,9, 1 ,9, 1 ,9, 1 ,9, 1 ,9,
Y real aY real Y imag aY imag
X real aX real X imag aX imag Y real aY real Y imag aY imag
X real aX real X imag aX imag Y real aY
R R R
1 ,9, 1 ,9,real Y imag aY imag
Fig. 14. The combination of the modal parameters in the right eigenvectors.
L L L L L L
,1 1 1 ,1, 1 ,1, 1 ,1, 1 ,1, 1 ,1, 1 ,1,
L L L L L L
,2 2 1 ,2, 1 ,2, 1 ,2, 1 ,2, 1 , 2, 1 ,2,
L L L L
,3 3 1 ,3, 1 ,3, 1 ,3, 1 ,3,
n Y real Y imag X real aX real X imag aX imag
n Y real Y imag X real aX real X imag aX imag
n Y real Y imag X real aX real
f
f
f
LL
1 ,3, 1 ,3,
L L L L L L
,4 4 1 ,4, 1 ,4, 1 ,4, 1 ,4, 1 , 4, 1 ,4,
L L L L L L
,5 5 1 ,5, 1 ,5, 1 ,5, 1 ,5, 1 ,5, 1 ,5,
LL
,6 6 1 ,6, 1 ,6, 1
X imag aX imag
n Y real Y imag X real aX real X imag aX imag
n Y real Y imag X real aX real X imag aX imag
n Y real Y imag
f
f
f
L L L L
,6, 1 ,6, 1 ,6, 1 , 6,
L L L L L L
,7 7 1 ,7, 1 ,7, 1 ,7, 1 ,7 , 1 , 7, 1 ,7,
L L L L L L
,8 8 1 ,8, 1 ,8, 1 ,8, 1 ,8, 1 ,8, 1 ,8,
,9 9 1
X real aX real X imag aX imag
n Y real Y imag X real aX real X imag aX imag
n Y real Y imag X real aX real X imag aX imag
n
f
f
f
L L L L L L
,9, 1 ,9, 1 ,9, 1 ,9, 1 ,9, 1 ,9,Y real Y imag X real aX real X imag aX imag
Fig. 15. The combination of the other modal parameters.
5-fold cross-validation [57] is performed to evaluate the accuracy and robustness
of the MTGP regression model for the prediction of the modal parameters [58]. The
STGP regression model, which was adopted in Ref. [21], is employed as the
benchmark method for comparison. For each modal parameter, the training data is
randomly partitioned into 5 equal subsets and the union set of 4 subsets is used for
model training while the remaining subset is used as the test dataset. Note that there
are m modal parameters combined within one MTGP model, as shown in Fig. 5(b).
When evaluating the MTGP models for the prediction of the modal parameter
j
, the
training data of other m-1 modal parameters in the same model is additionally used for
training. So the correlation between
j
and other m-1 modal parameters can be
mined in the training to improve the prediction performance.
The coefficient of determination R2 is used as the indicator of prediction
accuracy [24,58]. It is defined as:
2
21
2
1
ˆ
()
R =1-
()
D
D
N
ii
i
N
ii
i
=
=
−
−
, (31)
where
D
N
is the size of the dataset and
ˆi
is the predicted value of the regression
model.
i
is the actual value of the training data and
i
is the average value of
i
.
The accuracy of the regression model can be measured by the average value
2
R
of
R2 values for 5-fold cross-validation. The robustness of the regression model is
evaluated by the sample standard deviation
s
of R2 values for 5-fold
cross-validation, indicating the fluctuation and change of the R2 value of each model
established in 5-fold cross-validation. A larger average value
2
R
corresponds to
better accuracy and a smaller sample standard deviation
s
corresponds to better
22
robustness [59,60].
For two representative modal parameters (natural frequency
n
f
and damping
ratio
), the prediction accuracy and robustness are shown in Fig. 16 and Fig. 17,
respectively. For the modal parameters in two representative modes (mode 2 and
mode 9), the prediction accuracy and robustness are presented in Fig. 18 and Fig. 19,
respectively. The STGP regression model works poorly for some modal parameters,
which may be because the amount of experimental data is not enough for model
training. Similar to the idea of improving the prediction performance with additional
information in Ref. [23], the MTGP regression model introduces additional physical
knowledge into training through considering the correlation between the predicted
modal parameter and other parameters in the same MTGP model. Therefore, the
MTGP regression models have higher average values and lower sample standard
deviations than the STGP regression models in all the prediction cases. The results
demonstrate that the MTGP regression model outperforms the STGP regression
model with higher prediction accuracy and better robustness. The residual plots of
the MTGP regression models for the modal parameters in two representative modes
(mode 2 and mode 9) are shown in Fig. 20 and Fig. 21. The residuals are uncorrelated,
indicating that the MTGP models capture the main trends in the data well.
Fig. 16. The prediction accuracy for two representative modal parameters.
Fig. 17. The prediction robustness for two representative modal parameters.
23
Fig. 18. The prediction accuracy for the modal parameters in two representative modes.
Fig. 19. The prediction robustness for the modal parameters in two representative modes.
R6
1 ,2, ( s/ kg 10 )
X real
−
1 ,2, ( s/ kg )
L
aX real
R6
1 ,2, ( s/ kg 10 )
Y real
−
( )
,2 Hz
n
f
( )
2%
1 ,2, ( s/ kg )
L
X imag
1 ,2, ( s/ kg )
L
aX imag
1 ,2, ( s/ kg )
L
Y imag
R6
1 ,2, ( s/ kg 10 )
X imag
−
R6
1 ,2, ( s/ kg 10 )
aX imag
−
R6
1 ,2, ( s/ kg 10 )
Y imag
−
1 ,2, ( s/ kg )
L
X real
1 ,2, ( s/ kg )
L
Y real
R6
1 ,2, ( s/ kg 10 )
aX real
−
R6
1 ,2, ( s/ kg 10 )
aY real
−
R6
1 ,2, ( s/ kg 10 )
aY imag
−
20.96R=
20.93R=
20.91R=
20.93R=
20.90R=
20.93R=
20.85R=
20.94R=
20.81R=
20.99R=
20.82R=
20.99R=
20.99R=
20.97R=
20.99R=
20.97R=
Fig. 20. Cross validation residual plots for the modal parameters in mode 2.
24
( )
,9 Hz
n
f
1 ,9, ( s/ kg )
L
aX real
R5
1 ,9, ( s/ kg 10 )
X real
−
R5
1 ,9, ( s/ kg 10 )
Y real
−
( )
9%
1 ,9, ( s/ kg)
L
aX imag
R5
1 ,9, ( s/ kg 10 )
X imag
−
R5
1 ,9, ( s/ kg 10 )
Y imag
−
1 ,9, ( s/ kg )
L
X real
1 ,9, ( s/ kg )
L
Y real
R5
1 ,9, ( s/ kg 10 )
aX real
−
R5
1 ,9, ( s/ kg 10 )
aY real
−
1 ,9, ( s/ kg)
L
X imag
1 ,9, ( s/ kg)
L
Y imag
R5
1 ,9, ( s/ kg 10 )
aX imag
−
R5
1 ,9, ( s/ kg 10 )
aY imag
−
20.99R=
20.84R=
20.98R=
20.99R=
20.99R=
20.99R=
20.98R=
20.99R=
20.99R=
20.99R=
20.98R=
20.99R=
20.99R=
20.99R=
20.99R=
20.99R=
Fig. 21. Cross validation residual plots for the modal parameters in mode 9.
To further validate the trained MTGP models in the workspace, three different
robot postures not considered during model training are tested. In the three tested
postures, posture 2 (
1800 mm,200 mm,0
) is different from posture 1
(
1800 mm, 200 mm,0−
) on the tool tip position in the X-Y plane and posture 3
(
1800 mm,200 mm,60
) is different from posture 2 (
1800 mm,200 mm,0
) on
the redundant axis angle. As described in Section 4.3, the tool tip position in the X-Y
plane and the redundant axis angle determine the robot posture. So the difference
between the posture can be sufficiently reflected by the tested three postures. The
predicted results of two representative modal parameters (natural frequency
n
f
and
damping ratio
) for the first five dominant modes and the last four dominant
modes are listed in Table 2 and Table 3 respectively. The prediction accuracy is
evaluated with the relative error (RE):
ˆ
=RE
−
, (32)
where
and
ˆ
are the actual and predicted values respectively. It can be seen that
prediction errors of the natural frequency are less than the damping ratio and they are
all under 10%. Therefore, it indicates that the trained MTGP models can effectively
predict the posture-dependent modal parameters.
Taking the tool tip FRFs as the representative of the 16 displacement-to-force
FRFs, the measured FRFs and the FRFs fitted from the predicted modal parameters
are compared for the tested three postures. The tool tip FRFs of the first five
dominant modes and the last four dominant modes are shown in Fig. 22 and Fig. 23
respectively. As shown in the results, both the low-frequency and high-frequency
modes are posture-dependent, which is consistent with the results in Refs. [13,36]. It
is seen that the prediction errors are acceptable and the agreement between the
predicted and measured results is good, which indicates that the MTGP models
25
achieve effective predictions of the posture-dependent tool tip dynamics.
Table 2
Validation of the prediction model for the first five dominant modes.
Postures
Type
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Natural frequency
n
f
(Hz), Damping ratio
(%), RE (%)
,1n
f
1
,2n
f
2
,3n
f
3
,4n
f
4
,5n
f
5
Posture 1
Measured
8.19
13.90
10.51
1.17
18.03
4.08
17.62
1.45
20.63
7.36
Predicted
8.37
13.68
10.65
1.15
17.89
4.04
18.01
1.35
20.40
7.21
RE
2.17
1.59
1.37
2.07
0.80
0.92
2.19
7.31
1.12
2.05
Posture 2
Measured
9.68
14.07
10.74
1.60
18.15
4.14
18.42
1.52
20.53
6.09
Predicted
9.50
14.05
10.87
1.64
17.88
4.11
18.05
1.57
20.62
5.99
RE
1.85
0.12
1.13
2.96
1.47
0.70
1.98
3.37
0.41
1.52
Posture 3
Measured
9.29
16.63
10.52
0.88
18.95
5.53
19.31
1.94
21.18
5.13
Predicted
9.36
16.80
10.57
0.84
18.83
5.38
19.56
2.09
21.11
5.10
RE
0.67
1.02
0.48
5.35
0.63
2.75
1.28
7.87
0.34
0.66
Table 3
Validation of the prediction model for the last four dominant modes.
Postures
Type
Mode 6
Mode 7
Mode 8
Mode 9
Natural frequency
n
f
(Hz), Damping ratio
(%), RE (%)
,6n
f
6
,7n
f
7
,8n
f
8
,9n
f
9
Posture 1
Measured
824.10
6.62
851.34
6.91
877.75
2.90
888.82
1.50
Predicted
830.47
6.45
849.32
7.04
876.17
2.95
889.00
1.61
RE
0.77
2.48
0.24
1.89
0.18
1.56
0.02
7.35
Posture 2
Measured
869.35
12.84
872.98
2.79
876.87
4.69
895.50
1.36
Predicted
865.35
12.19
870.08
2.84
879.46
4.55
896.22
1.33
RE
0.46
5.10
0.33
1.76
0.30
3.02
0.08
2.13
Posture 3
Measured
844.90
9.22
873.90
4.97
880.67
1.60
901.64
1.37
Predicted
839.87
9.39
876.70
5.11
878.44
1.64
900.56
1.38
RE
0.59
1.85
0.32
2.93
0.25
2.79
0.12
0.48
26
H1x_1x H1x_1y
H1y_1y
H1y_1x
Fig. 22. The measured and predicted reference tool tip FRFs of the first five modes.
H1x_1x H1x_1y
H1y_1y
H1y_1x
Fig. 23. The measured and predicted reference tool tip FRFs of the last four modes.
5.3. Prediction of the posture-dependent target tool tip dynamics
In this section, the relationship of tool tip dynamics between the reference and
27
target tool is established in the experiment and combined with the posture-dependent
reference tool tip dynamics predicted in Section 5.2 to obtain the posture-dependent
target tool tip dynamics containing the cross coupling FRFs.
According to Eq. (4), the relationship of tool tip dynamics between the reference
and target tool depends on the difference in interface contact dynamics
1 1 1
21
=K K K
− − −
−
.
1
K−
is identified with the experimentally obtained FRFs
in high-frequency band for the reference and target tools at posture 1
(
1800 mm, 200 mm,0−
). In order to simplify the identification process, the
contact parameters of the reference tool T1 can be set to given fixed values, so the
difference in interface contact dynamics only depends on the contact parameters of
the target tool T2. The contact parameters of the reference tool T1 are set to the values
identified in Ref. [35]. The translational and rotational stiffness constants for the
reference tool are set as
7
6.25 10 N/m
and
6
3.86 10 Nm/rad
respectively. The
translational and rotational damping constants are set as
175Ns/m
and
0.14Nms/rad
respectively. Then the contact parameters of the target tool are
obtained by manually tuning the calculated target tool tip FRFs with respect to the
experimentally obtained tool point FRFs, as in Ref. [41]. The identified translational
and rotational stiffness constants for the target tool are
7
4.5 10 N/m
and
6
4.19 10 Nm/rad
respectively. The identified translational and rotational damping
constants are
210Ns/m
and
1.5Nms/rad
respectively. The measured and calculated
target tool tip FRFs in high-frequency band at posture 1 are shown in Fig. 24. It can be
seen that the measured and calculated FRFs match well with the identified contact
parameters.
H1x_1x H1x_1y
H1y_1y
H1y_1x
Fig. 24. The measured and calculated target tool tip FRFs in high-frequency band at posture 1.
The established relationship of tool tip dynamics is combined with the
posture-dependent reference tool tip dynamics predicted in Section 5.2 to obtain the
28
posture-dependent target tool tip dynamics containing the cross coupling FRFs. To
validate the prediction of the posture-dependent target tool tip dynamics, the target
tool tip FRFs are predicted at posture 2 (
1800 mm,200 mm,0
) and posture 3
(
1800 mm,200 mm,60
). Posture 2 reflects the change of the tool tip position in
the X-Y plane relative to posture 1, and posture 3 reflects the change of the
redundant axis angle relative to posture 2, which together reflect the change of
posture. The predicted and measured target tool tip FRFs in the low-frequency band
and high-frequency band at posture 2 are shown in Fig. 25 and Fig. 26 respectively.
And the predicted and measured target tool tip FRFs in the low-frequency band and
high-frequency band at posture 3 are shown in Fig. 27 and Fig. 28 respectively. It can
be seen that the FRFs in the high-frequency band match well, which indicates that
the identified contact parameters are correct and the prediction approach for the
target tool is effective. For the FRFs in the low-frequency band, tool changes have a
greater effect on cross coupling FRFs than direct FRFs, possibly due to tool
asymmetry introduced after the tool change. Though there are some differences
between the predicted and measured cross coupling FRFs, the predicted FRFs are able
to capture the trend and the error is acceptable, which indicates that the prediction
approach is effective to deal with the change of posture and tool.
H1x_1x H1x_1y
H1y_1y
H1y_1x
Fig. 25. The predicted and measured target tool tip FRFs in low-frequency band at posture 2.
29
H1x_1x H1x_1y
H1y_1y
H1y_1x
Fig. 26. The predicted and measured target tool tip FRFs in high-frequency band at posture 2.
H1x_1x H1x_1y
H1y_1y
H1y_1x
Fig. 27. The predicted and measured target tool tip FRFs in low-frequency band at posture 3.
30
H1x_1x H1x_1y
H1y_1y
H1y_1x
Fig. 28. The predicted and measured target tool tip FRFs in high-frequency band at posture 3.
5.4. The stability validation for robotic milling
With the target tool tip FRFs with cross coupling predicted in Section 5.3, the
dynamics of milling can be modeled by delayed differential equations (DDEs), as
described in Ref. [38]. Then the Stability Lobe Diagram (SLD) can be calculated
based on the full-discretization method (FDM) proposed by Ding et al [8]. The
stability is calculated for the low-frequency modes and high-frequency modes
separately using the FDM and the ultimate stability lobe is determined by taking the
lowest envelop [61]. Fig. 29 shows the calculated stability for the milling robot at
posture 2 (
1800 mm,200 mm,0
) with slot milling. A series of milling experiments
were performed to verify the predicted stability lobe diagram. The spectrums of the
vibrations recorded by the accelerometer were used to identify the stability during the
experiment. It can be seen that the predictions are in good agreement with the
experimental results. According to Fig. 29, the stability limits at low speed are
dominated by the low-frequency modes, which is consistent with the results in Refs.
[24,38]. At high speed milling, the stability limits are determined by both the
low-frequency modes and the high-frequency modes, which is different from the
results in Refs. [24,38]. This may be due to the fact that the low-frequency modes are
not fully damped out at high speed due to the change in milling conditions. As a result,
both the low-frequency modes and the high-frequency modes should be considered in
stability prediction for robotic milling. This indicates the necessity to pay attention to
both low-frequency modes and high-frequency modes in tool tip FRFs prediction.
31
2 mm
p
a=
⚫7200 rpm ⚫10800 rpm
chatter
(916 Hz) chatter
(935 Hz)
3 mm
p
a=
1 mm
p
a=
2 mm
p
a=
Fig. 29. The predicted stability diagrams at the selected posture.
6. Conclusion
In this paper, an effective approach is proposed to predict the posture-dependent
tool tip dynamics for different tools in robotic milling considering multi-mode and the
cross coupling FRFs. The approach combines the RCSA method with the MTGP
regression model. A more comprehensive RCSA procedure is adopted to consider the
non-negligible cross coupling FRFs and the impact test is designed to obtain the
measured FRF matrix. By fitting the measured FRF matrix with the MDOF model, the
number of modal parameters to be identified is significantly reduced. The MTGP
regression model is introduced to mine the intrinsic correlations between different
modal parameters. Compared to the ordinary Gaussian Process regression model, the
number of required regression models in MTGP is reduced and the prediction
performance is improved in terms of accuracy and robustness.
The proposed approach is validated through experiments on an industrial ABB
IRB6660 robot. The experiments show that both the low-frequency and
high-frequency modes are posture-dependent. Guided by the training time and the
correlation of modal parameters, the modal parameters are combined into groups to
build the MTGP regression models. The number of required MTGP regression
32
models has been reduced by 51.39% compared with the STGP regression models. The
MTGP regression models have higher accuracy and better robustness than the STGP
regression models in all the prediction cases by considering the correlation between
different modal parameters. The experiments for the target tool demonstrate that the
proposed prediction approach is effective to deal with the change of posture and tool.
Furthermore, the stability calculated with the predicted target tool tip FRFs is
validated through milling experiments.
The present work considers the main posture properties in the workspace, i.e.,
the tool tip position in the X-Y plane and the redundant axis angle. It is worth noting
that the number of required experiments can be further reduced and more posture
properties can be included by dimensionality reduction and non-uniform sampling,
which has been demonstrated in prior work [24]. The extension of the proposed
approach to higher dimensional workspaces for more complex machining conditions
is of interest for future works.
7. Acknowledgements
This work was supported by the National Natural Science Foundation of China
[grant numbers: 51935010, 51822506].
Appendix A
With the first-order finite-difference technology and the reciprocity of FRF, the
measured displacement-to-force FRFs in Eq. (5) can be used to estimate the
rotation-to-force receptances (N) and the displacement-to-moment receptances (L)
[46]:
1 _1 1 _1
1 _1 1 _1
==
x x ax x
x x x x
HH
NL
S
−
(A.1)
1 _1 1 _1
1 _1 1 _1
==
x ax ax ax
x ax ax x
HH
NL
S
−
(A.2)
1 _1 1 _1
1 _1 1 _1
==
y y ay y
y y y y
HH
NL
S
−
(A.3)
1 _1 1 _1
1 _1 1 _1
==
y ay ay ay
y ay ay y
HH
NL
S
−
(A.4)
1 _1 1 _1
1 _1 1 _1
==
x y ax y
x y y x
HH
NL
S
−
(A.5)
1 _1 1 _1
1 _1 1 _1
==
x ay ax ay
x ay ay x
HH
NL
S
−
(A.6)
1 _1 1 _1
1 _1 1 _1
==
y x ay x
y x x y
HH
NL
S
−
(A.7)
1 _1 1 _1
1 _1 1 _1
==
y ax ay ax
y ax ax y
HH
NL
S
−
(A.8)
Applying the first-order finite-difference technology again, the
rotation-to-moment receptances (P) can be derived from the displacement-to-moment
receptances (L) as follows [46]:
1 _1 1 _1
1
1 _1
1
==
x x ax x
x
xx
x
LL
PMS
−
(A.9)
33
1 1 _1 1 _1
1 _1
1
=y y y ay y
yy
y
LL
PMS
−
=
(A.10)
1 _1 1 _1
1
1 _1
1
=x y ax y
x
xy
y
LL
PMS
−
=
(A.11)
1 1 _1 1 _1
1 _1
1
=y y x ay x
yx
x
LL
PMS
−
=
(A.12)
Appendix B
Table B1
Identified modal parameters for the measured FRFs at the representative posture.
Mode
( )
Hz
n
f
( )
%
( s/ kg)
L
7
( s/ kg 10 )
R−
1
10.44
1.16
1.55-1.12i
0.54 + 0.70i
0.20 - 1.80i
1
19.04 - 8.83i
-2.26+ 5.44i
17.90 - 8.68i
-2.28 + 5.42i
2
13.11
7.98
0.13 + 0.04i
1.00+ 0.03i
0.16 - 0.001i
1
28.60 - 37.61i
-80.28 - 286.20i
19.77 - 52.51i
-83.08 - 289.94i
3
15.24
6.05
-7.73 + 11.96i
0.84 - 0.58i
-7.63 + 12.29i
1
-22.17 + 4.47i
1.56 + 2.70i
-21.01 + 5.14i
1.28+ 2.96i
4
15.96
2.08
-2.89 + 1.52i
0.66 + 0.05i
-2.73 + 1.39i
1
12.89 + 69.23i
-0.25- 14.60i
12.60 + 65.55i
-0.56 - 13.83i
5
21.17
5.04
0.08 + 0.02i
1.01 + 0.02i
0.13 + 0.03i
1
-4.27 - 27.81i
-63.86 - 326.41i
-8.77- 37.14i
-62.74 - 327.64i
6
844.46
9.92
0.15+0.48i
0.87 + 0.29i
0.11 + 0.34i
1
241.37 + 10.54i
578.91-1642.99i
299.70-240.88i
475.05-1405.91i
7
863.46
5.89
-1.72+2.31i
0.41+0.77i
-1.27+1.65i
1
-551.97+131.80i
215.69+ 277.06i
-458.39 +171.29i
131.43 + 245.93i
8
884.06
1.42
-0.06+ 2.61i
0.45 + 1.37i
-1.02 + 2.40i
1
-69.69 + 101.74i
24.62 + 46.83i
-59.94 + 96.89i
14.62 + 49.69i
9
895.58
2.30
-0.72+0.10i
1.19+0.69i
-0.69+ 0.03i
1
285.42 + 80.93i
-407.64 - 244.84i
194.78 + 34.62i
-319.32 - 205.79i
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