*Corr. Author’s Address: Anhui Polytechnic University, Wuhu, China, ahpulyy@163.com 529
Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 529-541 Received for review: 2022-01-17
© 2022 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2022-07-07
DOI:10.5545/sv-jme.2022.15 Original Scientific Paper Accepted for publication: 2022-08-26
MTPA- and MSM-based Vibration Transfer  
of 6-DOF Manipulator for Anchor Drilling
Liu, Y . – Ma, L. – Yang, S. – Liang Yuan, L. – Bo Chen
Youyu Liu
1,3
 – Liteng Ma
1,3
 – Siyang Yang
1,3
 – Liang Yuan
2
 – Bo Chen
1,3
1 
Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, China 
2 
Technology Department, Wuhu Yongyu Automobile Industry Co., China 
3 
School of Mechanical Engineering, Anhui Polytechnic University, China
An anchor drilling for a coal mine support system can liberate an operator from heavy work, but will cause serious vibration, which will 
be transmitted to the pedestal from the roof bolter along a manipulator. Based on the multi-level transfer path analysis (MTPA) and modal 
superposition method (MSM), a vibration transfer model for the subsystem composed of the joints of a manipulator with six degrees of 
freedom (DOF) was established. Moreover, its frequency response function matrix was also built. The 6-DOF excitation of the roof bolter was 
deduced. The exciting force on the roof bolter transmitted to the pedestal along the 6-DOF manipulator was analysed with a force Jacobian 
matrix, to identify the external loading on the pedestal. A case in engineering practice shows that the amplitude of each DOF of the pedestal 
from large to small is as follows: bending vibration (component 1), longitudinal vibration, torsional vibration, bending vibration (component 
2), rotational vibration around z-axis, rotational vibration around y-axis. The pedestal is mainly in the form of bending vibration. The theory 
of vibration transfer along the 6-DOF manipulator for anchor drilling proposed in this article can provide a theoretical foundation for the 
development of vibration-damping techniques and the design of absorbers.
Keywords: manipulator, multi-level transfer path analysis, modal superposition method, vibration transfer, force Jacobian matrix
Highlights
• 	 Based on MTPA and MSM, a mathematical model of vibration transfer of a 6-DOF manipulator for anchor drilling is established.
• 	 The external loading of the response point of the manipulator pedestal is analysed by using the force Jacobian matrix. 
• 	 The vibration responses on each DOF of the pedestal and the resonance frequency are obtained.
• 	 The case studied in an engineering practice shows that the pedestal is mainly in the form of bending vibration.
0  INTRODUCTION
Roof bolters are key mechanical equipment for a coal 
mine supporting system. In the past, drilling was done 
manually. Working in an area with high concentrations 
of dust for a long time, workers’ physical and mental 
health will be seriously threatened. At present, the 
development of a coal mine tunnel support tends to be 
automatic and intelligent. The manual labour of roof 
bolters has been gradually replaced by mechanical 
clamping [1]. An operator can control the roof bolter 
to drill automatically by human-computer interaction, 
which can liberate the operator from heavy work, 
and improve the stability and safety of the coal 
mine support. Due to the comprehensive excitation 
of different geotechnical parameters, axial thrust, 
torque and other factors, there are complex vibrations 
on drill strings during construction. The main forms 
include bending vibration, longitudinal vibration, 
and torsional vibration, which interact with each 
other to form a nonlinear coupled vibration [2]. The 
excitation vibration of each degree of freedom (DOF) 
of the roof bolter is transmitted to the pedestal along a 
manipulator, which causes the manipulator to vibrate 
violently, shorten its service life, and then affect the 
support effect.
Transfer path analysis (TPA) is a tool to study 
vibration transfer [3] and [4]. There are several methods, 
such as operational transfer path analysis (OTPA) [5] 
and [6], global transmissibility direct transmissibility 
(GTDT) [7] and [8], inverse substructure TPA (ITPA) 
[9], multi-level transfer path analysis (MTPA) 
[10] and [11], and so on. Lee and Lee [5] proposed 
the OTPA method using an emerging deep neural 
network model, which can successfully predict the 
path contributions using only operational responses. 
Yoshida and Tanaka [6] attempted to calculate the 
vibration mode contribution by modifying OTPA, and 
then considered the relationship between the principal 
component and the vibration mode, as well as the 
associated the principal components with the vibration 
modes of a test structure. High contributing vibration 
modes to the response point have been found. It is 
easily disturbed by factors such as excitation coupling 
and noise employing this method when calculating 
the transfer matrix. Wang [7] developed further the 
prediction capabilities of the GTDT method, which 
can predict a new response using measured variables 
of an original system, even though operational 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 529-541
530 Liu, Y. – Ma, L. – Yang, S. – Liang Yuan, L. – Bo Chen
forces are unknown. Guasch [8] addressed some 
issues concerning the prediction capabilities of the 
GTDT method when blocking transfer paths in a 
mechanical system and outlined differences with the 
more standard force TPA. Wang [9] developed the 
SDD method further by considering the mass effect of 
resilient links, which can identify decoupled transfer 
functions accurately, whilst eliminating the mass 
effect of resilient links. However, its manoeuvrability 
is poor for a serial system with many substructures. 
Gao [11] used MTPA to find the critical paths of seat 
jitter caused by dynamic unbalance excitation of the 
drive shaft. The key technology of this method is to 
identify the external excitation loading, which has 
good operability for series system.
For the vibration problem of pedestal from 
manipulator caused by the excitation of a roof bolter, 
a response amplitude matrix in pedestal is established 
by the modal superposition method (MSM) in 
this article. According to the excitation of the roof 
bolter, the external loading of the response point of 
the manipulator pedestal is analysed using the force 
Jacobian matrix. The 6-DOF frequency response 
function of each subsystem of the manipulator is 
derived by MTPA, and then the frequency response 
matrix is constructed, which can solve the problem 
of Transfer path analysis with low accuracy and poor 
operability. It will provide a theoretical foundation for 
the development of vibration damping techniques and 
the design of absorbers.
1  MULTI-LEVEL TRANSFER PATH ANALYSIS
To reduce the influence of non-important factors 
while analysing the vibration transfer of manipulator 
for anchor drilling, some simplifications are made 
as follows. 1) Each linkage of the manipulator is 
equivalent to a bar with uniform mass; 2) some transfer 
mechanisms, such as belt driving and harmonic 
decelerator in the manipulator, are equivalent to linear 
massless springs; 3) the modal parameters of the 
manipulator are linear, namely, the output caused by 
any combined input are equal to the combination of 
respective outputs; 4) it satisfies the assumption of 
time-invariance [12], namely, the dynamic properties 
of the system are not vary with time. According to the 
above simplifications, a vibration transfer model of 
the manipulator for anchor drilling is established, as 
shown in Fig. 1.
This is a multi-input and multi-output system, in 
which the six joints of the manipulator are connected 
in series. The vibration source is the roof bolter that 
provides excitation; the vibration receiver is the 
pedestal. The manipulator can be divided into six 
subsystems by the rotating joint. The external loading 
of the J
6
-subsystem is the excitation from the roof 
bolter; that of other subsystems is the output of the 
previous subsystem. The output (response point) 
of the subsystem is the input (exciting point) of the 
next subsystem. Nevertheless, the output of the J
1
-
subsystem acts on the pedestal. According to different 
excitations of spatial degrees of freedom, each 
subsystem has m inputs and n outputs.
According to MTPA, the transfer function of 
manipulator for anchor drilling can be expressed by 
the product of the transfer functions of all subsystems 
[11].
 HH HH  
JJ J
12 6
. (1)
The vibration response of the excitation from the 
roof bolter transmitted to the pedestal is expressed as 
follows.
 S=HF. (2)
2  MODAL SUPERPOSITION METHOD
2.1  Response Amplitude Matrix
The dynamic equation of the manipulator for anchor 
drilling is as follows.
Fig. 1.  Vibration Transfer model of the manipulator

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 529-541
531 MTPA- and MSM-based Vibration Transfer of 6-DOF Manipulator for Anchor Drilling 
 MS+CS+KS=F
 
. (3)
According to the linear hypothesis [12], the 
displacement column vectors of each subsystem of the 
manipulator can be expressed as the linear addition of 
each order of modal shapes.
 S


q
aa
a
n
 
1
. (4)
Substituting Eq. (4) into Eq. (3), then,
 MCK   qqqF
a
a
n
aa
a
n
aa
a
n
a



11 1
    . (5)
Both ends of Eq. (5) are multiplied by λ λ
b
T
, then,
 
    
   
b
a
n
aa b
a
n
aa
b
a
n
aa b
qq
q
TT
TT
MC
KF




 

 

 

11
1
 
. � (6)
According to the orthogonality of the main modal 
shape [13], Eq. (7) can be obtained from Eqs. (5) and 
(6).
 
  
  
  
ba
ba
ba
a
a
a
Mab
Cab
Kab
ab
T
T
T
M
C
K



















,
,
,
,
.
0
 (7)
Substituting Eq. (7) into Eq. (6), then,
 Mq Cq Kq
aa aa aa ab
     
TT
FF . (8)
The excitation loading and displacement response 
in Eq. (8) are expressed in complex form as follows.
 
Ff 






e
qQ e
jt
aa
jt


. (9)
Substituting Eq. (9) into Eq. (8),
 Q
Mj CK
a
a
aa a


 
T
f

2
. (10)
Substituting Eqs. (10) and (9) into Eq. (4), S
T
 can 
be obtained as Eq. (11).
 S
f
T
T




a
n
a
jt
a
aa a
e
Mj CK
1 2
  


. (11)
It is assumed that the system has two points: o 
and p, substitute Eqs. (11) and (9) into Eq. (4), the 
response amplitude of the point p can be expressed as 
follows.
S
F
Kj
MK M
K
M
K
p
a
n
oa op a
aa
aa a
a
a
a




















1
2
21



 
 /
. (12)
Their frequency response function (FRF) is as Eq. 
(13).
H
S
F
Kj
MK M
K
M
K
po
p
o
a
n
oa pa
aa
aa a
a
a
a

















1
2
21


 
/
 




. (13)
According to the linear superposition assumption 
[12], when F   FF F
N 12

T
, the response 
amplitude of each point of the system is as Eq. (14).
 S 















S
S
S
HH H
HH H
HH H
N
N
N
NN NN
1
2
11 12 1
21 22 2
12



 

 





















F
F
F
N
1
2

. (14)
2.2  Parameter Identification
Since the excitation of the roof bolter includes all DOF 
in space, and each subsystem has m(6) inputs and n(6) 
outputs, the frequency response of each subsystem of 
the manipulator is a 6×6
th
 order matrix, as follows.
 H
J
JJ J
JJ J
J
i
HH H
HH H
H
ii i
ii i
i

  
  

11 12 16
21 22 26
61


 
   














HH
ii 62 66
JJ 
. (15)
The parameters of the J
1
 to J
6
 frequency response 
curves are identified in the frequency domain [14] by 
the rational polynomial method [15]. Its mathematical 
model is a rational formula of frequency response 
function, as follows.
H
x xxxx
x xxxx
i mn
J  
 
 


1
5
2
4
3
3
4
2
56
1
5
2
4
3
3
4
2
5
 
6
. (16)

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 529-541
532 Liu, Y. – Ma, L. – Yang, S. – Liang Yuan, L. – Bo Chen
3  EXTERNAL EXCITATION LOADING
3.1  Excitation from Roof Bolter
Force and moment on roof bolter: F
g
, F
a
, F
z
, F
c
, and 
M
d
. The direction of F
g
, F
a
 and F
z
 is along the shaft 
of the roof bolter, and their vector expressions is as 
follows.
 
F
F
F
gg
zz
aa
F
F
F
 












00000
00000
00000
. (17)
The direction of M
d
 is along the shaft of the roof 
bolter, and its vector expression is as follows.
 M
dd
M   000 00. (18)
While the lateral displacement of the roof bolter 
is greater than the distance of them, the roof bolter 
will collide with rock-soil. The collision force in the z 
and y axes is as following [16], respectively.
 
F
kvtv tv t
F
kwtw tw t
cz
cy

   






   


sgn
sgn
0e lse
 















0e lse
. (19)
Eq. (19) is expressed in matrix form as follows.
 F
cc zc y
cz
cy
FF
F
F
D
D

 





00
22
. (20)
The excitation from the roof bolter at the tip of 
the manipulator is as Eq. (21).
F

 
 





FFFFFM
F
F
D
D
agzc zc yd
cz
cy
22
.(21)
3.2  Excitation to Pedestal
Force and torque on pedestal: F
g2
, τ
1
. Each joint 
of the manipulator can rotate independently. To 
accurately describe the mechanical properties of the 
excitation transmitted to the pedestal through the 
joints of the manipulator, a force Jacobian matrix 
of the manipulator is introduced [17]. The transfer 
relationship between the excitation and the joint 
generalized driving force is as follows [18]. 
    JF q
T
 (22)
The torques of each joint of the manipulator is as 
follows (Eq. (23)).
 





















   





1
2
3
4
5
6
12 4 456 46
JJ dc cc ss
xx
a as c
JJ dc cc ss as c
JJ dcs
yy
zz
35 6
12 4 456 46 35 6
12 445
00 0
00 0

   
a ac
sc cc ss cs cs cc cs sccc ss
36
23 456 46 23 56 456 46 456 46
00 0
    
5 56 6
23 456 46 23 56 4564 6 456 46 5
0 cs
sc cc ss cs cs cs cc sccc cs s

   
6 66
23 45 23 54 54 55
0
01






















c
scsc cs ss sc
F
F
F
M
x
y
z
x x
y
z
M
M




















, (23)
where Jd cc cc ss ss ca ca cd ss c
x 12 23 456 46 23 56 22 32 34 23 45
    



  c cc s
64 6
  ; c
ii
 cos( )  ; s
ii
 sin( )  ;
Jd cc cc ss ss ca ca cd ss c
y 12 23 456 46 23 56 22 32 34 23 4
    



 
5 56 46
cc s   ; s
23 23
 sin( )  ; 
c
23 23
 cos( )  ; Jd cc ss ca ca cd ss s
z 12 23 45 23 52 23 23 42 34 5
     ; 
Ja sc dc cc ss ascccs sc sc
x 23 56 4 456 46 23456 46 356
      



; 
Ja sc dc cc ss as cccs sc sc
y 23 56 4 456 46 23 456 46 356
        



; Ja cd cs as cs cc
z 23 64 45 23 45 36
   
;
FF
x
 sinc os  ; FF
y
 sinc os  ; FF
z
 cos .

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 529-541
533 MTPA- and MSM-based Vibration Transfer of 6-DOF Manipulator for Anchor Drilling 
4  CASE IN ENGINEERING PRACTICE
4.1 Essential Parameters
A 6-DOF manipulator for anchor drilling in a coal 
mine in Huainan, China, is taken as the research 
object. The size of the two-wing drill adopted is ϕ32 
mm; the length of drill string is 86 mm. The drilling 
object is sandstone, and its mechanical parameters 
[19] are as follows: ρ = 2600 kg/m
3
; R = 38 MPa;  
R
m
 = 0.34 MPa; E = 12 GPa; μ = 0.25; F
a
 = 6000 N; 
M
d
 =130 N; Ω = 0 mm; k = 10
9
 N·m
–1
; self-weight of 
the roof bolter is 40 kg; self-weight of the manipulator 
is 550 kg. Moreover, the parameters of the linkages of 
the manipulator are shown in Table 1 [20]. 
The three-dimensional model of the manipulator 
is shown in Fig. 2. Some finite element models of 
the manipulator are established, as shown in Fig. 3. 
The rotating joints of the manipulator are divided into 
some subsystems, and its exciting points and response 
points are determined.
Based on MTPA and MSM, the computation flow 
chart of the vibration transfer of 6-DOF manipulator 
for anchor drilling is shown Fig. 4.
Table 1.  Parameters of the linkages of the manipulator
Linkages Angle variable a
i–1
 [m] d
i
 [m] Angle range [rad]
1 0 0.56 –3.14 to 3.14
2 0.90 0 –2.27 to 1.22
3 0.16 0 –1.40 to 3.05
4 0 1.01 –6.28 to 6.28
5 0 0 –2.09 to 2.09
6 0 0.2 –6.28 to 6.28
Fig. 2.  The three-dimensional model of the manipulator
a)        b)        c) 
d)     e)     f) 
Fig. 3.  Grid division of subsystems and exciting points, response points; a) J
1
, b) J
2
, c) J
3
, d) J
4
, e) J
5
, f) J
6
4.2  Frequency Response Curves
The excitation of the roof bolter is high-frequency 
vibration; the frequency range in practice is 0 Hz 
to 200 Hz [21]. Substituting Eqs. (19) and (21) into 
Eq. (13), the frequency responses of subsystems are 
analysed using ABAQUS [22]. The acceleration 
frequency response curves of J
1
 to J
6
 are shown in 
Figs. 5 to 10.
According to Figs. 5 to 10, there are resonance 
peaks [23] in the frequency response curves of each 
subsystem of the manipulator for anchor drilling in the 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 529-541
534 Liu, Y. – Ma, L. – Yang, S. – Liang Yuan, L. – Bo Chen
Fig. 4.  Computation flow chart
a)             b) 
Fig. 5.  Frequency response curves of J
1
 ; a) translational acceleration, and b) rotational acceleration
a)            b)
Fig. 6.  Frequency response curves of J
2
; a) translational acceleration, and b) rotational acceleration
a)             b) 
Fig. 7.  Frequency responses curve of J
3
; a) translational acceleration, and b) rotational acceleration

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 529-541
535 MTPA- and MSM-based Vibration Transfer of 6-DOF Manipulator for Anchor Drilling 
frequency range of 0 Hz to 200 Hz. The frequencies 
corresponding to the peak values of the 6-DOF 
frequency response curves is shown in Table 2.
4.3  FRF Matrixes of Subsystems
The frequency response curves of each subsystem are 
imported into MATLAB, which are fitted according to 
Eq. (16) with the “Curve Fitting Tool” toolbox [24]. 
The coefficients of each element in the frequency 
response matrixes are shown in Tables 3 to 8.
a)             b) 
Fig. 8.  Frequency response curves of J
4
; a) translational acceleration, and b) rotational acceleration
a)             b) 
Fig. 9.  Frequency response curves of J
5
; a) translational acceleration, and b) rotational acceleration
a)             b) 
Fig. 10.  Frequency response curves of J
6
; a) translational acceleration, and b) rotational acceleration
Table 2.  Frequencies corresponding to peak values
Joints DOF Frequencies
x y z x
R
y
R
z
R
J
1
191.3 191.3 192.2 197.3 197.4 197.2
J
2
88.95 90.32 90.32 91.55 91.34 91.24
J
3
42.19 42.19 91.77 86.2 82.9 88.1
J
4
137.5 135.2 137.9 135.3 137.8 137.8
J
5
178.8 187.4 178.8 187.4 178.7 178.8
J
6
91.27 91.34 91.54 81.66 73.24 95.54

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 529-541
536 Liu, Y. – Ma, L. – Yang, S. – Liang Yuan, L. – Bo Chen
symmetrically with the change of joint angle of the 
manipulator. When θ
2
 is at the ultimate angle of –2.27 
rad and θ
4
 is at that of –6.28 rad, τ
1
 is only –2209 
N·m as θ
1
 and θ
3
 change. While θ
3
 ∈ (–1.24 ~ 1.13) 
rad, τ
1
 shows a trend of decay, when θ
3
 ∈ (1.13 ~ 
4.4  Torques of Joints
Substituting the data in Table 1 into Eq. (23), the 
values of τ
1
 change with θ
i
, γ and ϕ are obtained by 
MATLAB, as shown in Fig. 11. τ
1
 is distributed 
Table 3.  Coefficients of H
mn
(J
1
)
Coefficients H
11
(J
1
) H
12
(J
1
) H
13
(J
1
) H
14
(J
1
) H
15
(J
1
) H
16
(J
1
)
α
1
1.053×10
–5
–2.359×10
–4
0 –0.04122 –2.474 –3.082
α
2
18 7.91×10
–4
5.53×10
–4
9.311 569.6 708.6
α
3
–14.27 8.967 –2.796×10
–3
–99.29 –8613 –1.05×10
4
α
4
–6.116 –9.886 2.15 –988.8 18.74 –328.7
α
5
3.172 –0.2581 –3.32 –155.4 58.47 7.649
α
6
1.583 2.063 1.303 –22.72 11.5 3.809
β
1
0 0 0 1 1 1
β
2
1 0 0 –398.7 –398.8 –398.8
β
3
–0.793 1 0 3.98×10
4
3.983×10
4
3.982×10
4
β
4
–0.3398 –1.105 1 1979 376.4 1047
β
5
0.1762 –2.802×10
–2
–1.519 204.9 –200.2 –50.2
β
6
0.08792 0.2307 0.5861 30.14 –35.51 –10.31
Table 4.  Coefficients of H
mn
(J
2
)
Coefficients H
11
(J
2
) H
12
(J
2
) H
13
(J
2
) H
14
(J
2
) H
15
(J
2
) H
16
(J
2
)
α
1
14.37 0 0 0 0 0
α
2
–5269 18.7 1345 0 0.01195 8.488×10
–3
α
3
7.419×10
5
–4285 –2.042×10
5
0 –1.265 –1.264
α
4
–4.749×10
7
3.445×10
5
2.55×10
6
93.75 –30.33 28.8
α
5
1.163×10
9
–1.361×10
7
4.565×10
8
–1.865×10
4
5454 1803
α
6
4.135×10
7
3.364×10
8
–4.691×10
8
9.654×10
5
–2.66×10
4
–9660
β
1
1 0 1 0 0 0
β
2
–367.8 1 26.8 1 0 0
β
3
5.199×10
4
–221.8 –1.898×10
4
–406 1 1
β
4
–3.341×10
6
1.735×10
4
–2.815×10
5
6.166×10
4
–185.9 –185.8
β
5
8.21×10
7
–6.762×10
5
9.554×10
7
–4.15×10
6
9099 9093
β
6
3.09×10
6
1.654×10
7
–9.949×10
7
1.045×10
8
–4.014×10
4
–4.077×10
4
Table 5.  Coefficients of H
mn
(J
3
)
Coefficients H
11
(J
3
) H
12
(J
3
) H
13
(J
3
) H
14
(J
3
) H
15
(J
3
) H
16
(J
3
)
α
1
1.396×10
–2
2.342 0.4121 1.18×10
–3
8.639×10
–7
0.000374
α
2
–3.895 –276.7 –83.52 3.508 5.586 27.74
α
3
416.3 1.218×10
4
5613 –720 –1106 –5552
α
4
–1.863×10
4
–2.488×10
5
–1.427×10
5
3.802×10
4
5.885×10
4
3.039×10
5
α
5
2.958×10
5
2.244×10
6
1.487×10
6
–2.085×10
5
–4.029×10
5
–2.49×10
6
α
6
961 –1.232×10
6
–10.48 2.724×10
5
1.164×10
6
5.792×10
6
β
1
1 1 1 0 0 0
β
2
–183.2 –155.3 –242.3 1 1 1
β
3
1.311×10
4
9218 2.06×10
4
–194.5 –191.7 –193.6
β
4
–4.286×10
5
–2.496×10
5
–7.338×10
5
1.006×10
4
9927 1.028×10
4
β
5
5.332×10
6
2.666×10
6
1.026×10
7
–5.507×10
4
–6.78×10
4
–8.394×10
4
β
6
1.613×10
4
–1.849×10
6
–74.75 7.189×10
4
1.955×10
5
1.951×10
5

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 529-541
537 MTPA- and MSM-based Vibration Transfer of 6-DOF Manipulator for Anchor Drilling 
2.30) rad, τ
1 
shows a steady trend; and θ
3
 ∈ (2.30 ~ 
3.04) rad, τ
1
 showed a slight upward trend. When θ
6
 
is at the ultimate angle of –6.28 rad, τ
1
 is distributed 
symmetrically with the change of θ
5
; and τ
1 max
 = 6018 
N·m. As γ and φ change, τ
1
 is symmetrically 
distributed obviously; and τ
1 max
 = 9117 N·m. The 
maximum of τ
1
 applied to the pedestal is 9117 N·m, 
which is transmitted along the 6-DOF manipulator 
with any position and posture in space.
Table 6.  Coefficients of H
mn
(J
4
)
Coefficients H
11
(J
4
) H
12
(J
4
) H
13
(J
4
) H
14
(J
4
) H
15
(J
4
) H
16
(J
4
)
α
1
0 0 0 0 0.9853 –2.179
α
2
1.452 10.87 3.019 1.283 –174.3 1054
α
3
–542.2 –3655 –729.4 15.19 6824 –1.6×10
5
α
4
8.655×10
4
3.114×10
5
4.697×10
4
–9.851×10
4
8.145×10
4
8.051×10
6
α
5
–7.894×10
6
–6.236×10
5
–2.925×10
5
1.105×10
7
–6.495×10
5
–1.18×10
7
α
6
3.419×10
8
3.929×10
4
3.765×10
5
–3.454×10
4
6.07×10
5
5.426×10
4
β
1
0 0 0 1 1 1
β
2
1 1 1 –624.9 –256.5 –627.4
β
3
–538.6 –341.5 –277.6 1.458×10
5
1.548×10
4
1.606×10
5
β
4
1.113×10
5
2.958×10
4
2.022×10
4
–1.506×10
7
1.606×10
5
–1.892×10
7
β
5
–1.044×10
7
–5.924×10
4
–1.247×10
5
5.81×10
8
–1.351×10
6
8.4×10
8
β
6
3.746×10
8
3828 1.529×10
5
–1.813×10
6
1.274×10
6
–3.868×10
6
Table 7.  Coefficients of H
mn
(J
5
)
Coefficients H
11
(J
5
) H
12
(J
5
) H
13
(J
5
) H
14
(J
5
) H
15
(J
5
) H
16
(J
5
)
α
1
0 0 0 2.245 9.23×10
–3
0.0141
α
2
1.223×10
–2
4.005 12.8 –661.9 –3.182 –5.048
α
3
–4.295 –1464 –3564 3.512×10
4
312.4 481
α
4
388.1 1.366×10
5
6.619×10
4
2.645×10
6
–6312 –3513
α
5
–31.56 –4.933×10
5
3.108×10
7
–8.582×10
6
5.686×10
4
–3.849×10
4
α
6
46.17 3.554×10
5
–2.391×10
7
2.1×10
6
436 1.514×10
5
β
1
0 0 0 1 0 0
β
2
1 1 1 –23.44 1 1
β
3
–346.4 –366.1 696.9 –8.757×10
4
–364.4 –369.2
β
4
2.834×10
4
3.423×10
4
–3.441×10
5
1.099×10
7
3.424×10
4
3.642×10
4
β
5
3.039×10
5
–1.237×10
5
3.376×10
7
–4.608×10
7
–1.834×10
5
–4.209×10
5
β
6
6662 8.921×10
4
–2.461×10
7
4.1×10
7
–3.878×10
4
1.139×10
6
Table 8.  Coefficients of H
mn
(J
6
)
Coefficients H
11
(J
6
) H
12
(J
6
) H
13
(J
6
) H
14
(J
6
) H
15
(J
6
) H
16
(J
6
)
α
1
2.102×10
–3
–3.807 0.06361 –7.357×10
–6
–1.485×10
–5
4.141
α
2
–0.1981 1476 –14.32 3.304×10
–3
18.62 –834.1
α
3
–153.7 –1.796×10
5
1087 13.72 –3857 4.244×10
4
α
4
2.772×10
4
7.604×10
6
–3.055×10
4
–2855 2.287×10
5
7347
α
5
–1.185×10
6
–5.151×10
7
2.653×10
5
1.522×10
5
–2.966×10
6
–450.4
α
6
1.48×10
7
4.774×10
7
–4.187×10
5
–5.962×10
5
8.696×10
6
–211.3
β
1
0 1 0 0 0 1
β
2
1 –413.3 1 0 1 –195.1
β
3
–354.4 6.999×10
4
–162.5 1 –200.1 9678
β
4
4.741×10
4
–5.504×10
6
4823 –202.7 1.16×10
4
–1.156×10
4
β
5
–2.837×10
6
1.66×10
8
1.585×10
5
1.072×10
4
–1.497×10
5
5793
β
6
6.444×10
7
–1.577×10
8
–3.222×10
5
–4.198×10
4
4.382×10
5
1667

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 529-541
538 Liu, Y. – Ma, L. – Yang, S. – Liang Yuan, L. – Bo Chen
4.5  Vibration Response of the Pedestal
Substituting the frequency response matrix of each 
subsystem into Eq. (1) and substituting Eqs. (1) and 
(24) into Eq. (2), the vibration responses on each DOF 
of the pedestal are shown in Fig. 12, in which the 
positive and negative values of the vibration response 
only represent the direction.
As shown in Fig. 12a, the response of the 
longitudinal vibration (S
x
) of the pedestal reaches the 
maximum, being 1.65×10
–2
 
m, when the frequency is 
45 Hz. While the frequencies are 90 Hz and 180 Hz, 
the responses of the two components (S
y
 and S
z
) of the 
bending vibration of the pedestal reach the maximum, 
being 2.12×10
–2
 
m and 8.06×10
–3
 
m, respectively. At 
this time, the frequencies corresponding to the peaks 
are integer multiples of each other, so the phenomenon 
of resonance will happen.
As shown in Fig. 12b, the response of the torsional 
vibration (S
x
R) of the pedestal reaches the maximum, 
being 9×10
–3
 
m, when the frequency is 190 Hz. The 
two components (S
y
R and S
z
R) of rotational vibration 
a)          b) 
c)          d) 
Fig. 11.  τ
1
 changes with θ
i
 (i =1 to 6), γ and φ;  
a) τ
1
 change with θ
1
, θ
2
, b) τ
1
 change with θ
3
 θ
4
, c) τ
1
 change with θ
5
, θ
6
, d) τ
1
 change with γ, φ
a)           b) 
Fig. 12.  Vibration responses on each DOF of the pedestal; a) vibration responses of S
x
 , S
y
 , S
z
 , and b) vibration responses of S
x
R , S
y
R , S
z
R

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 529-541
539 MTPA- and MSM-based Vibration Transfer of 6-DOF Manipulator for Anchor Drilling 
around y and z axes sharply increase in the frequency 
interval [166-181] Hz and [151-180] Hz, and reach 
the maximum, being 2.08×10
–4
 
m and 2.46×10
–4
 
m 
respectively, at the frequencies of 180 Hz and 181 
Hz. At this time, the frequencies corresponding to 
the peaks are very similar, which is easy to induce 
a resonance. The amplitude of each DOF of the 
pedestal is from large to small is as follows: bending 
vibration (component 1) S
y
, longitudinal vibration S
x
, 
torsional vibration S
x
R, bending vibration (component 
2) S
z
, rotational vibration around z-axis S
z
R, rotational 
vibration around y-axis S
y
R.
5  CONCLUSIONS
(1)  Based on MTPA and MSM, a mathematical 
model of vibration transfer of 6-DOF manipulator 
for anchor drilling is established. The frequency 
response matrix of subsystems is derived under 
multi-DOF excitations. When the external 
excitation loading is determined, the frequency 
response function of each DOF at the response 
point can be calculated by the mathematical 
model, which is universal for series systems.
(2)  Within the frequency range (0 Hz to 200 Hz) of 
the excitation of the roof bolter, the corresponding 
frequency ranges of the peak values in the 6-DOF 
vibration direction of each subsystem are [191.3, 
197.4] Hz, [88.95, 91.55] Hz, [42.19, 91.77] Hz, 
[135.2, 137.8] Hz, [178.8, 187.4] Hz and [73.24, 
95.54] Hz, respectively. In the above frequency 
ranges, the vibration response will be the largest, 
and the resonance can be avoided by changing the 
excitation frequency of the roof bolter.
(3)  A case in engineering practice shows that the 
amplitude of each DOF of the pedestal is from 
large to small is as follows: bending vibration 
(component 1) S
y
 (2.12×10
–2
 
m) at 90 Hz, 
longitudinal vibration S
x
 (1.65×10
–2
 
m) at 45 
Hz, torsional vibration S
x
R (9×10
–3
 
m) at 190 Hz, 
bending vibration (component 2) S
z
 (8.06×10
–3
 
m) at 180 Hz, rotational vibration around z-axis 
S
z
R (2.46×10
–4
 
m) at 180 Hz, rotational vibration 
around y-axis S
y
R (2.08×10
–4
 
m) at 180 Hz. 
Obviously, the pedestal is mainly in the form of 
bending vibration.
(4)  The case also shows that a resonance will occur 
among the two components of the bending 
vibration at the frequencies of 90 Hz and 180 
Hz; a resonance among the two components of 
rotational vibration around the y and z axes is 
highly likely to occur at the frequencies of 180 
Hz and 181 Hz.
The theory of vibration Transfer along the 6-DOF 
manipulator for anchor drilling proposed in this 
article can provide a theoretical foundation for the 
development of vibration damping techniques and the 
design of absorbers.
6  NOMENCLATURES
λ
a
 the a
th
 order modal shape
λ
b
T
 the b
th
 order modal shape
λ
oa
 modal shape of the o
th
 DOF of the a
th
 modal 
vector
λ
pa
 modal shape of the p
th
 DOF of the a
th
 modal 
vector
q
a
 modal participation factors
o exciting point
p response point
n modal order
M
a
 modal mass coefficient
C
a
 modal damping coefficient
K
a
 modal stiffness coefficient
M mass matrix
C damping matrix
K stiffness matrix
S vibration displacement

S vibration speed

S vibration acceleration
J(q)
T
 force Jacobian matrix
J joint
J elements in Jacobian matrix 
d
i
 distance between two adjacent linkages along the 
common axis [m]
a
i–1
 common perpendicular length between joint i–1 
and joint i [m]
z moving DOF of z-axis
x moving DOF of x-axis
y moving DOF of y-axis
z
R
 rotational DOF of z-axis
x
R
 rotational DOF of x-axis
y
R
 rotational DOF of y-axis
S
y
 bending vibration (component 1) [m]
S
x
 longitudinal vibration [m]
S
z
 bending vibration (component 2) [m]
S
x
R rotational vibration around x-axis (torsional 
vibration) [m]
S
y
R rotational vibration around y-axis [m]
S
z
R rotational vibration around z-axis [m]
F external loading [N]
F
y
 force projected on the y-axis [N]
F
o
 loading at o point [N]
F
J
i
 excitation force transmitted to joints [N]
F
z
 force projected on the z-axis [N]
F
x
 force projected on the x-axis [N]

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)9, 529-541
540 Liu, Y. – Ma, L. – Yang, S. – Liang Yuan, L. – Bo Chen
F
g2
 gravity of manipulator [N]
F
g
 gravity [N]
F
a
 axial thrust [N]
F
z
 disturbing force [N]
F
c
 impact force [N]
M
d
 torque 
M
x
 torque projected on the x-axis [Nm]
M
y
 torque projected on the y-axis [Nm]
M
y
 torque projected on the z-axis [Nm]
v bending vibration displacement (component 2) 
[m]
w bending vibration displacement (component 1) 
[m]
μ Poisson’s ratio
D outer diameter of drill string [m]
ρ material density [kg·m
-3
]
R compressive strength [MPa]
R
m
 tensile strength [MPa]
E elastic modulus [GPa]
k elastic coefficient of impact force [N·m
-1
]
γ angle between force and z-axis [rad]
φ angle between force and x-axis [rad]
θ
i
 joint angle [rad]
Ω clearance between drill string and rock-soil [mm]
ξ
a
 damping ratio
τ torque matrix
τ
1
 torque from the J
1
-axis
α
i
 molecular coefficients, i = 1 to 6
β
i
 denominator coefficients, i = 1 to 6
7  ACKNOWLEDGEMENTS
This work was supported in part by Special Fund 
for Collaborative Innovation of Anhui Polytechnic 
University & Jiujiang District under Grant No. 
2021CYXTB3, and by and by Natural Science 
Research Project of Higher Education of Anhui 
Province of China under Grant No. KJ2020A0357.
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