MTPA- and MSM-based Vibration Transfer of 6-DOF Manipulator for Anchor Drilling

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. • Based on MTPA and MSM, a mathematical model of vibration transfer of a 6-DOF manipulator for anchor drilling is established. • Jacobian


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 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.

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 1subsystem 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]. 6 . (1) The vibration response of the excitation from the roof bolter transmitted to the pedestal is expressed as follows.

Response Amplitude Matrix
The dynamic equation of the manipulator for anchor drilling is as follows.
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.
Substituting Eq. (7) into Eq. (6), then, The excitation loading and displacement response in Eq. (8) are expressed in complex form as follows.
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.
According to the linear superposition assumption , the response amplitude of each point of the system is as Eq. (14).
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.

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.
The direction of M d is along the shaft of the roof bolter, and its vector expression is as follows.
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. Eq. (19) is expressed in matrix form as follows.
The excitation from the roof bolter at the tip of the manipulator is as Eq. (21).

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].
The torques of each joint of the manipulator is as follows (Eq. (23)). where

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.

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   Table 2.

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.  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 ~

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  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.

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  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.

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 (