Piezoelectric vibration suppression of a flexible mounting link for pre- cise space robot operating Feng Li 1 Shanghai Institute of Aerospace System Engineering, Shanghai 200240, P. R. China No.3888 Yuanjiang Road, Minhang District, Shanghai, China Lanqing Hu Shanghai Aerospace Equipment Manufacturer Co. , Ltd. , Shanghai 200240, China No.100 Huaning Road, Minhang District, Shanghai, China Ning Li Shanghai Institute of Aerospace System Engineering, Shanghai 200240, P. R. China No.3888 Yuanjiang Road, Minhang District, Shanghai, China Jinglong Liu Shanghai Institute of Aerospace System Engineering, Shanghai 200240, P. R. China No.3888 Yuanjiang Road, Minhang District, Shanghai, China Xuemei Ju Shanghai Institute of Aerospace System Engineering, Shanghai 200240, P. R. China No.3888 Yuanjiang Road, Minhang District, Shanghai, China

ABSTRACT In large-range manipulator operating mission, it is essential to connect an additional link with end- effectors to enlarge the operation space, which unavoidably introduces the vibration disturbance. An active piezoelectric vibration suppression method is proposed in this paper. The multiple piezoelec- tric patches shunted with time-variant resonant circuit are mounted on the link and an adaptive con- troller acting on circuit is utilized for suppressing the vibration due to excitation on various mass on the tip of the link during the robot operating. The electro-mechanical system of a kinematic manipu- lator model and an analytical flexible link model is established. The vibration controller is designed using the neural network which maps the relationship between the output voltage of piezoelectric sensors and the proper impedance of the circuit. The simulation study shows that the proposed method in this paper could be well applied in space robot operating.

1. INTRODUCTION

With the recent development in robot compliant precise control technology and robot vision, the types of robot executable tasks are increasing. Robots instead of human beings can perform specific tasks in complex environments instead, such as exploring deep-sea ecosystems or maintaining ma- chines in space environments. Furthermore, a part of these tasks require large operation space and higher precision for robot control. An effective method is to mount an additional slender long link to

1 flilqhu@163.com

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the manipulator. On account of flexibility of the slender long link compared to the rigid manipulator, the extra link is easily stimulated to vibrate during the operation which causes the end-effector to swing greatly. It is harmful to the image acquisition of the hand-eye camera and the precise operation of the manipulator. Meanwhile, the unknown mass of the end-effector influences the vibration char- acteristics of the flexible link. Therefore, a lightweight vibration control device and algorithm which can adapt to various vibration frequencies is required to reduce the vibration of end-effector.

The vibration control problem of an extra flexible link corresponds to the overall trajectory track- ing control problem of the tip position of a flexible-link robot. The analytical model for control system of the flexible link is always achieved by Euler-Bernoulli beam theory [1]. The control input can be chosen as motor torque, joint angel [2], frictional damper [3] or piezoelectric patches [4,5]. Moreover, combined trajectory tracking and vibration control with a motor torque and a voltage input applied to piezoelectric patches actuator leads to better performance [6]. Since a complex rigid-flexible coupling model is not easy to calculate the accurate end position through the joint angle or motor torque, pie- zoelectric actuators are utilized to suppress vibration of end-effector with consideration of lightweight and installation. The former studies show that piezoelectric vibration control is practical for light- weight link, by applying voltage to the PZT to produce external force actively or connecting reso- nance circuits to dissipate vibration energy passively [7]. The performance of the passive control is dependent on the corroboration between the natural frequency of the control system tuned by the electrical elements of circuits and that of the structure [8]. The tuning law of shunted circuit parame- ters with known system refers to [9]. For the system with structural changes, the optimal value can be searched by tuning the inductance and resistance parameters according to power consumption of the circuit, whereas lacks real-time performance [10].

In this paper, the dynamic equation governing the vibration of a flexible robot manipulator link which has a number of piezoelectric patches shunted with time-variant resonant circuit attached to the surface of the link is derived. And the frequency response characteristics of the end-effector are analyzed by simulation of the rigid-flex hybrid model of the manipulator with a flexible link, which determine the dominant natural frequency of the manipulator caused the large swing of the end-ef- fector during operation. Aiming at this frequency, the electro-mechanical coupling system model is simplified and an adaptive control algorithm is employed to adjust the impedance of shunted circuits to adapt to end-effector with various mass. The control algorithm constructs the mapping relationship between the amplitude of the piezoelectric sensor voltage in frequency domain and the natural fre- quency of the structure based on the artificial neural network (ANN) which can quickly obtain the control frequency from the sensor measurement. Combining with the optimal parameter relationship of the piezoelectric resonance circuit, the vibration of the end-effector can be effectively controlled within a certain frequency range.

2. THEORETICAL FORMULATIONS

A smart link model composed by a uniform continuous Euler-Bernoulli beam with piezoelectric patches under any external excitation of 𝑓(𝑥, 𝑡) from the robot arm are created. A tip mass to repre- sent the end-effector is equipped. The kinematic model of the robot manipulator is also discussed which contribute to guide the robot to move along a given path.

2.1. Analytical Model of the Smart Link

The electro-mechanical equations of motion for the response of the link with the shunted piezoe- lectric patch are shown in Figure 1. The left side of the link is clamped to the robot arm and the right side of the link is equipped with a mass. The thickness of the link is parallel to the ground to reduce

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the gravity effects. The piezoelectric patch number 𝑖 is bonded on the surface of the link. To be sim- plified, all actuator patches have same physical properties.

y

b h

1 ci Z  2 ci Z 

1 ci Z  ci Z

p h

–

t M

p t b t

x

z

p l

,1 i x

,2 i x

p l

Figure 1: Smart link with shunted piezoelectric patches.

The transverse motion 𝑤(𝑥, 𝑡) is parallel to the link thickness ℎ . The tip mass 𝑀 𝑡 is located at end of the link. Note that the subscripts b and p represent the link, piezoelectric patch respectively.

The analytical model is derived starting from the generalized form of Hamilton's principle:

𝑡 2

= 0 (1)

න𝛿(𝐾−𝑉+ 𝑊 𝑒 ) + 𝛿𝑊 𝑓 𝑑𝑡

𝑡 1

In the above equation, 𝐾 , 𝑉 , 𝑊 𝑒 are kinetic energy, the elastic potential energy and the electrical energy of the whole system, 𝑊 𝑓 is the total virtual work done by the non-conservative effects, are given by the following expressions:

𝑛

𝐾= 1

𝑑𝑉 𝑏 + ෍ 1

+ 1

2 (2)

2 න𝜌 𝑏 𝑤ሶ 2

2 න𝜌 𝑝𝑖 𝑤ሶ 2

𝑑𝑉 𝑝𝑖

2 𝑀 𝑡 𝑤ሶ൫𝑙 𝑝 , 𝑡൯

𝑉 𝑏

𝑉 𝑝𝑖

𝑖=1

𝑛

𝑉= 1

𝑑𝑉 𝑏 + ෍ 1

(3)

2 න𝑆 𝑏 𝜎 𝑏

2 න𝑆 11𝑖 𝜎 11𝑖

𝑑𝑉 𝑝𝑖

𝑉 𝑏

𝑉 𝑝𝑖

𝑖=1

𝑛

𝑊 𝑒 = ෍ 1

(4)

2 න𝐸 3 𝐷 3

𝑑𝑉 𝑝𝑖

𝑉 𝑝𝑖

𝑖=1

𝑛

𝑙 𝑝

𝛿𝑊 𝑓 = −න𝜇𝑤ሶ 2 𝛿𝑤𝑑𝑥

𝑑𝑥 (5)

+ න𝑓(𝑥, 𝑡)𝛿𝑤

−෍𝑞 𝑝𝑖 𝛿𝑣 𝑖

0

𝑙 𝑝

𝑖=1

𝜌 , 𝐴 , 𝑉 and 𝑙 are the density, surface area, volume, length respectively. 𝑡 with the subscript is the width. 𝜎 is stress and 𝑆 is strain. Moreover, 𝐸 3 and 𝐷 3 are the electrical field and electrical displace- ment of the piezoelectric patches respectively. 𝜇 is the viscous damping factor per unit length. 𝑞 𝑝𝑖 , 𝑣 𝑖 are the charge and voltage at the piezoelectric patch number 𝑖 . 𝑓(𝑥, 𝑡) is the transverse force at the location 𝑥 . Considering the constitutive equation of one-dimensional piezoelectric patches:

𝜎 11 ൨= ቂ 𝜀 𝑝 𝑒 𝑝 −𝑒 𝑝 𝑐 𝑝 ቃ൤ 𝐸 3

൤ 𝐷 3

𝑆 11 ൨ (6)

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𝑐 is Young's modulus assuming constant electrical field intensity, 𝑒 𝑝 is the piezoelectric stress/charge constants, 𝜀 𝑝 is the permittivity assuming constant strain.

Substituting Equation 2 – 6 into Equation 1 and considering the linearized strain–displacement relations and integration technique lead to Equation 7.

𝑛

𝑙 𝑏

𝑥 𝑖,2

𝑙 𝑏

𝑡 2

−න𝑀 𝑡 𝑤ሷ𝛿 𝑖 ൫𝑙 𝑝 ൯𝛿𝑤𝑑𝑥

න ൥−න𝜌 𝑏 𝐴 𝑏 𝑤ሷ𝛿𝑤𝑑𝑥

−෍න 𝜌 𝑝 𝐴 𝑝 𝑤ሷ𝛿𝑤𝑑𝑥

0

𝑥 𝑖,1

0

𝑡 1

𝑖=1

𝑛

𝑙 𝑏

𝑥 𝑖,2

𝑇 +𝑤 𝑥𝑥 𝛿𝑣) + 𝑣𝑡 𝑝 𝜀 𝑝 𝛿𝑣

(7)

+ 𝑧 𝑝 𝑒 𝑝 (𝑣𝛿𝑤 𝑥𝑥

−න𝐼 𝑏 𝑐 𝑏 𝑤 𝑥𝑥 𝛿𝑤 𝑥𝑥 𝑑𝑥

+ ෍න −𝐼 𝑝 𝑐 𝑝 𝑤 𝑥𝑥 𝛿𝑤 𝑥𝑥

𝑑𝑥

ℎ 𝑝

0

𝑥 𝑖,1

𝑖=1

𝑛

𝑙 𝑝

+ න−𝜇𝑤ሶ 2 𝛿𝑤+ 𝑓(𝑥, 𝑡)𝛿𝑤

൩= 0

𝑑𝑥−෍𝑞 𝑝𝑖 𝛿𝑣 𝑖

0

𝑖=1

3 12 Τ , 𝐼 𝑝 = 𝑡 𝑝 ℎ 𝑏

2 2 Τ + 𝑡 𝑝 ℎ 𝑝

3 3 Τ , 𝑧 𝑝 = 𝑡 𝑝 ൫ℎ 𝑏 + ℎ 𝑝 ൯2 Τ , 𝛿 𝑖 is the dirac delta function.

2 ℎ 𝑝 4 Τ + 𝑡 𝑝 ℎ 𝑏 ℎ 𝑝

where 𝐼 𝑏 = 𝑡 𝑏 ℎ 𝑏

2.2. Robot Kinematic Description

The basic knowledge about the robot operating is the forward kinematics which calculates the position and orientation of the end-effector with a sequence of the joint angles 𝜃 . The robot manipu- lator is considered as an n-link spatial open chain, the transformation matrix from base frame to end- effector frame is expressed as Equation 8 by applying the PoE (product of exponentials):

𝑇(𝜃) = 𝑒 ሾ𝑆 1 ሿ𝜃 1 𝑒 ሾ𝑆 2 ሿ𝜃 2 ⋯𝑒 ሾ𝑆 𝑛 ሿ𝜃 𝑛 𝑇 0 (8)

where 𝑆 𝑖 is the screw axis of the 𝑖 th joint, 𝜃 𝑖 is the 𝑖 th joint variable represents the joint angle between 𝑇(𝜃) and 𝑇 0 . In order to operate the robot to follow a given path, the Newton–Raphson method, one of the numerical inverse kinematics method is used for the PoE formula. The detailed theory of robot kinematics can be found in the robotics textbook which is not elaborated here.

The peg-in-hole assembly task is one of the common operation application in industrial field as well as space field. After the visual sensor determines the approximate position of the hole in the hole search stage, a random hole search is performed to find the accurate position and then insertion is carried out. The helix motion is a typical random search method, the vibration during the helix motion should be reduced to ensure that the end-effector tracks desired action perfectly.

2.3. Solution to the Equation of Motion

Considering the first m vibrational modes of the whole system, the transverse displacement of the link can be expressed in terms of the following modal summation:

𝑚

𝑤(𝑥, 𝑡) = ෍𝜑 𝑖 (𝑥)𝑞 𝑖 (𝑡) =

𝛗(𝑥)𝐪(𝑡) (9)

𝑖=1

where 𝜑 𝑖 (𝑥) = cosh𝛽 𝑖 𝑥−cos𝛽 𝑖 𝑥−𝛾 𝑖 (sinh𝛽 𝑖 𝑥−sin𝛽 𝑖 𝑥) are the natural modes of the link without piezoelectric patch considering the boundary condition. The constants 𝛾 𝑖 are the mode shape coeffi- cients. 𝑞 𝑖 (𝑡) are the generalized coordinates for the transverse vibration of the link.

Substituting Equation 9 in 7, the motion equation could be re-arranged in a matrix form as below:

𝐌𝐪ሷ(𝑡) + 𝐂𝐪ሶ(𝑡) + 𝐊𝐪(𝑡) + 𝚯𝐯(𝑡) = 𝐟(𝑡) (10)

−𝚯 𝑇 𝐪(𝑡) + 𝐂 𝒑 𝐯(𝑡) = 𝐐(𝑡) (11)

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in which 𝐌= 𝜌 𝑏 𝐴 𝑏 ׬ 𝛗 𝑇 𝛗𝑑𝑥 𝑙 𝑏 0 + 𝜌 𝑝 𝐴 𝑝 σ ׬ 𝛗 𝑇 𝛗𝑑𝑥 𝑥 𝑖,2 𝑥 𝑖,1 𝑛 𝑖=1 + ׬ 𝑀 𝑡 𝛗 𝑇 𝛗𝛿 𝑖 ൫𝑙 𝑝 ൯𝑑𝑥 𝑙 𝑏 0 is total mass ma-

trix, 𝐂= ׬ 𝜇𝛗 𝑇 𝛗𝑑𝑥 𝑙 𝑝 0 = 2ξ 𝑏 𝑀 𝑝 𝛚 𝑛 is total damping matrix, ξ 𝑏 is the modal damping ratio, 𝛚 𝑛 is a

diagonal matrix consists of the natural angular frequencies, 𝑀 𝑝 = 𝜌 𝑏 𝐴 𝑏 ׬ 𝜑 𝑖 𝜑 𝑖 𝑑𝑥 𝑙 𝑏 0 is the link

𝑇 𝛗 𝑥𝑥 𝑑𝑥 𝑙 𝑏 0 + 𝐼 𝑝 𝑐 𝑝 σ ׬ 𝛗 𝑥𝑥

𝑇 𝛗 𝑥𝑥 𝑑𝑥 𝑥 𝑖,2 𝑥 𝑖,1 𝑛 𝑖=1 is total stiffness matrix, 𝐟(𝑡) =

mass, 𝐊= 𝐼 𝑏 𝑐 𝑏 ׬ 𝛗 𝑥𝑥

𝑇 𝑑𝑥 𝑥 1,2 𝑥 1,1 … ׬ 𝑧 𝑝 𝑒 𝑝 𝛗 𝑥𝑥

𝑇 𝑑𝑥 𝑥 𝑛,2 𝑥 𝑛,1 ቃ is pie-

׬ 𝑓(𝑥, 𝑡)𝛗 𝑇 𝑑𝑥 𝑙 𝑝 0 is the external force vector, 𝚯= ቂ׬ 𝑧 𝑝 𝑒 𝑝 𝛗 𝑥𝑥

zoelectric coupling terms, 𝐂 𝒑 = 𝑑𝑖𝑎𝑔ቂ׬ 𝑡 𝑝 𝜀 𝑝 ℎ 𝑝 Τ 𝑥 𝑖,2 𝑥 𝑖,1 𝑑𝑥ቃ is the total capacitance matrix, 𝐐(𝑡) is the

total charge vector.

A common serial Resistance-Inductance (RL) circuit with negative capacitance parallel to the pi- ezoelectric patches is employed. The above equations can now be transforming into the state space form:

𝐱ሶ(𝑡) = 𝐀(𝑡)𝐱(𝑡) + 𝐁(𝑡)𝐟(𝑡) (12)

where

𝐪 𝐪ሶ 𝐪 𝑝 𝐪ሶ 𝑝

𝟎 𝐌 −1

቏

൪, 𝐁(𝑡) = ቎

𝐱= ൦

𝟎 𝟎

ۏ ێ ێ ێ ۍ 𝟎 𝐈 𝟎 𝟎 −𝐌 −1 ቀ𝐊+ 𝚯൫𝐂 𝑝 + 𝐂 𝑐 ൯

(13)

ۑ ۑ ۑ ې

−1 𝚯 𝑇 ቁ −𝐌 −1 𝐂 −𝐌 −1 𝚯൫𝐂 𝑝 + 𝐂 𝑐 ൯

−1 𝟎 𝟎 𝟎 𝟎 𝐈 −𝐋 𝑐

𝐀=

−1 𝚯 𝑇 𝟎 −𝐋 𝑐

−1 −𝐋 𝑐

−1 ൫𝐂 𝑝 + 𝐂 𝑐 ൯

−1 ൫𝐂 𝑝 + 𝐂 𝑐 ൯

−1 𝐑 𝑐 ے

Although the joint angle sequence of the desired path can be solved by the robot kinematic, the excitation transferred from the rotated robot arm is difficult to obtain by the robot dynamic. Hence the harmonic excitation is employed as the base excitation during the robot operation, which is con- sidered as the white noise (PSD = 1):

𝑛

𝑙 𝑏

𝑥 𝑖,2

+ 𝑀 𝑡 𝛗൫𝑙 𝑝 ൯൩𝐹 ሷ = −𝐻𝐹 ሷ 0 = −𝐻𝐹 0 𝑒 𝑖𝜔𝑡 (14)

𝐟(𝑡) = −൥𝜌 𝑏 𝐴 𝑏 න𝛗𝑑𝑥

+ 𝜌 𝑝 𝐴 𝑝 ෍න 𝛗𝑑𝑥

0

𝑥 𝑖,1

𝑖=1

The differential equations Equation 10, 11 for the response of the whole system in frequency do- main become:

(−𝜔 2 𝐌+ 𝑗𝜔𝐂+ 𝐊)𝐪(𝜔) + 𝚯𝐯(𝜔) = 𝜔 2 𝐻𝐹 0 (15)

−𝑗𝜔𝚯 𝑇 𝐪(𝜔) + 𝑗𝜔൫𝐂 𝑝 + 𝐂 𝑐 ൯𝐯(𝜔) = −(𝐑 𝑐 + 𝑗𝜔𝐋 𝑐 ) −1 𝐯(𝜔) (16)

The frequency response function (FRF) of tip displacement divided by base excitation can be writ- ten as:

𝑤 𝐹 0

−1 𝜔 2 𝐻 (17)

= 𝛗(𝑙)ൣ−𝜔 2 𝐌+ 𝑗𝜔𝐂+ 𝐊+ 𝑗𝜔𝚯𝐙 𝑝𝑐 𝚯 𝑇 ൧

−1 (18)

𝐙 𝑝𝑐 = ൣ(𝐑 𝑐 + 𝑗𝜔𝐋 𝑐 ) −1 + 𝑗𝜔൫𝐂 𝑝 + 𝐂 𝑐 ൯൧

The above formula is responsible for evaluating the control effect of the proposed algorithm.

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3. CONTROL ALGORITHM

In this study, a data-driven adaptive control algorithm is utilized, which consists of the ANN and optimal electrical parameter equation. The optimal electrical parameter with the variable mass and the voltage output of the sensor in frequency domain are the necessary data for the control algorithm.

3.1. Dataset Preparation

The optimal electrical parameter is used to corroborate natural frequency of the control system and the structure. The negative capacitance is constant, so the resonance frequency of the shunted circuit and the optimal inductance have a relationship: 𝑓 𝑐 ඥ𝐿 𝑐 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 . The natural frequency of the controlled structure without control can be obtained by the voltage signal of the sensor in frequency domain and the optimal inductance at this frequency can be found by trial and error, so the constant value is obtained. If the tip mass changes, the structural natural frequency is measurable and accord- ing to the inductance-frequency relationship the optimal inductance can be obtained. In order to ob- tain the optimal resistance, the circuit energy dissipation search is implemented. The thermal power 𝐷 𝑡 of the system in the time domain is averaged as follows ：

𝑛

𝑇 𝐑 𝑐 𝐂 𝑞ሶ 𝑝 𝐗ቃ (19)

𝑇 𝐑 𝑐 𝒒ሶ 𝑝 ൧= 𝑇𝑟ቂ𝐂 𝑞ሶ 𝑝

𝐷 𝑡 = 𝐸൥෍𝐷 𝑖 (𝑡)

൩= 𝐸ൣ𝒒ሶ 𝑝

𝑖=1

where 𝐂 𝑞ሶ 𝑝 = ሾ𝟎 𝟎 𝟎 𝐈ሿ is to extract the current of each circuit, 𝐗= 𝐸ሾ𝐱(𝑡)𝐱 𝑇 (𝑡)ሿ is the covari- ance matrix satisfied the Lyapunov equation under excitation ：

𝐀𝐗+ 𝐗𝐀 𝑇 + 𝐁𝐖𝐁 𝑇 = 𝟎 (20)

where 𝐖= 𝐸ሾ𝐟(𝑡)𝐟 𝑇 (𝑡)ሿ is a covariance matrix of the excitation. On the condition that the tip mass is fixed with the optimal inductance, the resistance that maximizes the thermal power of the circuit is the optimal resistance.

During the control process, when the inductance approaching the optimal value, the circuit with arbitrary resistance always performs a dynamic vibration absorber effect resulting in double-peak or flat phenomenon of sensor voltage in the frequency domain. It is difficult to determine the system natural frequency through the maximum peak value, so the ANN model is used to identify the system natural frequency. The training data of the ANN is the sensor voltage data in frequency domain under various tip mass with same resistance and various inductances control and the label is the current natural frequency with respect to the tip mass. The frequency to be controlled can be quickly identi- fied by ANN in spite of the unknown structure.

3.2. Optimal Parameter Solution

The control algorithm is depicted in Figure 2. The end-effector performs the given motion, the voltage amplitude in frequency domain in the range of 1–15 Hz is obtained by the measured value of the sensor through FFT. The voltage amplitude is the input to the ANN, and the ANN outputs the current control frequency. The optimal inductance and resistance is calculated by the computation module for optimal circuit parameter according to the current output frequency and parameters of the piezoelectric shunt circuit are updated. The frequency of the voltage signal changes according to the changeable tip mass, meanwhile the control frequency is updated by ANN in real time for effective vibration control.

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Figure 2: The scheme of the control system. 4. NUMERICAL SIMULATIONS

Numerical simulation is carried out to verify the effectiveness of the system model and control algorithm in this paper. Geometric parameters and properties for the PZT and link used are given in Table 1. Four piezoelectric actuators are tightly bonded on the surface of the link and a piezoelectric sensor is used to measure the voltage generated by the vibration near the root of the link.

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Table 1: Properties and geometry of the system.

Dimension

Young’s modulus

Density (kg/m 3 )

Piezoelectric coefficient 𝒆 𝒑

Capacitance

(nF)

l × b × h

(GPa)

(mm)

Link (Aluminum) 300 × 40 × 1 71.7 2740 - -

Actuator(PZT-5A) 60 × 30 × 0.3 60.5 5440 -11.6 51.88

Sensor(PZT-5A) 16 × 16 × 0.3 60.5 5440 -11.6 7.38

4.1. Modeling Comparison

In order to validate the accuracy of the flexible link modeling in this paper, a 6dof manipulator with an additional flexible link model is established in the commercial software Adams and the joint angle planning obtained by the inverse kinematic of the helical motion is regarded as the model input. The vibration frequency of the tip mass caused by the movement measured by the Adams is compared with the natural frequency of the established analytical model. The frequency of the max amplitude from simulation software is 3.418 Hz when the mass is 0.05 kg and it is the same as the first-order natural frequency 3.412 Hz of the analytical model. The change of natural frequency is caused by the change of the tip mass within a reasonable range, and the frequency corresponding to the max ampli- tude of the tip displacement in frequency domain is close to the first-order natural frequency of the

motion computation module joint angle input —> additional flexible manipulator link with tip mass

link. Therefore, the dominant control frequency during the movement of the manipulator is the first- order natural frequency of the link, which can simplify the analytical model proposed in this paper.

4.2. Vibration Suppression During the Helix Motion

The negative capacitance is set as 𝐶 𝑐 = −0.6𝐶 𝑝 . According to the analytical model, the optimal inductance and resistance are obtained for the first-order natural frequency with a tip mass range of 0.001–0.1 kg, and the relationship between the optimal resistance and first-order natural frequency is fitted in an exponential form 𝑅 𝑐 = a(𝑓 𝑐 ) b + c , where a, b, c are fitted values. Set the resistance to a large value, and the voltage signals of the sensor in frequency domain with various inductance and mass are prepared as the training data. Here, the voltage amplitudes of each 0.01Hz between 1 Hz and 15 Hz are utilized as training data. This paper constructs a four-layer ANN in PyTorch, including an input layer, two fully connected layers, and an output layer. The two fully connected layers have 1000 and 200 neurons respectively, and use 𝑡𝑎𝑛ℎ as the activation function. The optimizer uses Adam, and the loss function chooses the mean square loss function. The ANN input data is scaled between zero and one. A total of 12 epochs of training are performed, and the final error is 0.0001 on the validation set. After the ANN is constructed, the tip masses are taken as 0.038, 0.054, and 0.088 kg respectively to verify the control algorithm proposed in this paper. The control effect is shown in the Figure 3(a) and the relevant parameters of the ANN model for 0.088 kg are shown in the Figure 3(b).

M t = 88g

M t = 38g

M t = 54g

(a) Control effects through proposed method. (b) Control frequency identification at 𝑀 𝑡 =88g Figure 3: Applications of the proposed method with various mass.

Table 2: Parameters and control effects of the control system with various mass.

Mt (kg)

Structural Frequency

ANN Frequency

Inductance

Resistance

Reduction

(H)

(dB)

( 𝐤𝛀 )

(Hz)

(Hz)

0.038 5.0519 5.0492 4.78 × 10 4 27.24 8.18

0.054 4.3805 4.3855 6.34 × 10 4 39.13 9.49

0.088 3.5443 3.5504 9.68 × 10 4 66.46 11.27

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The results show that the ANN can accurately identify the first-order frequency of structure, and the shunted circuits are updated correctly through the optimal circuit parameter obtained by the com- putation module which suppress the vibration of the end-effector effectively.

5. CONCLUSIONS

In this paper, aiming at the vibration suppression problem of the manipulator connected to an additional flexible link during a movement, the motion equation of the flexible link with tip mass is established. The dominant control frequency of the manipulator is determined by simulation. An ANN model is constructed to identified the control frequency in real time through the input voltage signal in frequency domain, and the inductance and resistance in the circuit is updated by the identi- fied frequency. The simulation results show that proposed control algorithm effectively suppresses the vibration of the end-effector with unknown mass. 6. ACKNOWLEDGEMENTS

This research was supported by the National Natural Science Foundation of China with Grant No. U21B6002. 7. REFERENCES

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