# Motion Trajectory Control System for Production Line Robots Based on Variable Domain Fuzzy Control

Authors: Ling Pan
Journal: Advances in Multimedia
 (2022)

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## Abstract

Mechatronics production line is widely used in the production and operation of all walks of life, which greatly reduces the amount of labor of employees, improves the production level of enterprises, and creates great social and economic benefits for enterprises. The traditional manipulator system has modeling uncertainty and is vulnerable to external interference. In this paper, a compound control scheme is designed by combining the torque controller and fuzzy compensator, which effectively improves the system's response speed and greatly reduces the steady-state tracking error of the system. At the same time, the tracking process of the scheme is stable and flexible, and the chattering is minimal, which effectively improves the trajectory tracking performance of the system. Experiments show that the algorithm effectively solves the contradiction between computational complexity and control accuracy, improves the system's response speed, and reduces the system's steady-state tracking error.


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## Body

## 1. Introduction

“Mechatronics” is a composite comprehensive technology closely combined with various technologies, mainly including computer technology, information technology, mechanical technology, electronic technology, and other technologies. “Mechatronics” technology is a new integrated equipment control technology, which organically combines mechanical equipment with electronic equipment and supporting computer software [1]. The most important thing is to use mechanical and electrical technology to replace the most reliable electronic equipment [2]. The outstanding advantages of “mechatronics” are that it improves the equipment's safety, reliability, and performance. Mechatronics equipment has composite technology and composite function. Its self-inspection and monitoring function can automatically take measures against faults to make the work return to normal [3].Manipulator has the characteristics of large load, low energy consumption, and small volume. It can complete maintenance, butt joint, welding, spraying, handling, and other operations and has been widely used in mechanical manufacturing equipment [4]. The manipulator is composed of multiple rods. Through the manipulation function of the connecting rod structure and the movement function of the moving base, the manipulator has a large working space. The stability of the manipulator is affected by friction, dead zone, and saturation. Through the research on the manipulator's automatic control method, the manipulator's dynamic performance is improved and the adaptability of the manipulator to the production line environment is improved [5].With the development of the robot arm, it plays a more and more important role in the intelligent process of industrial manufacturing, aerospace, rail transit, and other fields. The increasingly customized and personalized production mode of the manufacturing industry puts forward higher requirements for the trajectory tracking control of the manipulator, which makes improving the trajectory tracking accuracy of the manipulator and enhancing its adaptability to different environments and tasks become the key technical problems to be solved in the application of the manipulator [6]. Due to the influence of uncertain factors such as modeling error, signal detection error, and external environmental interference, it brings great challenges to the control of manipulator. Therefore, scholars at home and abroad have proposed a variety of intelligent control strategies to improve the accuracy of manipulator trajectory tracking. The trajectory tracking control of manipulator based on the adaptive method has also been widely studied [7].The adaptive algorithm can solve the problem of online estimation of unknown parameters in the manipulator model. On this basis, combined with the fuzzy control method, it can further improve the response speed and trajectory tracking accuracy of the manipulator [8]. Fayazi et al. [9] proposed an impedance sliding mode control method for the position and force control of manipulator. Literature [10] aims at the posture adjustment of minimally invasive surgical robot. Through the combination of fuzzy theory and reinforcement learning, it can adaptively adjust the model to respond to the intention of the controller and has good controllability. Li and Adeli [11] designed a hierarchical fuzzy control algorithm to actively suppress the vibration of flexible rod and improve the calculation efficiency of fuzzy controller. Urrea et al. [12] proposed a variable universe fuzzy adaptive control method, which solves the problem of inaccurate control in traditional fuzzy control. Llama et al. [13] realized the control of nonlinear systems such as four-stage inverted pendulum by using the variable universe fuzzy method. Zhang et al. [14] used the variable universe method to realize the vehicle lateral autonomous control. Ziani [15] introduced the idea of variable universe into the membership function of fuzzy control and designed the adaptive law of input function, which is suitable for nonlinear, time-varying, and complex uncertain systems. Li et al. [16] used fuzzy neural network to adjust the universe of T-S fuzzy controller variables, which improves the stability of vehicles during driving. Li et al. [17] solved the lateral vibration problem of urban rail train suspension system by using the explicit genetic fuzzy control method. Kassem et al. [18] adopted the double-loop hybrid control method to control the load frequency in the power grid. The internal loop adopts variable universe fuzzy to reduce the load disturbance, and the external loop uses incremental genetic algorithm to optimize the control parameters online. Du et al. [19] applied genetic variable universe fuzzy controller to the control of train lateral semi-active suspension system and achieved good control effect. Hrama et al. [20] studied the temperature control of evaporation source by using the variable universe fuzzy adaptive PID method. Wang et al. [21] analyzed the approximation conditions of variable universe fuzzy controller.Due to the influence of uncertain factors such as modeling error, signal detection error, and external environmental interference, it brings great challenges to the control of manipulator. In this paper, the manipulator’s dynamic model and parameter uncertainty are considered. In order to improve the trajectory tracking accuracy of the manipulator, a composite control scheme is given by combining the nominal calculated torque controller and the variable universe fuzzy compensator method. The variable universe adaptive fuzzy controller is used to estimate and compensate the modeling uncertainty. The calculated torque controller is used to solve the nominal system control torque to reduce the mechanical trajectory tracking error, improve the trajectory tracking speed, and ensure the trajectory tracking performance. The experimental results show that the algorithm improves the system's response speed, reduces the system tracking error, and improves the overall performance.


## 2. Methodology

### 2.1. Variable Universe Fuzzy Adaptive Control

#### 2.1.1. Basic Principles

Under the premise that the fuzzy rules remain unchanged, the scope of the variable universe fuzzy control system changes according to the change of the feedback error information. When the feedback error decreases, the reduction of the universe is equivalent to adding the number of rules to improve the control accuracy. When the feedback error increases, the increase of the universe is equivalent to reducing the number of rules and improving the convergence speed. Its essence is a dynamic point-by-point convergence interpolator.LetG=Gx1≤x≤u be a set of fuzzy sets on a given domain I and ix be the peak of Gx, if for any i∈I,(1)∑x=0tGxi=1,is established, and when w≠t, iw≠it,w,t∈1,2…,u, Gx is called the primitive of G, and G is called a primitive of I. Further, if i≠ix and any adjacent primitives Gx and Gx+1 satisfy(2)Aix≠0≠Ai+1x,Gxi≠0≠Gx+1i,Aix+Ai+1x=1,Gxi+Gx+1i=1,then G is called biphasic base tuple on I. In particular, if Gi is a linear function, G is called a biphasic linear basis on I. Considering the dual-input single-output manipulator system, the two input domains are given as I=−E,E and J=−D,D, where I represents the trajectory tracking error domain, J represents the tracking error change rate domain, and J is affected by I. G=Gxx=1,2,…,u and H=Hyy=1,2,…,v are biphasic linear basis tuples on universe I and J, respectively. The single-output universe is K=−P,P, which represents the uncertainty compensation torque, and C=kxy is the linear base on it.The control structure is shown in Figure1. z is the sampling time, and ixz,jxz,kxz are the peak values of ix,jx and fuzzy single value element kx at z time, respectively. Gxiz,z and Hyiz,z are the membership functions of deviation and deviation change rate at time z; then, a set of dynamic control rules can be obtained:(3)If is and isHyzthenknisCxyz.Figure 1
Structure diagram of variable universe fuzzy control system.Then, the control function can be written as(4)Fiz,jz,z=∑x=1u∑y=1vGxiz,zHyiz,zkxyz.


#### 2.1.2. Dual-Input Single-Output Variable Universe Fuzzy Control Algorithm

The algorithm steps are as follows.(1)
Step 0: the initial value of the system input isi0∈I,j0∈J, and the output is(5)k1=Fi0,j0,0=∑x=0u∑y=0vGxi0,0Hyj0,0kxy0.(2)
Step 1: afterkx1 acts on the object, the system output is generated. After comparing the output with the reference input, it is obtained that the input of the system controller is ix1=αi1ix0,yi1=βi1,j1jx0, and the output is(6)z2=Fx1,y1,0=∑i=0p∑j=0qAix1,0Bjy1,0zij1.(3)
Step z:kz acts on the object to generate the system output. After the output is compared with the reference input, the input ixz=αizix0,jxz=βiz,jzjx0 of the system controller is obtained; then, the controller output is(7)kz+1=kzFizαiz,jzβiz,jz,=γkz∑x=0u∑y=0vGxizαizHyjzβiz,jzkxy0.Sincez,iz≠0 at any time, there is limz⟶∞iz≠0,limz⟶∞jz≠0, that is, there is limz⟶∞αiz−1/αiz=1,limz⟶∞γiz−1/γiz=1. Then, when iz⟶0,jz⟶0, that is, the dual-input single-output system based on variable universe fuzzy control is convergent.


### 2.2. Design of Manipulator Controller

#### 2.2.1. Controller Structure Design

In order to overcome the influence of uncertain factors and unknown disturbances on the trajectory tracking performance of manipulator, a composite controller was designed, composed of a computational torque controllerτc and a variable theory domain fuzzy compensation controller pfe,e˙. As shown in Figure 2, vd represents the expected value, v represents the actual value, and the input torque of the manipulator is τ=τc+pfe,e˙.Figure 2
Controller structure block diagram.As can be seen from Figure2, the controller's input is the trajectory tracking error e and its change rate ec becomes a dimensionless value et after corresponding quantization factors Ze and Zec, and their changes lead to the change of expansion factor αi,βi,j,γi. The controller's output is converted into pfe,e˙ by the proportional factor Zp, and the output τc of the calculated torque constitutes the control torque of the manipulator.


#### 2.2.2. Control Policy

Letv,v˙,v¨ be the actual measured values of angular displacement, angular velocity, and angular acceleration of the manipulator under consideration, respectively, and vd,v˙d are the desired angular displacement and angular velocity; then, the tracking error can be taken as(8)en=vn−vdn,e˙n=v˙n−v˙dn.Obviously, when there is no modeling uncertainty and external interference, if the error system(9)e¨+zue+zqe˙=0,is established, then each joint of the manipulator can accurately track the desired trajectory, where zu,zq are undetermined coefficients. In addition, the nominal dynamic model of the manipulator is(10)D0vv¨d+C0v,v˙v˙d+A0v=τc+τd.For nominal system (10), its control law is(11)τc=Dvv¨d−zue−zqe˙+Cv,v˙v˙d+Av.Then, the above formula can be obtained simultaneously.(12)e¨+zue+zqe˙=D0−1vΔDv¨d+ΔCv˙d+ΔA+τd.According to (12), the uncertainty contained in the controller will affect the stability of the system, so it needs to be compensated. The uncertain term in (12) is(13)a=D0−1vΔDv¨d+ΔCv˙d+ΔA+τd.To compensate for the influence on system stability,a was identified. Therefore, the control law given in (11) is modified as(14)τ=D0vv¨d−zue−zqe˙+C0v,v˙+A0v−D0va.On this basis, the variable theory domain adaptive fuzzy compensatorpfe,e˙ is used to approximate the uncertain term a. Let a^i be the estimate of a, and its expression is(15)aˆi≜aˆi|Θ=Θξ.The adaptive fuzzy compensation termpfe,e˙ in the control system can be defined as(16)pfe,e˙=−D0va^i,where ξ is a fuzzy basis function vector, Θ is a weight vector, and its optimal value Θ∗ satisfies(17)Θ∗=argminM∈Ωfsupi∈Pa^i|Θ−ai.According to (11) and (16), (14) can be further written as(18)τ=τc+pfe,e˙.Substitute it into (12) and obtain(19)e¨+zue+zqe˙=Θ˜ξ+ε,where ε is the approximation error term of the adaptive fuzzy compensator. If the output error vector of the manipulator is i=eN,e˙N, the state space expression of the closed-loop error equation of the system shown in (19) is(20)i˙=Gi+Ha.Among them,G=0X−zU−zQ,H=0X,a=Θ˜ξ+ε.Combining (18)–(20), the uncertain term a can be further expressed as(21)ai|Θ=Θ∗ξ+ε.Substituting (21) into (16), the fuzzy output factor γk in the variable theory domain affects the fuzzy basis function vector. Therefore, the variable theory domain adaptive compensator can be further expressed as(22)pfe,e˙=−D0vγkai|Θ≜−D0vΘ∗ξi|γk.Take the Lyapunov function as(23)Q=−12iNUi+12ηΘ˜NΘ˜,where U is a positive definite matrix and satisfies the following equation.(24)UG+GNU+V=0,where V≥0. Let Θ˜=Θ^−Θ∗, where Θ^ is an estimate of Θ∗. From the adaptive control law,(25)Θ˜˙=ηξi|γiNUH.The derivative of (23) can be obtained:(26)Q˙=−12iNVi−ξNΘ˜HNUi−εNHNUi+1ηΘ˜˙NΘ˜.Sinceε is the minimum approximation error, according to the approximation theory of fuzzy control, as the eigenvalue of V increases, the eigenvalue of U decreases, and ε can be sufficiently small, which can be obtained from (26).(27)Q˙≤−ξNi|γΘ˜HNUi+1ηΘ˜˙NΘ˜.Substitute adaptive control law (25) into (27) and obtain(28)Q˙≤−ξNi|γΘ˜HNUi+1ηηξi|γiNUHNΘ˜=0.So, whenΘ˜˙=ηξx|γiNUH, Q˙≤0. In other words, under this condition, the manipulator system based on variable theory domain adaptive fuzzy control is asymptotically stable, and the tracking effect is better with the smaller upper bound of fuzzy control error ε.


## 2.1. Variable Universe Fuzzy Adaptive Control

### 2.1.1. Basic Principles

Under the premise that the fuzzy rules remain unchanged, the scope of the variable universe fuzzy control system changes according to the change of the feedback error information. When the feedback error decreases, the reduction of the universe is equivalent to adding the number of rules to improve the control accuracy. When the feedback error increases, the increase of the universe is equivalent to reducing the number of rules and improving the convergence speed. Its essence is a dynamic point-by-point convergence interpolator.LetG=Gx1≤x≤u be a set of fuzzy sets on a given domain I and ix be the peak of Gx, if for any i∈I,(1)∑x=0tGxi=1,is established, and when w≠t, iw≠it,w,t∈1,2…,u, Gx is called the primitive of G, and G is called a primitive of I. Further, if i≠ix and any adjacent primitives Gx and Gx+1 satisfy(2)Aix≠0≠Ai+1x,Gxi≠0≠Gx+1i,Aix+Ai+1x=1,Gxi+Gx+1i=1,then G is called biphasic base tuple on I. In particular, if Gi is a linear function, G is called a biphasic linear basis on I. Considering the dual-input single-output manipulator system, the two input domains are given as I=−E,E and J=−D,D, where I represents the trajectory tracking error domain, J represents the tracking error change rate domain, and J is affected by I. G=Gxx=1,2,…,u and H=Hyy=1,2,…,v are biphasic linear basis tuples on universe I and J, respectively. The single-output universe is K=−P,P, which represents the uncertainty compensation torque, and C=kxy is the linear base on it.The control structure is shown in Figure1. z is the sampling time, and ixz,jxz,kxz are the peak values of ix,jx and fuzzy single value element kx at z time, respectively. Gxiz,z and Hyiz,z are the membership functions of deviation and deviation change rate at time z; then, a set of dynamic control rules can be obtained:(3)If is and isHyzthenknisCxyz.Figure 1
Structure diagram of variable universe fuzzy control system.Then, the control function can be written as(4)Fiz,jz,z=∑x=1u∑y=1vGxiz,zHyiz,zkxyz.


### 2.1.2. Dual-Input Single-Output Variable Universe Fuzzy Control Algorithm

The algorithm steps are as follows.(1)
Step 0: the initial value of the system input isi0∈I,j0∈J, and the output is(5)k1=Fi0,j0,0=∑x=0u∑y=0vGxi0,0Hyj0,0kxy0.(2)
Step 1: afterkx1 acts on the object, the system output is generated. After comparing the output with the reference input, it is obtained that the input of the system controller is ix1=αi1ix0,yi1=βi1,j1jx0, and the output is(6)z2=Fx1,y1,0=∑i=0p∑j=0qAix1,0Bjy1,0zij1.(3)
Step z:kz acts on the object to generate the system output. After the output is compared with the reference input, the input ixz=αizix0,jxz=βiz,jzjx0 of the system controller is obtained; then, the controller output is(7)kz+1=kzFizαiz,jzβiz,jz,=γkz∑x=0u∑y=0vGxizαizHyjzβiz,jzkxy0.Sincez,iz≠0 at any time, there is limz⟶∞iz≠0,limz⟶∞jz≠0, that is, there is limz⟶∞αiz−1/αiz=1,limz⟶∞γiz−1/γiz=1. Then, when iz⟶0,jz⟶0, that is, the dual-input single-output system based on variable universe fuzzy control is convergent.


## 2.1.1. Basic Principles

Under the premise that the fuzzy rules remain unchanged, the scope of the variable universe fuzzy control system changes according to the change of the feedback error information. When the feedback error decreases, the reduction of the universe is equivalent to adding the number of rules to improve the control accuracy. When the feedback error increases, the increase of the universe is equivalent to reducing the number of rules and improving the convergence speed. Its essence is a dynamic point-by-point convergence interpolator.LetG=Gx1≤x≤u be a set of fuzzy sets on a given domain I and ix be the peak of Gx, if for any i∈I,(1)∑x=0tGxi=1,is established, and when w≠t, iw≠it,w,t∈1,2…,u, Gx is called the primitive of G, and G is called a primitive of I. Further, if i≠ix and any adjacent primitives Gx and Gx+1 satisfy(2)Aix≠0≠Ai+1x,Gxi≠0≠Gx+1i,Aix+Ai+1x=1,Gxi+Gx+1i=1,then G is called biphasic base tuple on I. In particular, if Gi is a linear function, G is called a biphasic linear basis on I. Considering the dual-input single-output manipulator system, the two input domains are given as I=−E,E and J=−D,D, where I represents the trajectory tracking error domain, J represents the tracking error change rate domain, and J is affected by I. G=Gxx=1,2,…,u and H=Hyy=1,2,…,v are biphasic linear basis tuples on universe I and J, respectively. The single-output universe is K=−P,P, which represents the uncertainty compensation torque, and C=kxy is the linear base on it.The control structure is shown in Figure1. z is the sampling time, and ixz,jxz,kxz are the peak values of ix,jx and fuzzy single value element kx at z time, respectively. Gxiz,z and Hyiz,z are the membership functions of deviation and deviation change rate at time z; then, a set of dynamic control rules can be obtained:(3)If is and isHyzthenknisCxyz.Figure 1
Structure diagram of variable universe fuzzy control system.Then, the control function can be written as(4)Fiz,jz,z=∑x=1u∑y=1vGxiz,zHyiz,zkxyz.


## 2.1.2. Dual-Input Single-Output Variable Universe Fuzzy Control Algorithm

The algorithm steps are as follows.(1)
Step 0: the initial value of the system input isi0∈I,j0∈J, and the output is(5)k1=Fi0,j0,0=∑x=0u∑y=0vGxi0,0Hyj0,0kxy0.(2)
Step 1: afterkx1 acts on the object, the system output is generated. After comparing the output with the reference input, it is obtained that the input of the system controller is ix1=αi1ix0,yi1=βi1,j1jx0, and the output is(6)z2=Fx1,y1,0=∑i=0p∑j=0qAix1,0Bjy1,0zij1.(3)
Step z:kz acts on the object to generate the system output. After the output is compared with the reference input, the input ixz=αizix0,jxz=βiz,jzjx0 of the system controller is obtained; then, the controller output is(7)kz+1=kzFizαiz,jzβiz,jz,=γkz∑x=0u∑y=0vGxizαizHyjzβiz,jzkxy0.Sincez,iz≠0 at any time, there is limz⟶∞iz≠0,limz⟶∞jz≠0, that is, there is limz⟶∞αiz−1/αiz=1,limz⟶∞γiz−1/γiz=1. Then, when iz⟶0,jz⟶0, that is, the dual-input single-output system based on variable universe fuzzy control is convergent.


## 2.2. Design of Manipulator Controller

### 2.2.1. Controller Structure Design

In order to overcome the influence of uncertain factors and unknown disturbances on the trajectory tracking performance of manipulator, a composite controller was designed, composed of a computational torque controllerτc and a variable theory domain fuzzy compensation controller pfe,e˙. As shown in Figure 2, vd represents the expected value, v represents the actual value, and the input torque of the manipulator is τ=τc+pfe,e˙.Figure 2
Controller structure block diagram.As can be seen from Figure2, the controller's input is the trajectory tracking error e and its change rate ec becomes a dimensionless value et after corresponding quantization factors Ze and Zec, and their changes lead to the change of expansion factor αi,βi,j,γi. The controller's output is converted into pfe,e˙ by the proportional factor Zp, and the output τc of the calculated torque constitutes the control torque of the manipulator.


### 2.2.2. Control Policy

Letv,v˙,v¨ be the actual measured values of angular displacement, angular velocity, and angular acceleration of the manipulator under consideration, respectively, and vd,v˙d are the desired angular displacement and angular velocity; then, the tracking error can be taken as(8)en=vn−vdn,e˙n=v˙n−v˙dn.Obviously, when there is no modeling uncertainty and external interference, if the error system(9)e¨+zue+zqe˙=0,is established, then each joint of the manipulator can accurately track the desired trajectory, where zu,zq are undetermined coefficients. In addition, the nominal dynamic model of the manipulator is(10)D0vv¨d+C0v,v˙v˙d+A0v=τc+τd.For nominal system (10), its control law is(11)τc=Dvv¨d−zue−zqe˙+Cv,v˙v˙d+Av.Then, the above formula can be obtained simultaneously.(12)e¨+zue+zqe˙=D0−1vΔDv¨d+ΔCv˙d+ΔA+τd.According to (12), the uncertainty contained in the controller will affect the stability of the system, so it needs to be compensated. The uncertain term in (12) is(13)a=D0−1vΔDv¨d+ΔCv˙d+ΔA+τd.To compensate for the influence on system stability,a was identified. Therefore, the control law given in (11) is modified as(14)τ=D0vv¨d−zue−zqe˙+C0v,v˙+A0v−D0va.On this basis, the variable theory domain adaptive fuzzy compensatorpfe,e˙ is used to approximate the uncertain term a. Let a^i be the estimate of a, and its expression is(15)aˆi≜aˆi|Θ=Θξ.The adaptive fuzzy compensation termpfe,e˙ in the control system can be defined as(16)pfe,e˙=−D0va^i,where ξ is a fuzzy basis function vector, Θ is a weight vector, and its optimal value Θ∗ satisfies(17)Θ∗=argminM∈Ωfsupi∈Pa^i|Θ−ai.According to (11) and (16), (14) can be further written as(18)τ=τc+pfe,e˙.Substitute it into (12) and obtain(19)e¨+zue+zqe˙=Θ˜ξ+ε,where ε is the approximation error term of the adaptive fuzzy compensator. If the output error vector of the manipulator is i=eN,e˙N, the state space expression of the closed-loop error equation of the system shown in (19) is(20)i˙=Gi+Ha.Among them,G=0X−zU−zQ,H=0X,a=Θ˜ξ+ε.Combining (18)–(20), the uncertain term a can be further expressed as(21)ai|Θ=Θ∗ξ+ε.Substituting (21) into (16), the fuzzy output factor γk in the variable theory domain affects the fuzzy basis function vector. Therefore, the variable theory domain adaptive compensator can be further expressed as(22)pfe,e˙=−D0vγkai|Θ≜−D0vΘ∗ξi|γk.Take the Lyapunov function as(23)Q=−12iNUi+12ηΘ˜NΘ˜,where U is a positive definite matrix and satisfies the following equation.(24)UG+GNU+V=0,where V≥0. Let Θ˜=Θ^−Θ∗, where Θ^ is an estimate of Θ∗. From the adaptive control law,(25)Θ˜˙=ηξi|γiNUH.The derivative of (23) can be obtained:(26)Q˙=−12iNVi−ξNΘ˜HNUi−εNHNUi+1ηΘ˜˙NΘ˜.Sinceε is the minimum approximation error, according to the approximation theory of fuzzy control, as the eigenvalue of V increases, the eigenvalue of U decreases, and ε can be sufficiently small, which can be obtained from (26).(27)Q˙≤−ξNi|γΘ˜HNUi+1ηΘ˜˙NΘ˜.Substitute adaptive control law (25) into (27) and obtain(28)Q˙≤−ξNi|γΘ˜HNUi+1ηηξi|γiNUHNΘ˜=0.So, whenΘ˜˙=ηξx|γiNUH, Q˙≤0. In other words, under this condition, the manipulator system based on variable theory domain adaptive fuzzy control is asymptotically stable, and the tracking effect is better with the smaller upper bound of fuzzy control error ε.


## 2.2.1. Controller Structure Design

In order to overcome the influence of uncertain factors and unknown disturbances on the trajectory tracking performance of manipulator, a composite controller was designed, composed of a computational torque controllerτc and a variable theory domain fuzzy compensation controller pfe,e˙. As shown in Figure 2, vd represents the expected value, v represents the actual value, and the input torque of the manipulator is τ=τc+pfe,e˙.Figure 2
Controller structure block diagram.As can be seen from Figure2, the controller's input is the trajectory tracking error e and its change rate ec becomes a dimensionless value et after corresponding quantization factors Ze and Zec, and their changes lead to the change of expansion factor αi,βi,j,γi. The controller's output is converted into pfe,e˙ by the proportional factor Zp, and the output τc of the calculated torque constitutes the control torque of the manipulator.


## 2.2.2. Control Policy

Letv,v˙,v¨ be the actual measured values of angular displacement, angular velocity, and angular acceleration of the manipulator under consideration, respectively, and vd,v˙d are the desired angular displacement and angular velocity; then, the tracking error can be taken as(8)en=vn−vdn,e˙n=v˙n−v˙dn.Obviously, when there is no modeling uncertainty and external interference, if the error system(9)e¨+zue+zqe˙=0,is established, then each joint of the manipulator can accurately track the desired trajectory, where zu,zq are undetermined coefficients. In addition, the nominal dynamic model of the manipulator is(10)D0vv¨d+C0v,v˙v˙d+A0v=τc+τd.For nominal system (10), its control law is(11)τc=Dvv¨d−zue−zqe˙+Cv,v˙v˙d+Av.Then, the above formula can be obtained simultaneously.(12)e¨+zue+zqe˙=D0−1vΔDv¨d+ΔCv˙d+ΔA+τd.According to (12), the uncertainty contained in the controller will affect the stability of the system, so it needs to be compensated. The uncertain term in (12) is(13)a=D0−1vΔDv¨d+ΔCv˙d+ΔA+τd.To compensate for the influence on system stability,a was identified. Therefore, the control law given in (11) is modified as(14)τ=D0vv¨d−zue−zqe˙+C0v,v˙+A0v−D0va.On this basis, the variable theory domain adaptive fuzzy compensatorpfe,e˙ is used to approximate the uncertain term a. Let a^i be the estimate of a, and its expression is(15)aˆi≜aˆi|Θ=Θξ.The adaptive fuzzy compensation termpfe,e˙ in the control system can be defined as(16)pfe,e˙=−D0va^i,where ξ is a fuzzy basis function vector, Θ is a weight vector, and its optimal value Θ∗ satisfies(17)Θ∗=argminM∈Ωfsupi∈Pa^i|Θ−ai.According to (11) and (16), (14) can be further written as(18)τ=τc+pfe,e˙.Substitute it into (12) and obtain(19)e¨+zue+zqe˙=Θ˜ξ+ε,where ε is the approximation error term of the adaptive fuzzy compensator. If the output error vector of the manipulator is i=eN,e˙N, the state space expression of the closed-loop error equation of the system shown in (19) is(20)i˙=Gi+Ha.Among them,G=0X−zU−zQ,H=0X,a=Θ˜ξ+ε.Combining (18)–(20), the uncertain term a can be further expressed as(21)ai|Θ=Θ∗ξ+ε.Substituting (21) into (16), the fuzzy output factor γk in the variable theory domain affects the fuzzy basis function vector. Therefore, the variable theory domain adaptive compensator can be further expressed as(22)pfe,e˙=−D0vγkai|Θ≜−D0vΘ∗ξi|γk.Take the Lyapunov function as(23)Q=−12iNUi+12ηΘ˜NΘ˜,where U is a positive definite matrix and satisfies the following equation.(24)UG+GNU+V=0,where V≥0. Let Θ˜=Θ^−Θ∗, where Θ^ is an estimate of Θ∗. From the adaptive control law,(25)Θ˜˙=ηξi|γiNUH.The derivative of (23) can be obtained:(26)Q˙=−12iNVi−ξNΘ˜HNUi−εNHNUi+1ηΘ˜˙NΘ˜.Sinceε is the minimum approximation error, according to the approximation theory of fuzzy control, as the eigenvalue of V increases, the eigenvalue of U decreases, and ε can be sufficiently small, which can be obtained from (26).(27)Q˙≤−ξNi|γΘ˜HNUi+1ηΘ˜˙NΘ˜.Substitute adaptive control law (25) into (27) and obtain(28)Q˙≤−ξNi|γΘ˜HNUi+1ηηξi|γiNUHNΘ˜=0.So, whenΘ˜˙=ηξx|γiNUH, Q˙≤0. In other words, under this condition, the manipulator system based on variable theory domain adaptive fuzzy control is asymptotically stable, and the tracking effect is better with the smaller upper bound of fuzzy control error ε.


## 3. Result Analysis and Discussion

### 3.1. Establishment of Mechanical Arm Structure Model

FANUC100i six-degree-of-freedom manipulator is selected in the experiment, and J1–J6 are defined as six steering joints of the manipulator system. Firstly, the manipulator system is modeled and analyzed by MATLAB software, and the D-H parameters of the system are determined. In the manipulator link coordinate system,hi is the length of each link, bi is the offset of each link, and βi is the rotation angle of the joint. Limited by the range of motion of the manipulator, the corresponding range needs to be set for the variation of the offset distance of the connecting rod and the selection angle of the joint. The specific parameters of D-H are shown in Table 1.Table 1
D-H parameters of hybrid manipulator.
Connecting rodJoint angle (°)Offset distance (mm)Range of joint angle (°)1β1350(−180, 180)2β20(−180, − 90)3β30(−90, 120)4β4550(−180, 180)5β50(−120, 120)6β60(−90, 150)


### 3.2. Tracking Deviation of Manipulator Joint Curve

Input the interpolation point data of the manipulator generated by the interpolation algorithm into the MATLAB software to generate the motion trajectory of the manipulator, as shown in Figure3.Figure 3
Trajectory simulation of manipulator.During the movement of the manipulator, the compound control scheme is used to participate in the real-time deviation correction of the motion trajectory. In the experiment, the deviation degree between the actual movement curve and the expected curve of the six joints under different control algorithms is analyzed. The movement and rotation data of joints 1–6 are statistically analyzed, as shown inFigures4–6.Figure 4
Joint 1 and joint 2 curves of manipulator.Figure 5
Joint 3 and joint 4 curves of manipulator.Figure 6
Joint 5 and joint 6 curves of manipulator.Contrast to the actual control effect of various trajectory control algorithms, the control effect of the compound control scheme in this paper is closer to the theoretical expected curve trajectory, and the tracking deviation is obviously less than the control scheme in literature [22] and literature [23]. The higher the complexity of the manipulator, the larger the deviation of the traditional scheme in the complex dynamic displacement and joint rotation. Especially in the first joint and the end joint, there is a greater control error, which directly affects the operation accuracy of the end effector. The compound control scheme combining the nominal calculated torque controller and variable universe fuzzy compensator can realize the step-by-step control of multiple joints and ensure the control accuracy of the manipulator.


### 3.3. Algorithm Complexity Analysis

On the one hand, the complexity of the algorithm in this paper will affect the manipulator control's response time and affect the applicability of the algorithm. The lower the complexity of the algorithm, the shorter the response time. The specific verification process is as follows.(1)
Set the time point at the beginning of the manipulator trajectory control movement and start timing.(2)
Solve the time interval at the adjacent interpolation points and record the response time and interval time in 10 times. The faster the response time and the shorter the intermediate interval, the better the applicability of the composite control scheme and the lower the complexity.The data results based on MATLAB are shown in Tables2 and 3.Table 2
Comparison of response time of fault-tolerant algorithms.
Number of executionsAutomatic generation timeTime consumption between jointsProposedLiterature [24]Literature [25]ProposedLiterature [24]Literature [25]10.310.440.530.250.411.2220.290.650.460.130.461.3930.30.560.560.150.430.2540.320.430.230.260.360.6250.310.710.330.140.380.6960.290.550.860.260.610.9570.280.771.060.130.590.5380.301.051.320.160.610.5190.310.640.620.250.610.66100.310.740.540.20.560.57Average value0.3020.6540.6510.1930.5020.739Table 3
Comparison of maximum time intervals.
Number of executionsAutomatic generation timeTime consumption between jointsProposedLiterature [24]Literature [25]ProposedLiterature [24]Literature [25]10.010.071.010.060.220.1820.020.070.990.040.270.2730.030.111.010.050.130.2740.030.110.050.050.220.0750.020.080.060.060.140.1160.010.060.040.031.010.2470.030.090.070.040.920.2280.030.0860.050.040.810.4390.030.090.080.030.990.37100.040.111.010.050.670.25Average value0.0240.0890.4790.0460.5380.241Proposed average value of the ten executions.In terms of system response time, the automatic generation time and interjoint acceleration time of this algorithm are 0.304 and 0.194 s, respectively, which are greatly reduced compared with the traditional methods. This is mainly due to the better control effect of the manipulator in the initial moving state. The system has better training effect on motion position movement data and joint rotation data. The complexity of network parameter selection is lower, which ensures faster system response speed.Under this algorithm, the interval between information generation and instruction transmission between joints is very short. The automatic generation strategy of system instructions reduces the unnecessary parameter transmission between joints, reduces the burden of message transmission, and effectively makes up for the problem of weak real time and applicability of the system caused by the deviation correction of this algorithm.


### 3.4. Pose Accuracy Test

The pose accuracy of the method in this paper, the method in literature [24], and the method in literature [25] is tested, and the test results are shown in Figure 7.Figure 7
Pose accuracy test. (a)X-axis coordinate value of manipulator. (b) Y-axis coordinate value of manipulator. (c) Z-axis coordinate value of manipulator.
(a)(b)(c)It can be seen from Figure7 that the pose accuracy of the method in this paper is higher than that of the method in literature [24] and literature [25], the method in this paper is basically consistent with the real pose of the manipulator, and the result shows effective improvement of the pose accuracy.


## 3.1. Establishment of Mechanical Arm Structure Model

FANUC100i six-degree-of-freedom manipulator is selected in the experiment, and J1–J6 are defined as six steering joints of the manipulator system. Firstly, the manipulator system is modeled and analyzed by MATLAB software, and the D-H parameters of the system are determined. In the manipulator link coordinate system,hi is the length of each link, bi is the offset of each link, and βi is the rotation angle of the joint. Limited by the range of motion of the manipulator, the corresponding range needs to be set for the variation of the offset distance of the connecting rod and the selection angle of the joint. The specific parameters of D-H are shown in Table 1.Table 1
D-H parameters of hybrid manipulator.
Connecting rodJoint angle (°)Offset distance (mm)Range of joint angle (°)1β1350(−180, 180)2β20(−180, − 90)3β30(−90, 120)4β4550(−180, 180)5β50(−120, 120)6β60(−90, 150)


## 3.2. Tracking Deviation of Manipulator Joint Curve

Input the interpolation point data of the manipulator generated by the interpolation algorithm into the MATLAB software to generate the motion trajectory of the manipulator, as shown in Figure3.Figure 3
Trajectory simulation of manipulator.During the movement of the manipulator, the compound control scheme is used to participate in the real-time deviation correction of the motion trajectory. In the experiment, the deviation degree between the actual movement curve and the expected curve of the six joints under different control algorithms is analyzed. The movement and rotation data of joints 1–6 are statistically analyzed, as shown inFigures4–6.Figure 4
Joint 1 and joint 2 curves of manipulator.Figure 5
Joint 3 and joint 4 curves of manipulator.Figure 6
Joint 5 and joint 6 curves of manipulator.Contrast to the actual control effect of various trajectory control algorithms, the control effect of the compound control scheme in this paper is closer to the theoretical expected curve trajectory, and the tracking deviation is obviously less than the control scheme in literature [22] and literature [23]. The higher the complexity of the manipulator, the larger the deviation of the traditional scheme in the complex dynamic displacement and joint rotation. Especially in the first joint and the end joint, there is a greater control error, which directly affects the operation accuracy of the end effector. The compound control scheme combining the nominal calculated torque controller and variable universe fuzzy compensator can realize the step-by-step control of multiple joints and ensure the control accuracy of the manipulator.


## 3.3. Algorithm Complexity Analysis

On the one hand, the complexity of the algorithm in this paper will affect the manipulator control's response time and affect the applicability of the algorithm. The lower the complexity of the algorithm, the shorter the response time. The specific verification process is as follows.(1)
Set the time point at the beginning of the manipulator trajectory control movement and start timing.(2)
Solve the time interval at the adjacent interpolation points and record the response time and interval time in 10 times. The faster the response time and the shorter the intermediate interval, the better the applicability of the composite control scheme and the lower the complexity.The data results based on MATLAB are shown in Tables2 and 3.Table 2
Comparison of response time of fault-tolerant algorithms.
Number of executionsAutomatic generation timeTime consumption between jointsProposedLiterature [24]Literature [25]ProposedLiterature [24]Literature [25]10.310.440.530.250.411.2220.290.650.460.130.461.3930.30.560.560.150.430.2540.320.430.230.260.360.6250.310.710.330.140.380.6960.290.550.860.260.610.9570.280.771.060.130.590.5380.301.051.320.160.610.5190.310.640.620.250.610.66100.310.740.540.20.560.57Average value0.3020.6540.6510.1930.5020.739Table 3
Comparison of maximum time intervals.
Number of executionsAutomatic generation timeTime consumption between jointsProposedLiterature [24]Literature [25]ProposedLiterature [24]Literature [25]10.010.071.010.060.220.1820.020.070.990.040.270.2730.030.111.010.050.130.2740.030.110.050.050.220.0750.020.080.060.060.140.1160.010.060.040.031.010.2470.030.090.070.040.920.2280.030.0860.050.040.810.4390.030.090.080.030.990.37100.040.111.010.050.670.25Average value0.0240.0890.4790.0460.5380.241Proposed average value of the ten executions.In terms of system response time, the automatic generation time and interjoint acceleration time of this algorithm are 0.304 and 0.194 s, respectively, which are greatly reduced compared with the traditional methods. This is mainly due to the better control effect of the manipulator in the initial moving state. The system has better training effect on motion position movement data and joint rotation data. The complexity of network parameter selection is lower, which ensures faster system response speed.Under this algorithm, the interval between information generation and instruction transmission between joints is very short. The automatic generation strategy of system instructions reduces the unnecessary parameter transmission between joints, reduces the burden of message transmission, and effectively makes up for the problem of weak real time and applicability of the system caused by the deviation correction of this algorithm.


## 3.4. Pose Accuracy Test

The pose accuracy of the method in this paper, the method in literature [24], and the method in literature [25] is tested, and the test results are shown in Figure 7.Figure 7
Pose accuracy test. (a)X-axis coordinate value of manipulator. (b) Y-axis coordinate value of manipulator. (c) Z-axis coordinate value of manipulator.
(a)(b)(c)It can be seen from Figure7 that the pose accuracy of the method in this paper is higher than that of the method in literature [24] and literature [25], the method in this paper is basically consistent with the real pose of the manipulator, and the result shows effective improvement of the pose accuracy.


## 4. Conclusion

In this paper, a compound control scheme combining nominal calculated torque controller and variable universe fuzzy compensator is designed, in which the variable universe adaptive fuzzy controller is used to estimate and compensate the modeling uncertainty, and the calculated torque controller is used to solve the nominal system control torque. The adaptive law designed for the scaling factor can make the controller suitable for different regions of the state space at different times. By improving the system's response speed, the steady-state tracking error of the system is greatly reduced. At the same time, the minimal vibration also ensures that the tracking process is more stable and flexible and effectively improves the trajectory tracking performance of the system. By constructing the system model, the performance of the algorithm is verified. The results show that this model has obvious advantages in trajectory deviation control, algorithm complexity, and pose accuracy. In the future, how to ensure the stability of fuzzy control system, that is, how to solve the problems of stability and robustness in fuzzy control, will be the focus of the next research.



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*Source: 3893937-2022-06-14.xml*