# Trajectory Tracking Robust Control Method Based on Finite-Time Convergence of Manipulator with Nonsingular Fast Terminal Sliding Mode Surface

Authors: Hanqing Sun; Shijie Zhang; Qingle Quan; Zheng Liu; Guizhi Wang; Weimin Lian
Journal: Journal of Control Science and Engineering
 (2022)

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## Abstract

In order to make the manipulator track the desired trajectory in a finite time, a new control method based on fast nonsingular terminal sliding mode has been designed. This method combines the traditional fast terminal sliding mode with the nonsingular terminal sliding mode, has rapidity, nonsingularity, finite-time convergence, and strong robustness, and can effectively suppress the inherent chattering phenomenon of the sliding mode controller. The nonsingular fast terminal sliding mode surface is used to accelerate the convergence speed of manipulator trajectory tracking error, and the singularity problem in terminal sliding mode is solved. The finite-time convergence of the algorithm is proved by the Lyapunov function. The simulation results demonstrate that the proposed method can achieve accurate finite-time trajectory tracking characteristics and has robustness against external disturbances.


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## Body

## 1. Introduction

Realizing the fast and high-precision tracking of the desired trajectory has always been one of the key issues in the research of the trajectory tracking control of the manipulator. Considering that the dynamic model of the manipulator is a strongly coupled nonlinear system with time-varying parameters, and under the influence of uncertain factors such as external disturbances and friction, many traditional control methods, such as PD control, adaptive control, optimal control, model predictive control, etc., have limited tracking performance, making it difficult to meet the control requirements of the actual manipulator [1–7].The sliding mode control method is a nonlinear control algorithm with strong robustness that can eliminate internal coupling and realize decoupling. It is widely used in the control of manipulators [8–10]. However, the traditional sliding mode control can only ensure the progressive convergence of the system state and cannot guarantee that the system state error converges to zero within a finite time, which limits its application in actual industrial production. In order to overcome the above shortcomings, the terminal sliding mode control method has been proposed and developed in recent years to realize high-precision trajectory tracking in finite time and suppress uncertainty [11–17].At present, in the process of practical application, there are still some problems to be solved in the terminal sliding mode control method. First of all, terminal sliding mode control often brings chattering problems [18] and singular problems [19]. Chattering problem refers to the high-frequency chattering phenomenon that occurs when the controller passes through the sliding plane. This is due to the noncontinuous switching items in the sliding mode control signal. In practice, the switching operation time cannot be ignored and is easily affected by model parameters. Uncertainty and the influence of external interference. Chattering will excite the unmodeled high-frequency characteristics of the system, which will lead to a decrease in control performance; the singular problem means that the designed control signal will become infinite at certain moments, which is due to the fact that the terminal sliding mode control often contains negative fractional power. Since the actual driving function provided by the motor is bounded, the design of the control signal with singular problems is obviously inappropriate, which will not only deteriorate the control performance of the system but also seriously damage the driving equipment. In addition, the traditional terminal sliding mode control has a slower convergence speed when the system state is far from the equilibrium point.In response to the above problems, a fast nonsingular terminal sliding mode control strategy for manipulators is proposed in this paper. Firstly, an improved fast terminal sliding surface is designed to improve the convergence speed of the system state away from the equilibrium point based on the traditional terminal sliding surface. Secondly, considering the chattering phenomenon in the design of the sliding mode controller, an adaptive chattering suppression algorithm is proposed. Then, the sliding mode control law is designed based on the equivalent control theory, and the negative fractional power term in the control law is suppressed by the saturation function to avoid singularity. Finally, the simulation results verify the effectiveness of the designed control method.


## 2. Problem Formulation

A simplified dynamic model of then-joint manipulator considering the existence of external disturbances and inertia uncertainties can be expressed as(1)M0∗q¨+C0∗q,q˙q˙+Δq,q˙+d=τ,where(2)Δq,q˙=ΔMq¨+ΔCq,q˙q˙,M0∗ and C0∗q,q˙ are the estimation terms of the manipulator dynamics model, Δq,q˙ is the uncertainty term of the manipulator model, and d denotes the bounded unknown disturbances.Equation (1) can be rewritten as(3)M0∗q¨+C0∗q,q˙q˙=τ+ρ,where ρ=−ΔMq¨−ΔCq,q˙q˙+d denotes the external disturbance and modeling error of the system. Assuming ρ is a bounded function, and(4)ρ<b0+b1q+b2q˙2,where b0>0,b1>0 and b2>0 are constants. Let qd be the desired trajectory, then the error function can be written as(5)e=q−qd,e˙=q˙−q˙d.The state equation of the system can be written as(6)x˙=fx,u,where(7)x=e,e˙T=x1,x2T.The state equation of the system is(8)x˙=f=f1f2=e˙M0∗−1u+M0∗−1ρ−M0∗−1C∗q˙−q¨d,where(9)M0∗−1ρ≤M0∗−1b0+b1q+b2q˙2.Problem 1.
To find a feedback control lawu for the system (8) to guarantee that all the system states of the closed-loop system are uniformly ultimately bounded, and the tracking errors converge into small regions in finite time.


## 3. Design of Trajectory Tracking Controller

### 3.1. Analysis and Design of Sliding Surface

The sliding surfaces of terminal sliding mode control (TSM) and nonsingular terminal sliding mode control (NTSM) are designed respectively as follows:(9a)s=x˙+ksignax=0,(9b)s=x+k′signa′x˙=0,where, k>0,0<a<1, k′>0,1<a′<2, signax=signx·xa.Ifk′=k−a′ and a′=a−1, the above two sliding surfaces are equal. Then it can be concluded that the equilibrium point (9a) and (9b) is globally finite-time stable. For any given initial state x0, the system state converges to 0 in finite time(10a)TTSM=x01−ak1−a,(10b)TNTSM=k′1/a′·x01−1/a′1−1/a′,and when t>TTSM or t>TNTSM, x=0.The sliding surface of fast terminal sliding mode control (FTSM) is described by the following differential equation:(11)s=x˙+k1signa1x+k2signa2x=0.where k1>0,k2>0,a1≥1,0<a2<1, a2=p/q. p, and q is odd and p<q.By solving the differential equation, it can be seen that the system state reachesx=0 in finite time(12)TFTSM=∫0x01k1xa1+k2xa2dx=x01−a11−a1k11−a1/a1F1,a1−1a1−a2,2a1−a2−1a1−a2,−k2k1−1x0a2−a1.where F· is a Gaussian hypergeometric function. The parameters a1, a2, k1, and k2 can make F· converge.If the system state is far from the equilibrium point,k1signa1x has a greater impact on the system than k2signa2x. At this time, equation (11) can be written as s=x˙+k1signa1x=0 to ensure that the sliding surface can converge quickly.When the system state approaches the equilibrium point,k2signa2x has a greater impact on the system than k1signa1x. The term k2signa2x determines the finite-time convergence. Therefore, the fast sliding surface will converge at a faster speed.Equation (11) can be written as follows:(13)s=x˙+k¯1sign1/a¯2x+k¯2signa¯1x=0.If the system state is far from the equilibrium point, equation (13) can be approximately written as(14)s1=x˙+k¯1k¯2signa¯1/a¯2x=0.where, a¯1/a¯2>1.If the system state approaches the equilibrium point, equation (13) can be approximately written as(15)s2=x˙+k¯1sign1/a¯2x=0,which is the same form as (9a).Equation (13) reaches the equilibrium point within the time(16)TFTSM′=∫0x01k¯1x+k¯2xa¯11/a¯2dx=a¯2x01−1/a¯2k¯1a¯2−1F1a¯2,a¯2−1a¯1−1a¯2,1+a¯2−1a¯1−1a¯2,−k¯2x0a¯1−1.Using the sliding surface designed above can ensure faster convergence speed, but due to the negative exponential term in the derivation ofs, it will lead to singular problems. A nonsingular fast terminal sliding mode surface is designed as follows:The nonsingular fast terminal sliding mode surface (NFTSM) is constructed as(17)s=x˙+k1′signa1′x+k2′φx=0,φx=r1x+r2x2signxx<εsigna2′x˙x≥ε,where k1′>0,k2′>0,1<a2′<2,a1′>a2′, r1=2−a2′εa2′−1,r2=a2′−1εa2′−2, and ε>0 are constants.By a direct calculation, the time derivative ofs is(18)s˙=x¨+k1′a1′xa1′−1x˙+k2′r1x˙+2r2xx˙signxx<ε,x¨+k1′a1′xa1′−1x˙+k2′a2′xa2′−1x˙x≥ε..According to the above analysis, the nonsingular fast terminal sliding mode surface (NFTSM) (17) and the derivative are continuous.Remark 1.
Ifx≥ε, s=x˙+k1′signa1′x+k2′signa2′x˙, the sliding surface is transformed into a structure similar to equation (11) and has similar convergence characteristics. When the system states are far from the equilibrium point, the term k1′signa1′x in equation (17) plays a leading role to ensure rapid convergence. When the system states approach the equilibrium point, the term k2′signa2′x˙ in equation (17) plays a major role and the system is stable in a finite time. The system maintains a fast convergence rate throughout the dynamic process.Remark 2.
In a small neighborhood of the originx<ε, the sliding surface s=x˙+k1′signa1′x+k2′r1x+r2x2signx and the derivative of s do not contain negative exponential power terms, which avoids singularity.


### 3.2. Design and Stability Analysis of Tracking Controller

For Problem1, a nonsingular fast terminal sliding mode surface is constructed as follows:(19)s=x˙+Λ1signΓ1x1+Λ2φx˙1,φx=r1x1+r2x12signx1x1<εsignΓ2x˙1x1≥ε,Λ1=diagλ11,λ12,…,λ1n,Λ2=diagλ21,λ22,…,λ2n,Γ1=diagγ11,γ12,…,γ1n,Γ2=diagγ21,γ22,…,γ2n,signΓ1x1=diagsignx1x1Γ1=signγ11x11,signγ12x12,…,signγ1nx1n,signΓ2x˙1=diagsignx˙1x˙1Γ2=signγ21x˙11,signγ22x˙12,…,signγ2nx˙1n.where λ1i>0, λ2i>0, γ1i>γ2i, 1<γ2i<2, i=1,2,…,n. s is the switching function and s∈ℝn,Let(20)ddtsignΓ1x1=Γ1diagx1Γ1−In·x˙1,ddtsignΓ2x˙1=Γ2diagx˙1Γ2−In·x¨1where, In is the identity matrix.Using the sliding mode surface (17) of nonsingular fast terminal control, the control law is designed as follows:(21)u=−M0∗M2s+ρ′+M1ss+−M0∗−1C0∗q−q¨d+Γ2−1I2+Γ1diagx1Γ1−I2sign2Γ2−I2x2.where M1>0, M2>0, ρ′=M0∗−1b0+b1q+b2q˙2.Theorem 1.
For system (8), using the finite time nonsingular terminal sliding mode (14) designed in this paper and selecting the control law (20), the system state can reach the equilibrium point in finite time TNFTSM, which is related to the design parameters.(22)TNFTSM=∫0x0k′21/a2′x+k1′xa1′1/a2′dx=a2′x01−1/a2′k1′a2′−1·F1a2′,a2′−1a1′−1a2′,1+a2′−1a1′−1a2′,−k1′x0a1′−1.Proof.
The stability analysis is performed below. Firstly, the Lyapunov function is selected as(23)V=12sTs.
Derivation of equation (22) can be obtained(24)V˙=sTs˙=sTx˙1+Λ1Γ1diagx1Γ1−Inx˙1+Λ2Γ2diagx˙1Γ2−Inx¨1.
x¨1 can be expressed as(25)x¨1=ddtf1=df1dx1f1+df1dx2f2.
Then equation (23) can be expressed as(26)V˙=sTs˙=sTx˙1+Λ1Γ1diagx1Γ1−Inx˙1+Λ2Γ2diagx˙1Γ2−Indf1dx1f1+df1dx2f2.
Substituting the controller into equation (25), there is(27)V˙=−sT·Λ2Γ2·diagx˙1Γ2−InM1ss+M2s+ρ′ss−H0∗−1ρ≤−mini=1,…,nλ2iγ2ix˙iγ2−1M1s+M2sTs.
Let(28)σ1x˙1=mini=1,⋯,nλ2iγ2ix˙iγ2−12M1,σ2x˙1=mini=1,⋯,nλ2iγ2ix˙iγ2−12M2.
Then equation (26) can be rewritten as(29)V˙≤−σ1V1/2−σ2V.
It can be seen from equation (29), V˙≤0. The closed-loop system is stable.
Find the exponential reaching law satisfying the following conditions from the control law(30)s˙=−σ1∗ss−σ2∗s,where(31)σ1∗=M1Λ2Γ2·diagx˙1Γ2−In+ρ′−H0∗−1ρ·ss≥0,σ2∗=M2Λ2Γ2·diagx˙1Γ2−In≥0.
In order to suppress chattering, the following function can be used to replace the switching function in the control law:(32)ss+σ,where σ>0 is a small constant.


## 3.1. Analysis and Design of Sliding Surface

The sliding surfaces of terminal sliding mode control (TSM) and nonsingular terminal sliding mode control (NTSM) are designed respectively as follows:(9a)s=x˙+ksignax=0,(9b)s=x+k′signa′x˙=0,where, k>0,0<a<1, k′>0,1<a′<2, signax=signx·xa.Ifk′=k−a′ and a′=a−1, the above two sliding surfaces are equal. Then it can be concluded that the equilibrium point (9a) and (9b) is globally finite-time stable. For any given initial state x0, the system state converges to 0 in finite time(10a)TTSM=x01−ak1−a,(10b)TNTSM=k′1/a′·x01−1/a′1−1/a′,and when t>TTSM or t>TNTSM, x=0.The sliding surface of fast terminal sliding mode control (FTSM) is described by the following differential equation:(11)s=x˙+k1signa1x+k2signa2x=0.where k1>0,k2>0,a1≥1,0<a2<1, a2=p/q. p, and q is odd and p<q.By solving the differential equation, it can be seen that the system state reachesx=0 in finite time(12)TFTSM=∫0x01k1xa1+k2xa2dx=x01−a11−a1k11−a1/a1F1,a1−1a1−a2,2a1−a2−1a1−a2,−k2k1−1x0a2−a1.where F· is a Gaussian hypergeometric function. The parameters a1, a2, k1, and k2 can make F· converge.If the system state is far from the equilibrium point,k1signa1x has a greater impact on the system than k2signa2x. At this time, equation (11) can be written as s=x˙+k1signa1x=0 to ensure that the sliding surface can converge quickly.When the system state approaches the equilibrium point,k2signa2x has a greater impact on the system than k1signa1x. The term k2signa2x determines the finite-time convergence. Therefore, the fast sliding surface will converge at a faster speed.Equation (11) can be written as follows:(13)s=x˙+k¯1sign1/a¯2x+k¯2signa¯1x=0.If the system state is far from the equilibrium point, equation (13) can be approximately written as(14)s1=x˙+k¯1k¯2signa¯1/a¯2x=0.where, a¯1/a¯2>1.If the system state approaches the equilibrium point, equation (13) can be approximately written as(15)s2=x˙+k¯1sign1/a¯2x=0,which is the same form as (9a).Equation (13) reaches the equilibrium point within the time(16)TFTSM′=∫0x01k¯1x+k¯2xa¯11/a¯2dx=a¯2x01−1/a¯2k¯1a¯2−1F1a¯2,a¯2−1a¯1−1a¯2,1+a¯2−1a¯1−1a¯2,−k¯2x0a¯1−1.Using the sliding surface designed above can ensure faster convergence speed, but due to the negative exponential term in the derivation ofs, it will lead to singular problems. A nonsingular fast terminal sliding mode surface is designed as follows:The nonsingular fast terminal sliding mode surface (NFTSM) is constructed as(17)s=x˙+k1′signa1′x+k2′φx=0,φx=r1x+r2x2signxx<εsigna2′x˙x≥ε,where k1′>0,k2′>0,1<a2′<2,a1′>a2′, r1=2−a2′εa2′−1,r2=a2′−1εa2′−2, and ε>0 are constants.By a direct calculation, the time derivative ofs is(18)s˙=x¨+k1′a1′xa1′−1x˙+k2′r1x˙+2r2xx˙signxx<ε,x¨+k1′a1′xa1′−1x˙+k2′a2′xa2′−1x˙x≥ε..According to the above analysis, the nonsingular fast terminal sliding mode surface (NFTSM) (17) and the derivative are continuous.Remark 1.
Ifx≥ε, s=x˙+k1′signa1′x+k2′signa2′x˙, the sliding surface is transformed into a structure similar to equation (11) and has similar convergence characteristics. When the system states are far from the equilibrium point, the term k1′signa1′x in equation (17) plays a leading role to ensure rapid convergence. When the system states approach the equilibrium point, the term k2′signa2′x˙ in equation (17) plays a major role and the system is stable in a finite time. The system maintains a fast convergence rate throughout the dynamic process.Remark 2.
In a small neighborhood of the originx<ε, the sliding surface s=x˙+k1′signa1′x+k2′r1x+r2x2signx and the derivative of s do not contain negative exponential power terms, which avoids singularity.


## 3.2. Design and Stability Analysis of Tracking Controller

For Problem1, a nonsingular fast terminal sliding mode surface is constructed as follows:(19)s=x˙+Λ1signΓ1x1+Λ2φx˙1,φx=r1x1+r2x12signx1x1<εsignΓ2x˙1x1≥ε,Λ1=diagλ11,λ12,…,λ1n,Λ2=diagλ21,λ22,…,λ2n,Γ1=diagγ11,γ12,…,γ1n,Γ2=diagγ21,γ22,…,γ2n,signΓ1x1=diagsignx1x1Γ1=signγ11x11,signγ12x12,…,signγ1nx1n,signΓ2x˙1=diagsignx˙1x˙1Γ2=signγ21x˙11,signγ22x˙12,…,signγ2nx˙1n.where λ1i>0, λ2i>0, γ1i>γ2i, 1<γ2i<2, i=1,2,…,n. s is the switching function and s∈ℝn,Let(20)ddtsignΓ1x1=Γ1diagx1Γ1−In·x˙1,ddtsignΓ2x˙1=Γ2diagx˙1Γ2−In·x¨1where, In is the identity matrix.Using the sliding mode surface (17) of nonsingular fast terminal control, the control law is designed as follows:(21)u=−M0∗M2s+ρ′+M1ss+−M0∗−1C0∗q−q¨d+Γ2−1I2+Γ1diagx1Γ1−I2sign2Γ2−I2x2.where M1>0, M2>0, ρ′=M0∗−1b0+b1q+b2q˙2.Theorem 1.
For system (8), using the finite time nonsingular terminal sliding mode (14) designed in this paper and selecting the control law (20), the system state can reach the equilibrium point in finite time TNFTSM, which is related to the design parameters.(22)TNFTSM=∫0x0k′21/a2′x+k1′xa1′1/a2′dx=a2′x01−1/a2′k1′a2′−1·F1a2′,a2′−1a1′−1a2′,1+a2′−1a1′−1a2′,−k1′x0a1′−1.Proof.
The stability analysis is performed below. Firstly, the Lyapunov function is selected as(23)V=12sTs.
Derivation of equation (22) can be obtained(24)V˙=sTs˙=sTx˙1+Λ1Γ1diagx1Γ1−Inx˙1+Λ2Γ2diagx˙1Γ2−Inx¨1.
x¨1 can be expressed as(25)x¨1=ddtf1=df1dx1f1+df1dx2f2.
Then equation (23) can be expressed as(26)V˙=sTs˙=sTx˙1+Λ1Γ1diagx1Γ1−Inx˙1+Λ2Γ2diagx˙1Γ2−Indf1dx1f1+df1dx2f2.
Substituting the controller into equation (25), there is(27)V˙=−sT·Λ2Γ2·diagx˙1Γ2−InM1ss+M2s+ρ′ss−H0∗−1ρ≤−mini=1,…,nλ2iγ2ix˙iγ2−1M1s+M2sTs.
Let(28)σ1x˙1=mini=1,⋯,nλ2iγ2ix˙iγ2−12M1,σ2x˙1=mini=1,⋯,nλ2iγ2ix˙iγ2−12M2.
Then equation (26) can be rewritten as(29)V˙≤−σ1V1/2−σ2V.
It can be seen from equation (29), V˙≤0. The closed-loop system is stable.
Find the exponential reaching law satisfying the following conditions from the control law(30)s˙=−σ1∗ss−σ2∗s,where(31)σ1∗=M1Λ2Γ2·diagx˙1Γ2−In+ρ′−H0∗−1ρ·ss≥0,σ2∗=M2Λ2Γ2·diagx˙1Γ2−In≥0.
In order to suppress chattering, the following function can be used to replace the switching function in the control law:(32)ss+σ,where σ>0 is a small constant.


## 4. Simulation

In this paper, sliding modes NFTSM and FTSM are simulated under the same controller parameters, initial conditions, and external disturbance conditions. A simplified dynamic model of the manipulator is expressed as(33)M0∗q¨+C0∗q,q˙q˙=τ+ρ,where(34)M0∗q=v+q01+2q02cosq2q01+q02cosq2q01+q02cosq2q01,C0∗q,q˙=−q02q˙2sinq2−q02q˙1+q˙2sinq2q02q˙1sinq20,ρ=3sin2πt2cosπt.and v=13.33,q01=8.72,q02=8.74.The desired tracking trajectory isq1d=cosπt,q2d=sinπt, and the initial states are q1q˙1q2q˙2=0.4−0.2−0.30.4.The sliding mode parameters and controller parameters are shown in Table1.Table 1
Sliding mode and controller design parameters.
Sliding modea1a2k1k2εFTSM21.52.52—NFTSM21.52.520.1Based on the proposed nonsingular fast terminal sliding mode control law (16), the parameters of the controller are selected as follows:(35)b0=b1=b2=2,Γ1=1.8001.8,Γ2=1.6001.6.The block diagram of the control system is shown in Figure1.Figure 1
Block diagram of the closed-loop control scheme.The simulation results are shown in Figures2–4 by using the NFTSM method.Figure 2
Position tracking of joint 1 and joint 2 (NFTSM).Figure 3
The tracking error of joint 1 and joint 2 (NFTSM).Figure 4
Control torque of joint1 and joint 2 (NFTSM).The position tracking of joint 1 and joint 2 are shown in Figure2. It can be observed that it takes 0.5 s to track the desired signal on each joint angle. The tracking error and the control input of joint 1 and joint 2 are shown in Figures 3 and 4, respectively. It can be seen that the manipulator can track the reference trajectory well, and the error signals converge to zero rapidly. The control torques are exhibited in Figure 4. From the simulation results, it can be concluded that the proposed method has robustness against external disturbances.The simulation results are shown in Figures5–7 by using the FTSM method.Figure 5
Position tracking of joint 1 and joint 2 (FTSM).Figure 6
The tracking error of joint 1 and joint 2 (FTSM).Figure 7
Control torque of joint1 and joint 2 (FTSM).The position tracking and the tracking error of joint 1 and joint 2 are shown in Figures5 and 6. It can be seen that it takes 2.1 s to track the desired signal on each joint angle. The time of tracking the desired trajectory in the FTSM method is longer than that in the NFTSM method. The control inputs of joint 1 and joint 2 are shown in Figure 7.Compared with the two control methods, although the manipulator can track the desired trajectory in a limited time, the convergence speed of NFTSM is obviously better than FTSM. It can also be seen that the proposed method can achieve better control performance with a lower control energy cost.


## 5. Conclusion

In this paper, a fast-nonsingular terminal sliding mode control method is designed to meet the requirements of manipulator trajectory tracking. The new sliding mode controller has the following advantages:(1)
Compared with the traditional sliding mode, it has fast convergence and solves the control singularity problem well.(2)
The control system can converge to the expected value in a finite time and has robustness.(3)
The new sliding mode controller can effectively suppress the inherent chattering phenomenon of the traditional sliding mode controller.The nonsingular finite-time sliding mode control method established in this paper has fast convergence speed, low energy consumption, and no chattering. It has good control quality and is convenient for practical application.



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*Source: 2271804-2022-04-07.xml*