# Fault-Tolerant Control forN-Link Robot Manipulator via Adaptive Nonsingular Terminal Sliding Mode Control Technology

Authors: Nannan Shi; Fanghui Luo; Zhikuan Kang; Lihui Wang; Zhuo Zhao; Qiang Meng; Weiqin Hou
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

An adaptive nonsingular terminal sliding mode control (ANTSMC) scheme for then-link robot manipulator is presented in this study, which can achieve faster convergence and higher precision tracking compared with the linear hyperplane-based sliding mode control. Novel adaptive updating laws based on the actual tracking error are employed to online adjust the upper bound of uncertainty, which comprehensively consider both the tracking performance and chattering eliminating problem. The stability analysis of the proposed ANTSMC is verified using the Lyapunov method with the existence of the parameter uncertainty and the actuator faults. Numerical simulation studies the comparison of performance between ANTSMC and the conventional nonsingular terminal sliding mode control (NTSMC) scheme to validate the advantages of the proposed control algorithm.


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## Body

## 1. Introduction

Robot manipulators play a pivotal role in the modern industrial field, whose dynamics are typically multi-input multioutput (MIMO) nonlinear systems. In practice, parameter variations and external disturbances are inevitable for the mechanical systems. Moreover, high accuracy and safety demands of the robot manipulator cause a greatly increasing attention to the unpredictable faults. Recently, many fault detection methods have been successfully employed for the robot manipulators [1–6], such as the prediction-error-based approach [1], second-order sliding mode observer [2, 3], nonlinear observer [4], optimal unknown input observer [6], and power consumption modeling based method [5]. Besides, the fault-tolerant control research applied in the robot manipulators has been a hot topic. Song introduced a multilayered feed-forward neural network to identify the faults for robot manipulator and then achieved the tracking performance in the presence of uncertainty [7]. Siqueira et al. [8] proposed an output-feedback H∞ fault-tolerant controller for robot manipulators by linearizing the dynamic model around the operation point. In 2009, Siqueira and Terra [9] further derived H2, H∞, and mixed H2/H∞ Markovian algorithms for the robot manipulator, but their drawbacks are still the linearization problems. Liu et al. [10] published a novel robust fixed-time fault-tolerant tracking controller for the uncertain robot manipulator. Hagh et al. [11] gave an active fault-tolerant control design for actuator fault mitigation in robotic manipulators. In this paper, we aim to develop a simple and effective passive fault-tolerant control algorithm for the robot manipulator without the fault detection and diagnosis (FDD) module.Sliding mode control (SMC) schemes are well known for their robustness to the parameter variation and disturbances and have been successfully adopted in many nonlinear systems [12–16]. To assure the finite-time convergence, the terminal sliding mode control (TSMC) has been derived to reach the equilibrium point within a finite time to realize high accuracy control [17]. Besides, the singularity problem of the conventional TSMC has been overcome by indirect [17, 18] and direct approaches [19]. Nowadays, researchers focus on the unknown functions estimated or designed for the TSMC. Since neural networks and fuzzy logic are universal function approximations, the mixed control algorithms combining them and the TSMC technologies have been extensively employed. Lin proposed an adaptive nonsingular TSMC method for the robotic systems using fuzzy wavelet networks and verified the corresponding controller on a six-link robot manipulator [20]. A dynamics estimation method based on an adaptive algorithm and fuzzy logic is introduced to the nonsingular TSMC for a class of MIMO uncertain nonlinear systems [21]. Xu et al. derived a TSMC based on adaptive fuzzy-neural observer for nonaffine nonlinear system [22]. Cao et al. also used the adaptive fuzzy algorithm to achieve an adaptive nonsingular TSMC for the fault-tolerant small satellite attitude control [23]. Moreover, other adaptive TSMC schemes using different adaptive updating laws without the neural networks and fuzzy logic have also been successfully adopted in many nonlinear systems, such as the electromechanical actuator [24], spacecraft formation flying [25], uncertain nonlinear single-input single-output (SISO) systems [26], and nonlinear differential inclusion systems [27]. However, for the robot manipulator, only the nonsingular TSMC control schemes have been developed [18, 28], though the upper bound of the uncertainty is usually unknown in most of the real systems (especially in the faulty conditions). In fact, the upper bound uncertainty of the nonsingular TSMC for the robot manipulator is conventionally designed as a fixed one, which will bring a trade-off problem between the control accuracy and the control chattering. Therefore, we hope to derive a novel adaptive nonsingular TSMC for the robot manipulator by combining an adaptive updating law to consider both the control accuracy and the control chattering.Motivated by the above discussion, an adaptive nonsingular TSMC scheme for the robot manipulators with the existence of the parameter uncertainty and the actuator faults is derived in this paper, which can online tune the upper bound of the uncertainty to ensure the high precise tracking performance, finite time convergence, and Lyapunov stability.


## 2. Problem Formulation

Then-link robot manipulator dynamics can be formulated as(1)Mqq¨+Cq,q˙+Gq=τ,where Mq, Cq,q˙, and Gq denote the inertia matrix, Coriolis and centrifugal forces, and gravitational torque, respectively, q represents the generalized coordinates, and τ is the joint torque.Considering the uncertainties and actuator faults, the dynamics of the uncertain robot manipulator can be rewritten as(2)M0+ΔMq¨+C0+ΔC+G0+ΔG=δaτ+δf,where M0, C0, and G0 represent the estimated terms; ΔM, ΔC, and ΔG denote the uncertain terms; δa and δf are the multiplicative and additive faults.Equation (2) can be further rearranged as(3)M0q¨+C0+G0=τ+δtotal,(4)δtotal=δaτ+δf−τ−ΔG−ΔC−ΔMq¨,where δtotal denotes the total uncertainty of the robot manipulator.


## 3. Nonsingular Terminal Sliding Mode Control (NTSMC)

In this section, the nonsingular terminal sliding mode control (NTSMC) proposed by Man and Yu [18] is briefly described, which solves the singular problem of the TSMC by direct approach.Lemma 1 (see [18]).
For the uncertainn-link manipulator (described by equation (3)), if the NTSMC manifold and its input command are formulated by equations (5)–(8) and Assumption 1 holds, then the system tracking error ε will converge to zero in finite time.(5)s=ε+Λε˙p/q,where(6)τntsmc=M0q¨d+C0+G0−qpM0Λ−1ε˙2−p/q−sTWTsTW2sWΓconstant,(7)Γconstant=b1+b2q+b3q˙2,(8)W=Λdiagε˙p/q−1M0−1,where ε=q−qd; q and qd represent the actual input and desired input, respectively; p and q are positive odd integers, which satisfy 1<p/q<2; and Λ=diagλ1,…,λn denotes a design matrix.Assumption 1.
The total uncertaintyδtotal of the robot manipulator can be bounded, which can be formulated as [18](9)δtotal<b1+b2q+b3q˙2,where b1, b2, and b3 are positive numbers.Remark 1.
Lemma1 gives a conventional NTSMC scheme for the robot manipulator based on Assumption 1, and the design parameters (b1, b2, and b3) of the upper bound Γconstant are constant. In practice, the high upper bound can obtain the high-quality tracking performance at the cost of serious chattering, and the low upper bound owns the opposite characteristic. Moreover, unpredictable faults may cause the great change of the uncertainty upper bound satisfying Assumption 1. Therefore, it is a hard task to choose an appropriate upper bound Γconstant for the NTSMC.


## 4. Adaptive Nonsingular Terminal Sliding Mode Control

To appropriately select the uncertainty upper bound of the NTSMC, adaptive updating methods are natural choices. Considering the mechanical systems, there are some special stages: (1) in the start-up phase, the initial trajectory error causes the large starting torque, which appears to be serious chattering phenomenon for the SMC control schemes; (2) in the stability phase, the tracking performance is affected by the system uncertainty; and (3) in the faulty condition, the uncertainty of the nonlinear system may generate great change. Therefore, the uncertainty upper bound of the NTSMC should be chosen to be a low one, a moderate one, and a high one for the above three stages, respectively.In this section, we introduce adaptive updating laws based on the tracking errors to online adjust the uncertainty upper bound, which are summarized as follows.Theorem 1.
For the uncertainn-link robot manipulator (described by equation (3)), if the adaptive nonsingular terminal sliding mode control (ANTSMC) manifold and its input command are formulated by equations (10)–(16) and Assumption 1 holds, then the system tracking error ε will converge to zero in finite time.(10)s=ε+Λε˙p/q,(11)τantsmc=M0q¨d+C0+G0−qpM0Λ−1ε˙2−p/q−sTWTsTW2sWΓ^adaptive,(12)W=Λdiagε˙p/q−1M0−1.Moreover, the updating laws are chosen as(13)Γ^adaptive=b^1+b^2q+b^3q˙2,(14)b^˙1=d1⋅pqWs,(15)b^˙2=d2⋅pqWsq,(16)b^˙3=d3⋅pqWsq˙2,where Γ^adaptive, b^1, b^2, and b^3 denote the adaptive estimated values and d1, d2, and d3 are constants.Proof.
Consider the following Lyapunov function:(17)Vantsmc=12sTs+1d1b˜1Tb˜1+1d2b˜2Tb˜2+1d3b˜3Tb˜3,where b˜1, b˜2, and b˜3 represent the mismatch between the actual and estimated parameters.(18)b˜1=b1−b^1,(19)b˜2=b2−b^2,(20)b˜3=b3−b^3.
DifferentiatingVantsmc with respect to time and substituting equation (10) and equations (18)–(20) into it yield(21)V˙antsmc=sTε˙+pqΛdiagε˙p/q−1ε¨−1d1b˜1Tb^˙1−1d2b˜2Tb^˙2−1d3b˜3Tb^˙3.
Substituting equation (3) into equation (21) gives(22)V˙antsmc=sTε˙+pqΛdiagε˙p/q−1M0−1τantsmc+δtotal−C0−G0−q¨d−1d1b˜1Tb^˙1−1d2b˜2Tb^˙2−1d3b˜3Tb^˙3.
Substituting equation (11) into equation (22), equation (22) can be further rearranged as(23)V˙antsmc=sTε˙+pqΛdiagε˙p/q−1−qpΛ−1ε˙2−p/qM0−1−sTWTsTW2sWΓ^adaptive+δtotal−1d1b˜1Tb^˙1−1d2b˜2Tb^˙2−1d3b˜3Tb^˙3=sTpqΛdiagε˙p/q−1M0−1−sTWTsTW2sWΓ^adaptive+δtotal−1d1b˜1Tb^˙1−1d2b˜2Tb^˙2−1d3b˜3Tb^˙3.
Substituting equation (12) into equation (23) yields(24)V˙antsmc=sTpqW−sTWTsTW2sWΓ^adaptive+δtotal−1d1b˜1Tb^˙1−1d2b˜2Tb^˙2−1d3b˜3Tb^˙3=−pqsWΓ^adaptive+pqsTWδtotal−1d1b˜1Tb^˙1−1d2b˜2Tb^˙2−1d3b˜3Tb^˙3.
Substituting equations (14)–(16) into equation (24), equation (24) can be rewritten as(25)V˙antsmc=−pqsWΓ^adaptive+b˜1T+b˜2Tq+b˜3Tq˙2+pqsTWδtotal.
Substituting equation (13) and equations (18)–(20) into equation (25) gives(26)V˙antsmc=−pqsWb1+b2q+b3q˙2+pqsTWδtotal.
Equation (25) can be further rearranged as(27)V˙antsmc≤pqsWδtotal−b1−b2q−b3q˙2.
Therefore,V˙antsmc<0 can be satisfied using Assumption 1. This completes the proof.Remark 2.
The initial uncertainty upper bound of the ANSMC can be designed as a low value, and it can adaptively change based on the actual tracking errors to consider both of the tracking performance and chattering phenomenon.Remark 3.
The convergence speed of the ANSMC is mainly affected by the choice of the parametersp, q, and Λ=diagλ1,…,λn. The controller tracking performances including the accuracy and chattering are mainly designed by the choice of the parameters (b1, b2, b3, b^1, b^2, and b^3 of the upper bound Γconstant).
Moreover, the finite-time converge characteristic of the NTSMC has been approved for trajectory stabilization of the SISO nonlinear systems in [18]; we further extended the corresponding study to the trajectory tracking of the MIMO systems.
Equation (10) leads to(28)si=εi+λiε˙ip/q,where ∗i denotes the ith element of ∗ and εi=qi−qd,i.
The final timets,i that is taken to travel from tr,i to εitr,i+ts,i=0 for the ith element of system states can be formulated by(29)ts,i=−λiq/p∫εitr,i0εi−q/pdεi=λiq/ppp−qεitr,i1−q/p,where tr,i is the time when the sliding mode sitr,i=0 is reached.
Therefore, the convergence time for the MIMO nonlinear systems can be given by(30)ts=maxts,1,…,ts,n.
This means that the system statesq will converge to the desired qd in finite time ts after the sliding mode s=0 is reached.Remark 4.
To eliminate chattering, the following algorithm is introduced for the NTSMC and ANTSMC control schemes at the cost of introducing tracking errors and deteriorating the perfect stability conditions.(31)sWsTW2=sWsTW+ξ2,where ξ is a small constant.Remark 5.
For the robot manipulator, the implementation complexity of the proposed ANSMC is mainly decided by the calculation accuracy of the estimated termsM0, C0, and G0. The estimated terms M0, C0, and G0 are calculated by the multibody dynamics of the robot manipulators. The more components of the robot manipulator are considered in the dynamic analysis of the robot system and will induce the higher implementation complexity of the proposed ANSMC controller. Generally speaking, the main components of the robot manipulator should be computed, such as the thirteen moving rigid bodies of the Stewart Platform [29]. Fortunately, the controller proposed in this paper can obtain a satisfactory performance by introducing adaptive updating laws to compensate the unmodeled dynamics influences. In practice, the calculation accuracy of the estimated terms M0, C0, and G0 can be chosen by comprehensively considering the difficulty of the robot multibody dynamic analysis and the implementation hardware platform capability.


## 5. Simulation Studies

A two-link robot manipulator is employed to evaluate the performance of the proposed ANTSMC control scheme. Its dynamic model can be formulated by equation (32) with the following parameters: r1, r2, J1, J2, m1, and m2 are 1 m, 0.8 m, 5 kg·m, 5 kg·m, 0.5 kg, and 1.5 kg, respectively [18].(32)Mrobqrobq¨rob+Crobqrob,q˙rob+Grobqrob=τrob,where(33)qrob=q1,q2T,M11=m1+m2r12+m2r22+2m2r1r2cosq2+J1,M12=M21=m2r22+m2r1r2cosq2,M22=m2r22+J2,C1=−m2r1r2sinq2q˙12+2q˙1q˙2,C2=m2r1r2sinq2q˙22,G1=m1+m2r1cosq2+m2r2cosq1+q2g,G2=m2r2cosq1+q2g,where Mij denotes the (i, j) element of the matrix Mrob and Ci and Gi represent the ith elements of Crob and Grob.The desired trajectoryqrob,r=q1,r,q2,rT and initial trajectory qrob,0 can be given by(34)q1,r=a1sinω1t+a2cosω2t+a3sinω3t+a4cosω4t+a5sinω5t+a6cosω6t,(35)q2,r=b1sinω1t+b2cosω2t+b3sinω3t+b4cosω4t+b5sinω5t+b6cosω6t,(36)qrob,0=1.5,−1.5T,where a1∼a6, b1∼b6, and ω1∼ω6 are selected as (1, 1, 0.01, 1, 0.01, 0.001), (−1, −1, −0.01, −1, −0.01, −0.001), and (0.1, 0.2, 0.4, 0.8, 1.6, 3.2), respectively.The estimated values of the dynamic parameters are assumed to be(37)m^1=0.4kg,m^2=1.2kg,J^1=4kg⋅m,J^2=4kg⋅m.Therefore, three controllers (NTSMC1, NTSMC2, and ANTSMC) are selected to verify the advantages of the proposed ANTSMC. To fairly compare these controllers, their control parameters except the uncertainty upper bound are chosen as the same.(38)Λ=15000150,q=5,p=9,ξ=0.001,τmax=40,where τmax denotes the maximum torque of the robot manipulator.Moreover, the appropriate constant uncertainty upper bound satisfying Assumption1 is hard to determine in practice, especially for the existing kinds of uncertainties. Two uncertainty upper bound designs of NTSMC are selected: the lower one (NTSMC1) and the higher one (NTSMC2). Additionally, the initial value of the uncertainty upper bound of the ANTSMC is chosen as a low value, and it will adaptively adjust according to the actual tracking errors during the motion period. Therefore, the uncertainty upper bounds of the three controllers are designed as follows.The uncertainty upper bounds of NTSMC1 are given by(39)b1=1,b2=8,b3=3.The uncertainty upper bounds of NTSMC2 are given by(40)b1=10,b2=80,b3=30.The uncertainty upper bounds of ANTSMC are given by(41)b^1,0=1,b^2,0=1,b^3,0=1,d1=5,d2=0.1,d3=1,where b^1,0, b^2,0, and b^3,0 are the initial values of b^1, b^2, and b^3.


### 5.1. Fault-Free Conditions

The numerical simulations of the robot manipulator under NTSMC1, NTSMC2, and ANTSMC are achieved in MATLAB 6.5 with the step size of 0.0005 s. The corresponding simulation results are illustrated in Figure1, where “q1∗,” “q2∗,” “eq1∗,” “eq2∗,” “u1∗,” and “u2∗” represent the tracking of joint 1, tracking of joint 2, tracking error of joint 1, tracking error of joint 2, control input of joint 1, and control input of joint 2 under the ∗ control scheme, respectively; “q1_desired” and “q2_desired” denote the desired trajectories of joint 1 and joint 2.Figure 1
Fault-free condition. (a) Tracking of joint 1; (b) tracking of joint 2; (c) tracking error of joint 1; (d) tracking error of joint 2; (e) control input of NTSMC1; (f) control input of NTSMC2; (g) control input of ANTSMC.
(a)(b)(c)(d)(e)(f)(g)The system state errors of the robot manipulator can converge to zero in finite time (2 s) under all of the three controllers, which can be observed in Figures1(a)–1(d). However, owing to its chosen uncertainty, upper bound is lower than the actual uncertainty, and the tracking condition cannot be maintained under NTSMC1, which can be seen in 2.5∼5.8 s of Figures 1(c) and 1(d). Moreover, the tracking performances of NTSMC2 and ANTSMC are satisfactory in all of the simulation time.In the initial time, there are torque variations under the three controllers, which are always unavoidable for the mechanical systems. However, the torque command under NTSMC2 has serious chattering and saturation problems in the first 1.5 s, and there even exists slight chattering in the stabilization stage (6.7∼8.7 s), shown in Figure1(f). Besides, the torque command under NTSMC1 generates short-time chattering to compensate the tracking error in 4.9∼5.2 s, which makes the tracking performance be improved after 6 s. Furthermore, the torque command under ANTSMC is always smooth in 1∼20 s. Therefore, ANTSMC is the best control scheme owing to its high-accuracy tracking and smooth input command.


### 5.2. Faulty Conditions

To further research the influence of the unpredictable faults, the actuator faults are assumed to be(42)δa=1,t<10,0.9−0.1cos0.1t−10,t≥10,(43)δa=diagδa,δa,(44)δf=00T,t<10,−6−0.2sin0.2t−10−3−0.1sin0.2t−10T,t≥10.Besides, the parameter uncertainty, desired trajectory, and controller parameters of the robot manipulator are assumed as same as the fault-free conditions. Then, the corresponding results are presented in Figure2.Figure 2
Faulty condition. (a) Tracking of joint 1; (b) tracking of joint 2; (c) tracking error of joint 1; (d) tracking error of joint 2; (e) control input of NTSMC1; (f) control input of NTSMC2; (g) control input of ANTSMC.
(a)(b)(c)(d)(e)(f)(g)According to equations (42)–(44), actuator faults occur after 10 s, which will cause the torque mutations under the three controllers (shown in Figures 2(e)–2(g)). Moreover, owing to the simultaneous appearances of actuator faults and disturbances, the torque command is calculated to be saturated in a short time to generate the tracking errors for the robot manipulator. After the torque saturation state, the actual trajectories will well track the desired ones under NTSMC2 and ANTSMC at the cost of short-period torque chattering, but the tracking errors will be still existing under NTSMC1.In summary, ANTSMC is the best control scheme considering both the tracking errors and torque chattering.


## 5.1. Fault-Free Conditions

The numerical simulations of the robot manipulator under NTSMC1, NTSMC2, and ANTSMC are achieved in MATLAB 6.5 with the step size of 0.0005 s. The corresponding simulation results are illustrated in Figure1, where “q1∗,” “q2∗,” “eq1∗,” “eq2∗,” “u1∗,” and “u2∗” represent the tracking of joint 1, tracking of joint 2, tracking error of joint 1, tracking error of joint 2, control input of joint 1, and control input of joint 2 under the ∗ control scheme, respectively; “q1_desired” and “q2_desired” denote the desired trajectories of joint 1 and joint 2.Figure 1
Fault-free condition. (a) Tracking of joint 1; (b) tracking of joint 2; (c) tracking error of joint 1; (d) tracking error of joint 2; (e) control input of NTSMC1; (f) control input of NTSMC2; (g) control input of ANTSMC.
(a)(b)(c)(d)(e)(f)(g)The system state errors of the robot manipulator can converge to zero in finite time (2 s) under all of the three controllers, which can be observed in Figures1(a)–1(d). However, owing to its chosen uncertainty, upper bound is lower than the actual uncertainty, and the tracking condition cannot be maintained under NTSMC1, which can be seen in 2.5∼5.8 s of Figures 1(c) and 1(d). Moreover, the tracking performances of NTSMC2 and ANTSMC are satisfactory in all of the simulation time.In the initial time, there are torque variations under the three controllers, which are always unavoidable for the mechanical systems. However, the torque command under NTSMC2 has serious chattering and saturation problems in the first 1.5 s, and there even exists slight chattering in the stabilization stage (6.7∼8.7 s), shown in Figure1(f). Besides, the torque command under NTSMC1 generates short-time chattering to compensate the tracking error in 4.9∼5.2 s, which makes the tracking performance be improved after 6 s. Furthermore, the torque command under ANTSMC is always smooth in 1∼20 s. Therefore, ANTSMC is the best control scheme owing to its high-accuracy tracking and smooth input command.


## 5.2. Faulty Conditions

To further research the influence of the unpredictable faults, the actuator faults are assumed to be(42)δa=1,t<10,0.9−0.1cos0.1t−10,t≥10,(43)δa=diagδa,δa,(44)δf=00T,t<10,−6−0.2sin0.2t−10−3−0.1sin0.2t−10T,t≥10.Besides, the parameter uncertainty, desired trajectory, and controller parameters of the robot manipulator are assumed as same as the fault-free conditions. Then, the corresponding results are presented in Figure2.Figure 2
Faulty condition. (a) Tracking of joint 1; (b) tracking of joint 2; (c) tracking error of joint 1; (d) tracking error of joint 2; (e) control input of NTSMC1; (f) control input of NTSMC2; (g) control input of ANTSMC.
(a)(b)(c)(d)(e)(f)(g)According to equations (42)–(44), actuator faults occur after 10 s, which will cause the torque mutations under the three controllers (shown in Figures 2(e)–2(g)). Moreover, owing to the simultaneous appearances of actuator faults and disturbances, the torque command is calculated to be saturated in a short time to generate the tracking errors for the robot manipulator. After the torque saturation state, the actual trajectories will well track the desired ones under NTSMC2 and ANTSMC at the cost of short-period torque chattering, but the tracking errors will be still existing under NTSMC1.In summary, ANTSMC is the best control scheme considering both the tracking errors and torque chattering.


## 6. Conclusions

In this paper, an adaptive nonsingular terminal sliding mode control (ANTSMC) scheme is proposed for then-link robot manipulator with parameter uncertainties and actuator faults. The characteristics of the conventional nonsingular terminal sliding mode control (NTSMC) scheme, such as the global nonsingular and finite time reaching, are maintained for the proposed ANTSMC. Moreover, numerical simulations are achieved by comparing the performances of NTSMC and ANTSMC, and the advantages of ANTSMC can be summarized as follows:(1)
Tracking Performance. The actual trajectory can track the desired one under ANTSMC and NTSMC2 (higher uncertainty upper bound), even in the conditions of disturbances, faults, and saturation. However, for the trajectory under NTSMC1 (lower uncertainty upper bound), there may exist short-time or long-time errors in the fault-free and faulty conditions, respectively.(2)
Chattering Phenomenon. In the initial time, for the torque command under NTSMC2, there exists serious chattering and saturation problem, and the torque commands of NTSMC1 and ANTSMC are satisfactory. In the stabilization period, many factors may cause short-time chattering for the three controllers, such as large tracking error and faults occurrence.Therefore, according to the results of the simulations, the drawback of existing tracking errors of NTSMC1, the occurrence of serious chattering and saturation for the NTSMC2, and the effectiveness of ANTSMC considering both of the trajectory performance and chattering elimination are observed, respectively.


## 7. Future Recommendation

In this paper, we proposed an adaptive nonsingular terminal sliding mode control (ANTSMC) scheme for the general robot manipulator. Further research will be focused on the implementation to the special robot systems in practice.



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