# Terminal Sliding-Mode Control of Uncertain Robotic Manipulator System with Predefined Convergence Time

Authors: Yang Wang; Mingshu Chen; Yu Song
Journal: Complexity
 (2021)

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## Abstract

This paper concentrates on the predefined-time trajectory tracking for an uncertain robotic manipulator system. First, a modified predefined-time control (PTC) algorithm is proposed. Subsequently, with the help of proposed modified PTC algorithm and the nonsingular design method of terminal sliding mode, a novel nonsingular terminal sliding-mode control (NTSMC) scheme is proposed for ensuring the predefined-time convergence of tracking errors. The advantages of the newly proposed control scheme are as follows. (i) Unlike the conventional predefined-time sliding-mode control (SMC) which only guarantees the predefined-time convergence of sliding-mode surface, the proposed scheme can guarantee the predefined-time convergence of tracking errors. (ii) Compared with the conventional PTC algorithm, the proposed modified PTC algorithm can reduce the initial control peaking and enhance the precision of convergence time. The performance and effectiveness of the proposed control scheme are illustrated by comparing with the existing methods.


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## Body

## 1. Introduction

Trajectory tracking control of robot manipulators has been recognized as one of the most important control modes in the robot control field [1, 2]. As a classical control method, proportional-integral-derivative (PID) control has been widely used in engineering [3–6]. However, the tracking precision of these PID-based control schemes may be dramatically degraded by uncertainty. With the development of technology, the robotic manipulator is not only required to accomplish the repetitive labor missions in the structured factory environment but also expected to service in many unstructured environments, such as underwater environment [7], man-machine interaction environment [8], and space environment [9]. In these unstructured environments, many complex uncertainties may be brought by the unknown knowledge of the dynamic model, the model linearization error, and external environmental disturbance. To suppress the uncertainties, many classical robust control methods have been introduced, including linear matrix inequality (LMI) control [10], composite nonlinear feedback control [11], modified PID [12], and SMC [13]. Among these robust methods, based on the sample structure and the insensitivity property [14, 15], SMC was widely studied in literatures. However, the conventional SMC adopted the linear sliding-mode surface, which implies that the convergence time is infinite.Combining with finite-time control [16–18] and SMC, the terminal sliding-mode control (TSMC) has been explored to achieve the finite convergence time [19–26]. In [19], for a one-order chaotic system, a global TSMC scheme was proposed. And the reaching period of the sliding-mode surface can be omitted. Thus, the high robust performance can be guaranteed from the initial time. In [20], for the uncertain robotic manipulator, a traditional TSMC was designed. However, for such two-order robotic manipulator system or higher-order uncertain systems, the control input of the traditional TSMC in [20] diverges to infinity when the tracking errors converge to zero. This is known as the singular problem of TSMC. To overcome this problem, the NTSMC schemes have been proposed by several different methods, including restricted sliding-mode surface method [21, 22], transformed sliding-mode surface method [23, 24], and integral TSMC method [25, 26]. However, these NTSMC schemes in [19–26] are typical finite-time control methods, which means that the convergence times of these NTSMC schemes in [19–26] relate to the initial tracking errors. Especially, the convergence performance may be affected seriously when the initial tracking errors are large.In recent years, to overcome the above shortage of finite-time control, the fixed-time control has been developed [27–30]. Unlike the finite-time stability achieved in [19–26], the convergence time of fixed-time control is bounded by a setting time constant, which is unrelated to the initial tracking errors [27–30]. Recognizing this advantage of fixed-time stability, the conventional TSMC has been extended to fixed-time TSMC in several contributions [31, 32]. And the design idea of fixed-time TSMC has been introduced to the trajectory tracking of a robotic manipulator system in [33–35]. Like the conventional TSMC, the singular problem also needs to be eliminated in the fixed-time TSMC. In [33, 34], the fixed-time NTSMC was proposed for the robotic manipulators and the approximate piecewise functions were utilized to avoid the potential singular problem. In [35], based on the bi-limit homogeneous technique, a fixed-time NTSMC with prescribed performance was developed for a n-DOF uncertain manipulator, and the singular problem was eliminated by designing integral sliding-mode surface. Although the fixed-time NTSMC has significantly improved the convergence performance of conventional TSMC, the setting time constant of fixed-time NTSMC cannot be simply adjusted by using control parameters [33, 34] or is completely unclear [35]. Actually, for many practice engineering cases, it is necessary to define the settling time constant in advance.With the purpose of predefining the settling time constant of fixed-time control by parameters, the predefined-time control (PTC) has been introduced in [36–39]. Among these methods, the Lyapunov-function-based PTC algorithm proposed in [39] has the advantages of simple structure and easy implementation. Thus, in recent two years, the PTC algorithm of [39] has been applied to synchronization of chaotic systems [40–43] and trajectory tracking control of robotic manipulator systems [44]. In particular, for the uncertain robotic manipulator system, the authors in [44] combined the PTC algorithm of [39] with the SMC to construct a robust predefined-time SMC which has greatly promoted the development of predefined-time control for the uncertain robotic manipulator system. However, the control performance and the actual engineering feasibility of [44] can be improved if following two limitations can be eliminated: (i) the sliding-mode surface of [44] is linear, which means that predefined-time SMC of [44] only can guarantee the predefined-time convergence of sliding-mode surface rather than the tracking errors; (ii) the PTC algorithm proposed in [44] is too conservative, which does not consider any information of initial system conditions. According to engineering practice of the robotic manipulator system, even if the exact value of initial system conditions is unknown, we can know the upper bound of initial system conditions.By designing a novel NTSMC and modifying the conventional PTC algorithm, a novel predefined-time NTSMC which can eliminate above two limitations is proposed in this paper. The contributions are as follows:(1)
Compared with the recent predefined-time SMC in [44], the main contribution of the proposed method is that the proposed NTSMC guarantees the predefined-time convergence of tracking errors rather than the predefined-time convergence of sliding-mode surface.(2)
In comparison with the existing PTC algorithm in [44], the proposed modified PTC algorithm can achieve higher precision of convergence time and smaller initial control peaking.The rest of this paper is organized as follows. In Section2, the preliminaries including the dynamic model of robot manipulators, assumption, design objective, and motivation are given. In Section 3, the controller and its stability analysis are presented. In Section 4, the simulation is given to compare the proposed control scheme and the related works. In Section 5, the conclusion is given.Notations: in this paper, t denotes the time and the initial time is 0.


## 2. Preliminaries

### 2.1. Dynamic Model of Uncertain Robotic Manipulator System

An uncertain robotic manipulator system with n-DOF can be described by the following Euler–Lagrange equation:(1)Jϕϕ¨+Lϕ,ϕ˙ϕ˙+Gϕ=τ+D,where ϕ∈Rn×1 is the joint angle, ϕ˙∈Rn×1 is the angular velocity, ϕ¨ is the angular acceleration, Jϕ∈Rn×n is the inertia matrix, Lϕ,ϕ˙∈Rn×n is the centrifugal-Coriolis forces matrix, Gϕ∈Rn×1 is the gravitational torque, τ∈Rn×1 is the control torque, and D is the uncertainty, respectively.Let the position tracking error bee1=ϕ−ϕd=e11,…,e1nT. Then, we have(2)e¨1=ϕ¨−ϕ¨d.Lete˙1=e2=e21,…,e2nT, φ=−Jϕ−1Lϕ,ϕ˙ϕ+Gϕ−ϕ¨d, Jϕ−1τ=u, and D¯=Jϕ−1D; then, the error dynamic equation can be established as(3)e˙1=e2,e˙2=φ+u+D¯,where(4)φ=φ1…φnT∈Rn×1,(5)u=u1…unT∈Rn×1,(6)D¯=D¯1…D¯nT∈Rn×1.Assumption 1.
The uncertain vectorD¯=D¯1…D¯nT∈Rn×1 is bounded as D¯i≤D¯imax,i=1,…,n, where D¯imax is a positive constant.
Fore1i and e2i,i=1,…,n, according to (3)–(6), the error dynamic equation can be rewritten as(7)e˙1i=e2i,e˙2i=φi+ui+D¯i.


### 2.2. Predefined-Time Stability

Consider following dynamic system:(8)y˙=fy,where y∈Rn×1 is the system state and fy denotes a nonlinear function. For system (8), the system state y can reach the equilibrium point at the time Tyy0. Tyy0 denotes the convergence time function and y0 is the initial system state. Then, the related stability definitions are given as follows.Definition 1 (finite-time stability).
For system (8), the finite-time stability is guaranteed if Tyy0 is finite.Definition 2 (fixed-time stability).
For system (8), the fixed-time stability is guaranteed if Tyy0 is bounded by a constant TF, i.e., ∃TF>0:∀y0∈Rn:Tyy0≤TF. TF is the settling time constant.Definition 3 (predefined-time stability).
For system (8), the predefined-time stability is guaranteed if Tyy0 is bounded by a settling time constant Tc, i.e., ∀y0∈Rn:Tyy0≤Tc. And the settling time constant Tc is a predefined control parameter.


### 2.3. Motivation of This paper

As stated in the Introduction section, for uncertain robotic manipulator system (1), the NTSMC schemes in [19–26] can not only suppress the uncertainties but also guarantee that the convergence time is finite. However, according to Definition 1, we know that the convergence time function of finite-time stability is related to the initial system state. Especially, the convergence performance may be affected seriously when the initial system state is large. According to Definition 2, we know that the convergence time function of fixed-time stability is bounded by a settling time constant. Thus, the fixed-time stability can guarantee the fast convergence performance even if the initial system state is large. Based on fixed-time stability and SMC, several fixed-time NTSMC schemes have been proposed in [33–35] for uncertain robotic manipulator systems. However, the settling time constant of fixed-time NTSMC schemes in [33–35] cannot be simply adjusted by using control parameters or is completely unclear. Recently, with the purpose of predefining the settling time constant of fixed-time control by control parameters, the predefined-time SMC scheme has been developed in [44]. The predefined-time SMC in [44] provides a novel way to predefine the convergence rate of the uncertain robotic manipulator system. Inspired by the result of [44], to further improve the control performance of predefined-time control, we mainly focus on two research objectives: (i) designing a new NTSMC scheme to achieve the predefined-time stability of tracking errors rather than the predefined-time stability of sliding-mode surface in [44] and (ii) modifying the PTC algorithm used in [44] to reduce initial control peaking and enhance the precision of convergence time of conventional PTC algorithm.


## 2.1. Dynamic Model of Uncertain Robotic Manipulator System

An uncertain robotic manipulator system with n-DOF can be described by the following Euler–Lagrange equation:(1)Jϕϕ¨+Lϕ,ϕ˙ϕ˙+Gϕ=τ+D,where ϕ∈Rn×1 is the joint angle, ϕ˙∈Rn×1 is the angular velocity, ϕ¨ is the angular acceleration, Jϕ∈Rn×n is the inertia matrix, Lϕ,ϕ˙∈Rn×n is the centrifugal-Coriolis forces matrix, Gϕ∈Rn×1 is the gravitational torque, τ∈Rn×1 is the control torque, and D is the uncertainty, respectively.Let the position tracking error bee1=ϕ−ϕd=e11,…,e1nT. Then, we have(2)e¨1=ϕ¨−ϕ¨d.Lete˙1=e2=e21,…,e2nT, φ=−Jϕ−1Lϕ,ϕ˙ϕ+Gϕ−ϕ¨d, Jϕ−1τ=u, and D¯=Jϕ−1D; then, the error dynamic equation can be established as(3)e˙1=e2,e˙2=φ+u+D¯,where(4)φ=φ1…φnT∈Rn×1,(5)u=u1…unT∈Rn×1,(6)D¯=D¯1…D¯nT∈Rn×1.Assumption 1.
The uncertain vectorD¯=D¯1…D¯nT∈Rn×1 is bounded as D¯i≤D¯imax,i=1,…,n, where D¯imax is a positive constant.
Fore1i and e2i,i=1,…,n, according to (3)–(6), the error dynamic equation can be rewritten as(7)e˙1i=e2i,e˙2i=φi+ui+D¯i.


## 2.2. Predefined-Time Stability

Consider following dynamic system:(8)y˙=fy,where y∈Rn×1 is the system state and fy denotes a nonlinear function. For system (8), the system state y can reach the equilibrium point at the time Tyy0. Tyy0 denotes the convergence time function and y0 is the initial system state. Then, the related stability definitions are given as follows.Definition 1 (finite-time stability).
For system (8), the finite-time stability is guaranteed if Tyy0 is finite.Definition 2 (fixed-time stability).
For system (8), the fixed-time stability is guaranteed if Tyy0 is bounded by a constant TF, i.e., ∃TF>0:∀y0∈Rn:Tyy0≤TF. TF is the settling time constant.Definition 3 (predefined-time stability).
For system (8), the predefined-time stability is guaranteed if Tyy0 is bounded by a settling time constant Tc, i.e., ∀y0∈Rn:Tyy0≤Tc. And the settling time constant Tc is a predefined control parameter.


## 2.3. Motivation of This paper

As stated in the Introduction section, for uncertain robotic manipulator system (1), the NTSMC schemes in [19–26] can not only suppress the uncertainties but also guarantee that the convergence time is finite. However, according to Definition 1, we know that the convergence time function of finite-time stability is related to the initial system state. Especially, the convergence performance may be affected seriously when the initial system state is large. According to Definition 2, we know that the convergence time function of fixed-time stability is bounded by a settling time constant. Thus, the fixed-time stability can guarantee the fast convergence performance even if the initial system state is large. Based on fixed-time stability and SMC, several fixed-time NTSMC schemes have been proposed in [33–35] for uncertain robotic manipulator systems. However, the settling time constant of fixed-time NTSMC schemes in [33–35] cannot be simply adjusted by using control parameters or is completely unclear. Recently, with the purpose of predefining the settling time constant of fixed-time control by control parameters, the predefined-time SMC scheme has been developed in [44]. The predefined-time SMC in [44] provides a novel way to predefine the convergence rate of the uncertain robotic manipulator system. Inspired by the result of [44], to further improve the control performance of predefined-time control, we mainly focus on two research objectives: (i) designing a new NTSMC scheme to achieve the predefined-time stability of tracking errors rather than the predefined-time stability of sliding-mode surface in [44] and (ii) modifying the PTC algorithm used in [44] to reduce initial control peaking and enhance the precision of convergence time of conventional PTC algorithm.


## 3. Main Result

### 3.1. Modified Predefined-Time Control

For convenience, considering the vectorξ=ξ1,…,ξnT∈Rn and constant α>0, in this paper, the vector function Sig⟶ξα∈Rn and function Sigξiα∈Ri=1,2…,n are defined as follows:(9)Sig⟶ξα=ξ1α+ξ1,…,ξnα+ξnT,Sigξα=ξiα+ξi.The PTC algorithm used in [44] is given as follows.Lemma 1 (see [44]).
Consider a dynamic system as follows:(10)x˙=v,where x∈R denotes the system state and v denotes the control input. If the control input is designed as(11)v=v1=−π/2TsαSigx1−α+Sigx1+α,where Ts is the predefined time parameter and 1>α>0, then x=0 is predefined-time stable with the predefined time Ts, and x converges to zero at time tf1:(12)xt=0,ift≥tf1,(13)tf1=Tsπ/2arctanx0α,tf1≤Ts.Proof.
See [44].
Then, the PTC algorithm is modified as follows.Lemma 2.
Consider a dynamic system given as follows:(14)x˙=v,where x∈R denotes the system state and v denotes the control input. If the control input is designed as(15)v=v2=−arctanxmaxα+κxTsαSigx1−α+Sigx1+α,where Ts is the predefined time parameter, 1>α>0, and κx>0 (the parameter xmax is selected as x0≤xmax), then x=0 is predefined-time stable with the predefined time Ts, and x converges to zero at time tf2:(16)xt=0,ift≥tf2,(17)tf2=arctanx0αarctanxmaxα+κxTs,tf2≤Ts.Proof.
A Lyapunov functionVx is defined as(18)Vx=x2.
Calculating the time derivative ofVx and substituting the expressions of v given in (15) into (19), we get(19)V˙x=2xx˙=−2arctanxmaxα+κxTsαVx1−α/2+Vx1+α/2.
Then, we have(20)Tsα2arctanxmaxα+κxVxα/2−11+VxαdVx=−dt.
The convergence time is defined astf2. Then, integrating (20) from t=0 to t=tf2, we have(21)tf2=−Tsα2arctanxmaxα+κx∫Vx0Vxtf2Vxα/2−11+VxαdVx=−Tsarctanxmaxα+κxarctanVxα/2tf2−arctanVxα/20=−Tsarctanxmaxα+κxarctanxtf2α−arctanx0α.
ConsideringVxtf2=xtf2=0, κx>0 and x0≤xmax, then (16) and (17) can be satisfied.
The proof is finished.Remark 1.
According to (13) and (17), for the exact convergence time tf1 and tf2, we have(22)tf1tf2=arctanxmaxα+κxπ/2<1.
Then, consideringtf1≤Ts and tf2≤Ts, we have(23)tf1<tf2≤Ts.
Thus, we know thattf2 is closer to the predefined time Ts. And according to (11) and (15), we have(24)v10v20=π/2arctanxmaxα+κx>1.
Thus, we know that initial control inputv20 is smaller than v10. Although it is difficult to achieve the initial system conditions in advance, we can achieve the upper bound of initial system condition according to the engineering practice. The parameter xmax is closer to the exact value of x0, and more accurate convergence time and smaller initial control peaking can be achieved.


### 3.2. Predefined-Time Nonsingular Sliding-Mode Control

The PTC algorithm (11) is adopted in [44] to design the reaching law to guarantee the predefined-time convergence of sliding-mode surface. To extend the result of Muñoz-Vázquez et al. [44] to achieve the predefined-time convergence of tracking errors, it is necessary not only to use PT algorithm to design the reaching law but also to use PTC algorithm to design the sliding-mode surface. However, the sliding-mode surface is directly designed by the PTC algorithm (11) or (15) can cause the singular problem. To avoid this problem, a novel predefined-time NTSMC scheme will be developed in this section.First, inspired by the nonsingular method of Zhao et al. [21] and based on the modified PTC algorithm (15), the predefined-time nonsingular sliding-mode surface is developed for system (7) as follows:(25)si=e2i+arctane1imaxαei+κeiTei−Tsiαeife1i+Sige1i1+αei,where Tei and Tsi are the predefined time parameters, and they satisfy Tei>Tsi>0. κei is a positive constant. e1imax and αei satisfy e1imax≥e1iTsi and 0<αei<1. The nonlinear function fe1i is defined as(26)fe1i=−1−lnδei1−αei−δeilnδeiSige1i2−αei+δei−2αeiδeie1i1−αei−δeilnδeie1i,ife1i≤δei,Sige1i1−αei,ife1i>δei,where δei∈0,1/e=0,0.36 and δei=αei.It can be known thatfe1i is differentiable and its derivative is given as(27)dfe1idt=−1−lnδei1−αei−δeilnδei2−αeie1i1−αeie2i+δei−2αeiδeie1i1−αei−δeilnδeie1ilnδei+1e2i,ife1i≤δei,1−αeie1i−αeie2i,ife1i>δei.From (27), we have dfe1i/dt=δei−2αei/1−αei−δeilnδeie2i if e1i=0. Thus, we can know that there is no singular term existing in dfe1i/dt if e1i=0. We also can know that dfe1i/dt is continuous according to (27).Then, the control input is designed based on the sliding-mode surface (25), and the modified PTC algorithm (15) is given as follows:(28)ui=−φi+arctane1imaxαei+κeiTei−Tsiαeidfe1idt+1+αeie1iαeie2i−arctansimaxαsi+κsiTsiαsiSigsi1−αsi+Sigsi1+αsi−D¯imaxsi,where D¯imax is given by Assumption 1. The parameters satisfy simax≥si0, κsi>0 and 0<αsi<1.The input torqueτ can be calculated by following equation:(29)τ=Jφu.Then, the predefined-time stability analysis of the proposed controller (28) is given as follows.Theorem 1.
For uncertain robotic manipulator system (7), if Assumption 1 is satisfied, then control scheme (28) can guarantee that the tracking errors e1i,i=1,2 converge to the following small region in predefined time Tei:(30)e1i≤δei,ift≥Tei.
Then,e1i can arrive at the origin asymptotically.Proof.
Calculating the time derivative ofsi, we have(31)s˙i=φi+ui+D¯i+arctane1imaxαei+κeiTei−Tsiαeidfe1idt+1+αeie1iαeie2i.
Substituting controller (28) into (31), we have(32)s˙i=−arctansimaxαsi+κsiTsiαsiSigsi1−αsi+Sigsi1+αsi−D¯imaxsi+D¯i.
A Lyapunov functionV1 is defined as(33)V1=si2.
Calculating the time derivative ofV1, we get(34)V˙1=2sis˙i.
Substituting (32) into (34), we get(35)V˙1=2−arctansimaxαsi+κsiTsiαsiSigsi1−αsi+Sigsi1+αsi−D¯imaxsi+D¯isi=−2arctansimaxαsi+κsiTsiαsiV11−αsi/2+V11+αsi/2+D¯i−D¯imaxsisi≤−2arctansimaxαsi+κsiTsiαsiV11−αsi/2+V11+αsi/2+D¯i−D¯imaxsi.
Considering Assumption1, we have(36)V˙1≤−2arctansimaxαsi+κsiTsiαsiV11−αsi/2+V11+αsi/2.
Then, according to the proof of Lemma2, we know that(37)V1=si=0,ift≥arctansi0αsiarctansimaxαsi+κsiTsi.
Then, combining (25) and (37), we have(38)e2i=−arctane1imaxαei+κeiTei−Tsiαeife1i+Sige1i1+αei,ift≥σsiTsi,where σsi=arctansi0αsi/arctansimaxαsi+κsi.
Fore1i≥δei, we have(39)e2i=−arctane1imaxαeiTei−TsiαeiSige1i1−αei+Sige1i1+αei,ift≥σsiTsi.
Then, according to Lemma2, we know that e1i will converge to following region:(40)e1i<δei,ift≥σsiTsi+σeiTei−Tsi,where σei=arctane1iTsiαei/arctane1imaxαei+κei. Since simaxαsi+κsi>si0αsi and e1imaxαei+κei>e1iTsiαei, we have 0<σei<1 and 0<σsi<1. Then, combining Tei>Tsi, we have(41)e1i≤δei,ift≥Tei.
According to (37), 0<σsi<1 and Tei>Tsi, and we have(42)si=0,ift≥Tei.
Then, combining (25), (26), (41), and (42), we have(43)e2i=−arctane1imaxαeiTei−Tsiαei−1−lnδei1−αei−δeilnδeiSige1i2−αei−arctane1imaxαeiTei−Tsiαeiδei−2αeiδeie1ie1i1−αei−δeilnδei+Sige1i1+αei,ift≥Tei.
We define a new Lyapunov function as(44)V2=e1i22.
Then, calculating the time derivative ofV2 and considering (43), we get(45)V˙2=e1ie2i=−arctane1imaxαeiTei−Tsiαei−1−lnδei1−αei−δeilnδeie1i3−αei−arctane1imaxαeiTei−Tsiαeiδei−2αeiδeie1i1−αei−δeilnδeie1i2+e1i2+αei,ift≥Tei.
Sinceδei∈0,1/e and 0<αei<1, we have(46)−1−lnδei1−αei−δeilnδei>0,(47)δei−2αei1−αei−δeilnδei>0.
Considering (41) and δei∈0,1/e, we have(48)δeie1i≥δeiδei>0,ift≥Tei.
Then, combining (45), (46), (47), and (48), we have(49)V˙2≤−keiV2,ift≥Tei,where the constant kei=δei−2αei/21−αei−δeilnδeiδeiδei>0.
From (49), we have(50)V2≤V2Teie−keit−Tei,ift≥Tei.
Thus, we haveV2∞=0. Combining e1i≤δei, ift≥Tei (see (41)), and V2∞=0, the proof is finished.Remark 2.
From the above analysis, to guarantee the predefined-time stability, we needsimax>si0 and e1imax>e1iTsi. In the practical cases, the upper bounds of initial states e1i0 and e2i0 can be obtained by considering physical constraint. However, the upper bounds of e1iTsi and si0 cannot be obtained directly. To obtain the upper bound parameters simax and e1imax, an estimation method is given in the Appendix.Remark 3.
The predefined-time performance (setting time constant isTei) must be satisfied by selecting Tei>Tsi>0, κei>0, δei∈0,1/e=0,0.36, δei=αei, 0<αei<1, κsi>0, and 0<αsi<1, and the upper bound parameters simax and e1imax satisfy the estimation methods given in Appendix. Although the predefined-time convergence performance must be guaranteed if parameters satisfy these conditions, we can select the parameters appropriately to achieve a smaller initial control input. For example, κei is a positive constant, and the purpose of defining κei is to estimate the upper bound of e1i (see (b) of Appendix) and achieve small initial control input. To achieve exact upper bound of e1i and small initial control input, κei should be large. However, large κei may cause a large initial control input, and thus we need to balance the choice of κei. Fortunately, even if the selection of κei can affect the initial control input, by defining an arbitrary κei still can achieve a smaller initial control input than conventional PTC (see Remark 1).Remark 4.
Although this paper focuses on the predefined-time control of rigid manipulator system, the proposed predefined-time NTSMC scheme can be directly applied to the one-order or two-order matched uncertain system. For example, since many chaotic systems are typical one-order or two-order systems (such as Lorenz system, Rossler system, and chaotic gyros system) and we usually only consider that these systems are affected by matched uncertainties, the proposed method can directly enhance the convergence performance of the synchronization control and the chaos anti-control for these complex chaotic systems. Similarly, the proposed scheme also can be applied to many complex engineering systems, including the formation control system of autonomous underwater vehicles, the aircraft attitude control system, and the angle constrained guidance system.


## 3.1. Modified Predefined-Time Control

For convenience, considering the vectorξ=ξ1,…,ξnT∈Rn and constant α>0, in this paper, the vector function Sig⟶ξα∈Rn and function Sigξiα∈Ri=1,2…,n are defined as follows:(9)Sig⟶ξα=ξ1α+ξ1,…,ξnα+ξnT,Sigξα=ξiα+ξi.The PTC algorithm used in [44] is given as follows.Lemma 1 (see [44]).
Consider a dynamic system as follows:(10)x˙=v,where x∈R denotes the system state and v denotes the control input. If the control input is designed as(11)v=v1=−π/2TsαSigx1−α+Sigx1+α,where Ts is the predefined time parameter and 1>α>0, then x=0 is predefined-time stable with the predefined time Ts, and x converges to zero at time tf1:(12)xt=0,ift≥tf1,(13)tf1=Tsπ/2arctanx0α,tf1≤Ts.Proof.
See [44].
Then, the PTC algorithm is modified as follows.Lemma 2.
Consider a dynamic system given as follows:(14)x˙=v,where x∈R denotes the system state and v denotes the control input. If the control input is designed as(15)v=v2=−arctanxmaxα+κxTsαSigx1−α+Sigx1+α,where Ts is the predefined time parameter, 1>α>0, and κx>0 (the parameter xmax is selected as x0≤xmax), then x=0 is predefined-time stable with the predefined time Ts, and x converges to zero at time tf2:(16)xt=0,ift≥tf2,(17)tf2=arctanx0αarctanxmaxα+κxTs,tf2≤Ts.Proof.
A Lyapunov functionVx is defined as(18)Vx=x2.
Calculating the time derivative ofVx and substituting the expressions of v given in (15) into (19), we get(19)V˙x=2xx˙=−2arctanxmaxα+κxTsαVx1−α/2+Vx1+α/2.
Then, we have(20)Tsα2arctanxmaxα+κxVxα/2−11+VxαdVx=−dt.
The convergence time is defined astf2. Then, integrating (20) from t=0 to t=tf2, we have(21)tf2=−Tsα2arctanxmaxα+κx∫Vx0Vxtf2Vxα/2−11+VxαdVx=−Tsarctanxmaxα+κxarctanVxα/2tf2−arctanVxα/20=−Tsarctanxmaxα+κxarctanxtf2α−arctanx0α.
ConsideringVxtf2=xtf2=0, κx>0 and x0≤xmax, then (16) and (17) can be satisfied.
The proof is finished.Remark 1.
According to (13) and (17), for the exact convergence time tf1 and tf2, we have(22)tf1tf2=arctanxmaxα+κxπ/2<1.
Then, consideringtf1≤Ts and tf2≤Ts, we have(23)tf1<tf2≤Ts.
Thus, we know thattf2 is closer to the predefined time Ts. And according to (11) and (15), we have(24)v10v20=π/2arctanxmaxα+κx>1.
Thus, we know that initial control inputv20 is smaller than v10. Although it is difficult to achieve the initial system conditions in advance, we can achieve the upper bound of initial system condition according to the engineering practice. The parameter xmax is closer to the exact value of x0, and more accurate convergence time and smaller initial control peaking can be achieved.


## 3.2. Predefined-Time Nonsingular Sliding-Mode Control

The PTC algorithm (11) is adopted in [44] to design the reaching law to guarantee the predefined-time convergence of sliding-mode surface. To extend the result of Muñoz-Vázquez et al. [44] to achieve the predefined-time convergence of tracking errors, it is necessary not only to use PT algorithm to design the reaching law but also to use PTC algorithm to design the sliding-mode surface. However, the sliding-mode surface is directly designed by the PTC algorithm (11) or (15) can cause the singular problem. To avoid this problem, a novel predefined-time NTSMC scheme will be developed in this section.First, inspired by the nonsingular method of Zhao et al. [21] and based on the modified PTC algorithm (15), the predefined-time nonsingular sliding-mode surface is developed for system (7) as follows:(25)si=e2i+arctane1imaxαei+κeiTei−Tsiαeife1i+Sige1i1+αei,where Tei and Tsi are the predefined time parameters, and they satisfy Tei>Tsi>0. κei is a positive constant. e1imax and αei satisfy e1imax≥e1iTsi and 0<αei<1. The nonlinear function fe1i is defined as(26)fe1i=−1−lnδei1−αei−δeilnδeiSige1i2−αei+δei−2αeiδeie1i1−αei−δeilnδeie1i,ife1i≤δei,Sige1i1−αei,ife1i>δei,where δei∈0,1/e=0,0.36 and δei=αei.It can be known thatfe1i is differentiable and its derivative is given as(27)dfe1idt=−1−lnδei1−αei−δeilnδei2−αeie1i1−αeie2i+δei−2αeiδeie1i1−αei−δeilnδeie1ilnδei+1e2i,ife1i≤δei,1−αeie1i−αeie2i,ife1i>δei.From (27), we have dfe1i/dt=δei−2αei/1−αei−δeilnδeie2i if e1i=0. Thus, we can know that there is no singular term existing in dfe1i/dt if e1i=0. We also can know that dfe1i/dt is continuous according to (27).Then, the control input is designed based on the sliding-mode surface (25), and the modified PTC algorithm (15) is given as follows:(28)ui=−φi+arctane1imaxαei+κeiTei−Tsiαeidfe1idt+1+αeie1iαeie2i−arctansimaxαsi+κsiTsiαsiSigsi1−αsi+Sigsi1+αsi−D¯imaxsi,where D¯imax is given by Assumption 1. The parameters satisfy simax≥si0, κsi>0 and 0<αsi<1.The input torqueτ can be calculated by following equation:(29)τ=Jφu.Then, the predefined-time stability analysis of the proposed controller (28) is given as follows.Theorem 1.
For uncertain robotic manipulator system (7), if Assumption 1 is satisfied, then control scheme (28) can guarantee that the tracking errors e1i,i=1,2 converge to the following small region in predefined time Tei:(30)e1i≤δei,ift≥Tei.
Then,e1i can arrive at the origin asymptotically.Proof.
Calculating the time derivative ofsi, we have(31)s˙i=φi+ui+D¯i+arctane1imaxαei+κeiTei−Tsiαeidfe1idt+1+αeie1iαeie2i.
Substituting controller (28) into (31), we have(32)s˙i=−arctansimaxαsi+κsiTsiαsiSigsi1−αsi+Sigsi1+αsi−D¯imaxsi+D¯i.
A Lyapunov functionV1 is defined as(33)V1=si2.
Calculating the time derivative ofV1, we get(34)V˙1=2sis˙i.
Substituting (32) into (34), we get(35)V˙1=2−arctansimaxαsi+κsiTsiαsiSigsi1−αsi+Sigsi1+αsi−D¯imaxsi+D¯isi=−2arctansimaxαsi+κsiTsiαsiV11−αsi/2+V11+αsi/2+D¯i−D¯imaxsisi≤−2arctansimaxαsi+κsiTsiαsiV11−αsi/2+V11+αsi/2+D¯i−D¯imaxsi.
Considering Assumption1, we have(36)V˙1≤−2arctansimaxαsi+κsiTsiαsiV11−αsi/2+V11+αsi/2.
Then, according to the proof of Lemma2, we know that(37)V1=si=0,ift≥arctansi0αsiarctansimaxαsi+κsiTsi.
Then, combining (25) and (37), we have(38)e2i=−arctane1imaxαei+κeiTei−Tsiαeife1i+Sige1i1+αei,ift≥σsiTsi,where σsi=arctansi0αsi/arctansimaxαsi+κsi.
Fore1i≥δei, we have(39)e2i=−arctane1imaxαeiTei−TsiαeiSige1i1−αei+Sige1i1+αei,ift≥σsiTsi.
Then, according to Lemma2, we know that e1i will converge to following region:(40)e1i<δei,ift≥σsiTsi+σeiTei−Tsi,where σei=arctane1iTsiαei/arctane1imaxαei+κei. Since simaxαsi+κsi>si0αsi and e1imaxαei+κei>e1iTsiαei, we have 0<σei<1 and 0<σsi<1. Then, combining Tei>Tsi, we have(41)e1i≤δei,ift≥Tei.
According to (37), 0<σsi<1 and Tei>Tsi, and we have(42)si=0,ift≥Tei.
Then, combining (25), (26), (41), and (42), we have(43)e2i=−arctane1imaxαeiTei−Tsiαei−1−lnδei1−αei−δeilnδeiSige1i2−αei−arctane1imaxαeiTei−Tsiαeiδei−2αeiδeie1ie1i1−αei−δeilnδei+Sige1i1+αei,ift≥Tei.
We define a new Lyapunov function as(44)V2=e1i22.
Then, calculating the time derivative ofV2 and considering (43), we get(45)V˙2=e1ie2i=−arctane1imaxαeiTei−Tsiαei−1−lnδei1−αei−δeilnδeie1i3−αei−arctane1imaxαeiTei−Tsiαeiδei−2αeiδeie1i1−αei−δeilnδeie1i2+e1i2+αei,ift≥Tei.
Sinceδei∈0,1/e and 0<αei<1, we have(46)−1−lnδei1−αei−δeilnδei>0,(47)δei−2αei1−αei−δeilnδei>0.
Considering (41) and δei∈0,1/e, we have(48)δeie1i≥δeiδei>0,ift≥Tei.
Then, combining (45), (46), (47), and (48), we have(49)V˙2≤−keiV2,ift≥Tei,where the constant kei=δei−2αei/21−αei−δeilnδeiδeiδei>0.
From (49), we have(50)V2≤V2Teie−keit−Tei,ift≥Tei.
Thus, we haveV2∞=0. Combining e1i≤δei, ift≥Tei (see (41)), and V2∞=0, the proof is finished.Remark 2.
From the above analysis, to guarantee the predefined-time stability, we needsimax>si0 and e1imax>e1iTsi. In the practical cases, the upper bounds of initial states e1i0 and e2i0 can be obtained by considering physical constraint. However, the upper bounds of e1iTsi and si0 cannot be obtained directly. To obtain the upper bound parameters simax and e1imax, an estimation method is given in the Appendix.Remark 3.
The predefined-time performance (setting time constant isTei) must be satisfied by selecting Tei>Tsi>0, κei>0, δei∈0,1/e=0,0.36, δei=αei, 0<αei<1, κsi>0, and 0<αsi<1, and the upper bound parameters simax and e1imax satisfy the estimation methods given in Appendix. Although the predefined-time convergence performance must be guaranteed if parameters satisfy these conditions, we can select the parameters appropriately to achieve a smaller initial control input. For example, κei is a positive constant, and the purpose of defining κei is to estimate the upper bound of e1i (see (b) of Appendix) and achieve small initial control input. To achieve exact upper bound of e1i and small initial control input, κei should be large. However, large κei may cause a large initial control input, and thus we need to balance the choice of κei. Fortunately, even if the selection of κei can affect the initial control input, by defining an arbitrary κei still can achieve a smaller initial control input than conventional PTC (see Remark 1).Remark 4.
Although this paper focuses on the predefined-time control of rigid manipulator system, the proposed predefined-time NTSMC scheme can be directly applied to the one-order or two-order matched uncertain system. For example, since many chaotic systems are typical one-order or two-order systems (such as Lorenz system, Rossler system, and chaotic gyros system) and we usually only consider that these systems are affected by matched uncertainties, the proposed method can directly enhance the convergence performance of the synchronization control and the chaos anti-control for these complex chaotic systems. Similarly, the proposed scheme also can be applied to many complex engineering systems, including the formation control system of autonomous underwater vehicles, the aircraft attitude control system, and the angle constrained guidance system.


## 4. Simulation Results

The effectiveness of proposed predefined-time NTSMC will be illustrated by simulating in this section. We consider that system (1) is a two-DOF robot manipulator. The system parameters are given as follows:(51)J⟶ϕ⟶=ε1+ε2σ12+ε2σ22+2ε2σ1σ2cosϕ2+P1ε2σ22+ε2σ1σ2cosϕ2ε2σ22+ε2σ1σ2cosϕ2ε2σ22+P2,L⟶ϕ⟶,ϕ⟶˙=−ε2σ1σ2sinϕ2ϕ˙1−2ε2σ1σ2sinϕ2ϕ˙10ε2σ1σ2sinϕ2ϕ˙2,G⟶ϕ⟶=ε1+ε2σ1g1cosϕ1+ε2σ2g1cosϕ1+ϕ2,ε2σ2g1cosϕ1+ϕ2T,where ε1=0.7, ε2=0.7, σ1=0.5, σ2=0.5, P1=P2=1.5, and g1=9.8. The desired joint position ϕ⟶d is chosen as(52)ϕ⟶d=0.7−cost/3,−0.3−sint/2Trad.The initial joint angle and angular velocity are chosen as(53)ϕ0=11Trad,ϕ˙0=0−0.5Trad/s.The uncertainty is chosen as(54)D=3cost/3,2.5sint/2T.To remove the chattering problem, the chattering term in the proposed control (28) is replaced by the sigmoid function sigmf⋅ (see [37]) given as(55)D¯imaxsignsi=D¯imaxsigmfsi,sigmfsi=211+exp−ϖsi−12,where ϖ is chosen as 30.


### 4.1. Comparison of the Proposed Predefined-Time NTSMC and Existing Methods

For comparison, the finite-time NTSMC in [27], the fixed-time NTSMC in [33], the predefined-time linear sliding-mode control (LSMC) in [44], and the proposed predefined-time NTSMC (28) will be considered in this section.According to [27], the finite-time NTSMC scheme can be designed as(56)τ=τ0+τeq+τr,where the sliding-mode surface sT and the control components τ0, τeq, and τr are given as(57)sT=e2+∫0tΩ1Se1ζ1+Ω2Se2ζ2dυ,τ0=Jϕϕ¨d+Lϕ,ϕ˙ϕ˙+Gϕ,τeq=−MqΩ1Se1υ1+Ω2Se2υ2,τr=−11−δ¯fkf+b0+b2q˙2+ϖfτ0+τeqsTsT+sf.Se1ζ1 and Se2ζ2 are defined as(58)Se1ζ1=Se11ζ1Se12ζ1T,Se2ζ2=Se21ζ2Se22ζ2T.According to [33], the fixed-time NTSMC can be given as(59)τ=τ1+τ2+τ3,where(60)τ1=Jϕϕ¨d+Lϕ,ϕ˙ϕ˙+Gϕ,τ2=−K0SigsFr−JϕK1Fυ1e1+υ2K2Hυ2e1e2,τ3=−11−δ¯k0+k1+k2ϕ˙2+ϖτ1+τ2sFsF+sF0,sF=e2+K1Sυ1e1+K2Sig⟶υ2e1.The vectorS⟶υ1e⟶1 and the matrixes H⟶υ2e⟶1 and F⟶υ1e⟶1 are defined as(61)Sυ1e1=Se11υ1Se12υ1T,Hυ2e1=diage11υ2−1,e12υ2−1,Fυ1e1=diagfe11υ1,fe12υ1,where the functions Se1iυ1 and fe1iυ1,i=1,2, are defined as(62)Se1iυ1=e1iυ1signe1i,e1i≥δFi,δFiυ1−1e1i,e1i<δFi,(63)fe1iυ1=υ1e1iυ1−1,e1i≥δFi,δFiυ1−1,e1i<δFi.According to [44], the predefined-time LSMC is designed as(64)τ=τ¯1+τ¯2+Y¯^r,where(65)τ⟶¯1=−πηTcγ1−η/2sr−η+γ1+η/2srηsr,ifsr≥δr,−πηTcγ1−η/2δr−η+γ1+η/2δrηsr,ifsr<δr,τ¯2=−kτsrs⟶r,ifsr≥δr,−kτsrδr,ifsr<δr,Y^r=J^ϕϕ¨m+L^ϕ,ϕ˙ϕ˙m+G^ϕ,ϕ˙m=ϕ˙d−kαϕ−ϕd,ϕ¨m=ϕ¨d−kαϕ˙−ϕ˙d,sr=e2+kαe1.Then, we consider the following two cases.Case 1-1: we desire the convergence time which is bounded by 2.5 s. In this case, we consider four control schemes including finite-time NTSMC, fixed-time NTSMC, predefined-time LSMC, and predefined-time NTSMC. First, we consider the following initial system state.


#### 4.1.1. The First Kind

The initial joint angle and angular velocity are chosen as(66)ϕ0=11Trad,(67)ϕ˙0=0−0.5Trad/s.Then, the initial tracking error can be calculated as(68)e1=1.31.3Trad,(69)e2=00Trad/s.It is assumed that we can use the trial-and-error method to adjust the parameters. The predefined-time parameters of predefined-time LSMC and predefined-time NTSMC are set asTc=2.5 and Te1=Te2=2.5, respectively. For the fixed-time NTSMC, the gain matrix K⟶2 is selected as K⟶2=2002.4. The other parameters of the three schemes are given in Table 1. The simulation results are shown in Figures 1–3 . From Figure 1, we know that the four schemes can guarantee that the convergence time is bounded by the desired time (2.5s) under these parameters. From Figure 3, it can be known that the input torques of fixed-time NTSMC have the problem of large switching. The reason is that the derivative of nonsingular function (63) in fixed-time NTSMC is discontinuous. The nonsingular function and its derivative of the proposed NTSMC are continuous, and thus the problem can be avoided by the proposed scheme.Table 1
Control parameters.
Control schemeControl parametersFinite-time NTSMCζ1=0.5, ζ2=2/3, Ω⟶1=230019, Ω⟶2=160012, kf=2, b0=12, b2=2.8, sf=0.01, δ¯f=0.5, ϖf=0.4Fixed-time NTSMCυ1=0.5, υ2=3/2, K0=1001, K1=1001, v1=0.5, v2=1.5, δF1=δF2=0.2, k0=2, k1=12, k2=2.8, r=1.5, sF0=0.01, δ¯=0.5, ϖ=0.4Predefined-time LSMCη=0.5, δr=0.2, kα=5, kτ=5, γ=0.03Proposed predefined-time NTSMCαe1=αe2=0.35, δe1=δe2=0.35, αs1=αs2=0.55, e11max=e12max=3, κs1=κs2=1, κe1=κe2=1, D¯1max=D¯2max=3, Ts1=Ts2=0.3, s1max=s2max=35Figure 1
Position tracking errorse11 and e12 in Case 1-1 (first kind of initial system state).
(a)(b)Figure 2
Sliding-mode surfacess1 and s2 in Case 1-1 (first kind of initial system state).
(a)(b)Figure 3
Input torquesτ1 and τ2 in Case 1-1 (first kind of initial system state).
(a)(b)Then, without changing the control parameters, we chose a new kind of initial system state.


#### 4.1.2. The Second Kind

The initial joint angle and angular velocity are chosen as(70)ϕ0=3.63.6Trad,ϕ˙0=0−0.5Trad/s.Then, the initial tracking error can be calculated as(71)e1=3.93.9T,(72)e2=00T.By comparing equations (68) and (71), we can know that the second kind of initial tracking error is much bigger than the first kind of initial tracking error. The simulation results are shown in Figures 4–6 . From Figure 4, we can know that the convergence performance of finite-time NTSMC is greatly affected by the increase of initial system state. The convergence performances of fixed-time NTSMC, predefined-time LSMC, and proposed predefined-time NTSMC are not affected even if the initial system states are greatly increased. As stated in Introduction section and Section 2, the reason is that convergence time function of finite-time stability is related to the initial system state.Figure 4
Position tracking errorse11 and e12 in Case 1-1 (second kind of initial system state).
(a)(b)Figure 5
Sliding-mode surfacess1 and s2 in Case 1-1 (second kind of initial system state).
(a)(b)Figure 6
Input torquesτ1 and τ2 in Case 1-1 (second kind of initial system state).
(a)(b)Case 1-2: compared with Case 1-1, we desire a new convergence time which is bounded by 0.7s. It is assumed that we cannot use the trial-and-error method. We can only change the parameters once in this case. In this case, we consider three control schemes including fixed-time NTSMC, predefined-time LSMC, and predefined-time NTSMC. The initial system states are chosen as in (66) and (67). Then, the predefined-time parameters of predefined-time LSMC and proposed predefined-time NTSMC are set as Tc=0.7 and Te1=Te2=0.7. However, for the fixed-time NTSMC, the relationship between the parameters and the settling time constant is complex and nonlinear. Since the new desired time is smaller than the value of Case 1-1, we try to increase the gain matrix K⟶2 to K⟶2=5006. The other parameters of the three schemes are given in Table 1. Then, the simulation results of Case 1-2 are shown in Figures 7–9. From Figure 7, we know that only the proposed predefined-time NTSMC can guarantee that the convergence time is bounded by the new desired time (0.7s). Thus, according to Definition 3, the predefined-time stability is only achieved by the proposed scheme. As stated in Introduction section and Section 2, the reasons are as follows. (i) The sliding-mode surface of predefined-time LSMC is linear, which means that predefined-time LSMC only can guarantee the predefined-time convergence of sliding-mode surface (see Figure 8) rather than the predefined-time convergence of tracking errors. (ii) For the fixed-time NTSMC, the relationship between parameters and the setting time constant is complex and nonlinear. Thus, for the fixed-time NTSMC, it is difficult to adjust the parameters to achieve the predefined-time convergence without using the trial-and-error method.Figure 7
Position tracking errorse11 and e12 in Case 1-2.
(a)(b)Figure 8
Sliding-mode surfacess1 and s2 in Case 1-2.
(a)(b)Figure 9
Input torquesτ1 and τ2 in Case 1-2.
(a)(b)


### 4.2. Comparison of the Proposed Modified PTC and Conventional PTC Scheme

In this section, we consider the initial system state chosen as in (66) and (67). The predefined-time parameters are set as Te1=Te2=2. In this section, we consider four kinds of upper bound of initial tracking errors e1i0 and e2i0. We assume that e1i0≤k1i and e2i0≤k2i. From Case 2-1 to Case 2-4, the upper bounds are gradually approaching the exact values:Case 2-1 (conventional PTC algorithm):k11=k12=∞, k21=k22=∞.Case 2-2:k11=k12=20, k21=k22=10.Case 2-3:k11=k12=5, k21=k22=2.5.Case 2-4:k11=k12=2, k21=k22=1.For Case 2-1, the proposed PTC algorithm is equivalent to the conventional PTC algorithm. We selectκe1=κe2=1. Then, according to Appendix, for the above cases, the upper bound parameters can be estimated as follows.Case 2-1 (conventional PTC algorithm):e11max=e12max=∞, s1max=s2max=∞.Case 2-2:e11max=e12max=37.9979, s1max=s2max=253.7187.Case 2-3:e11max=e12max=10.9008, s1max=s2max=46.2597.Case 2-4:e11max=e12max=5.0891, s1max=s2max=15.6292.The other parameters are chosen asαe1=αe2=0.35, δe1=δe2=0.35, αs1=αs2=0.55, κs1=κs2=1, D¯1max=D¯2max=3, and Ts1=Ts2=0.3. The simulation results are shown in Figures 7 and 8. From Figures 10 and 11 , compared with the conventional PTC algorithm, the proposed modified PTC algorithm can achieve more accurate convergence time and smaller initial control input torques. Moreover, if the upper bounds k1i and k2i are closer to the exact values of initial values e1i0 and e2i0, more accurate convergence time and smaller initial control input torques can be achieved.Figure 10
Position tracking errorse11 and e12 described in Section 4.2.
(a)(b)Figure 11
Position tracking errorse11 and e12 described in Section 4.2.
(a)(b)


## 4.1. Comparison of the Proposed Predefined-Time NTSMC and Existing Methods

For comparison, the finite-time NTSMC in [27], the fixed-time NTSMC in [33], the predefined-time linear sliding-mode control (LSMC) in [44], and the proposed predefined-time NTSMC (28) will be considered in this section.According to [27], the finite-time NTSMC scheme can be designed as(56)τ=τ0+τeq+τr,where the sliding-mode surface sT and the control components τ0, τeq, and τr are given as(57)sT=e2+∫0tΩ1Se1ζ1+Ω2Se2ζ2dυ,τ0=Jϕϕ¨d+Lϕ,ϕ˙ϕ˙+Gϕ,τeq=−MqΩ1Se1υ1+Ω2Se2υ2,τr=−11−δ¯fkf+b0+b2q˙2+ϖfτ0+τeqsTsT+sf.Se1ζ1 and Se2ζ2 are defined as(58)Se1ζ1=Se11ζ1Se12ζ1T,Se2ζ2=Se21ζ2Se22ζ2T.According to [33], the fixed-time NTSMC can be given as(59)τ=τ1+τ2+τ3,where(60)τ1=Jϕϕ¨d+Lϕ,ϕ˙ϕ˙+Gϕ,τ2=−K0SigsFr−JϕK1Fυ1e1+υ2K2Hυ2e1e2,τ3=−11−δ¯k0+k1+k2ϕ˙2+ϖτ1+τ2sFsF+sF0,sF=e2+K1Sυ1e1+K2Sig⟶υ2e1.The vectorS⟶υ1e⟶1 and the matrixes H⟶υ2e⟶1 and F⟶υ1e⟶1 are defined as(61)Sυ1e1=Se11υ1Se12υ1T,Hυ2e1=diage11υ2−1,e12υ2−1,Fυ1e1=diagfe11υ1,fe12υ1,where the functions Se1iυ1 and fe1iυ1,i=1,2, are defined as(62)Se1iυ1=e1iυ1signe1i,e1i≥δFi,δFiυ1−1e1i,e1i<δFi,(63)fe1iυ1=υ1e1iυ1−1,e1i≥δFi,δFiυ1−1,e1i<δFi.According to [44], the predefined-time LSMC is designed as(64)τ=τ¯1+τ¯2+Y¯^r,where(65)τ⟶¯1=−πηTcγ1−η/2sr−η+γ1+η/2srηsr,ifsr≥δr,−πηTcγ1−η/2δr−η+γ1+η/2δrηsr,ifsr<δr,τ¯2=−kτsrs⟶r,ifsr≥δr,−kτsrδr,ifsr<δr,Y^r=J^ϕϕ¨m+L^ϕ,ϕ˙ϕ˙m+G^ϕ,ϕ˙m=ϕ˙d−kαϕ−ϕd,ϕ¨m=ϕ¨d−kαϕ˙−ϕ˙d,sr=e2+kαe1.Then, we consider the following two cases.Case 1-1: we desire the convergence time which is bounded by 2.5 s. In this case, we consider four control schemes including finite-time NTSMC, fixed-time NTSMC, predefined-time LSMC, and predefined-time NTSMC. First, we consider the following initial system state.


### 4.1.1. The First Kind

The initial joint angle and angular velocity are chosen as(66)ϕ0=11Trad,(67)ϕ˙0=0−0.5Trad/s.Then, the initial tracking error can be calculated as(68)e1=1.31.3Trad,(69)e2=00Trad/s.It is assumed that we can use the trial-and-error method to adjust the parameters. The predefined-time parameters of predefined-time LSMC and predefined-time NTSMC are set asTc=2.5 and Te1=Te2=2.5, respectively. For the fixed-time NTSMC, the gain matrix K⟶2 is selected as K⟶2=2002.4. The other parameters of the three schemes are given in Table 1. The simulation results are shown in Figures 1–3 . From Figure 1, we know that the four schemes can guarantee that the convergence time is bounded by the desired time (2.5s) under these parameters. From Figure 3, it can be known that the input torques of fixed-time NTSMC have the problem of large switching. The reason is that the derivative of nonsingular function (63) in fixed-time NTSMC is discontinuous. The nonsingular function and its derivative of the proposed NTSMC are continuous, and thus the problem can be avoided by the proposed scheme.Table 1
Control parameters.
Control schemeControl parametersFinite-time NTSMCζ1=0.5, ζ2=2/3, Ω⟶1=230019, Ω⟶2=160012, kf=2, b0=12, b2=2.8, sf=0.01, δ¯f=0.5, ϖf=0.4Fixed-time NTSMCυ1=0.5, υ2=3/2, K0=1001, K1=1001, v1=0.5, v2=1.5, δF1=δF2=0.2, k0=2, k1=12, k2=2.8, r=1.5, sF0=0.01, δ¯=0.5, ϖ=0.4Predefined-time LSMCη=0.5, δr=0.2, kα=5, kτ=5, γ=0.03Proposed predefined-time NTSMCαe1=αe2=0.35, δe1=δe2=0.35, αs1=αs2=0.55, e11max=e12max=3, κs1=κs2=1, κe1=κe2=1, D¯1max=D¯2max=3, Ts1=Ts2=0.3, s1max=s2max=35Figure 1
Position tracking errorse11 and e12 in Case 1-1 (first kind of initial system state).
(a)(b)Figure 2
Sliding-mode surfacess1 and s2 in Case 1-1 (first kind of initial system state).
(a)(b)Figure 3
Input torquesτ1 and τ2 in Case 1-1 (first kind of initial system state).
(a)(b)Then, without changing the control parameters, we chose a new kind of initial system state.


### 4.1.2. The Second Kind

The initial joint angle and angular velocity are chosen as(70)ϕ0=3.63.6Trad,ϕ˙0=0−0.5Trad/s.Then, the initial tracking error can be calculated as(71)e1=3.93.9T,(72)e2=00T.By comparing equations (68) and (71), we can know that the second kind of initial tracking error is much bigger than the first kind of initial tracking error. The simulation results are shown in Figures 4–6 . From Figure 4, we can know that the convergence performance of finite-time NTSMC is greatly affected by the increase of initial system state. The convergence performances of fixed-time NTSMC, predefined-time LSMC, and proposed predefined-time NTSMC are not affected even if the initial system states are greatly increased. As stated in Introduction section and Section 2, the reason is that convergence time function of finite-time stability is related to the initial system state.Figure 4
Position tracking errorse11 and e12 in Case 1-1 (second kind of initial system state).
(a)(b)Figure 5
Sliding-mode surfacess1 and s2 in Case 1-1 (second kind of initial system state).
(a)(b)Figure 6
Input torquesτ1 and τ2 in Case 1-1 (second kind of initial system state).
(a)(b)Case 1-2: compared with Case 1-1, we desire a new convergence time which is bounded by 0.7s. It is assumed that we cannot use the trial-and-error method. We can only change the parameters once in this case. In this case, we consider three control schemes including fixed-time NTSMC, predefined-time LSMC, and predefined-time NTSMC. The initial system states are chosen as in (66) and (67). Then, the predefined-time parameters of predefined-time LSMC and proposed predefined-time NTSMC are set as Tc=0.7 and Te1=Te2=0.7. However, for the fixed-time NTSMC, the relationship between the parameters and the settling time constant is complex and nonlinear. Since the new desired time is smaller than the value of Case 1-1, we try to increase the gain matrix K⟶2 to K⟶2=5006. The other parameters of the three schemes are given in Table 1. Then, the simulation results of Case 1-2 are shown in Figures 7–9. From Figure 7, we know that only the proposed predefined-time NTSMC can guarantee that the convergence time is bounded by the new desired time (0.7s). Thus, according to Definition 3, the predefined-time stability is only achieved by the proposed scheme. As stated in Introduction section and Section 2, the reasons are as follows. (i) The sliding-mode surface of predefined-time LSMC is linear, which means that predefined-time LSMC only can guarantee the predefined-time convergence of sliding-mode surface (see Figure 8) rather than the predefined-time convergence of tracking errors. (ii) For the fixed-time NTSMC, the relationship between parameters and the setting time constant is complex and nonlinear. Thus, for the fixed-time NTSMC, it is difficult to adjust the parameters to achieve the predefined-time convergence without using the trial-and-error method.Figure 7
Position tracking errorse11 and e12 in Case 1-2.
(a)(b)Figure 8
Sliding-mode surfacess1 and s2 in Case 1-2.
(a)(b)Figure 9
Input torquesτ1 and τ2 in Case 1-2.
(a)(b)


## 4.1.1. The First Kind

The initial joint angle and angular velocity are chosen as(66)ϕ0=11Trad,(67)ϕ˙0=0−0.5Trad/s.Then, the initial tracking error can be calculated as(68)e1=1.31.3Trad,(69)e2=00Trad/s.It is assumed that we can use the trial-and-error method to adjust the parameters. The predefined-time parameters of predefined-time LSMC and predefined-time NTSMC are set asTc=2.5 and Te1=Te2=2.5, respectively. For the fixed-time NTSMC, the gain matrix K⟶2 is selected as K⟶2=2002.4. The other parameters of the three schemes are given in Table 1. The simulation results are shown in Figures 1–3 . From Figure 1, we know that the four schemes can guarantee that the convergence time is bounded by the desired time (2.5s) under these parameters. From Figure 3, it can be known that the input torques of fixed-time NTSMC have the problem of large switching. The reason is that the derivative of nonsingular function (63) in fixed-time NTSMC is discontinuous. The nonsingular function and its derivative of the proposed NTSMC are continuous, and thus the problem can be avoided by the proposed scheme.Table 1
Control parameters.
Control schemeControl parametersFinite-time NTSMCζ1=0.5, ζ2=2/3, Ω⟶1=230019, Ω⟶2=160012, kf=2, b0=12, b2=2.8, sf=0.01, δ¯f=0.5, ϖf=0.4Fixed-time NTSMCυ1=0.5, υ2=3/2, K0=1001, K1=1001, v1=0.5, v2=1.5, δF1=δF2=0.2, k0=2, k1=12, k2=2.8, r=1.5, sF0=0.01, δ¯=0.5, ϖ=0.4Predefined-time LSMCη=0.5, δr=0.2, kα=5, kτ=5, γ=0.03Proposed predefined-time NTSMCαe1=αe2=0.35, δe1=δe2=0.35, αs1=αs2=0.55, e11max=e12max=3, κs1=κs2=1, κe1=κe2=1, D¯1max=D¯2max=3, Ts1=Ts2=0.3, s1max=s2max=35Figure 1
Position tracking errorse11 and e12 in Case 1-1 (first kind of initial system state).
(a)(b)Figure 2
Sliding-mode surfacess1 and s2 in Case 1-1 (first kind of initial system state).
(a)(b)Figure 3
Input torquesτ1 and τ2 in Case 1-1 (first kind of initial system state).
(a)(b)Then, without changing the control parameters, we chose a new kind of initial system state.


## 4.1.2. The Second Kind

The initial joint angle and angular velocity are chosen as(70)ϕ0=3.63.6Trad,ϕ˙0=0−0.5Trad/s.Then, the initial tracking error can be calculated as(71)e1=3.93.9T,(72)e2=00T.By comparing equations (68) and (71), we can know that the second kind of initial tracking error is much bigger than the first kind of initial tracking error. The simulation results are shown in Figures 4–6 . From Figure 4, we can know that the convergence performance of finite-time NTSMC is greatly affected by the increase of initial system state. The convergence performances of fixed-time NTSMC, predefined-time LSMC, and proposed predefined-time NTSMC are not affected even if the initial system states are greatly increased. As stated in Introduction section and Section 2, the reason is that convergence time function of finite-time stability is related to the initial system state.Figure 4
Position tracking errorse11 and e12 in Case 1-1 (second kind of initial system state).
(a)(b)Figure 5
Sliding-mode surfacess1 and s2 in Case 1-1 (second kind of initial system state).
(a)(b)Figure 6
Input torquesτ1 and τ2 in Case 1-1 (second kind of initial system state).
(a)(b)Case 1-2: compared with Case 1-1, we desire a new convergence time which is bounded by 0.7s. It is assumed that we cannot use the trial-and-error method. We can only change the parameters once in this case. In this case, we consider three control schemes including fixed-time NTSMC, predefined-time LSMC, and predefined-time NTSMC. The initial system states are chosen as in (66) and (67). Then, the predefined-time parameters of predefined-time LSMC and proposed predefined-time NTSMC are set as Tc=0.7 and Te1=Te2=0.7. However, for the fixed-time NTSMC, the relationship between the parameters and the settling time constant is complex and nonlinear. Since the new desired time is smaller than the value of Case 1-1, we try to increase the gain matrix K⟶2 to K⟶2=5006. The other parameters of the three schemes are given in Table 1. Then, the simulation results of Case 1-2 are shown in Figures 7–9. From Figure 7, we know that only the proposed predefined-time NTSMC can guarantee that the convergence time is bounded by the new desired time (0.7s). Thus, according to Definition 3, the predefined-time stability is only achieved by the proposed scheme. As stated in Introduction section and Section 2, the reasons are as follows. (i) The sliding-mode surface of predefined-time LSMC is linear, which means that predefined-time LSMC only can guarantee the predefined-time convergence of sliding-mode surface (see Figure 8) rather than the predefined-time convergence of tracking errors. (ii) For the fixed-time NTSMC, the relationship between parameters and the setting time constant is complex and nonlinear. Thus, for the fixed-time NTSMC, it is difficult to adjust the parameters to achieve the predefined-time convergence without using the trial-and-error method.Figure 7
Position tracking errorse11 and e12 in Case 1-2.
(a)(b)Figure 8
Sliding-mode surfacess1 and s2 in Case 1-2.
(a)(b)Figure 9
Input torquesτ1 and τ2 in Case 1-2.
(a)(b)


## 4.2. Comparison of the Proposed Modified PTC and Conventional PTC Scheme

In this section, we consider the initial system state chosen as in (66) and (67). The predefined-time parameters are set as Te1=Te2=2. In this section, we consider four kinds of upper bound of initial tracking errors e1i0 and e2i0. We assume that e1i0≤k1i and e2i0≤k2i. From Case 2-1 to Case 2-4, the upper bounds are gradually approaching the exact values:Case 2-1 (conventional PTC algorithm):k11=k12=∞, k21=k22=∞.Case 2-2:k11=k12=20, k21=k22=10.Case 2-3:k11=k12=5, k21=k22=2.5.Case 2-4:k11=k12=2, k21=k22=1.For Case 2-1, the proposed PTC algorithm is equivalent to the conventional PTC algorithm. We selectκe1=κe2=1. Then, according to Appendix, for the above cases, the upper bound parameters can be estimated as follows.Case 2-1 (conventional PTC algorithm):e11max=e12max=∞, s1max=s2max=∞.Case 2-2:e11max=e12max=37.9979, s1max=s2max=253.7187.Case 2-3:e11max=e12max=10.9008, s1max=s2max=46.2597.Case 2-4:e11max=e12max=5.0891, s1max=s2max=15.6292.The other parameters are chosen asαe1=αe2=0.35, δe1=δe2=0.35, αs1=αs2=0.55, κs1=κs2=1, D¯1max=D¯2max=3, and Ts1=Ts2=0.3. The simulation results are shown in Figures 7 and 8. From Figures 10 and 11 , compared with the conventional PTC algorithm, the proposed modified PTC algorithm can achieve more accurate convergence time and smaller initial control input torques. Moreover, if the upper bounds k1i and k2i are closer to the exact values of initial values e1i0 and e2i0, more accurate convergence time and smaller initial control input torques can be achieved.Figure 10
Position tracking errorse11 and e12 described in Section 4.2.
(a)(b)Figure 11
Position tracking errorse11 and e12 described in Section 4.2.
(a)(b)


## 5. Conclusion

A novel predefined-time NTSMC was developed for the uncertain robotic manipulator system. The conventional PTC algorithm was modified such that the initial control peaking was reduced and the precision of convergence time was enhanced. Moreover, the proposed control scheme constructed a predefined-time nonsingular sliding-mode surface rather than the linear sliding-mode surface used in the conventional predefined-time SMC scheme. Thus, the proposed scheme can strictly guarantee the predefined-time stability of tracking error rather than only the predefined-time convergence of sliding-mode surface. The performance of the proposed control scheme has been illustrated by simulation. The proposed method of this paper focuses on the predefined-time convergence of the rigid manipulator system with uncertainties. The rigid manipulator system is a typical two-order matched uncertain system. However, some manipulator systems such as the flexible joint manipulator system are high-order systems (four order) and are affected by mismatched uncertainties. In the future work, we will extend the proposed predefined-time NTSMC scheme to higher-order and mismatched uncertain systems.



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*Source: 9991989-2021-06-10.xml*