# Adaptive Fast Nonsingular Fixed-Time Tracking Control for Robot Manipulators

Authors: Huihui Pan; Guangming Zhang
Journal: Complexity
 (2021)

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## Abstract

This paper studies the fixed-time trajectory tracking control problem of robot manipulators in the presence of uncertain dynamics and external disturbances. First, a novel nonsingular fixed-time sliding mode surface is presented, which can ensure that the convergence time of the suggested surface is bounded regardless of the initial states. Subsequently, a novel fast nonsingular fixed-time sliding mode control (NFNFSMC) is developed so that the closed-loop system is fixed-time convergent to the equilibrium. By applying the proposed NFNFSMC method and the adaptive technique, a novel adaptive nonsingular fixed-time control scheme is proposed, which can guarantee fast fixed-time convergence of the tracking errors to small regions around the origin. With the proposed control method, the lumped disturbance is compensated by the adaptive technique, whose prior information about the upper bound is not needed. The fixed-time stability of the trajectory tracking control under the proposed controller is proved by the Lyapunov stability theory. Finally, corresponding simulations are given to illustrate the validity and superiority of the proposed control approach.


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## Body

## 1. Introduction

In the past decades, many research efforts [1–6] have been devoted to addressing the trajectory tracking problem of robot manipulators in the presence of uncertain dynamics and external disturbances. Several advanced control strategies have been employed in robot manipulator control systems, such as sliding mode control (SMC) [7–11], feedback linearization [12], backstepping control [13], neural network control [14], and model predictive control [15]. Among them, SMC is a popular method owing to its fast convergence, insensitivity property to uncertain dynamics, and strong robustness to external disturbances [16–18].Among the SMC category, terminal sliding mode control (TSMC) [19, 20] can achieve the robust finite-time tracking of robot manipulators. However, standard TSMC may cause the singularity problem [21]. To remedy the singularity of the TSMC, many effective methods have been put forward. One method [22, 23] to overcome the singularity problem was given by developing a new type of TSM without singularity known as nonsingular TSM (NTSM). Another method [24] to avoid the singularity was presented by switching the terminal sliding mode surface to a general sliding mode surface with a nonlinear function. Additionally, fast nonsingular terminal sliding mode control (FNTSMC) has been widely investigated for the robot manipulator control system to enhance the convergence rate [25–28]. Reference [25] proposed a new fast nonsingular terminal sliding mode manifold combining the satisfactory characteristics of the linear SM and the NTSM to achieve fast finite-time stable tracking. A FNTSMC scheme with the adaptive technique was presented for robot manipulators to ensure that the position tracking errors could converge to zero within finite time in [26]. In [27], an improved NTSMC based on a nonlinear function was proposed for robot manipulators, which could guarantee system performance and fast finite-time stability. A novel adaptive second-order FNTSMC was introduced to achieve fast finite-time convergence and good tracking precision in [28]. Because both the sliding variables and the tracking errors can be stabilized to the equilibrium in finite time, the aforementioned SMCs are known as the finite-time controls.These finite-time controls have a drawback that the convergence time is related to the initial conditions of robotic systems. That is to say, the settling time of trajectory tracking cannot be acquired priorly. Recently, as an extension of the finite-time control, the fixed-time control has received a lot of attention [29–31]. Compared with the finite-time control, the fixed-time control can ensure that the convergence time is upper bounded by a fixed time and independent of initial conditions. The fixed-time control has been widely employed in many nonlinear control systems. More specifically, [32] presented a novel fixed-time output feedback control which could be employed to double integrator systems. In [33], an adaptive nonsingular fixed-time control strategy was proposed for the tracking control of the rigid spacecraft, which could guarantee the fixed-time stability of both the attitude and angular velocity. In [34], an adaptive fast nonsingular terminal sliding mode guidance law was designed, which could achieve system stabilization within a fixed time. A novel fixed-time NTSMC method was applied to a single inverted pendulum control system in [35]. Reference [30] developed a fixed-time convergent guidance law with impact angle control so that the impact angle error could be stabilized to zero before the interception within a fixed time. The above literature review indicates that the fixed-time controls are applicable to some physical control systems and can obtain system stabilization with fixed-time convergence. As far as the authors know, little attention has been paid to the fixed-time tracking control of uncertain robot manipulators. Moreover, further accelerating the convergence rate of tracking control is worth being considered in the controller design.This paper focuses on the development of an adaptive fast nonsingular fixed-time tracking control for uncertain robot manipulators so that satisfactory features including fast fixed-time convergence and high steady-state tracking precision are provided. The main contributions can be summarized as follows:(1)
A novel fixed-time nonsingular fast terminal sliding mode manifold (NFNFSM) is developed to shorten the time during which the system states arrive at the equilibrium.(2)
Based on the proposed NFNFSM, a novel fixed-time nonsingular fast terminal sliding mode controller (NFNFSMC) is designed. Moreover, the proof of fixed-time stability is provided in detail.(3)
An adaptive NFNFSMC (ANFNFSMC) scheme is presented by combining the proposed NFNFSMC and the adaptive technique. The lumped disturbance is compensated by the designed adaptive law, whose prior information about the upper bound is not needed. The proposed ANFNFSMC can not only obtain strong robustness to uncertain disturbances but also achieve fast fixed-time convergence of robot manipulator systems.The rest of this paper is organized as follows: Section2 provides the dynamic model of robot manipulators and some lemmas. The NFNFSMC algorithm is presented in Section 3. In Section 4, the ANFNFSMC strategy is proposed for the tracking control of robot manipulators. Simulation results illustrate the feasibility and superiority of the proposed control scheme in Section 5. Finally, the conclusion is shown in Section 6.Throughout the paper,sigxα denotes xαsignx.


## 2. Mathematical Preliminaries

### 2.1. Model of Robot Manipulators

The dynamic equation of the n-link robot manipulator model with uncertain disturbances can be described as [1](1)Mqq¨+Cq,q˙q˙+Gq=u+d,whereq,q˙,q¨∈Rn stand for the vectors of joint positions, velocities, and accelerations, respectively.Mq is the positive definite inertia matrix, Cq,q˙ is the centripetal Coriolis matrix, and Gq is the gravitational vector. u is the control input vector and d is the external disturbance vector.Assumption 1.
The model parameters in (1) are given by(2)Mq=M0q+ΔMq,Cq,q˙=C0q,q˙+ΔCq,q˙,Gq=G0q+ΔGq,where M0q, C0q,q˙, and G0q represent the nominal terms. ΔMq, ΔCq,q˙, and ΔGq represent the uncertain terms.
According to Assumption1, system (1) can be expressed as follows:(3)M0qq¨+C0q,q˙q˙+G0q=u+ρ,in which the lumped disturbance ρ is defined as(4)ρ=−ΔMqq¨−ΔCq,q˙q˙−ΔGq+d.Assumption 2.
The lumped disturbanceρ is bounded by(5)ρ<a0+a1q+a2q˙2,where a0, a1, and a2 are unknown positive constants.Assumption 3.
The norms of desired vectors are bounded by(6)qd≤b,0q˙d≤b,1where qd and q˙d denote the vector of desired position and velocity, respectively. b0 and b1 are positive constants.
The research focus of this paper is to propose a novel fast nonsingular fixed-time sliding mode control strategy for robot manipulators in the existence of uncertainties and disturbances such that fast fixed-time stability can be guaranteed.


### 2.2. Fundamental Facts

Definition 1 (see [36]).
Consider a dynamic system(7)z˙=fz,z0=z0,f0=0,wherez∈Rn, fz:D∈Rn is a continuous nonlinear function that is an open neighborhood D of the origin. The system is fixed-time stable if the convergence time is a bounded function Tz0, that is, there exists a time constant Tmax such that Tz0<Tmax.Lemma 1 (see [37]).
If system (7) is fixed-time stable, then there exists a continuous positive Lyapunov function Vz, α,β,a,b∈R+, a<1, b>1, and 0<Η<∞ satisfying that V˙z≤−αVza−βVzb+H. Then, the system can reach the residual set which is bounded as limt⟶Tz|Vz≤minH/α1−θ1/a,H/β1−θ1/b, where θ is scalar and satisfies 0<θ≤1. The time to reach the neighborhood of the origin is upper bounded by T≤1/αθ1−a+1/βθb−1.Lemma 2 (see [38, 39]).
For any nonnegative real numbersζ1,ζ2,…ζn,p>1, and 0<q≤1, the following two inequalities hold:(8)∑i=1nζip≥n1−p∑i=1nζip,∑i=1nζiq≥∑i=1nζiq.Lemma 3.
Consider the following nonlinear system:(9)z˙=−αsigzγ−csigzη,where α>0, c>0, γ=0.5r+1+0.5r−1signz−1, η=0.5r+φ+0.5r−φsignz−1, r>1, 0<φ<1. System (9) is fixed-time stable with the convergence time T bounded by(10)T≤Tmax=1α+cr−1+1α1−φln1+αc.Proof.
The differential equation for system (9) can be converted into the following form:(11)z˙=−αsigzr−csigzr,z>1,z˙=−αz−csigzφ,z≤1.
Denoteχ=1+Inzforz>1. Denote χ=z1−φ forz≤1. The initial values of z and χ are defined as z0 and χ0. Equation (11) can be rewritten as(12)χ˙=−αer−1χ−1−cer−1χ−1,χ>1,χ˙=−αχ1−φ−c1−φ,0<χ≤1.
Solving (12), the upper bound of convergence time can be calculated as(13)Tmax=limz0⟶∞Tz0=limχ0⟶∞∫1χ01α+cer−1χ−1dχ+∫0111−φαχ+cdχ=limχ0⟶∞∫1χ01α+cer−1χ−1dχ+1α1−φln1+αc.
Defineξ=er−1χ−1 and denote ξ0 as the initial value of ξ; we have(14)limχ0⟶∞∫1χ01α+cer−1χ−1dχ=1r−1limξ0⟶∞∫1ξ01ξαξ+cξdξ=1α+cr−1.
Thus, for system (9), the upper bound of convergence time can be expressed as(15)Tmax=1α+cr−1+1α1−φln1+αc.
The proof of Lemma3 is completed.Remark 1.
Reference [40] presented a fixed-time stable system z˙=−αsigzr−csigzφ, and the system can be stabilized within a fixed time T<1/αr−1+1/c1−φ. A fast fixed-time stable system z˙=−αsigzγ−csigzφwith γ=0.5r+1+0.5r−1signz−1 was introduced in [35], and the system can be stabilized within a fixed time T<1/αr−1+ln1+α/c/α1−φ. Since the relations 1/α+c<1/α and ln1+α/c<α/c hold, the proposed system (9) can offer a fast convergence rate than the systems presented in [35, 40].


## 2.1. Model of Robot Manipulators

The dynamic equation of the n-link robot manipulator model with uncertain disturbances can be described as [1](1)Mqq¨+Cq,q˙q˙+Gq=u+d,whereq,q˙,q¨∈Rn stand for the vectors of joint positions, velocities, and accelerations, respectively.Mq is the positive definite inertia matrix, Cq,q˙ is the centripetal Coriolis matrix, and Gq is the gravitational vector. u is the control input vector and d is the external disturbance vector.Assumption 1.
The model parameters in (1) are given by(2)Mq=M0q+ΔMq,Cq,q˙=C0q,q˙+ΔCq,q˙,Gq=G0q+ΔGq,where M0q, C0q,q˙, and G0q represent the nominal terms. ΔMq, ΔCq,q˙, and ΔGq represent the uncertain terms.
According to Assumption1, system (1) can be expressed as follows:(3)M0qq¨+C0q,q˙q˙+G0q=u+ρ,in which the lumped disturbance ρ is defined as(4)ρ=−ΔMqq¨−ΔCq,q˙q˙−ΔGq+d.Assumption 2.
The lumped disturbanceρ is bounded by(5)ρ<a0+a1q+a2q˙2,where a0, a1, and a2 are unknown positive constants.Assumption 3.
The norms of desired vectors are bounded by(6)qd≤b,0q˙d≤b,1where qd and q˙d denote the vector of desired position and velocity, respectively. b0 and b1 are positive constants.
The research focus of this paper is to propose a novel fast nonsingular fixed-time sliding mode control strategy for robot manipulators in the existence of uncertainties and disturbances such that fast fixed-time stability can be guaranteed.


## 2.2. Fundamental Facts

Definition 1 (see [36]).
Consider a dynamic system(7)z˙=fz,z0=z0,f0=0,wherez∈Rn, fz:D∈Rn is a continuous nonlinear function that is an open neighborhood D of the origin. The system is fixed-time stable if the convergence time is a bounded function Tz0, that is, there exists a time constant Tmax such that Tz0<Tmax.Lemma 1 (see [37]).
If system (7) is fixed-time stable, then there exists a continuous positive Lyapunov function Vz, α,β,a,b∈R+, a<1, b>1, and 0<Η<∞ satisfying that V˙z≤−αVza−βVzb+H. Then, the system can reach the residual set which is bounded as limt⟶Tz|Vz≤minH/α1−θ1/a,H/β1−θ1/b, where θ is scalar and satisfies 0<θ≤1. The time to reach the neighborhood of the origin is upper bounded by T≤1/αθ1−a+1/βθb−1.Lemma 2 (see [38, 39]).
For any nonnegative real numbersζ1,ζ2,…ζn,p>1, and 0<q≤1, the following two inequalities hold:(8)∑i=1nζip≥n1−p∑i=1nζip,∑i=1nζiq≥∑i=1nζiq.Lemma 3.
Consider the following nonlinear system:(9)z˙=−αsigzγ−csigzη,where α>0, c>0, γ=0.5r+1+0.5r−1signz−1, η=0.5r+φ+0.5r−φsignz−1, r>1, 0<φ<1. System (9) is fixed-time stable with the convergence time T bounded by(10)T≤Tmax=1α+cr−1+1α1−φln1+αc.Proof.
The differential equation for system (9) can be converted into the following form:(11)z˙=−αsigzr−csigzr,z>1,z˙=−αz−csigzφ,z≤1.
Denoteχ=1+Inzforz>1. Denote χ=z1−φ forz≤1. The initial values of z and χ are defined as z0 and χ0. Equation (11) can be rewritten as(12)χ˙=−αer−1χ−1−cer−1χ−1,χ>1,χ˙=−αχ1−φ−c1−φ,0<χ≤1.
Solving (12), the upper bound of convergence time can be calculated as(13)Tmax=limz0⟶∞Tz0=limχ0⟶∞∫1χ01α+cer−1χ−1dχ+∫0111−φαχ+cdχ=limχ0⟶∞∫1χ01α+cer−1χ−1dχ+1α1−φln1+αc.
Defineξ=er−1χ−1 and denote ξ0 as the initial value of ξ; we have(14)limχ0⟶∞∫1χ01α+cer−1χ−1dχ=1r−1limξ0⟶∞∫1ξ01ξαξ+cξdξ=1α+cr−1.
Thus, for system (9), the upper bound of convergence time can be expressed as(15)Tmax=1α+cr−1+1α1−φln1+αc.
The proof of Lemma3 is completed.Remark 1.
Reference [40] presented a fixed-time stable system z˙=−αsigzr−csigzφ, and the system can be stabilized within a fixed time T<1/αr−1+1/c1−φ. A fast fixed-time stable system z˙=−αsigzγ−csigzφwith γ=0.5r+1+0.5r−1signz−1 was introduced in [35], and the system can be stabilized within a fixed time T<1/αr−1+ln1+α/c/α1−φ. Since the relations 1/α+c<1/α and ln1+α/c<α/c hold, the proposed system (9) can offer a fast convergence rate than the systems presented in [35, 40].


## 3. Novel Nonsingular Fixed-Time Control and Stability Analysis

In this section, a novel nonsingular fixed-time control and the related stability analysis are presented.


### 3.1. Novel Nonsingular Fixed-Time Control

For a clear interpretation of the key idea, we first consider the novel nonsingular fixed-time control of a single second-order system given by(16)x˙1=x2,x˙2=u,where x1 and x2 are system states. u is the control input.Based on Lemma3, for system (16), a new form of fixed-time sliding mode surface is designed as(17)s=x2+α1sigx1γ1+c1sigx1η1,where r1>1, 0<φ1<1, γ1=0.5r1+1+0.5r1−1signx1−1, η1=0.5r1+φ1+0.5r1−φ1signx1−1.Differentiating (17) results in(18)s˙=x˙2+α1γ1x1γ1−1x2+c1η1x1η1−1x2.For (18), if x1=0 and x2≠0, it may suffer from the singularity problem due to φ1−1<0.To circumvent the singularity problem, a novel fast nonsingular fixed-time sliding mode surface (NFNFSM) is proposed as(19)s=x1+βsigα1sigx1γ1+x21/η1,where α1>0, c1>0, r1>1, 0<φ1<1, β=c1−1/η1,(20)γ1=0.5r1+1+0.5r1−1signx1−1,η1=0.5r1+φ1+0.5r1−φ1signα1sigx1γ1+x2c1−1.Remark 2.
Whens=0 is achieved, it can be deduced that x2=−α1sigx1γ1−c1sigx1η1 and x1η1=α1sigx1γ1+x2/c1.
Whenα1sigx1γ1+x2/c1≥1, then η1=r1 and x1≥1. When α1sigx1γ1+x2/c1<1, thenη1=φ1 and x1<1. Therefore, onces=0 is satisfied, η1 is equivalent to the equality η1=0.5r1+φ1+0.5r1−φ1signx1−1.Remark 3.
Note that the proposed NFNFSM can solve the singularity problem without switching the terminal sliding mode surface into a general sliding mode surface, which is different from some existing nonsingular fixed-time sliding mode manifolds, such as in [24, 27].
Next, to illustrate the superiority of the proposed sliding surface, the convergence performance of NFSM in [39], FNFSM in [34], and the proposed NFNFSM are compared in the sliding motion. The sliding surface NFSM [39] is(21)s=x1+sigkx2q/p,with(22)k=1α1x1m/n−p/q+c1,where α1,c1>0 and m,n,p, and q are odd integers satisfying that p<q<2p, m/n−p/q>1.
The sliding surface FNFSM [34] is(23)s=x2+α1sigx1γ1+c1μx1,with(24)μx1=sigx1φ1,x1≥δ,β1x1+β2sigx12,x1<δ,where γ1=0.5r+1+0.5r−1signx1−1, 0<δ<1, β1=2−φ1δφ1−1, β2=φ1−1δφ1−2.
The parameters of the three sliding surfaces are selected asα1=c1=1, r=2, φ1=0.6, m/n=2, p/q=0.6, δ=0.01.
The same initial condition is thatx10=5. We illustrate the convergence of the three sliding surfaces in Figure 1. It can be noted from Figure 1 that the proposed NFNFSM offers a faster convergence rate than NFSM and FNFSM.
Based on the proposed sliding surface (19), a novel fixed-time controller is designed as(25)u=−η1βsigα1sigx1γ1+x22−1/η1+α1γ1x1γ1−1x2+α1η1x1γ1−1α1sigx1γ1+x2+η1βα1sigx1γ1+x21−1/η1⋅ψεα1sigx1γ1+x21/η1−1⋅α2sigsr2+c2sigsφ2,in which ψε⋅:0,+∞⟶0,1 as(26)ψεx=sinπ2⋅xε,x≤ε,1,x>ε,whereα2>0,c2>0，r2>1，0<φ2<1，x>0, 0<ε<1, and ψεx/x⟶π/2ε as x⟶0.Figure 1
Comparison of the three sliding surfaces.


### 3.2. Stability Analysis

Theorem 1.
Consider system (16) with the proposed fixed-time controller defined by (25). Then the system states can converge to the origin within a fixed time and the convergence time is expressed as(27)T<Tmax=T1+T2+ϕε,where(28)T1=1α2r2−1+1c21−φ2,T2=1α1+c1r1−1+ln1+α1/c1α11−φ1,and ϕεrepresents a small time margin associated with ε.Proof.
Select the following Lyapunov function:(29)V1=0.5s2.
The derivative ofV1 to time is(30)V˙1=sx2+βη1α1sigx1γ1+x21/η1−1α1γ1x1γ1−1x2+x˙2.
Substituting (25) into (30), there is(31)V˙1=−sψεα1sigx1γ1+x21/η1−1α2sigsr2+c2sigsφ2−sα1x1γ1−1s≤−ψεα1sigx1γ1+x21/η1−1α2sr2+1+c2sφ2+1=−ψεα1sigx1γ1+x21/η1−1α22V1r2+1/2+c22V1φ2+1/2.
Denoteϖ=α1sigx1γ1+x21/η1−1, g=α1sigx1γ1+x2. According to (31),ψεϖ>0 if g≠0. To facilitate analysis, the state spacex∈R2 is divided into two separate regions as Ω1=x1,x2|ϖ=g1/η1−1≥ε,Ω2=x1,x2|ϖ=g1/η1−1<ε.Case 1. When the statesx1,x2enter into the region Ω1, then the function ψεϖ=1. If V1≠0, let y0=2V1 and y˙0=V˙1/2V1, and we obtain y˙0≤−α2sigy0r2−c2sigy0φ2. According to [39], the states x1,x2will arrive at the sliding surface s=0or enter into the region Ω2within fixed time T1 .Case 2. In the region Ω2, 0<ψεϖ<1if g≠0. It can be deduced from (31) that s=0is still attractive. Next, we need to prove that g=0is not attractive except for the origin x1,x2=0,0. When g is very close to 0, the control law (25) reduces to the following form:(32)u=−α1γ1x1γ1−1x2−πη12βεα2sigsr2+c2sigsφ2,where the fact ψεϖ/ϖ⟶π/2ε, as g⟶0is used.
Differentiatinggand combining (32), we have(33)g˙=α1γ1x1γ1−1x2+x˙2=−πη12βεα2sigsr2+c2sigsφ2.
From (33), it can be concluded that g˙<0 for s>0 and g˙>0for s<0. This means that g=0 is not an attractor. Then, the system states x1,x2will transgress Ω2 into Ω1 monotonically in a small time marginϕε.
Accordingly, the sliding surfaces=0can be arrived at within time Tr<T1+ϕε. Once the sliding surface s=0is reached, it can be known from Lemma 3 that the system states x1,x2can converge to the origin within fixed time Ts<T2. Then, the total convergence time is upper bounded by (27).
The proof of Theorem1 is completed.Remark 4.
Note that the timeϕε across Ω2 cannot be calculated precisely. Nevertheless, for enough small ε, it can be regarded as g=0. From (19), it can be obtained that s=x1. Integrating both sides of (33) yields(34)∫−εη1/1−η1εη1/1−η1dg=∫0ϕεπη12βεα2sigsr2+c2sigsφ2dt.
Solving (34) obtains(35)ϕε=4βε1/1−η1πη1α2x1r2+c2x1φ2,where 1/1−η1>1. This means that ϕεcan be small enough by selecting sufficiently small ε. Then, the finite time ϕεcan be ignored for smallε because of the estimation conservativeness of Τ1.


## 3.1. Novel Nonsingular Fixed-Time Control

For a clear interpretation of the key idea, we first consider the novel nonsingular fixed-time control of a single second-order system given by(16)x˙1=x2,x˙2=u,where x1 and x2 are system states. u is the control input.Based on Lemma3, for system (16), a new form of fixed-time sliding mode surface is designed as(17)s=x2+α1sigx1γ1+c1sigx1η1,where r1>1, 0<φ1<1, γ1=0.5r1+1+0.5r1−1signx1−1, η1=0.5r1+φ1+0.5r1−φ1signx1−1.Differentiating (17) results in(18)s˙=x˙2+α1γ1x1γ1−1x2+c1η1x1η1−1x2.For (18), if x1=0 and x2≠0, it may suffer from the singularity problem due to φ1−1<0.To circumvent the singularity problem, a novel fast nonsingular fixed-time sliding mode surface (NFNFSM) is proposed as(19)s=x1+βsigα1sigx1γ1+x21/η1,where α1>0, c1>0, r1>1, 0<φ1<1, β=c1−1/η1,(20)γ1=0.5r1+1+0.5r1−1signx1−1,η1=0.5r1+φ1+0.5r1−φ1signα1sigx1γ1+x2c1−1.Remark 2.
Whens=0 is achieved, it can be deduced that x2=−α1sigx1γ1−c1sigx1η1 and x1η1=α1sigx1γ1+x2/c1.
Whenα1sigx1γ1+x2/c1≥1, then η1=r1 and x1≥1. When α1sigx1γ1+x2/c1<1, thenη1=φ1 and x1<1. Therefore, onces=0 is satisfied, η1 is equivalent to the equality η1=0.5r1+φ1+0.5r1−φ1signx1−1.Remark 3.
Note that the proposed NFNFSM can solve the singularity problem without switching the terminal sliding mode surface into a general sliding mode surface, which is different from some existing nonsingular fixed-time sliding mode manifolds, such as in [24, 27].
Next, to illustrate the superiority of the proposed sliding surface, the convergence performance of NFSM in [39], FNFSM in [34], and the proposed NFNFSM are compared in the sliding motion. The sliding surface NFSM [39] is(21)s=x1+sigkx2q/p,with(22)k=1α1x1m/n−p/q+c1,where α1,c1>0 and m,n,p, and q are odd integers satisfying that p<q<2p, m/n−p/q>1.
The sliding surface FNFSM [34] is(23)s=x2+α1sigx1γ1+c1μx1,with(24)μx1=sigx1φ1,x1≥δ,β1x1+β2sigx12,x1<δ,where γ1=0.5r+1+0.5r−1signx1−1, 0<δ<1, β1=2−φ1δφ1−1, β2=φ1−1δφ1−2.
The parameters of the three sliding surfaces are selected asα1=c1=1, r=2, φ1=0.6, m/n=2, p/q=0.6, δ=0.01.
The same initial condition is thatx10=5. We illustrate the convergence of the three sliding surfaces in Figure 1. It can be noted from Figure 1 that the proposed NFNFSM offers a faster convergence rate than NFSM and FNFSM.
Based on the proposed sliding surface (19), a novel fixed-time controller is designed as(25)u=−η1βsigα1sigx1γ1+x22−1/η1+α1γ1x1γ1−1x2+α1η1x1γ1−1α1sigx1γ1+x2+η1βα1sigx1γ1+x21−1/η1⋅ψεα1sigx1γ1+x21/η1−1⋅α2sigsr2+c2sigsφ2,in which ψε⋅:0,+∞⟶0,1 as(26)ψεx=sinπ2⋅xε,x≤ε,1,x>ε,whereα2>0,c2>0，r2>1，0<φ2<1，x>0, 0<ε<1, and ψεx/x⟶π/2ε as x⟶0.Figure 1
Comparison of the three sliding surfaces.


## 3.2. Stability Analysis

Theorem 1.
Consider system (16) with the proposed fixed-time controller defined by (25). Then the system states can converge to the origin within a fixed time and the convergence time is expressed as(27)T<Tmax=T1+T2+ϕε,where(28)T1=1α2r2−1+1c21−φ2,T2=1α1+c1r1−1+ln1+α1/c1α11−φ1,and ϕεrepresents a small time margin associated with ε.Proof.
Select the following Lyapunov function:(29)V1=0.5s2.
The derivative ofV1 to time is(30)V˙1=sx2+βη1α1sigx1γ1+x21/η1−1α1γ1x1γ1−1x2+x˙2.
Substituting (25) into (30), there is(31)V˙1=−sψεα1sigx1γ1+x21/η1−1α2sigsr2+c2sigsφ2−sα1x1γ1−1s≤−ψεα1sigx1γ1+x21/η1−1α2sr2+1+c2sφ2+1=−ψεα1sigx1γ1+x21/η1−1α22V1r2+1/2+c22V1φ2+1/2.
Denoteϖ=α1sigx1γ1+x21/η1−1, g=α1sigx1γ1+x2. According to (31),ψεϖ>0 if g≠0. To facilitate analysis, the state spacex∈R2 is divided into two separate regions as Ω1=x1,x2|ϖ=g1/η1−1≥ε,Ω2=x1,x2|ϖ=g1/η1−1<ε.Case 1. When the statesx1,x2enter into the region Ω1, then the function ψεϖ=1. If V1≠0, let y0=2V1 and y˙0=V˙1/2V1, and we obtain y˙0≤−α2sigy0r2−c2sigy0φ2. According to [39], the states x1,x2will arrive at the sliding surface s=0or enter into the region Ω2within fixed time T1 .Case 2. In the region Ω2, 0<ψεϖ<1if g≠0. It can be deduced from (31) that s=0is still attractive. Next, we need to prove that g=0is not attractive except for the origin x1,x2=0,0. When g is very close to 0, the control law (25) reduces to the following form:(32)u=−α1γ1x1γ1−1x2−πη12βεα2sigsr2+c2sigsφ2,where the fact ψεϖ/ϖ⟶π/2ε, as g⟶0is used.
Differentiatinggand combining (32), we have(33)g˙=α1γ1x1γ1−1x2+x˙2=−πη12βεα2sigsr2+c2sigsφ2.
From (33), it can be concluded that g˙<0 for s>0 and g˙>0for s<0. This means that g=0 is not an attractor. Then, the system states x1,x2will transgress Ω2 into Ω1 monotonically in a small time marginϕε.
Accordingly, the sliding surfaces=0can be arrived at within time Tr<T1+ϕε. Once the sliding surface s=0is reached, it can be known from Lemma 3 that the system states x1,x2can converge to the origin within fixed time Ts<T2. Then, the total convergence time is upper bounded by (27).
The proof of Theorem1 is completed.Remark 4.
Note that the timeϕε across Ω2 cannot be calculated precisely. Nevertheless, for enough small ε, it can be regarded as g=0. From (19), it can be obtained that s=x1. Integrating both sides of (33) yields(34)∫−εη1/1−η1εη1/1−η1dg=∫0ϕεπη12βεα2sigsr2+c2sigsφ2dt.
Solving (34) obtains(35)ϕε=4βε1/1−η1πη1α2x1r2+c2x1φ2,where 1/1−η1>1. This means that ϕεcan be small enough by selecting sufficiently small ε. Then, the finite time ϕεcan be ignored for smallε because of the estimation conservativeness of Τ1.


## 4. Adaptive Fixed-Time Control of Robot Manipulators

### 4.1. Nonsingular Fixed-Time Sliding Mode Surface

Definee=q−qdand e˙=q˙−q˙d. The error equation of system (1) can be expressed as follows:(36)e¨=fe,e˙+be,e˙u+ρ,where(37)fe,e˙=−M0−1qC0q,q˙q˙−M0−1qG0q−q¨d,be,e˙=−M0−1q.A vector of novel fast nonsingular fixed-time sliding mode surface (NFNFSM) is constructed as(38)s=e+βsigα1sigeγ1+e˙1/η1=e+βυe,e˙,where β is a diagonal matrix: (39)βi=c1−1/η1i,α1>0,c1>0,r1>1,0<φ1<1,υe,e˙=υe1,e˙1,υe2,e˙2,…,υen,e˙nT,υei,e˙i=sigα1sigeiγ1i+e˙i1/η1i,i=1,2…,n,γ1i=0.5r1+1+0.5r1−1signei−1,η1i=0.5r1+φ1+0.5r1−φ1signα1sigeiγ1i+e˙ic1−1.From (38), e˙i=−α1sigeiγ1i−c1sigeiη1iwhen si=0. Consider the Lyapunov function V2=0.5ei2, and its time derivative isV˙2=eie˙i=−α1eiγ1i+1−c1eiη1i+1. Whenei≥1, there is V˙2≤−α12V2r1+1/2−c12V2r1+1/2. Whenei<1, there is V˙2≤−α12V2−c12V2φ1+1/2. According to Lemma 3, the system states can reach the designed sliding surface within a fixed time, given by(40)Ts≤22r1+1/2α1+2r1+1/2c1r1−1+1α11−φ1ln1+2α12φ1+1/2c1.


### 4.2. Controller Design

To achieve the fixed-time tracking, an adaptive NFNFSM controller (ANFNFSMC) is designed as(41)u=u0+u1+u2,with(42)u0=C0q,q˙q˙+G0q+M0qq¨d,(43)u1=−M0qβ−1η1sigα1sigeγ1+e˙2−1/η1+diagα1η1ieiγ1i−1α1sigeγ1+e˙+diagα1γ1ieiγ1i−1e˙+β−1diagη1iα1sigeiγ1i+e˙i1−1/η1i⋅diagψεα1sigeiγ1i+e˙i1/η1i−1⋅α2sigsr2+c2sigsφ2,(44)u2=−M0qa^0+a^1q+a^2q˙2s¯s¯,in which α2,c2>0,r2>1 , 0<φ2<1, s¯=sTϑT=ϑTs , ϑ=diagϑ1,ϑ2,…,ϑn,ϑi=βi/η1iα1sigeiγ1i+e˙i1/η1i−1. a^iis the estimation of ai, and the adaptive laws are given by(45)a^˙0=s¯,a^˙1=s¯q,a^˙2=s¯q˙2.


### 4.3. Stability Analysis

The fixed-time stability of the error system (36) in both the reaching phase and the sliding phase is stated in Theorem 2.Theorem 2.
For the error system (36), using the proposed sliding mode surface given by (38) and the novel adaptive nonsingular fixed-time controller defined by (41)–(45), the following performance of the closed-loop system can be achieved:(1)
The variabless,a˜0, a˜1, and a˜2 are all bounded(2)
The proposed sliding mode manifoldscan converge to the neighborhood of s=0 within a fixed time(3)
The state variableseiand e˙i can converge into a small region near the origin within a fixed timeProof.
(1)
Denotea˜i=ai−a^i,i=0,1,2 and choose a positive Lyapunov function as(46)V3=0.5sTs+a˜02+a˜12+a˜22.DifferentiatingV3 with respect to time and taking into account (38), (41)–(45), we obtain(47)V˙3=−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2−sTdiagα1eiγ1i−1s+sTρ−a^0+a^1q+a^2q˙2s¯s¯−a0−a^0a^˙0−a1−a^1a^˙1−a2−a^2a^˙2≤−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2+s¯ρ−a^0+a^1q+a^2q˙2−a0−a^0s¯−a1−a^1s¯q−a2−a^2s¯q˙2≤−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2+s¯ρ−a0+a1q+a2q˙2.According to Assumption2, there is(48)V˙3≤−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1⋅α2sigsr2+c2sigsφ2.From (48), we know that V˙3≤0, which implies that V3 is bounded. Then, s, a˜0, a˜1, and a˜2 are all bounded. It can be noted from (38) that the boundedness of ei,e˙i are guaranteed. And according to Assumption 3, it can be known that both q and q˙are bounded. Thus, there exists a positive constant σ0 such that s¯a˜0+a˜1q+a˜2q˙2≤σ0.(2)
To examine the fixed-time stability, consider the following positive definite Lyapunov function:(49)V4=0.5sTs.Taking the derivative ofV4 results in(50)V˙4≤−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2−sTdiagα1eiγ1i−1s+s¯ρ−a^0+a^1q+a^2q˙2≤−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2+s¯a0+a1q+a2q˙2−s¯a^0+a^1q+a^2q˙2=−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2+s¯a˜0+a˜1q+a˜2q˙2.Based on the above analysis, we know thats, a˜0, a˜1, a˜2,q, and q˙ are all bounded; thus, there exists a positive constant σ satisfying that s¯a˜0+a˜1q+a˜2q˙2+sTI2−diagψεα1isigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2≤σ. Then, according to Lemma 2, (50) is simplified as(51)V˙4≤−sTα2sigsr2+c2sigsφ2+σ≤−n1−r2/2α22V4r2+1/2−c22V4φ2+1/2+σ.Applying Lemma1, the system is fixed-time stable with the following convergence region:(52)Δ=limt⟶Trs|V2≤minσ2r2+1/2n1−r2/2α21−θ2/r2+1,σ2φ2+1/2c21−θ2/φ2+1.According to Lemma1, the convergence time of the reaching motion can be estimated as(53)Tr≤22r2+1/2n1−r2/2α2θr2−1+22φ2+1/2c2θ1−φ2.(3)
Once the sliding variablesi converges to the region si≤Δ, i=1,2,…,n, there is(54)si=ei+βisigα1sigeiγ1i+e˙i1/η1i,si≤Δ.
Equation (54) can be rewritten as(55)ei+βi−sisigα1sigeiγ1i+e˙i1/η1i⋅sigα1sigeiγ1i+e˙i1/η1i=0.
Whenβi−si/sigα1sigeiγ1i+e˙i1/η1i>0, (55) still maintains the form of NFNFSM as (38). Thus, the system trajectory will persistently converge to the NFNFSM (38) until it satisfies the following condition: (56)α1sigeiγ1i+e˙i1/η1i≤Δβi.
From (38) and (56), the tracking error will converge to the region(57)ei≤βiα1sigeiγ1i+e˙i1/η1i+si≤2Δ.
From (56), we have(58)α1sigeiγ1i−e˙i≤α1sigeiγ1i+e˙i≤Δβiη1i.
Then,(59)α1sigeiγ1i−e˙i≥−Δβiη1i.
Thus, the fixed-time convergence region ofei is(60)e˙i≤α1sigeiγ1i+Δβiη1i≤α12Δγ1i+Δβiη1i.
It is concluded that the system states will converge to the setR=ei,e˙i:ei≤2Δ,e˙i≤α12Δγ1i+Δ/βiη1i in fixed time.
Accordingly, the upper bound of the total convergence time can be estimated as(61)T=Tr+Ts≤12r2−1/2n1−r2/2α2θr2−1+12φ2−1/2c2θ1−φ2+12r1−1/2α1+c1r1−1+1α11−φ1ln1+α12φ1−1/2c1.
The proof of Theorem2 is completed.


## 4.1. Nonsingular Fixed-Time Sliding Mode Surface

Definee=q−qdand e˙=q˙−q˙d. The error equation of system (1) can be expressed as follows:(36)e¨=fe,e˙+be,e˙u+ρ,where(37)fe,e˙=−M0−1qC0q,q˙q˙−M0−1qG0q−q¨d,be,e˙=−M0−1q.A vector of novel fast nonsingular fixed-time sliding mode surface (NFNFSM) is constructed as(38)s=e+βsigα1sigeγ1+e˙1/η1=e+βυe,e˙,where β is a diagonal matrix: (39)βi=c1−1/η1i,α1>0,c1>0,r1>1,0<φ1<1,υe,e˙=υe1,e˙1,υe2,e˙2,…,υen,e˙nT,υei,e˙i=sigα1sigeiγ1i+e˙i1/η1i,i=1,2…,n,γ1i=0.5r1+1+0.5r1−1signei−1,η1i=0.5r1+φ1+0.5r1−φ1signα1sigeiγ1i+e˙ic1−1.From (38), e˙i=−α1sigeiγ1i−c1sigeiη1iwhen si=0. Consider the Lyapunov function V2=0.5ei2, and its time derivative isV˙2=eie˙i=−α1eiγ1i+1−c1eiη1i+1. Whenei≥1, there is V˙2≤−α12V2r1+1/2−c12V2r1+1/2. Whenei<1, there is V˙2≤−α12V2−c12V2φ1+1/2. According to Lemma 3, the system states can reach the designed sliding surface within a fixed time, given by(40)Ts≤22r1+1/2α1+2r1+1/2c1r1−1+1α11−φ1ln1+2α12φ1+1/2c1.


## 4.2. Controller Design

To achieve the fixed-time tracking, an adaptive NFNFSM controller (ANFNFSMC) is designed as(41)u=u0+u1+u2,with(42)u0=C0q,q˙q˙+G0q+M0qq¨d,(43)u1=−M0qβ−1η1sigα1sigeγ1+e˙2−1/η1+diagα1η1ieiγ1i−1α1sigeγ1+e˙+diagα1γ1ieiγ1i−1e˙+β−1diagη1iα1sigeiγ1i+e˙i1−1/η1i⋅diagψεα1sigeiγ1i+e˙i1/η1i−1⋅α2sigsr2+c2sigsφ2,(44)u2=−M0qa^0+a^1q+a^2q˙2s¯s¯,in which α2,c2>0,r2>1 , 0<φ2<1, s¯=sTϑT=ϑTs , ϑ=diagϑ1,ϑ2,…,ϑn,ϑi=βi/η1iα1sigeiγ1i+e˙i1/η1i−1. a^iis the estimation of ai, and the adaptive laws are given by(45)a^˙0=s¯,a^˙1=s¯q,a^˙2=s¯q˙2.


## 4.3. Stability Analysis

The fixed-time stability of the error system (36) in both the reaching phase and the sliding phase is stated in Theorem 2.Theorem 2.
For the error system (36), using the proposed sliding mode surface given by (38) and the novel adaptive nonsingular fixed-time controller defined by (41)–(45), the following performance of the closed-loop system can be achieved:(1)
The variabless,a˜0, a˜1, and a˜2 are all bounded(2)
The proposed sliding mode manifoldscan converge to the neighborhood of s=0 within a fixed time(3)
The state variableseiand e˙i can converge into a small region near the origin within a fixed timeProof.
(1)
Denotea˜i=ai−a^i,i=0,1,2 and choose a positive Lyapunov function as(46)V3=0.5sTs+a˜02+a˜12+a˜22.DifferentiatingV3 with respect to time and taking into account (38), (41)–(45), we obtain(47)V˙3=−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2−sTdiagα1eiγ1i−1s+sTρ−a^0+a^1q+a^2q˙2s¯s¯−a0−a^0a^˙0−a1−a^1a^˙1−a2−a^2a^˙2≤−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2+s¯ρ−a^0+a^1q+a^2q˙2−a0−a^0s¯−a1−a^1s¯q−a2−a^2s¯q˙2≤−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2+s¯ρ−a0+a1q+a2q˙2.According to Assumption2, there is(48)V˙3≤−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1⋅α2sigsr2+c2sigsφ2.From (48), we know that V˙3≤0, which implies that V3 is bounded. Then, s, a˜0, a˜1, and a˜2 are all bounded. It can be noted from (38) that the boundedness of ei,e˙i are guaranteed. And according to Assumption 3, it can be known that both q and q˙are bounded. Thus, there exists a positive constant σ0 such that s¯a˜0+a˜1q+a˜2q˙2≤σ0.(2)
To examine the fixed-time stability, consider the following positive definite Lyapunov function:(49)V4=0.5sTs.Taking the derivative ofV4 results in(50)V˙4≤−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2−sTdiagα1eiγ1i−1s+s¯ρ−a^0+a^1q+a^2q˙2≤−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2+s¯a0+a1q+a2q˙2−s¯a^0+a^1q+a^2q˙2=−sTdiagψεα1sigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2+s¯a˜0+a˜1q+a˜2q˙2.Based on the above analysis, we know thats, a˜0, a˜1, a˜2,q, and q˙ are all bounded; thus, there exists a positive constant σ satisfying that s¯a˜0+a˜1q+a˜2q˙2+sTI2−diagψεα1isigeiγ1i+e˙i1/η1i−1α2sigsr2+c2sigsφ2≤σ. Then, according to Lemma 2, (50) is simplified as(51)V˙4≤−sTα2sigsr2+c2sigsφ2+σ≤−n1−r2/2α22V4r2+1/2−c22V4φ2+1/2+σ.Applying Lemma1, the system is fixed-time stable with the following convergence region:(52)Δ=limt⟶Trs|V2≤minσ2r2+1/2n1−r2/2α21−θ2/r2+1,σ2φ2+1/2c21−θ2/φ2+1.According to Lemma1, the convergence time of the reaching motion can be estimated as(53)Tr≤22r2+1/2n1−r2/2α2θr2−1+22φ2+1/2c2θ1−φ2.(3)
Once the sliding variablesi converges to the region si≤Δ, i=1,2,…,n, there is(54)si=ei+βisigα1sigeiγ1i+e˙i1/η1i,si≤Δ.
Equation (54) can be rewritten as(55)ei+βi−sisigα1sigeiγ1i+e˙i1/η1i⋅sigα1sigeiγ1i+e˙i1/η1i=0.
Whenβi−si/sigα1sigeiγ1i+e˙i1/η1i>0, (55) still maintains the form of NFNFSM as (38). Thus, the system trajectory will persistently converge to the NFNFSM (38) until it satisfies the following condition: (56)α1sigeiγ1i+e˙i1/η1i≤Δβi.
From (38) and (56), the tracking error will converge to the region(57)ei≤βiα1sigeiγ1i+e˙i1/η1i+si≤2Δ.
From (56), we have(58)α1sigeiγ1i−e˙i≤α1sigeiγ1i+e˙i≤Δβiη1i.
Then,(59)α1sigeiγ1i−e˙i≥−Δβiη1i.
Thus, the fixed-time convergence region ofei is(60)e˙i≤α1sigeiγ1i+Δβiη1i≤α12Δγ1i+Δβiη1i.
It is concluded that the system states will converge to the setR=ei,e˙i:ei≤2Δ,e˙i≤α12Δγ1i+Δ/βiη1i in fixed time.
Accordingly, the upper bound of the total convergence time can be estimated as(61)T=Tr+Ts≤12r2−1/2n1−r2/2α2θr2−1+12φ2−1/2c2θ1−φ2+12r1−1/2α1+c1r1−1+1α11−φ1ln1+α12φ1−1/2c1.
The proof of Theorem2 is completed.


## 5. Simulation Study

In this section, based on MATLAB (R2014a)/Simulink, simulations are carried out to confirm the effectiveness and superiority of the proposed ANFNFSMC method.The dynamics of a typical two-link robot manipulator can be expressed as [1](62)Mq=N1+2N2cosq2N3+N2cosq2N3+N2cosq2N4,Cq,q˙=−N2sinq2q˙1−2N2sinq2q˙10N2sinq2q˙2,Gq=N5cosq1+N6cosq1+q2N6cosq1+q2,with(63)N1=m1+m2l12+m2l22+J1,N2=m2l1l2,N3=m2l22,N4=N3+J2,N5=m1+m2gl1,N6=m2gl2.The system parameters in (63) are listed as m1=0.5kg, m2=1.5kg, l1=1m, l2=0.8m, J1=5kg⋅m2, J2=5kg⋅m2, g=9.8N/s2. The normal values of m1 and m2 are set as m10=0.4kg and m20=1.2kg. The external disturbances are assumed as d=5cos2t+2sin2πt,3sin2t+2sinπtT.The controller and adaptive law parameters in (43) are set as α1=c1=1, r1=2, φ1=0.6, α2=c2=2, r2=2.5, φ2=0.6.


### 5.1. Tracking Control with Different Initial States

In this case, to confirm the fixed-time tracking performance of the proposed control approach, simulations are performed for trajectory tracking of a robot manipulator with different initial states, as shown in Figures2–4. The desired signals are q1d=0.5e−4t−1.2e−t+1.25and q2d=e−t−0.5e−4t+1.25. These different initial states are given by(64)I1:q0=1,1T,q˙0=0,0T,I2:q0=2.5,−0.5T,q˙0=0,0T,I3:q0=6,5T,q˙0=0,0T,I4:q0=−2,3T,q˙0=0,0T.Figure 2
Position tracking performance with different initial states.
(a)(b)Figure 3
Position tracking errors with different initial states.
(a)(b)Figure 4
Sliding surfaces versus time with different initial states.
(a)(b)According to Theorem2, the system can be stabilized with a unique bounded time for all initial conditions. It is seen from Figure 2 that the total convergence time of the proposed ANFNFSMC with four different initial conditions is bounded by 2s, which is in line with Theorem 2. This means that the upper bounded convergence time of the proposed control scheme can be estimated without relying on the initial states and robot model. Tracking error signals are displayed in Figure 3. Figure 4 exhibits the time responses of the proposed sliding surface. It is observed from Figure 4 that the reaching time under the proposed ANFNFSMC for all different initial conditions are upper bounded by 1s, which do not exceed the theoretical maximum in Theorem 2. Simulation results show that the tracking errors and the sliding variables can converge to the equilibrium within a fixed time, which implies that the upper bound of convergence time is only related to the design parameters and can be acquired in advance.


### 5.2. Tracking Control with Different Desired Signals

In this case, the proposed ANFNFSMC method is employed in the trajectory tracking of a robot manipulator with different desired reference signals. The initial conditions are set asq0=1.5,1T, q˙0=0,0T. These different desired signals qd=q1d,q2dTare given by(65)D1:q1d=0.5e−4t−1.2e−t+1.25,q2d=e−t−0.5e−4t+1.25,D2:q1d=0.5e−4t−1.2e−t+2,q2d=e−t−0.5e−4t+2,D3:q1d=0.5e−4t−1.2e−t+3,q2d=e−t−0.5e−4t+3,D4:q1d=0.5e−4t−1.2e−t+5,q2d=e−t−0.5e−4t+5.The simulation results with different desired reference signals under the proposed control scheme are displayed in Figures5–7. The positions of joints 1 and 2 are shown in Figure 5. Figure 6 illustrates the tracking error signals. The time responses of the proposed nonsingular fixed-time sliding surfaces are depicted in Figure 7. Observed by Figures 5–7, the proposed ANFNFSMC scheme can guarantee the robot manipulator’s track of different desired reference trajectories with a bounded time. Since the uncertain lumped disturbances are compensated by using the adaptive technique, the sliding surfaces and the position tracking errors can be stabilized to the equilibrium within a fixed time. The upper bound of the convergence time is independent of desired reference signals and can be known priorly.Figure 5
Position tracking performance with different desired signals.
(a)(b)Figure 6
Position tracking errors with different desired signals.
(a)(b)Figure 7
Sliding surfaces versus time with different desired signals.
(a)(b)


### 5.3. Various Control Parameters

To illustrate the influence of control parameters on the tracking performance of the proposed control method, the simulations with four sets of parameters are accomplished in Figures8–10. The initial states are set as q0=1.5,1.5T, q˙0=0,0T. These sets of parameters are listed as follows:Figure 8
Position tracking performance with different parameters.
(a)(b)Figure 9
Position tracking errors with different parameters.
(a)(b)Figure 10
Sliding surfaces versus time with different parameters.
(a)(b)(66)P1:α1=c1=1,r1=2,φ1=0.6,α2=c2=2,r2=2.5,φ2=0.6,P2:α1=c1=1,r1=2.5,φ1=0.55,α2=c2=2,r2=3,φ2=0.55,P3:α1=c1=2,r1=2,φ1=0.6,α2=c2=2,r2=2.5,φ2=0.6,P4:α1=c1=1,r1=2,φ1=0.6,α2=c2=8,r2=2.5,φ2=0.6.As shown in Figures8–10, the proposed ANFNFSMC with four sets of control parameters have different convergence times that are bounded by their upper bound of the convergence time. The simulation results show that the convergence time is only related to the control parameters, which further confirms the theoretical analysis in Theorem 2. It is noted from Figure 9 that since α1and c1increase, the total convergence time of P3 is less than those of the other three cases. Given in Figure 10, the reaching time of P4 is shorter than those of the other three cases because α2and c2increase. Additionally, it is obvious that the parameters α1, c1, α2, and c2 have more effect on the tracking performance than the parameters r1, φ1, r2, and φ2. Although, the parameters α1, c1, α2, and c2can be set sufficiently large so that both the convergence regions and convergence time can be as small as desired. However, the design parameters cannot be large enough because of the control saturation constraint. Accordingly, the parameters selection should be considered with a trade-off between the control input and the tracking performance.


### 5.4. Comparison with Other Control Methods

To show the superiority of the proposed control method, other control methods are also utilized in comparative simulations. The initial conditions are set asq0=2,0.5T, q˙0=0,0T. The desired reference signals are q1d=0.5e−4t−1.2e−t+1.25 and q2d=e−t−0.5e−4t+1.25.Based on the above simulation conditions, comparative simulations are performed with the other three sliding mode control methods.According to [41], controller 1 is designed as(67)s=e+sigeκ1+sige˙κ2,u=u0+u1,u1=−M0qκ2−1I2+κ1diageκ1−I2⋅sige˙2I2−κ2+ℓ2s+ℓ1+uauxss,uaux=−M0−1qa0+a1q+a2q˙2,where u0is defined as (42). The parameters are chosen as κ1=2, κ2=5/3, ℓ2=2, ℓ1=1, a0=12, a1=2.2, a2=2.8.According to [9], controller 2 is designed as(68)Ω=e˙+le,s=ΓΩ−Ω0e−νt,u=u0+u1+u2,u1=−M0qle˙+νΩ0e−νt,u2=−M0qksΓT+ω0sigsΓT0.5+ω1,ω˙1=ω^signsΓT,ω^˙=ω2sΓT,where u0 is defined as (42). The parameters are chosen as Γ=1, l=2, ν=4, k=2, ω0=20, ω2=1.Using the control method presented in [26], controller 3 is designed as(69)s=e+ι1sigeϕ1+ι2sige˙ϕ2,u=u0+u1+u2,u1=−M0qι2ϕ2e˙2−ϕ21+ϕ1ι1eϕ1−1signe˙,u2=−M0qκs+a^0+a^1q+a^2q˙2+ξsigns,a^˙0=υ0se˙ϕ2−1,a^˙1=υ1sqe˙ϕ2−1,a^˙2=υ2sq˙2e˙ϕ2−1,where u0 is defined as (42). The parameters are chosen as(70)ι1=ι2=1,ϕ1=2,ϕ2=53,κ=2,ξ=1,υ0=υ1=υ2=1.Four controllers are applied for tracking control of the robot manipulator. The position tracking performance and position tracking errors under these four controllers are shown in Figures11 and 12, respectively. Figures 11 and 12 illustrate the faster convergence of the control system by the proposed controller compared to the other three controllers. The zoomed position tracking errors are plotted in Figure 13. It is seen from Figure 13 that the proposed controller achieves higher steady-state tracking accuracy and smaller fluctuation in comparison with the other three controllers. The comparative simulations make clear that the tracking performance of the proposed fixed-time control method is better than the other three control methods.Figure 11
Position tracking performance under four controllers.
(a)(b)Figure 12
Position tracking errors under four controllers.
(a)(b)Figure 13
Zoomed position tracking errors under four controllers.
(a)(b)For quantitative analysis, two comparison criteria are given as follows: the integral of the absolute value of the error (IAE) defined asIAE=∫edt and the integral of the time multiplied by the absolute value of the error (ITAE) defined as ITAE=∫tedt. Under these four controllers, the values of two comparison criteria in t=6s are shown in Table 1. It is found from Table 1 that the values of IAE and ITAE for the proposed controller are smaller than the other three controllers, which indicates that the proposed controller has better control performance.Table 1
Comparison of control performance.
ControllersJointsIAEITAEController 1Joint 11.13770.6199Joint 20.97250.5155Controller 2Joint 10.95690.5091Joint 20.84290.4496Controller 3Joint 11.16600.6402Joint 21.26570.7525Proposed controllerJoint 10.68420.2555Joint 20.60090.2208On the basis of all the above simulations and analysis, it can be concluded that the proposed ANFNFSMC method provides excellent tracking performance with regard to fast fixed-time convergence and high steady-state tracking precision.


## 5.1. Tracking Control with Different Initial States

In this case, to confirm the fixed-time tracking performance of the proposed control approach, simulations are performed for trajectory tracking of a robot manipulator with different initial states, as shown in Figures2–4. The desired signals are q1d=0.5e−4t−1.2e−t+1.25and q2d=e−t−0.5e−4t+1.25. These different initial states are given by(64)I1:q0=1,1T,q˙0=0,0T,I2:q0=2.5,−0.5T,q˙0=0,0T,I3:q0=6,5T,q˙0=0,0T,I4:q0=−2,3T,q˙0=0,0T.Figure 2
Position tracking performance with different initial states.
(a)(b)Figure 3
Position tracking errors with different initial states.
(a)(b)Figure 4
Sliding surfaces versus time with different initial states.
(a)(b)According to Theorem2, the system can be stabilized with a unique bounded time for all initial conditions. It is seen from Figure 2 that the total convergence time of the proposed ANFNFSMC with four different initial conditions is bounded by 2s, which is in line with Theorem 2. This means that the upper bounded convergence time of the proposed control scheme can be estimated without relying on the initial states and robot model. Tracking error signals are displayed in Figure 3. Figure 4 exhibits the time responses of the proposed sliding surface. It is observed from Figure 4 that the reaching time under the proposed ANFNFSMC for all different initial conditions are upper bounded by 1s, which do not exceed the theoretical maximum in Theorem 2. Simulation results show that the tracking errors and the sliding variables can converge to the equilibrium within a fixed time, which implies that the upper bound of convergence time is only related to the design parameters and can be acquired in advance.


## 5.2. Tracking Control with Different Desired Signals

In this case, the proposed ANFNFSMC method is employed in the trajectory tracking of a robot manipulator with different desired reference signals. The initial conditions are set asq0=1.5,1T, q˙0=0,0T. These different desired signals qd=q1d,q2dTare given by(65)D1:q1d=0.5e−4t−1.2e−t+1.25,q2d=e−t−0.5e−4t+1.25,D2:q1d=0.5e−4t−1.2e−t+2,q2d=e−t−0.5e−4t+2,D3:q1d=0.5e−4t−1.2e−t+3,q2d=e−t−0.5e−4t+3,D4:q1d=0.5e−4t−1.2e−t+5,q2d=e−t−0.5e−4t+5.The simulation results with different desired reference signals under the proposed control scheme are displayed in Figures5–7. The positions of joints 1 and 2 are shown in Figure 5. Figure 6 illustrates the tracking error signals. The time responses of the proposed nonsingular fixed-time sliding surfaces are depicted in Figure 7. Observed by Figures 5–7, the proposed ANFNFSMC scheme can guarantee the robot manipulator’s track of different desired reference trajectories with a bounded time. Since the uncertain lumped disturbances are compensated by using the adaptive technique, the sliding surfaces and the position tracking errors can be stabilized to the equilibrium within a fixed time. The upper bound of the convergence time is independent of desired reference signals and can be known priorly.Figure 5
Position tracking performance with different desired signals.
(a)(b)Figure 6
Position tracking errors with different desired signals.
(a)(b)Figure 7
Sliding surfaces versus time with different desired signals.
(a)(b)


## 5.3. Various Control Parameters

To illustrate the influence of control parameters on the tracking performance of the proposed control method, the simulations with four sets of parameters are accomplished in Figures8–10. The initial states are set as q0=1.5,1.5T, q˙0=0,0T. These sets of parameters are listed as follows:Figure 8
Position tracking performance with different parameters.
(a)(b)Figure 9
Position tracking errors with different parameters.
(a)(b)Figure 10
Sliding surfaces versus time with different parameters.
(a)(b)(66)P1:α1=c1=1,r1=2,φ1=0.6,α2=c2=2,r2=2.5,φ2=0.6,P2:α1=c1=1,r1=2.5,φ1=0.55,α2=c2=2,r2=3,φ2=0.55,P3:α1=c1=2,r1=2,φ1=0.6,α2=c2=2,r2=2.5,φ2=0.6,P4:α1=c1=1,r1=2,φ1=0.6,α2=c2=8,r2=2.5,φ2=0.6.As shown in Figures8–10, the proposed ANFNFSMC with four sets of control parameters have different convergence times that are bounded by their upper bound of the convergence time. The simulation results show that the convergence time is only related to the control parameters, which further confirms the theoretical analysis in Theorem 2. It is noted from Figure 9 that since α1and c1increase, the total convergence time of P3 is less than those of the other three cases. Given in Figure 10, the reaching time of P4 is shorter than those of the other three cases because α2and c2increase. Additionally, it is obvious that the parameters α1, c1, α2, and c2 have more effect on the tracking performance than the parameters r1, φ1, r2, and φ2. Although, the parameters α1, c1, α2, and c2can be set sufficiently large so that both the convergence regions and convergence time can be as small as desired. However, the design parameters cannot be large enough because of the control saturation constraint. Accordingly, the parameters selection should be considered with a trade-off between the control input and the tracking performance.


## 5.4. Comparison with Other Control Methods

To show the superiority of the proposed control method, other control methods are also utilized in comparative simulations. The initial conditions are set asq0=2,0.5T, q˙0=0,0T. The desired reference signals are q1d=0.5e−4t−1.2e−t+1.25 and q2d=e−t−0.5e−4t+1.25.Based on the above simulation conditions, comparative simulations are performed with the other three sliding mode control methods.According to [41], controller 1 is designed as(67)s=e+sigeκ1+sige˙κ2,u=u0+u1,u1=−M0qκ2−1I2+κ1diageκ1−I2⋅sige˙2I2−κ2+ℓ2s+ℓ1+uauxss,uaux=−M0−1qa0+a1q+a2q˙2,where u0is defined as (42). The parameters are chosen as κ1=2, κ2=5/3, ℓ2=2, ℓ1=1, a0=12, a1=2.2, a2=2.8.According to [9], controller 2 is designed as(68)Ω=e˙+le,s=ΓΩ−Ω0e−νt,u=u0+u1+u2,u1=−M0qle˙+νΩ0e−νt,u2=−M0qksΓT+ω0sigsΓT0.5+ω1,ω˙1=ω^signsΓT,ω^˙=ω2sΓT,where u0 is defined as (42). The parameters are chosen as Γ=1, l=2, ν=4, k=2, ω0=20, ω2=1.Using the control method presented in [26], controller 3 is designed as(69)s=e+ι1sigeϕ1+ι2sige˙ϕ2,u=u0+u1+u2,u1=−M0qι2ϕ2e˙2−ϕ21+ϕ1ι1eϕ1−1signe˙,u2=−M0qκs+a^0+a^1q+a^2q˙2+ξsigns,a^˙0=υ0se˙ϕ2−1,a^˙1=υ1sqe˙ϕ2−1,a^˙2=υ2sq˙2e˙ϕ2−1,where u0 is defined as (42). The parameters are chosen as(70)ι1=ι2=1,ϕ1=2,ϕ2=53,κ=2,ξ=1,υ0=υ1=υ2=1.Four controllers are applied for tracking control of the robot manipulator. The position tracking performance and position tracking errors under these four controllers are shown in Figures11 and 12, respectively. Figures 11 and 12 illustrate the faster convergence of the control system by the proposed controller compared to the other three controllers. The zoomed position tracking errors are plotted in Figure 13. It is seen from Figure 13 that the proposed controller achieves higher steady-state tracking accuracy and smaller fluctuation in comparison with the other three controllers. The comparative simulations make clear that the tracking performance of the proposed fixed-time control method is better than the other three control methods.Figure 11
Position tracking performance under four controllers.
(a)(b)Figure 12
Position tracking errors under four controllers.
(a)(b)Figure 13
Zoomed position tracking errors under four controllers.
(a)(b)For quantitative analysis, two comparison criteria are given as follows: the integral of the absolute value of the error (IAE) defined asIAE=∫edt and the integral of the time multiplied by the absolute value of the error (ITAE) defined as ITAE=∫tedt. Under these four controllers, the values of two comparison criteria in t=6s are shown in Table 1. It is found from Table 1 that the values of IAE and ITAE for the proposed controller are smaller than the other three controllers, which indicates that the proposed controller has better control performance.Table 1
Comparison of control performance.
ControllersJointsIAEITAEController 1Joint 11.13770.6199Joint 20.97250.5155Controller 2Joint 10.95690.5091Joint 20.84290.4496Controller 3Joint 11.16600.6402Joint 21.26570.7525Proposed controllerJoint 10.68420.2555Joint 20.60090.2208On the basis of all the above simulations and analysis, it can be concluded that the proposed ANFNFSMC method provides excellent tracking performance with regard to fast fixed-time convergence and high steady-state tracking precision.


## 6. Conclusion

In this paper, the fixed-time tracking control problem of uncertain robot manipulators is researched. The NFNFSM is constructed to circumvent the singularity and achieve fast convergence. Based on the proposed NFNFSM and the adaptive technique, a novel adaptive nonsingular fixed-time sliding mode control method is presented to guarantee the properties of disturbance suppression and fixed-time convergence. The convergence time of the system can be estimated regardless of the initial states, and the system states can converge into a small vicinity of the equilibrium point within a fixed time. Compared with the existing control methods [9, 26, 41], the proposed control scheme can provide a faster convergence rate and higher control accuracy. Moreover, the control method can be also applicable to some other complicated second-order nonlinear systems. It is noted that the proposed controller design requires the velocity states of robot manipulator systems. However, the velocity states are often not easily obtained or measured. Thus, a velocity-free fixed-time controller design for uncertain robot manipulators will be included in our future investigation.



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*Source: 6629993-2021-05-08.xml*