# Robust Fixed-Time Inverse Dynamic Control for Uncertain Robot Manipulator System

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

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## Abstract

This paper proposes a novel robust fixed-time control for the robot manipulator system with uncertainties. Based on the uniform robust exact differentiator (URED) algorithm, a robust control term is constructed. Then, a robust fixed-time inverse dynamics control (IDC) is proposed. For the proposed control method, the fixed-time stability of a closed-loop system with uncertainties is strictly proved. The newly proposed method exhibits the following two attractive features. First, the proposed control scheme extends the existing fixed-time IDC for the robot manipulator system to the robust control scheme. Second, the proposed method is strictly nonsingular rather than the commonly used approximate approach. Simulation result demonstrates the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

Based on the development of control theory, measurement technology, and material science, the performance of robot manipulator has been greatly improved, and it has been widely used in the engineering areas, such as undersea exploration, space exploration, repetitive labor, and emergency rescue [1–6]. Most of the robot manipulators adpoted the trajectory tracking control to complete the desired mission. Thus, the study on the high-precision trajectory tracking control has become one of the most representative control problems in the research field of robot manipulator. As a classical control method, proportional-integral-derivative control (PID) [7] has the advantage of easy implementation in engineering. Many PID-based methods have been applied to design the tracking controller of the robot manipulator, such as fuzzy PID [8], neural PID [9], robust nonlinear PID, [10] and adaptive PID [11]. However, the conventional PID and these PID-based methods in [8–11] are asymptotically stable, which means that the convergence time of tracking error is infinite.Actually, the fast convergence rate is a very important desired control performance of the robot manipulator. Thus, it is necessary to design a finite-time controller for the robot manipulator. Since the 1990s, based on the development of finite-time stability theories [12–15], the study on finite-time convergence method has become a hotspot. And, to guarantee the robustness of system in the presence of uncertainties, the sliding-mode control (SMC) is a good candidate [16]. Combining the finite-time convergence and SMC, many terminal sliding-mode control (TSMC) schemes have been developed. In [17], for the first-order chaotic system, the authors adopted the FTSMC (fast terminal sliding-mode control) and the composite nonlinear feedback approach to guarantee the finite-time and high-performance synchronization of the chaotic systems in the presence of the external disturbances, parametric uncertainties, Lipschitz nonlinearities, and time delays. For the high-order system, there is a singular problem in the conventional TSMC method [18]. To avoid the singular problem and guarantee the finite-time stability, in [19], a novel recursive singularity-free FTSMC for finite-time tracking control of nonholonomic systems was proposed. In [20], a nonsingular FTSMC was developed, and the uncertainties were compensated by the estimation of disturbance observer. Recently, the development of TSMC also continuously changed the control design of a robot manipulator in engineering [21]. In [22], the authors adopted conventional TSMC to design the robust MIMO finite-time controller. However, an undesirable singular problem can be caused by the conventional TSMC. To avoid the singular problem, in [23–25], the nonsingular terminal sliding-mode surface was applied to design the nonsingular TSMC scheme. In [26, 27], the authors adopted the integral terminal sliding mode to design controller, and then the singular problem also was eliminated. Besides the TSMC-based methods in [22–27], based on finite-time convergence theory of the homogeneous system, the results in [28, 29] have been developed and also can guarantee the finite-time convergence. These results in [28, 29] not only guarantee the finite-time stability but also eliminate the above singular problem in the conventional TSMC. Although these FTC methods in [22–29] can achieve the finite-time convergence of tracking error, the finite convergence times in [22–29] are heavily dependent on the initial system conditions. Thus, the convergence time may be an unacceptable long time for the large initial system condition cases.In recent years, an advanced stability notion, that is, fixed-time stability, has been developed [30, 31]. The fixed-time stability control method can not only derive the system trajectories to equilibrium in finite time but also guarantee that the convergence time is independent on the initial system conditions. Thus, the above initial system condition problem in finite-time control can be eliminated. So far, for the robot manipulator, some fixed-time control methods have been developed in [32–34]. In [32, 33], the fixed-time controllers were developed based on the fixed-time sliding-mode control. However, to avoid the singular problem, the schemes in [32, 33] adopted a nonlinear function to approximate the singular control term. Thus, the schemes in [32, 33] can only guarantee that the tracking error converges to a neighborhood of zero. Recently, in [34], a fixed-time inverse dynamics control (IDC) scheme was proposed by using the bilimit homogeneity technique. Unlike the approximate fixed-time schemes in [32, 33], the method in [34] is nonsingular and can strictly guarantee that the tracking error converges to zero. In addition, for the control scheme in [34], since the control design is based on the commonly used IDC, the error dynamics are completely decoupled and the tracking performance is very easy to quantify [34]. However, the fixed-time IDC in [34] does not consider uncertainties.For the robot manipulator system, the fixed-time control can eliminate the initial system condition problem of conventional finite-time control. However, existing fixed-time control results in [32, 33] which cannot drive the tracking error to zero, which implies that the manipulator cannot accomplish some high-precision operation tasks. The IDC results in [34] can guarantee that the tracking error converges to zero, but the uncertainties are not considered in [34]. Since the actual robot manipulator is a typical uncertain system, the fixed-time IDC in [34] may not achieve the desired fixed-time convergence performance in practical situations. Thus, it is necessary to design a new fixed-time control scheme which can not only strictly drive the tracking error to zero but also compensate the uncertainties in fixed time. The main motivation of this paper is to propose a new scheme to eliminate the above problems of existing fixed-time control for the robot manipulator in [32–34].The main difficulties to avoid the above two problems of existing results is how to design a scheme which can not only strictly drive the tracking error to zero in fixed time but also fully compensate the uncertainties in fixed time. In this paper, firstly, a robust control term based on the uniform robust exact differentiator (URED) algorithm is designed to compensate the uncertainties in fixed time. Then, a robust time fixed-time IDC is developed by using the IDC, the bilimit homogeneity technique in [34], and the proposed robust control term. In contrast to the aforementioned results, the contributions of this paper are as follows:(1)
The main theoretical contributions of this paper are that the proposed method can not only extend the commonly used IDC for trajectory tracking of robot manipulators to fixed-time stable scheme but also suppress the bounded uncertainties of the robot manipulator system in a fixed time. Thus, the proposed method extends the recent fixed-time IDC result in [34] to a robust scheme.(2)
Since the proposed scheme is designed based on the bilimit homogeneity technique in [34] and the additional URED is nonsingular, the proposed scheme does not require the commonly used approximate approach used in existing fixed-time control schemes in [32, 33]. Thus, the proposed method can drive the tracking error to zero rather than a neighborhood of zero in [32, 33].The remaining parts of this paper are as follows. In Section2, the dynamic model, the fundamental facts, and the motivation of this paper are expounded. The main results are presented in Section 3. In Section 3.1, a novel robust fixed-time IDC is developed for an uncertain robot manipulator system. Then, in Section 3.2, fixed-time stability of the proposed controller is presented. In Section 4, a simulation process verifies the effectiveness of the proposed scheme. In Section 5, the conclusion of the whole paper is presented.Notations: the following notations will be used in this paper:t denotes the time and the initial time is 0. Let · denote the Euclidean norm of a vector and its induced norm of a matrix.


## 2. Preliminaries

### 2.1. Dynamic Model of Robotic Manipulator with Uncertainties

The dynamics of a rigid robotic manipulator with uncertainties andn-degree of freedom are given as [32, 33](1)Mqq¨+Cq,q˙q˙+Gq=τ+d,where q,q˙,q¨∈Rn denote the vector of joint position, vector of joint velocity, and vector of joint acceleration, respectively. Mq∈Rn×n is the positive definite symmetric inertia matrix. Cq,q˙∈Rn×n is the centrifugal-Coriolis forces matrix. Gq∈Rn is the vector of gravitational torque. τ=τ1,…,τnT∈Rn is the input torque. d∈Rn denotes the uncertainties, which can be caused by the unmodeled dynamics, external environment, and parameter uncertainties. The desired joint position is defined as q⟶d∈Rn. The tracking error is defined as(2)e1=q−qd,where e1=e11,…,e1nT∈Rn.


#### 2.1.1. Control Objective

The objective is to design the input torqueτ in such a way that the tracking error e1=0 in fixed time despite the presence uncertainties. And the convergence time is always bounded regardless of system initial conditions.Lete˙1=e2, where e2=e21,…,e2nT∈Rn. Then, calculating the time derivative of along the trajectories (1), we have(3)e˙1=e2,e˙2=fq,q˙+bqτ+Δ,where(4)fq,q˙=−Mq−1Cq,q˙q˙+Gq−q¨d,bq=Mq−1,Δ=Mq−1d=δ1,…,δnT.Assumption1 is assumed to be valid throughout this paper.Assumption 1.
The uncertaintiesδij=1,…,n is differentiable and satisfies δi≤Dmax and δ˙i≤Dmaxd, where Dmax and Dmaxd are positive constants.


### 2.2. Fundamental Facts

Before giving the control scheme, some useful definitions are recalled for convenience.Definition 1 (finite-time stability [35]).
For the following uncertain system,(5)x⟶˙=px,ψ,where x∈Rn is the system state, ψ∈Rn is the system uncertainty, and px,ψ:Rn⟶Rn is a nonlinear function, the origin of (5) is globally finite-time stable if it is globally asymptotically stable and any solution xt,x0 of (5) reaches the equilibrium at some finite-time moment, i.e., ∀t≥Tx0:xt,x0=0, where Tx0:Rn⟶R+∪0 is the convergence time function.Definition 2 (fixed-time stability [35]).
For uncertain system (5), the origin of (5) is fixed-time stable if it is globally finite-time stable and the convergence time function Tx0 is bounded, i.e., ∃Tmax≥0:∀x0∈Rn, Tx0≤Tmax, where Tmax is a positive constant and independent on the initial system condition x⟶0.Lemma 1 (see [36], URED algorithm).
Consider a dynamic system in the presence of uncertaintyf0t:(6)σ˙0=−k¯1ϕ¯1σ0+σ1,σ˙1=−k¯2ϕ¯2σ0−f0t,where σjj=0,1 is the system state and f0t is bounded as f0t≤L. The functions ϕ¯1σ0 and ϕ¯2σ0 are given as(7)ϕ¯1σ0=σ01/2signσ0+μ¯σ03/2signσ0,ϕ¯2σ0=12signσ0+2μ¯σ0+32μ¯2σ02signσ0,where μ¯>0 is a scalar. If k¯1 and k¯2 are in the set(8)κ¯=k¯1,k¯2∈R2|0<k¯1≤2L,k¯2>k¯124+4L2k¯12∪k¯1,k¯2∈R2|k¯1>2L,k¯2>2L,then we have the following two propositions.Proposition 1.
The Lyapunov function is defined as(9)V1=σTPσσ,where σ⟶=ϕ¯1σ0,σ1T. P⟶σ is a symmetric and positive definite matrix. For some ε¯>0, the derivative of the Lyapunov function V1 satisfies the inequality(10)V˙1≤−κ1Pσ,ε¯V11/2−κ¯2Pσ,ε¯μ¯σ01/2V1,where κ¯jP⟶σ,ε¯j=1,2 is a positive scalar.Proposition 2.
The statesσ0 and σ1 will converge to the origin uniformly in time tf. And the convergence time tf is bounded as tf≤Tfmax, where Tfmax is a positive constant and independent on the initial conditions σ00 and σ10.Lemma 2.
(see [34]). Consider a dynamic system as follows:(11)e¨0=−Kpe0α1signe0−Kde˙0α2signe˙0−Lpe0β1signe0−Lde˙0β2signe˙0,where e0∈R and e˙0∈R are the system state. If the constants Kp>0, Kd>0, Lp>0, Ld>0, 0<α1<1, α2=2α1/α1+1, β1=2α1+1, and β2=2α1+1/α1+1 and the initial system condition e00 is bounded, then the system state is fixed-time stable, i.e.,(12)e0=0,e˙0=0,ift≥te,te≤Te,where Te is a positive constant and independent on the initial system conditions.Remark 1.
The proof of Lemma1 can be referred to Appendixes A and B of [36]. And the expression of Pσ, κ¯1Pσ,ε¯ and κ¯2Pσ,ε¯ are given in Appendix A of [36]. The expression of Te is given in Appendix B of [36]. The proof of Lemma 2 can be referred to the proof of Theorem 3.1 in [34].


### 2.3. Problem Description and Purposes of this Paper

#### 2.3.1. Problem Description

For the robotic manipulator system (1), the conventional finite-time control scheme can achieve a finite convergence time. However, the convergence time is affected by the initial system conditions e10 and e20. Thus, the convergence time may be large in the case that e10 and e20 are large. Recently, based on the fixed-time control, the control schemes proposed in [32–34] can eliminate the effect brought by initial system conditions. However, to avoid the singular problem, the schemes in [32, 33] adopted the approximate method. Thus, these schemes in [32, 33] only guaranteed the tracking error e10 converges to a neighborhood of zero. The results in [34] adopted the IDC method and bilimit homogeneity technique to design a scheme which can solve the singular problem without using approximate method in [32, 33]. However, the method in [34] assumed that the robotic manipulator system is certain, i.e., the uncertainties vector d=0. Obviously, d is not equal to zero in practical case. The uncertainties d may be caused by the unmodeled dynamics, external environment, and parameter uncertainties.


#### 2.3.2. Purpose of This Paper

To further improve the performance of fixed-time control for the robotic manipulator system, the research topic of this paper is to designing a new robust fixed-time IDC which not only can strictly derive the tracking errore10 to zero but also compensate the uncertainties d⟶=0⟶ in fixed time.


## 2.1. Dynamic Model of Robotic Manipulator with Uncertainties

The dynamics of a rigid robotic manipulator with uncertainties andn-degree of freedom are given as [32, 33](1)Mqq¨+Cq,q˙q˙+Gq=τ+d,where q,q˙,q¨∈Rn denote the vector of joint position, vector of joint velocity, and vector of joint acceleration, respectively. Mq∈Rn×n is the positive definite symmetric inertia matrix. Cq,q˙∈Rn×n is the centrifugal-Coriolis forces matrix. Gq∈Rn is the vector of gravitational torque. τ=τ1,…,τnT∈Rn is the input torque. d∈Rn denotes the uncertainties, which can be caused by the unmodeled dynamics, external environment, and parameter uncertainties. The desired joint position is defined as q⟶d∈Rn. The tracking error is defined as(2)e1=q−qd,where e1=e11,…,e1nT∈Rn.


### 2.1.1. Control Objective

The objective is to design the input torqueτ in such a way that the tracking error e1=0 in fixed time despite the presence uncertainties. And the convergence time is always bounded regardless of system initial conditions.Lete˙1=e2, where e2=e21,…,e2nT∈Rn. Then, calculating the time derivative of along the trajectories (1), we have(3)e˙1=e2,e˙2=fq,q˙+bqτ+Δ,where(4)fq,q˙=−Mq−1Cq,q˙q˙+Gq−q¨d,bq=Mq−1,Δ=Mq−1d=δ1,…,δnT.Assumption1 is assumed to be valid throughout this paper.Assumption 1.
The uncertaintiesδij=1,…,n is differentiable and satisfies δi≤Dmax and δ˙i≤Dmaxd, where Dmax and Dmaxd are positive constants.


## 2.1.1. Control Objective

The objective is to design the input torqueτ in such a way that the tracking error e1=0 in fixed time despite the presence uncertainties. And the convergence time is always bounded regardless of system initial conditions.Lete˙1=e2, where e2=e21,…,e2nT∈Rn. Then, calculating the time derivative of along the trajectories (1), we have(3)e˙1=e2,e˙2=fq,q˙+bqτ+Δ,where(4)fq,q˙=−Mq−1Cq,q˙q˙+Gq−q¨d,bq=Mq−1,Δ=Mq−1d=δ1,…,δnT.Assumption1 is assumed to be valid throughout this paper.Assumption 1.
The uncertaintiesδij=1,…,n is differentiable and satisfies δi≤Dmax and δ˙i≤Dmaxd, where Dmax and Dmaxd are positive constants.


## 2.2. Fundamental Facts

Before giving the control scheme, some useful definitions are recalled for convenience.Definition 1 (finite-time stability [35]).
For the following uncertain system,(5)x⟶˙=px,ψ,where x∈Rn is the system state, ψ∈Rn is the system uncertainty, and px,ψ:Rn⟶Rn is a nonlinear function, the origin of (5) is globally finite-time stable if it is globally asymptotically stable and any solution xt,x0 of (5) reaches the equilibrium at some finite-time moment, i.e., ∀t≥Tx0:xt,x0=0, where Tx0:Rn⟶R+∪0 is the convergence time function.Definition 2 (fixed-time stability [35]).
For uncertain system (5), the origin of (5) is fixed-time stable if it is globally finite-time stable and the convergence time function Tx0 is bounded, i.e., ∃Tmax≥0:∀x0∈Rn, Tx0≤Tmax, where Tmax is a positive constant and independent on the initial system condition x⟶0.Lemma 1 (see [36], URED algorithm).
Consider a dynamic system in the presence of uncertaintyf0t:(6)σ˙0=−k¯1ϕ¯1σ0+σ1,σ˙1=−k¯2ϕ¯2σ0−f0t,where σjj=0,1 is the system state and f0t is bounded as f0t≤L. The functions ϕ¯1σ0 and ϕ¯2σ0 are given as(7)ϕ¯1σ0=σ01/2signσ0+μ¯σ03/2signσ0,ϕ¯2σ0=12signσ0+2μ¯σ0+32μ¯2σ02signσ0,where μ¯>0 is a scalar. If k¯1 and k¯2 are in the set(8)κ¯=k¯1,k¯2∈R2|0<k¯1≤2L,k¯2>k¯124+4L2k¯12∪k¯1,k¯2∈R2|k¯1>2L,k¯2>2L,then we have the following two propositions.Proposition 1.
The Lyapunov function is defined as(9)V1=σTPσσ,where σ⟶=ϕ¯1σ0,σ1T. P⟶σ is a symmetric and positive definite matrix. For some ε¯>0, the derivative of the Lyapunov function V1 satisfies the inequality(10)V˙1≤−κ1Pσ,ε¯V11/2−κ¯2Pσ,ε¯μ¯σ01/2V1,where κ¯jP⟶σ,ε¯j=1,2 is a positive scalar.Proposition 2.
The statesσ0 and σ1 will converge to the origin uniformly in time tf. And the convergence time tf is bounded as tf≤Tfmax, where Tfmax is a positive constant and independent on the initial conditions σ00 and σ10.Lemma 2.
(see [34]). Consider a dynamic system as follows:(11)e¨0=−Kpe0α1signe0−Kde˙0α2signe˙0−Lpe0β1signe0−Lde˙0β2signe˙0,where e0∈R and e˙0∈R are the system state. If the constants Kp>0, Kd>0, Lp>0, Ld>0, 0<α1<1, α2=2α1/α1+1, β1=2α1+1, and β2=2α1+1/α1+1 and the initial system condition e00 is bounded, then the system state is fixed-time stable, i.e.,(12)e0=0,e˙0=0,ift≥te,te≤Te,where Te is a positive constant and independent on the initial system conditions.Remark 1.
The proof of Lemma1 can be referred to Appendixes A and B of [36]. And the expression of Pσ, κ¯1Pσ,ε¯ and κ¯2Pσ,ε¯ are given in Appendix A of [36]. The expression of Te is given in Appendix B of [36]. The proof of Lemma 2 can be referred to the proof of Theorem 3.1 in [34].


## 2.3. Problem Description and Purposes of this Paper

### 2.3.1. Problem Description

For the robotic manipulator system (1), the conventional finite-time control scheme can achieve a finite convergence time. However, the convergence time is affected by the initial system conditions e10 and e20. Thus, the convergence time may be large in the case that e10 and e20 are large. Recently, based on the fixed-time control, the control schemes proposed in [32–34] can eliminate the effect brought by initial system conditions. However, to avoid the singular problem, the schemes in [32, 33] adopted the approximate method. Thus, these schemes in [32, 33] only guaranteed the tracking error e10 converges to a neighborhood of zero. The results in [34] adopted the IDC method and bilimit homogeneity technique to design a scheme which can solve the singular problem without using approximate method in [32, 33]. However, the method in [34] assumed that the robotic manipulator system is certain, i.e., the uncertainties vector d=0. Obviously, d is not equal to zero in practical case. The uncertainties d may be caused by the unmodeled dynamics, external environment, and parameter uncertainties.


### 2.3.2. Purpose of This Paper

To further improve the performance of fixed-time control for the robotic manipulator system, the research topic of this paper is to designing a new robust fixed-time IDC which not only can strictly derive the tracking errore10 to zero but also compensate the uncertainties d⟶=0⟶ in fixed time.


## 2.3.1. Problem Description

For the robotic manipulator system (1), the conventional finite-time control scheme can achieve a finite convergence time. However, the convergence time is affected by the initial system conditions e10 and e20. Thus, the convergence time may be large in the case that e10 and e20 are large. Recently, based on the fixed-time control, the control schemes proposed in [32–34] can eliminate the effect brought by initial system conditions. However, to avoid the singular problem, the schemes in [32, 33] adopted the approximate method. Thus, these schemes in [32, 33] only guaranteed the tracking error e10 converges to a neighborhood of zero. The results in [34] adopted the IDC method and bilimit homogeneity technique to design a scheme which can solve the singular problem without using approximate method in [32, 33]. However, the method in [34] assumed that the robotic manipulator system is certain, i.e., the uncertainties vector d=0. Obviously, d is not equal to zero in practical case. The uncertainties d may be caused by the unmodeled dynamics, external environment, and parameter uncertainties.


## 2.3.2. Purpose of This Paper

To further improve the performance of fixed-time control for the robotic manipulator system, the research topic of this paper is to designing a new robust fixed-time IDC which not only can strictly derive the tracking errore10 to zero but also compensate the uncertainties d⟶=0⟶ in fixed time.


## 3. Main Result

### 3.1. Control Design

To facilitate the design and analysis, for the vectorξ=ξ1,…,ξnT∈Rn and constant α>0, the vector Sξα∈Rn,ϕ1ξ⟶∈Rn,ϕ2ξ∈Rn is defined as(13)Sξα=ξ1αsignξ1,…,ξnαsignξnT,ϕ1ξ=ϕ1ξ1,…,ϕ1ξnT,ϕ2ξ=ϕ2ξ1,…,ϕ2ξnT,where ϕ1ξi and ϕ2ξii=1,2,…,n are defined as(14)ϕ1ξi=ξi1/2signξi+μξi3/2signξi,ϕ2ξi=12signξi+2μξi+32μ2ξi2signξi,where μ>0 is a scalar.Then, the robust fixed-time IDC for the uncertain robot manipulator system (3) is design as(15)τ=bq−1−fq,q˙+He1,e2+Re1,e2,where He1,e2∈Rn is the fixed-time convergence term and designed as in [33]:(16)He1,e2=−K1Se1ω1−K2Se2ω2−L1Se1γ1−L2Se2γ2.The parameters satisfy the following conditions:(17)K1>0,K2>0,L1>0,L2>0,0<ω1<1,ω2=2ω1ω1+1,γ1=2ω1+1,γ2=2ω1+1ω1+1,where Re1,e2 is the robust term and designed as(18)Re1,e2=−k1ϕ1H¯⟶e1,e2+∫0t−k2ϕ2H¯⟶e1,e2dυ,H¯⟶e1,e2=e2−∫0tHe1,e2dυ.The parameters satisfy the following conditions:(19)k1,k2∈R2|0<k1≤2Dmaxd,k2>k124+4Dmaxd2k12∪k1,k2∈R2|k1>2Dmaxd,k2>2Dmaxd.


### 3.2. Stability Analysis

Theorem 1.
Consider the uncertain robot manipulator system (3) adopts the controller (15). If Assumption 1 is valid and the control parameters are chosen as in (17) and (19), then system (3) is fixed-time stable.Proof.
Substituting the proposed controller (15) into (3), we have(20)e¨1=He1,e2+Re1,e2+Δ.
For each control errore1ii=1,2,…,n, we have(21)e¨1i=Hi+Ri+δi,where(22)Hi=−K1e1iω1signe1i−K2e2iω2signe2i−L1e1iγ1signe1i−L2e2iγ2signe2i,(23)Ri=−k1ϕ1e2i−∫0tHidυ+∫0t−k2ϕ2e2i−∫0tHidυdυ.
Letη0i=e2i−∫0tHidυ and η1i=∫0t−k2ϕ2e2i−∫0tHidυdυ+δi. Then, from (21), we have(24)η˙0i=−k1ϕ1η0i+η1i,η˙1i=−k2ϕ2η0i+δ˙i.
According to the Proposition2 of Lemma 1, it can be known that η0i=0 can be satisfied in a fixed time if δ˙i satisfies the Assumption 1. Then, the fixed-time convergence of control error e1i can be analyzed. However, the convergence stability of e1i may be affected during the convergence period of η0i. Thus, to consider the whole convergence dynamics, in the following, the proof of Theorem 1 consists of two steps: in the first step, it will be proved that the control error e1i is bounded before η0i converges to zero. In the second step, it will be proved that the control error e1i will converge to zero in fixed time after η0i converges to zero, i.e., e1i=0 is fixed-time stable.Step 1.
A Lyapunov functionV2 is defined as (9)(25)V2=ηTPηη,where η=ϕ1η0i,η1iT. Pη is a symmetric and positive definite matrix. Then, according to Proposition 1 of Lemma 1, if k1 and k2 are in the set (19), for some ε>0, the derivative of the Lyapunov function V2 satisfies the inequality(26)V˙2≤−κ1Pη,εV21/2−κ2Pη,εμη0i1/2V2,where κjPη,εj=1,2 is a positive scalar. And the expression of Pη, κ1Pη,ε and κ2Pη,ε is given in Appendix A in [34]. From (26), it is clear that V˙2≤0 for t≥0; thus, ϕ1η0i and η1i are always bounded:(27)ϕ1η0i≤ϕ1max,(28)η1i≤ϕ2max,where ϕ1max and ϕ2max are positive constants.
Then, a Lyapunov functionV3 is defined as(29)V3=12e1i2+e2i2.
Calculating the time derivative ofV3 along the trajectories of (21), we get(30)V˙3=e1ie2i+e2ie˙2i=e1ie2i+e2iHi+Ri+δi.
By combining (22) and (23), (30) can be rewritten as(31)V˙3=e1ie2i+e2i−K1e1iω1signe1i−K2e2iω2signe2i−L1e1iγ1signe1i−L2e2iγ2signe2i+e2i−k1ϕ1e2i−∫0tHidυ+∫0t−k2ϕ2e2i−∫0tHidυdυ+δi.
Consideringη0i=e2i−∫0tHidυ and η1i=∫0t−k2ϕ2e2i−∫0tHidυdυ+δi, we have(32)V˙3=e1ie2i+e2i−K1e1iω1signe1i−K2e2iω2signe2i−L1e1iγ1signe1i−L2e2iγ2signe2i+e2i−k1ϕ1η0i+η1i,=e1ie2i−K1e1iω1signe1ie2i−K2e2iω2+1−L1e1iγ1signe1ie2i−L2e2iγ2+1+e2i−k1ϕ1η0i+η1i.
SinceK2>0 and L2>0, we have(33)V˙3≤e1ie2i−K1e1iω1signe1ie2i−L1e1iγ1signe1ie2i+e2i−k1ϕ1η0i+η1i.
Considering the upper bound ofϕ1η0i and η1i given in (27) and (28), (33) can be rewritten as(34)V˙3≤e1ie2i−K1e1iω1signe1ie2i−L1e1iγ1signe1ie2i+e2ik1ϕ1max+ϕ2max.
We consider the following two cases:Case 1: ife1ie2i<0, we have e1ie˙1i<0. It is clear that e1i is bounded in this case.Case 2: Ife1ie2i≥0. From (34), we have(35)V˙3≤e1ie2i+e2ik1ϕ1max+ϕ2max.
Then, according to Young’s inequality [37], (35) can be rewritten as(36)V˙3≤e1i2+e2i22+e2i2+k1ϕ1max+ϕ2max22=e1i22+e2i2+k1ϕ1max+ϕ2max22.
LetCmax=k1ϕ1max+ϕ2max2/2, we have(37)V˙3≤2V3+Cmax.
From (37), for any initial time t=tg, we have(38)V3≤V3tge2t+Cmax1−e2t2,ift≥tg,where e is the natural constant. From (38), it is clear that V3 and e1i are bounded in arbitrary finite time. From the discussions in case 1 and case 2, it is clear that the control error e1i is always bounded in arbitrary finite time.Step 2.
According to Proposition2 of Lemma 1, it can be known from (24) that the following equation can be satisfied in a fixed-time T1:(39)η0i=e2i−∫0tHidυ=0,ift≥T1,η˙0i=e˙2i−Hi=0,ift≥T1,where T1 is independent on the initial system conditions. Then, we have(40)e˙2i=Hi=−K1e1iω1signe1i−K2e2iω2signe2i−L1e1iγ1signe1i−L2e2iγ2signe2i,ift≥T1.
It has been proved thate1i is always bounded in arbitrary finite time in Step 1. Thus, it is clear that e1iT1 is bounded. Then, according to Lemma 2, it can be known from (40) that e1i=0 can be guaranteed in a fixed-time T2+T1:(41)e1i=0,ift≥T2+T1,where T2+T1 is also independent on the initial system conditions.
The proof is finished.


## 3.1. Control Design

To facilitate the design and analysis, for the vectorξ=ξ1,…,ξnT∈Rn and constant α>0, the vector Sξα∈Rn,ϕ1ξ⟶∈Rn,ϕ2ξ∈Rn is defined as(13)Sξα=ξ1αsignξ1,…,ξnαsignξnT,ϕ1ξ=ϕ1ξ1,…,ϕ1ξnT,ϕ2ξ=ϕ2ξ1,…,ϕ2ξnT,where ϕ1ξi and ϕ2ξii=1,2,…,n are defined as(14)ϕ1ξi=ξi1/2signξi+μξi3/2signξi,ϕ2ξi=12signξi+2μξi+32μ2ξi2signξi,where μ>0 is a scalar.Then, the robust fixed-time IDC for the uncertain robot manipulator system (3) is design as(15)τ=bq−1−fq,q˙+He1,e2+Re1,e2,where He1,e2∈Rn is the fixed-time convergence term and designed as in [33]:(16)He1,e2=−K1Se1ω1−K2Se2ω2−L1Se1γ1−L2Se2γ2.The parameters satisfy the following conditions:(17)K1>0,K2>0,L1>0,L2>0,0<ω1<1,ω2=2ω1ω1+1,γ1=2ω1+1,γ2=2ω1+1ω1+1,where Re1,e2 is the robust term and designed as(18)Re1,e2=−k1ϕ1H¯⟶e1,e2+∫0t−k2ϕ2H¯⟶e1,e2dυ,H¯⟶e1,e2=e2−∫0tHe1,e2dυ.The parameters satisfy the following conditions:(19)k1,k2∈R2|0<k1≤2Dmaxd,k2>k124+4Dmaxd2k12∪k1,k2∈R2|k1>2Dmaxd,k2>2Dmaxd.


## 3.2. Stability Analysis

Theorem 1.
Consider the uncertain robot manipulator system (3) adopts the controller (15). If Assumption 1 is valid and the control parameters are chosen as in (17) and (19), then system (3) is fixed-time stable.Proof.
Substituting the proposed controller (15) into (3), we have(20)e¨1=He1,e2+Re1,e2+Δ.
For each control errore1ii=1,2,…,n, we have(21)e¨1i=Hi+Ri+δi,where(22)Hi=−K1e1iω1signe1i−K2e2iω2signe2i−L1e1iγ1signe1i−L2e2iγ2signe2i,(23)Ri=−k1ϕ1e2i−∫0tHidυ+∫0t−k2ϕ2e2i−∫0tHidυdυ.
Letη0i=e2i−∫0tHidυ and η1i=∫0t−k2ϕ2e2i−∫0tHidυdυ+δi. Then, from (21), we have(24)η˙0i=−k1ϕ1η0i+η1i,η˙1i=−k2ϕ2η0i+δ˙i.
According to the Proposition2 of Lemma 1, it can be known that η0i=0 can be satisfied in a fixed time if δ˙i satisfies the Assumption 1. Then, the fixed-time convergence of control error e1i can be analyzed. However, the convergence stability of e1i may be affected during the convergence period of η0i. Thus, to consider the whole convergence dynamics, in the following, the proof of Theorem 1 consists of two steps: in the first step, it will be proved that the control error e1i is bounded before η0i converges to zero. In the second step, it will be proved that the control error e1i will converge to zero in fixed time after η0i converges to zero, i.e., e1i=0 is fixed-time stable.Step 1.
A Lyapunov functionV2 is defined as (9)(25)V2=ηTPηη,where η=ϕ1η0i,η1iT. Pη is a symmetric and positive definite matrix. Then, according to Proposition 1 of Lemma 1, if k1 and k2 are in the set (19), for some ε>0, the derivative of the Lyapunov function V2 satisfies the inequality(26)V˙2≤−κ1Pη,εV21/2−κ2Pη,εμη0i1/2V2,where κjPη,εj=1,2 is a positive scalar. And the expression of Pη, κ1Pη,ε and κ2Pη,ε is given in Appendix A in [34]. From (26), it is clear that V˙2≤0 for t≥0; thus, ϕ1η0i and η1i are always bounded:(27)ϕ1η0i≤ϕ1max,(28)η1i≤ϕ2max,where ϕ1max and ϕ2max are positive constants.
Then, a Lyapunov functionV3 is defined as(29)V3=12e1i2+e2i2.
Calculating the time derivative ofV3 along the trajectories of (21), we get(30)V˙3=e1ie2i+e2ie˙2i=e1ie2i+e2iHi+Ri+δi.
By combining (22) and (23), (30) can be rewritten as(31)V˙3=e1ie2i+e2i−K1e1iω1signe1i−K2e2iω2signe2i−L1e1iγ1signe1i−L2e2iγ2signe2i+e2i−k1ϕ1e2i−∫0tHidυ+∫0t−k2ϕ2e2i−∫0tHidυdυ+δi.
Consideringη0i=e2i−∫0tHidυ and η1i=∫0t−k2ϕ2e2i−∫0tHidυdυ+δi, we have(32)V˙3=e1ie2i+e2i−K1e1iω1signe1i−K2e2iω2signe2i−L1e1iγ1signe1i−L2e2iγ2signe2i+e2i−k1ϕ1η0i+η1i,=e1ie2i−K1e1iω1signe1ie2i−K2e2iω2+1−L1e1iγ1signe1ie2i−L2e2iγ2+1+e2i−k1ϕ1η0i+η1i.
SinceK2>0 and L2>0, we have(33)V˙3≤e1ie2i−K1e1iω1signe1ie2i−L1e1iγ1signe1ie2i+e2i−k1ϕ1η0i+η1i.
Considering the upper bound ofϕ1η0i and η1i given in (27) and (28), (33) can be rewritten as(34)V˙3≤e1ie2i−K1e1iω1signe1ie2i−L1e1iγ1signe1ie2i+e2ik1ϕ1max+ϕ2max.
We consider the following two cases:Case 1: ife1ie2i<0, we have e1ie˙1i<0. It is clear that e1i is bounded in this case.Case 2: Ife1ie2i≥0. From (34), we have(35)V˙3≤e1ie2i+e2ik1ϕ1max+ϕ2max.
Then, according to Young’s inequality [37], (35) can be rewritten as(36)V˙3≤e1i2+e2i22+e2i2+k1ϕ1max+ϕ2max22=e1i22+e2i2+k1ϕ1max+ϕ2max22.
LetCmax=k1ϕ1max+ϕ2max2/2, we have(37)V˙3≤2V3+Cmax.
From (37), for any initial time t=tg, we have(38)V3≤V3tge2t+Cmax1−e2t2,ift≥tg,where e is the natural constant. From (38), it is clear that V3 and e1i are bounded in arbitrary finite time. From the discussions in case 1 and case 2, it is clear that the control error e1i is always bounded in arbitrary finite time.Step 2.
According to Proposition2 of Lemma 1, it can be known from (24) that the following equation can be satisfied in a fixed-time T1:(39)η0i=e2i−∫0tHidυ=0,ift≥T1,η˙0i=e˙2i−Hi=0,ift≥T1,where T1 is independent on the initial system conditions. Then, we have(40)e˙2i=Hi=−K1e1iω1signe1i−K2e2iω2signe2i−L1e1iγ1signe1i−L2e2iγ2signe2i,ift≥T1.
It has been proved thate1i is always bounded in arbitrary finite time in Step 1. Thus, it is clear that e1iT1 is bounded. Then, according to Lemma 2, it can be known from (40) that e1i=0 can be guaranteed in a fixed-time T2+T1:(41)e1i=0,ift≥T2+T1,where T2+T1 is also independent on the initial system conditions.
The proof is finished.


## 4. Simulation Results

In this section, to illustrate the effectiveness of the proposed method, the mathematical simulation is presented. The dynamic parameters in system (1) are given as a two-DOFs robot manipulator in [34]:(42)Mq=p1+2p2cosq2p3+p2cosq2p3+p2cosq2p4,Cq,q˙=−p2sinq2q˙1−2p2sinq2q˙10p2sinq2q˙2,Gq=p5cosq1+p6cosq1+q2,p6cosq1+q2T,p1=m1+m2r12+m2r22+J1,p2=m2r1r2,p3=m2r22,p4=p3+J2,p5=m1+m2r1g1,p6=m2r2g1,where m1=0.5, m2=1.5, r1=1, r2=0.8, J1=J2=5, and g1=9.8. The desired joint position vector qd is selected as(43)qd=−1−sint3,−0.6−cost2Trad.The parameters of the proposed fixed-time IDC scheme are selected as(44)ω1=0.3,K1=83,K2=53,L1=83,L2=53,k1=3,μ=1,k2=2.2.In this section, MATLAB 2016 is chosen as simulation software. The simulation method is chosen as the fixed-step dormand-prince method. The step size of simulation is set as 0.001 s.For the comparison, the FTSMC scheme in [21], fixed-time IDC scheme in [34], and the proposed robust fixed-time IDC (15) are considered in this section.According to [21], for system (1), the FTSMC scheme can be designed as(45)τ=τ0+τeq+τr,where the sliding-mode surface sT, the control component τ0, τeq, and τr are given as(46)τ0=Mqq¨d+Cq,q¨q˙+Gq,τeq=−MqΩ1Se1υ1+Ω2Se2υ2,τr=−11−δ¯k0+b0+b2q˙2+ϖτ0+τeqsTsT+s0,sT=e2+∫0tΩ1Se1υ1+Ω2Se2υ2dυ,where the parameters are chosen as υ1=0.5, υ2=2/3, Ω1=230019, Ω2=160012, k0=2, b0=12, b2=2.8, s0=0.01, δ¯=0.5, and ϖ=0.4.According to [33], for system (1), the fixed-time IDC can be designed by removing the robust term Re1,e2 in (15):(47)τ=bq−1−fq,q˙−He1,e2.The parameters of (47) are chosen as in (44).For verification, we consider the following five cases:Case 1 (system with none uncertainties, comparison of small and large initial condition): this case shows the performance comparison of finite-time and fixed-time schemes for different initial conditions. We consider there are no uncertaintiesd in the manipulator system (1). We consider two kinds of initial system conditions:(48)small initial condition:q0=0,0Trad,q˙0=0,0Trads,(49)large initial condition:q0=6,2Trad,q˙0=0,0Trads.The simulation results are shown in Figures1 and 2. From Figure 1(a), for the small initial conditions (48), the three methods can guarantee that the position tracking errors converge to zero in 1.5 s. However, from Figure 2(a), with the same parameters, it can be known that the convergence time of FTSMC is increased to 2.3 s for the large initial condition (49). As stated in the Introduction section and Section 2.3, the reason for the simulation results is that the FTSMC is designed based on finite-time stability, and then the convergence time of FTSMC is affected by different initial system conditions. And, the fixed-time IDC and proposed robust fixed-time IDC can eliminate the negative effect brought by different initial conditions. The input torques of the three methods are given in Figures 1(b) and 2(b), respectively.Then, the following cases 2 and 5 will show the comparison of the fixed-time IDC and the proposed robust fixed-time IDC for the robot manipulator with uncertainties. We consider four kinds of uncertainties: (1) case 2: uncertainties have small amplitude and low frequency; (2) case 3: uncertainties have large amplitude and low frequency; (3) case 4: uncertainties have small amplitude and high frequency; and (3) case 5: uncertainties have large amplitude and high frequency. And in the following four cases, the initial conditions are chosen as in (48).Case 2 (uncertainties have small amplitude and low frequency): in this case, the uncertainties are chosen as(50)d=30cos0.3t,40cos0.3tTNm.The simulation results are shown in Figure3. From Figure 3(a), it is clear that the convergence precision and convergence rate of the robot manipulator with the fixed-time IDC are affected by the uncertainties (50). And, it also can be observed that the proposed robust fixed-time IDC still can guarantee a fast convergence rate and high convergence precision like case 1. As stated in Section 2.2, the robust term of the proposed method can effectively suppress the uncertainties.Case 3 (uncertainties have large amplitude and low frequency): in this case, we increase the amplitude of uncertainties in case 1. The uncertainties are chosen as(51)d=90cos0.3t,120cos0.3tTNm.The simulation results are shown in Figure4. From Figure 4(a), for the uncertainties with larger amplitude, it is clear that the convergence performance of fixed-time IDC is affected more seriously than that in Case 1. And, the proposed robust fixed-time IDC still can achieve a similar excellent control performance like Case 1.Case 4 (uncertainties have small amplitude and high frequency): in this case, we increase the frequency of uncertainties in case 1. The uncertainties are chosen as(52)d=30cos2t,40cos2tTNm.The simulation results are shown in Figure5. From Figure 5(a), for the uncertainties with these kind of uncertainties, it is clear that the convergence performance of the proposed robust fixed-time IDC can achieve a better convergence performance than fixed-time IDC in this case.Case 5 (uncertainties have large amplitude and high frequency): in this case, we simultaneously increase the amplitude and frequency of uncertainties in case 1. The uncertainties are chosen as(53)d=90cos2t,120cos2tTNm.The simulation results are shown in Figure6. From Figure 6(a), with this kind of uncertainties, it clear that the convergence performance of fixed-time IDC is greatly affected. From Figure 6(a), for the fixed-time IDC, an undesirable large oscillation of the position tracking errors can be observed. This undesirable oscillation may lead to the failure of robot manipulator’s control mission. Figure 6(a) also shows that the proposed robust fixed-time IDC still can achieve a fast convergence rate and high control precision despite the presence of this kind of large amplitude and fast-varying uncertainties.Figure 1
Responses of small initial condition in case 1q0=0,0rad. (a) Position tracking errors. (b) Input torques.
(a)(b)Figure 2
Responses of large initial condition in case 1q0=6,2Trad. (a) Position tracking errors. (b) Input torques.
(a)(b)Figure 3
Responses in case 2. (a) Position tracking errors. (b) Input torques.
(a)(b)Figure 4
Responses in case 3. (a) Position tracking errors. (b) Input torques.
(a)(b)Figure 5
Responses in case 4. (a) Position tracking errors. (b) Input torques.
(a)(b)Figure 6
Responses in case 5. (a) Position tracking errors. (b) Input torques.
(a)(b)According to the simulation results, the following can be concluded:(1)
The convergence rate of finite-time IDC can be lowered gradually with the increase of initial system conditions (Figures1 and 2). The convergence rate of fixed-time IDC and proposed robust fixed-time IDC is independent of the initial system conditions, which means that the fixed-time schemes can provide a fast convergence rate without changing the control parameters for the different initial system conditions (Figures 1 and 2).(2)
The fixed-time IDC only can guarantee the desired excellent control performance when the robotic manipulator system does not have uncertainties (Figures1 and 2). For the different kinds of uncertainties, the convergence rate and control precision of fixed-time IDC are significantly affected (Figures 3–6). Since the actual robot manipulator system is a typical uncertain system, the fixed-time IDC may not achieve the desired convergence performance. By designing a robust term to improve the fixed-time IDC, the proposed scheme can not only maintain the desired convergence performance of fixed-time control but also eliminate the affected by different uncertainties (Figures 3–6).


## 5. Conclusion

In this paper, a novel robust fixed-time control scheme has been proposed for the robotic manipulator system with uncertainties. By designing a robust control term based on the uniform robust exact differentiator (URED) algorithm, a robust fixed-time IDC extended the existing fixed-time IDC to the robust scheme. Moreover, the proposed method eliminated the singular problem without using approximate approach. It has been strictly proved that the convergence time of the closed-loop system with the proposed control method is fixed-time and independent of the initial conditions. Finally, the simulation results have demonstrated the effectiveness of the proposed method. In the future work, we will focus on the extension of the proposed robust fixed-time control method to more complex manipulator system, such as the flexible joint manipulator. Unlike the rigid manipulator considered in this paper has two orders and is matched uncertain, the flexible joint manipulator system is a high-order system (four orders) and is a typically mismatched uncertain system.



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*Source: 6664750-2021-01-20.xml*