#### 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.

#### 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: Section 2 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, denotes .

#### 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]where stand for the vectors of joint positions, velocities, and accelerations, respectively. is the positive definite inertia matrix, is the centripetal Coriolis matrix, and is the gravitational vector. is the control input vector and is the external disturbance vector.

*Assumption 1. *The model parameters in (1) are given bywhere , , and represent the nominal terms. , , and represent the uncertain terms.

According to Assumption 1, system (1) can be expressed as follows:in which the lumped disturbance is defined as

*Assumption 2. *The lumped disturbance is bounded bywhere , , and are unknown positive constants.

*Assumption 3. *The norms of desired vectors are bounded bywhere and denote the vector of desired position and velocity, respectively. and 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 systemwhere, is a continuous nonlinear function that is an open neighborhood of the origin. The system is fixed-time stable if the convergence time is a bounded function , that is, there exists a time constant such that .

Lemma 1 (see [37]). *If system (7) is fixed-time stable, then there exists a continuous positive Lyapunov function , , , , and satisfying that . Then, the system can reach the residual set which is bounded as , where is scalar and satisfies . The time to reach the neighborhood of the origin is upper bounded by .*

Lemma 2 (see [38, 39]). *For any nonnegative real numbers , and , the following two inequalities hold:*

Lemma 3. *Consider the following nonlinear system:where , , , , , . System (9) is fixed-time stable with the convergence time bounded by*

*Proof. *The differential equation for system (9) can be converted into the following form:Denote . Denote for. The initial values of and are defined as and . Equation (11) can be rewritten asSolving (12), the upper bound of convergence time can be calculated asDefine and denote as the initial value of ; we haveThus, for system (9), the upper bound of convergence time can be expressed asThe proof of Lemma 3 is completed.

*Remark 1. *Reference [40] presented a fixed-time stable system , and the system can be stabilized within a fixed time . A fast fixed-time stable system with was introduced in [35], and the system can be stabilized within a fixed time . Since the relations and 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 bywhere and are system states. is the control input.

Based on Lemma 3, for system (16), a new form of fixed-time sliding mode surface is designed aswhere , , , .

Differentiating (17) results in

For (18), if and , it may suffer from the singularity problem due to .

To circumvent the singularity problem, a novel fast nonsingular fixed-time sliding mode surface (NFNFSM) is proposed aswhere , , , , ,

*Remark 2. *When is achieved, it can be deduced that and .

When , then and . When , then and . Therefore, once is satisfied, is equivalent to the equality .

*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] iswithwhere and , and are odd integers satisfying that , .

The sliding surface FNFSM [34] iswithwhere , , , .

The parameters of the three sliding surfaces are selected as , , , , , .

The same initial condition is that . 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 asin which aswhere,，，，, , and as .

##### 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 aswhereand represents a small time margin associated with .*

*Proof. *Select the following Lyapunov function:The derivative of to time isSubstituting (25) into (30), there isDenote, . According to (31), if . To facilitate analysis, the state space is divided into two separate regions as ,. *Case 1.* When the statesenter into the region , then the function . If , let and , and we obtain . According to [39], the states will arrive at the sliding surface or enter into the region within fixed time . *Case 2.* In the region , if . It can be deduced from (31) that is still attractive. Next, we need to prove that is not attractive except for the origin . When is very close to 0, the control law (25) reduces to the following form:where the fact , as is used.

Differentiating and combining (32), we haveFrom (33), it can be concluded that for and for . This means that is not an attractor. Then, the system states will transgress into monotonically in a small time margin.

Accordingly, the sliding surface can be arrived at within time . Once the sliding surface is reached, it can be known from Lemma 3 that the system states can converge to the origin within fixed time . Then, the total convergence time is upper bounded by (27).

The proof of Theorem 1 is completed.

*Remark 4. *Note that the time across cannot be calculated precisely. Nevertheless, for enough small , it can be regarded as . From (19), it can be obtained that . Integrating both sides of (33) yieldsSolving (34) obtainswhere . This means that can be small enough by selecting sufficiently small . Then, the finite time can be ignored for small because of the estimation conservativeness of .

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

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

Define and . The error equation of system (1) can be expressed as follows:where

A vector of novel fast nonsingular fixed-time sliding mode surface (NFNFSM) is constructed aswhere is a diagonal matrix:

From (38), when . Consider the Lyapunov function , and its time derivative is. When, there is . When, there is . According to Lemma 3, the system states can reach the designed sliding surface within a fixed time, given by

##### 4.2. Controller Design

To achieve the fixed-time tracking, an adaptive NFNFSM controller (ANFNFSMC) is designed aswithin which , , , , ,. is the estimation of , and the adaptive laws are given by

##### 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 variables ,, , and are all bounded*(2)*The proposed sliding mode manifold can converge to the neighborhood of within a fixed time*(3)*The state variables and can converge into a small region near the origin within a fixed time*

*Proof. *(1)Denote and choose a positive Lyapunov function as Differentiating with respect to time and taking into account (38), (41)–(45), we obtain According to Assumption 2, there is From (48), we know that , which implies that is bounded. Then, , , , and are all bounded. It can be noted from (38) that the boundedness of are guaranteed. And according to Assumption 3, it can be known that both and are bounded. Thus, there exists a positive constant such that .(2)To examine the fixed-time stability, consider the following positive definite Lyapunov function: Taking the derivative of results in Based on the above analysis, we know that , , , ,, and are all bounded; thus, there exists a positive constant satisfying that . Then, according to Lemma 2, (50) is simplified as Applying Lemma 1, the system is fixed-time stable with the following convergence region: According to Lemma 1, the convergence time of the reaching motion can be estimated as(3)Once the sliding variable converges to the region , , there isEquation (54) can be rewritten asWhen , (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: From (38) and (56), the tracking error will converge to the regionFrom (56), we haveThen,Thus, the fixed-time convergence region of isIt is concluded that the system states will converge to the set in fixed time.

Accordingly, the upper bound of the total convergence time can be estimated asThe proof of Theorem 2 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]with

The system parameters in (63) are listed as , , , , , , . The normal values of and are set as and . The external disturbances are assumed as .

The controller and adaptive law parameters in (43) are set as , , , , , .

##### 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 Figures 2–4. The desired signals are and . These different initial states are given by

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According to Theorem 2, 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 as , . These different desired signals are given by

The simulation results with different desired reference signals under the proposed control scheme are displayed in Figures 5–7. 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.

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##### 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 Figures 8–10. The initial states are set as , . These sets of parameters are listed as follows:

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