Fixed-Time Event-Triggered Control of Nonholonomic Mobile Robots with Uncertain Dynamics and Preassigned Transient Performance

Abstract

In this paper, a novel adaptive control scheme is proposed for the path-following problem of a nonholonomic mobile robot with uncertain dynamics based on barrier functions. To optimize communication resources, we integrate an event-triggered mechanism that avoids Zeno behavior and ensures that the tracking error of the closed-loop system converges to a small neighborhood around zero within a fixed time, while consistently satisfying predefined transient performance requirements. Extensive simulation studies demonstrate the effectiveness of the proposed approach and validate the theoretical results.

1. Introduction

The field of robotics has witnessed rapid advancements in recent decades. These developments have expanded the capabilities of robots, allowing them to perform increasingly complex tasks in diverse environments. Among the different types of robots, wheeled mobile robots have gained particular prominence due to their efficiency, simplicity, and adaptability [1]. These robots are often used in a variety of applications ranging from industrial automation to service robots, where precise and reliable movement is essential. Wheeled mobile robots have advantages such as energy saving and high-speed operation due to their mechanical design.

Unlike aerial or legged robots, which have a greater range of motion due to their greater number of degrees of freedom, wheeled mobile robots are subject to nonholonomic constraints that prevent them from instantaneously moving in certain directions. For example, a nonholonomic mobile robot can move forward or backward, but cannot slide sideways without first changing direction. The study of nonholonomic systems has attracted extensive attention in the past three decades [2]. The interest in nonholonomic systems stems from their wide range of potential applications and technical challenges. Specifically, due to the Brockett necessary condition, it is impossible to design a smooth, time-invariant controller that can stabilize a nonholonomic system [3]. Numerous effective strategies have been proposed to address the stabilization problem of nonholonomic systems, including switching-based state-scaling design [4], smooth time-varying methods [5], and hybrid feedback methods [6]. In addition to stabilization control, the interesting tracking control problem of nonholonomic mobile robots has also been widely investigated using the dynamic feedback linearization [7], the backstepping technique [8], dynamic feedback control [9], fixed-time control [10], predefined-time control [11], and model predictive control [12]. Moreover, the cooperative control of multiple nonholonomic mobile robots remains an attractive research area [13,14].

The two-wheeled mobile robot is a typical underactuated system with two inputs and three outputs. Recently, considerable research has been devoted to solving the control problem of underactuated systems. One approach involves a coordinate transformation with an auxiliary variable to transform the underactuated system into a fully actuated system [15]. Another method divides the error system into two cascaded subsystems, consisting of a third-order and a second-order subsystem [16]. However, these methods often overlook position constraints. Addressing system output constraints during operation is a critical practical consideration. For instance, in real-world applications, the position of an autonomous vehicle should be controlled to ensure it remains on the road and accurately follows the lane [17]. Barrier Lyapunov functions have been introduced in the literature to design controllers for nonholonomic mobile robots with state constraints [18,19]. However, most existing approaches only address symmetric constraint requirements. In [20], a tracking control method for strict-feedback nonlinear systems was proposed for asymmetric constraints using an asymmetric barrier Lyapunov function. In practical engineering, nonholonomic mobile robots usually require shorter and more precise convergence times. Unfortunately, in most finite-time or Lyapunov stabilization control frameworks, the convergence time is affected by the initial conditions of the system [21] and is therefore often unknown when these conditions change. In contrast to finite-time stability, fixed-time stability has faster convergence and allows for the a priori estimation of an upper bound on the settling time. In addition, the convergence time in fixed-time stability does not depend on the initial conditions [22].

To ensure a predictable convergence time for the system, control algorithms based on fixed-time and prescribed finite-time convergence have been developed in recent years [23,24]. Compared with traditional finite-time control methods [25], these algorithms allow the convergence time estimate to be independent of the initial conditions and can be predetermined according to user specifications. The fixed-time stability issue is considered in [26] for impulsive systems. Two fixed-time consensus control protocols are constructed using event-triggered mechanisms in [27]. The self-triggered strategy is combined with the fixed-time cooperative control to tackle the leader–follower consensus issue [28]. In practical applications, the microprocessor is often constrained by limited energy resources and communication bandwidth. High-frequency data transmission can lead to congestion in the communication channel [29]. To address these issues, event-triggered schemes have gained popularity in recent years. Unlike time-triggered approaches, event-triggered mechanisms introduce a new paradigm for control systems with reduced data transmission. These mechanisms determine state updates or controller actions based on specific event conditions [30]. In [31], it demonstrates that implementing a dynamic event-triggered mechanism provides greater design flexibility to eliminate the Zeno phenomenon. Additionally, in [32], the event-triggered mechanism is integrated with an adaptive control strategy for mobile robots, efficiently managing unknown parameters while conserving communication bandwidth. Despite significant advancements in event-triggered control across various domains, achieving adaptive fixed-time convergence under limited communication resources remains challenging.

Inspired by the above discussions, this work addresses the problem of designing a path-following controller for a nonholonomic mobile robot under asymmetric state constraints. A novel adaptive fixed-time event-triggered control scheme is proposed for nonholonomic mobile robot systems. Theoretical analysis and simulation results show that the proposed controller ensures fixed-time convergence of the tracking error and maintains the boundedness of all signals in the closed-loop system without Zeno behavior. The major contributions of this work are as follows:

1. This paper introduces a barrier Lyapunov function control technique to address the fixed-time path-following control problem of nonholonomic mobile robots, effectively accommodating the asymmetric state constraints imposed by the system.

2. The proposed method enables the nonholonomic mobile robot to accurately track the reference trajectory within a short time, while efficiently utilizing limited communication resources and preventing the occurrence of Zeno phenomena.

The remainder of the paper is organized as follows. Section 2 presents the dynamics model of the nonholonomic mobile robot system and provides the problem formulation. Section 3 develops the control scheme for the system and analyzes its stability. Section 4 validates the effectiveness of the proposed scheme through extensive simulations. Finally, the conclusions are summarized in Section 5.

2. Problem Formulation and Preliminaries

2.1. Preliminaries

Lemma 1

([33]). Consider the system of differential equations

$$\dot{x}(t)=f(t,x),x(0)=x_0$$

(1)

where $f:{\mathbb{R}}_{+}\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ is a continuous function and $x\in {\mathbb{R}}^n$. Suppose the origin is an equilibrium point. If there exists a positive definite function $V(x)$, which satisfies

$$\dot{V}(x)\le -\alpha V^p(x)-\beta V^q(x)$$

(2)

where $\alpha ,\beta >0,p>1,0<q<1$, then the origin of the system is globally fixed-time stable and the setting time function $T_{\mathrm{fd}}$ can be estimated by

$$T_{\mathrm{fd}}\le T_{max}={\displaystyle \frac{1}{\alpha (p-1)}}+{\displaystyle \frac{1}{\beta (1-q)}}.$$

(3)

Lemma 2

([34]). For any real numbers $x_1,\cdots ,x_n$, and $0\le b\le 1$, we have:

$$(|x_1|+\cdots +|x_n{|)}^b\le |x_1{|}^b+\cdots +{|x_n|}^b.$$

(4)

Lemma 3

([21]). For $x,y\in \mathbb{R}$, the following inequality holds:

$$xy\le {\displaystyle \frac{{\iota }^p}{p}}{|x|}^p+{\displaystyle \frac{1}{q{\iota }^q}}{|y|}^q$$

(5)

where $\iota >0,p>1,q>1,$ and $(p-1)(q-1)=1$.

2.2. Universal Barrier Function

In this paper, we consider the problem of asymmetric error constraints. To address this, we introduce a universal asymmetric barrier function to accommodate the asymmetric constraint requirements. This function is defined as follows [35]:

$$V_b={\displaystyle \frac{1}{2}}{\eta }^2,\eta ={\displaystyle \frac{b_h{b}_l{e}_r}{(b_h-e_r)(b_l+e_r)}}$$

(6)

where $b_h$ and $b_l$ are the pre-defined constraint requirement functions on the system output, $-b_l(0)<e_r(0)<b_h(0)$. It is easy to see that $\eta =0$ if and only if $e_r=0$. Besides, when $e_r\to b_h$, we have $\eta \to +\infty$, and hence $V_b\to +\infty$. Alternatively, when $e_r\to -b_l$, we have $\eta \to -\infty$, and therefore $V_b\to +\infty$. If the constraint functions are symmetric, namely $b_h=b_l=b$, the barrier function in (6) becomes

$$\eta ={\displaystyle \frac{b^2{e}_r}{b^2-e_r^2}}.$$

(7)

This implies that systems with symmetric output constraint requirements can be viewed as a special case of those with asymmetric constraint requirements.

2.3. Robot Dynamics

We consider a two-wheeled mobile robot as shown in Figure 1, which can be described by the following dynamic model [36]:

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta }\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{r}{2}}cos\theta & {\displaystyle \frac{r}{2}}cos\theta \\ {\displaystyle \frac{r}{2}}sin\theta & {\displaystyle \frac{r}{2}}sin\theta \\ {\displaystyle \frac{r}{2R}}& -{\displaystyle \frac{r}{2R}}\end{array}\right]\omega$$

(8)

$$M\dot{\omega }+C\omega +D\omega =u$$

(9)

where $(x,y)$ denotes the position of the robot, and $\theta$ represents the orientation. $\omega ={[{\omega }_1,{\omega }_2]}^T$ denotes the angular velocities of the left and right wheels, respectively. $u={[u_1,u_2]}^T$ represents the control torques acting on the two wheels, M is a symmetric positive definite inertia matrix, C is the centripetal and coriolis matrix and D denotes the surface friction. M, C, and D are defined as

$$M=\left[\begin{array}{cc}{m}_1& m_2\\ m_2& m_1\end{array}\right]D=\left[\begin{array}{cc}{d}_1& 0\\ 0& d_2\end{array}\right]$$

(10)

$$C={\displaystyle \frac{1}{2R}}{r}^2{m}_c\dot{\theta }\overline{d}\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$$

(11)

where $m_1={\displaystyle \frac{r^2}{4R^2}}(m{R}^2+I)+I_{\omega }$, $m_2={\displaystyle \frac{r^2}{4R^2}}(m{R}^2-I)$, $m=m_c+2m_{\omega }$, and $I=m_c{\overline{d}}^2+2m_{\omega }{R}^2+I_c+2I_m$, R is the half width of the mobile robot, r is the radius of the wheel, $m_c$ and $m_w$ are the masses of the body and the wheel of the robot, respectively. $\overline{d}$ is the distance between the center of the mass of the robot (i.e., point $P_c$) and the middle point between the left and right wheels (i.e., $P_o$), $I_c$, $I_w$, and $I_m$ are the moment of inertia of the body about the vertical axis through $P_c$, the wheel with a motor about the wheel axis, and the wheel with a motor about the diameter, respectively. The positive constants $d_1$ and $d_2$ in (10) are the damping coefficients. System parameters including r, R, $m_c$, $m_w$, $I_c$, $I_w$, $I_m$, $\overline{d}$, $d_1$, and $d_2$ are assumed to be unknown.

2.4. Control Objective

In this work, the control objective is to design control torques $u_1$ and $u_2$ such that the position (x, y) of the mobile robot (8) follows a reference path $q_d=(x_d(s),y_d(s))$, where s is a path parameter. Specifically, our goal is to guide the mobile robot to track a virtual robot that moves along this path. By effectively steering the robot towards the desired position and minimizing the distance between the robot and the virtual robot, we can achieve the control objective.

From the system dynamics (8), it is evident that the number of control inputs ($u_1$ and $u_2$) is less than the configuration variables (x, y, and $\theta$). This indicates that the robot has degrees of freedom that cannot be directly controlled. As a result, the system is classified as an underactuated mechanical system, where the motion can be achieved by leveraging the system’s inherent dynamics or through indirect control strategies. As shown in Figure 2, we define the following variables to mathematically formulate the control objective [37]:

$$x_e=x_d-x,y_e=y_d-y$$

(12)

$$z_e=\sqrt{x_e^2+y_e^2}$$

(13)

$${\theta }_e=\theta -{\theta }_d$$

(14)

where ${\theta }_d=arcsin{\displaystyle \frac{y_e}{z_e}}$. By utilizing the variables $z_e$ and ${\theta }_e$, the control variables of the target nonholonomic mobile robot are reduced to two ($z_e$ and ${\theta }_e$). As a result, the initially underactuated control problem can be transformed into a fully actuated control one. Moreover, $z_e$ represents the distance between the robot and the reference trajectory, which helps to subsequently control the transient error performance of the robot’s position and orientation. From (12)–(14), we have

$$\begin{array}{cc}\hfill sin({\theta }_e)& =sin(\theta -{\theta }_d)\hfill \\ \hfill & =sin\theta cos{\theta }_d-cos\theta sin{\theta }_d\hfill \\ \hfill & ={\displaystyle \frac{x_e}{z_e}}sin\theta -{\displaystyle \frac{y_e}{z_e}}cos\theta \hfill \\ \hfill cos({\theta }_e)& =cos(\theta -{\theta }_d)\hfill \\ \hfill & =cos\theta cos{\theta }_d+sin\theta sin{\theta }_d\hfill \\ \hfill & ={\displaystyle \frac{x_e}{z_e}}cos\theta +{\displaystyle \frac{y_e}{z_e}}sin\theta .\hfill \end{array}$$

(15)

Combining (8)–(11) and (12)–(15), the derivatives of $z_e$ and ${\theta }_e$ can be computed as

$$\begin{array}{cc}\hfill {\dot{z}}_e& ={\displaystyle \frac{d}{dt}}\sqrt{x_e^2+y_e^2}\hfill \\ \hfill & ={\displaystyle \frac{x_e{\dot{x}}_e+y_e{\dot{y}}_e}{\sqrt{x_e^2+y_e^2}}}\hfill \\ \hfill & ={\displaystyle \frac{x_e(\frac{d{x}_d}{ds}\dot{s}-\dot{x})+y_e(\frac{d{y}_d}{ds}\dot{s}-\dot{y})}{\sqrt{x_e^2+y_e^2}}}\hfill \\ \hfill & ={\displaystyle \frac{x_e(x_d^{\prime }\dot{s}-\dot{x})+y_e(y_d^{\prime }\dot{s}-\dot{y})}{z_e}}\hfill \\ \hfill & =({\displaystyle \frac{x_e}{z_e}}{x}_d^{\prime }+{\displaystyle \frac{y_e}{z_e}}{y}_d^{\prime })\dot{s}-{\displaystyle \frac{x_e\dot{x}+y_e\dot{y}}{z_e}}\hfill \\ \hfill & =({\displaystyle \frac{x_e}{z_e}}{x}_d^{\prime }+{\displaystyle \frac{y_e}{z_e}}{y}_d^{\prime })\dot{s}-cos{\theta }_{e}r{\tau }_1\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\dot{\theta }}_e& =\dot{\theta }-{\dot{\theta }}_d\hfill \\ \hfill & =r{R}^{-1}{\tau }_2-{\displaystyle \frac{z_e}{x_e}}·{\displaystyle \frac{{\dot{y}}_e{z}_e-y_e\left(\frac{x_e{\dot{x}}_e+y_e{\dot{y}}_e}{z_e}\right)}{z_e^2}}\hfill \\ \hfill & =r{R}^{-1}{\tau }_2-{\displaystyle \frac{{\dot{y}}_e{z}_e^2-y_e(x_e{\dot{x}}_e+y_e{\dot{y}}_e)}{x_e{z}_e^2}}\hfill \\ \hfill & =r{R}^{-1}{\tau }_2-{\displaystyle \frac{{\dot{y}}_e{x}_e-y_e{\dot{x}}_e}{z_e^2}}\hfill \\ \hfill & =r{R}^{-1}{\tau }_2+{\displaystyle \frac{{\dot{x}}_e{y}_e}{z_e^2}}-{\displaystyle \frac{{\dot{y}}_e{x}_e}{z_e^2}}\hfill \end{array}$$

(17)

where ${\tau }_1={\displaystyle \frac{1}{2}}({\omega }_1+{\omega }_2),{\tau }_2={\displaystyle \frac{1}{2}}({\omega }_1-{\omega }_2)$, ${\dot{x}}_e=x_d^{\prime }\dot{s}-r{\tau }_{1}cos(\theta )$, ${\dot{y}}_e=y_d^{\prime }\dot{s}-r{\tau }_{1}sin(\theta )$, $x_d^{\prime }={\displaystyle \frac{d{x}_d}{ds}}$, and $y_d^{\prime }={\displaystyle \frac{d{y}_d}{ds}}$. s is the path parameter of $q_d$, while $\dot{s}$, its time derivative, is closely related to the speed of the robot. This speed is carefully determined to ensure the feasibility of the reference path for real-world robotic systems. Similar to [37,38], we select $\dot{s}$ as

$$\dot{s}={(x_d^{\prime 2}+y_d^{\prime 2})}^{-{\displaystyle \frac{1}{2}}}(p_1-p_2{e}^{-p_{3}t})$$

(18)

where $p_1,p_2<1$, and $p_3$ are positive constants.

From (16) and (17), it is noted that the error system will be ill defined if $z_e=0$ and ${\theta }_e=\pm {\displaystyle \frac{\pi }{2}}$. Therefore, the following constraints should be satisfied when the control inputs are designed:

$$z_e(t)>0,-{\displaystyle \frac{\pi }{2}}<{\theta }_e(t)<{\displaystyle \frac{\pi }{2}},\forall t\ge 0.$$

(19)

To ensure that the system error remains within the desired bounds and to guarantee its convergence behavior, we introduce a prescribed performance function, which is a decreasing smooth function as follows [39]:

$$\phi (t)=(\phi (0)-{\phi }_{\infty })e^{-\lambda t}+{\phi }_{\infty }$$

(20)

where the parameters ${\phi }_{\infty }$ and $\lambda$ are constants that satisfy $\phi (0)>|e(0)|$ and $\lambda >0$. For the convenience of discussion, $e(t)$ represents the error variable that needs to be controlled. The parameter $\lambda$ determines the convergence rate, while ${\phi }_{\infty }$ represents the steady-state error value. The performance function $\phi (t)$ is a continuous, bounded, strictly positive, and monotonically decreasing function that satisfies ${lim}_{t\to \infty }\phi (t)={\phi }_{\infty }>0$. If the tracking error $e(t)$ remains within the performance envelope defined by $\phi (t)$ and $-\phi (t)$, the preset performance requirements are met, i.e., $-\phi (t)<e(t)<\phi (t),t\ge 0$. Since the decay rate of $\phi (t)$ is related to the constant, $\lambda$ also determines the convergence rate of $e(t)$. Based on the above discussion, the error performance constraints for the system in this work are specified as follows:

$${\rho }_L<z_e<{\rho }_H,-{\kappa }_L<{\theta }_e<{\kappa }_H$$

(21)

where ${\rho }_L(t)=a_1\phi (t)$, ${\rho }_H(t)=a_2\phi (t)$, ${\kappa }_L(t)=b_1\phi (t)$, ${\kappa }_H(t)=b_2\phi (t)$, $a_1$, $a_2$, $b_1$, and $b_2$ are positive constants.

In this work, the tracking error $z_e$ should always be positive and bounded away from zero, to avoid singularity problems for the tracking error. From (19) and (21), we can obtain

$$0<{\rho }_L(t)<z_e(t)<{\rho }_H(t).$$

(22)

In this case, define ${\overline{z}}_e(t)\triangleq z_e(t)-\alpha (t)$, where $\alpha$ is an intermediate variable, and ${\rho }_L(t)<\alpha (t)<{\rho }_H(t)$. We use ${\overline{z}}_e$ in the universal barrier function design such that

$$-{\mathsf{\Omega }}_L(t)<{\overline{z}}_e(t)<{\mathsf{\Omega }}_H(t)$$

(23)

where ${\mathsf{\Omega }}_L(t)=\alpha (t)-{\rho }_L(t)$ and ${\mathsf{\Omega }}_H(t)={\rho }_H(t)-\alpha (t)$.

2.5. Time-Varying Relative Threshold Event-Triggered

Frequent triggering of the controller often places a considerable computational load on the system, making it essential to implement a dynamic mechanism to decide when to send updated control inputs to the plant. To address this, we propose an event-triggered control scheme with a time-varying relative threshold as follows [40]:

$$u_i(t)={\mu }_i(t_k),\forall t\in [t_k,t_{k+1})$$

(24)

$$t_{k+1}=inf\{t>t_k||{\mu }_i(t)-u_i(t)|\ge m_i|u_i(t)|+n_i\}$$

(25)

where $m_i$ and $n_i$ are positive design parameters, $u_i(t)$ and ${\mu }_i(t_k)$ are the ith element of $u(t)$ and $\mu (t_k)$, respectively. $t_k$, $k\in {\mathbb{Z}}_{+}$ is the controller update time. Whenever (25) is triggered, the time will be marked as $t_{k+1}$ and the control value $u(t_{k+1})$ will be applied to the robot. During the time $t\in [t_k,t_{k+1})$, the control signal holds constant. Thus, for $t\in [t_k,t_{k+1})$, $u_i(t)$ is ${\mu }_i(t_k)$ such that $|{\mu }_i(t)-u_i(t)|\le m_i|u_i(t)|+n_i$, which further indicates $u_i(t)={\displaystyle \frac{{\mu }_i(t)}{1+{\varrho }_{i,1}(t)m_i}}-{\displaystyle \frac{{\varrho }_{i,2}(t)n_i}{1+{\varrho }_{i,1}(t)m_i}}$, where ${\varrho }_{i,1}(t),{\varrho }_{i,2}(t)\in [-1,1]$ are the time-varying threshold parameters.

Remark 1.

In practical applications, especially with experimental platforms and actual nonholonomic mobile robots, direct control using torque may not be feasible. Instead, the available input channel is the voltage applied to the motor. Therefore, an actuator model should be incorporated, where voltages serve as the inputs to amplifiers to control the robot, as shown in [41]. The fixed-time backstepping technique can be used to design the control input such that the required control torque in (24) is achieved through the motor output. However, a rigorous analysis of this method is required and is beyond the scope of this paper. This topic represents a promising direction for future research.

3. Control Design and Stability Analysis

We utilize the backstepping design procedure to develop our controller. Building upon the previous discussions and taking into account the control objectives related to steady-state tracking errors, we define ${\overline{z}}_e=z_e-\alpha$. The tracking errors need to satisfy

$$\begin{array}{cc}\hfill -{\mathsf{\Omega }}_L& <{\overline{z}}_e<{\mathsf{\Omega }}_H\hfill \\ \hfill -{\kappa }_L& <\theta e<{\kappa }_H\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\eta }_z& ={\displaystyle \frac{{\mathsf{\Omega }}_H{\mathsf{\Omega }}_L}{({\mathsf{\Omega }}_H-{\overline{z}}_e)({\mathsf{\Omega }}_L+{\overline{z}}_e)}}\hfill \\ \hfill {\eta }_{\theta }& ={\displaystyle \frac{{\kappa }_L{\kappa }_H}{({\kappa }_H-{\theta }_e)({\kappa }_L+{\theta }_e)}}.\hfill \end{array}$$

(27)

Step 1. Define two error variables:

$${\tilde{\tau }}_1={\tau }_1-{\tau }_{1d},{\tilde{\tau }}_2={\tau }_2+{\tau }_{2d}$$

(28)

where ${\tau }_{1d}$ and ${\tau }_{2d}$ are the virtual controls of ${\tau }_1$ and ${\tau }_2$. The system dynamics can be transformed to

$$\begin{array}{cc}\hfill {\dot{\eta }}_z& ={\mathsf{{\rm Y}}}_z+{\upsilon }_z{\dot{\overline{z}}}_e\hfill \\ \hfill {\dot{\eta }}_{\theta }& ={\mathsf{{\rm Y}}}_{\theta }+{\upsilon }_{\theta }{\dot{\theta }}_e\hfill \end{array}$$

(29)

where

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}_z& ={\displaystyle \frac{\partial {\eta }_z}{\partial {\mathsf{\Omega }}_H}}·{\dot{\mathsf{\Omega }}}_H+{\displaystyle \frac{\partial {\eta }_z}{\partial {\mathsf{\Omega }}_L}}·{\dot{\mathsf{\Omega }}}_L\hfill \\ \hfill {\mathsf{{\rm Y}}}_{\theta }& ={\displaystyle \frac{\partial {\eta }_{\theta }}{\partial {\kappa }_H}}·{\dot{\kappa }}_H+{\displaystyle \frac{\partial {\eta }_{\theta }}{\partial {\kappa }_L}}·{\dot{\kappa }}_L\hfill \end{array}$$

(30)

and

$$\begin{array}{c}\hfill {\upsilon }_z={\displaystyle \frac{\partial {\eta }_z}{\partial {\overline{z}}_e}}={\displaystyle \frac{{\mathsf{\Omega }}_H{\mathsf{\Omega }}_L({\overline{z}}_e^2+{\mathsf{\Omega }}_H{\mathsf{\Omega }}_L)}{{({\mathsf{\Omega }}_H-{\overline{z}}_e)}^2{({\mathsf{\Omega }}_L+{\overline{z}}_e)}^2}}\\ \hfill {\upsilon }_{\theta }={\displaystyle \frac{\partial {\eta }_z}{\partial {\theta }_e}}={\displaystyle \frac{{\kappa }_H{\kappa }_L({\theta }_e^2+{\kappa }_H{\kappa }_L)}{{({\kappa }_H-{\theta }_e)}^2{({\kappa }_L+{\theta }_e)}^2}}.\end{array}$$

(31)

Now, define the barrier Lyapunov functions as

$$V_z={\displaystyle \frac{1}{2}}{\eta }_z^2,V_{\theta }={\displaystyle \frac{1}{2}}{\eta }_{\theta }^2.$$

(32)

Let ${\xi }_1=r^{-1}$, ${\xi }_2=r^{-1}R$, and ${\xi }_3=R$, which are all unknown parameters. By introducing ${\widehat{\xi }}_1$, ${\widehat{\xi }}_2$, and ${\widehat{\xi }}_3$ as the estimates of ${\xi }_1$, ${\xi }_2$, and ${\xi }_3$, respectively, with ${\tilde{\xi }}_1={\xi }_1-{\widehat{\xi }}_1$, ${\tilde{\xi }}_2={\xi }_2-{\widehat{\xi }}_2$, and ${\tilde{\xi }}_3={\xi }_3-{\widehat{\xi }}_3$ are the estimation errors. The Lyapunov function at this step is designed as

$$V_1=r^{-1}{V}_z+R{r}^{-1}{V}_{\theta }+{\displaystyle \frac{1}{2{\lambda }_1}}{\tilde{\xi }}_1^2+{\displaystyle \frac{1}{2{\lambda }_2}}{\tilde{\xi }}_2^2+{\displaystyle \frac{1}{2{\lambda }_3}}{\tilde{\xi }}_3^2$$

(33)

where ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ are positive constants. We obtain

$$\begin{array}{cc}\hfill {\dot{V}}_1=& {\eta }_z{\mathsf{{\rm Y}}}_z+{\eta }_z{\upsilon }_z{\tilde{\xi }}_1(({\displaystyle \frac{x_e}{z_e}}{\dot{x}}_d+{\displaystyle \frac{y_e}{z_e}}{\dot{y}}_d)\dot{s}-\dot{\alpha })-cos({\theta }_e){\eta }_z{\upsilon }_{z}r{\tau }_1+{\eta }_{\theta }{\mathsf{{\rm Y}}}_{\theta }\hfill \\ & +{\eta }_{\theta }{\upsilon }_{\theta }({\tilde{\xi }}_2{\tau }_2+{\displaystyle \frac{y_e}{z_e^2}}\left[x_d^{\prime }\dot{s}-r{\tau }_{1}cos(\theta )\right]-{\displaystyle \frac{x_e}{z_e^2}}\left[y_d^{\prime }\dot{s}-r{\tau }_{1}sin(\theta )\right]\hfill \\ & -{\lambda }_1^{-1}{\tilde{\xi }}_1{\dot{\widehat{\xi }}}_1-{\lambda }_2^{-1}{\tilde{\xi }}_2{\dot{\widehat{\xi }}}_2-{\lambda }_3^{-1}{\tilde{\xi }}_3{\dot{\widehat{\xi }}}_3\hfill \end{array}$$

(34)

which implies

$$\begin{array}{cc}\hfill {\dot{V}}_1\le & -K_{1,1}{\left({\displaystyle \frac{1}{2}}{\eta }_z^2\right)}^{{\displaystyle \frac{3}{4}}}-K_{1,2}{\left({\displaystyle \frac{1}{2}}{\eta }_z^2\right)}^2+{\eta }_z{\tilde{\tau }}_1-K_{1,1}{\left({\displaystyle \frac{1}{2}}{\eta }_{\theta }^2\right)}^{{\displaystyle \frac{3}{4}}}-K_{1,2}{\left({\displaystyle \frac{1}{2}}{\eta }_{\theta }^2\right)}^2\hfill \\ & +{\eta }_{\theta }{\tilde{\tau }}_2-{\lambda }_1^{-1}{\tilde{\xi }}_1{\widehat{\xi }}_1-{\lambda }_2^{-1}{\tilde{\xi }}_2{\widehat{\xi }}_2-{\lambda }_3^{-1}{\tilde{\xi }}_3{\widehat{\xi }}_3.\hfill \end{array}$$

(35)

where $K_{1,1}$ and $K_{1,2}$ are positive constants. The virtual controls ${\tau }_{1d}$ and ${\tau }_{2d}$ can be designed as

$$\begin{array}{cc}\hfill {\tau }_{1d}& ={\displaystyle \frac{k_1{\eta }_z}{{\upsilon }_{z}cos({\theta }_e)}}+{\displaystyle \frac{\dot{s}{\widehat{\xi }}_1}{cos({\theta }_e)}}({\displaystyle \frac{x_e}{z_e}}{x}_d^{\prime }+{\displaystyle \frac{y_e}{z_e}}{y}_d^{\prime })-{\displaystyle \frac{{\mathsf{{\rm Y}}}_z}{{\upsilon }_{z}cos({\theta }_e)}}\hfill \\ \hfill {\tau }_{2d}& =-{\displaystyle \frac{k_2{\eta }_{\theta }}{{\upsilon }_{\theta }}}-{\widehat{\xi }}_2({\displaystyle \frac{y_e}{z_e^2}}{\dot{x}}_e-{\displaystyle \frac{x_e}{z_e^2}}{\dot{y}}_e)-{\displaystyle \frac{{\widehat{\xi }}_3{\mathsf{{\rm Y}}}_{\theta }}{{\upsilon }_{\theta }r}}.\hfill \end{array}$$

(36)

Design the parameter update laws as

$$\begin{array}{cc}& {\dot{\widehat{\xi }}}_1={\lambda }_1{\eta }_z{\upsilon }_z\left[\left({\displaystyle \frac{x_e}{z_e}}{x}_d^{\prime }+{\displaystyle \frac{y_e}{z_e}}{y}_d^{\prime }\right)\dot{s}-\alpha \right]\hfill \\ & {\dot{\widehat{\xi }}}_2={\lambda }_2{\eta }_{\theta }({\displaystyle \frac{y_e}{z_e^2}}{x}_d^{\prime }\dot{s}-{\displaystyle \frac{x_e}{z_e^2}}{y}_d^{\prime }\dot{s}-\alpha )\hfill \\ & {\dot{\widehat{\xi }}}_3=-{\lambda }_3{\eta }_{\theta }{\tau }_1({\displaystyle \frac{y_e}{z_e^2}}cos(\theta )-{\displaystyle \frac{x_e}{z_e^2}}sin(\theta )).\hfill \end{array}$$

(37)

Step 2. Recalling ${\tau }_1={\displaystyle \frac{1}{2}}({\omega }_1+{\omega }_2)$ and ${\tau }_2={\displaystyle \frac{1}{2}}({\omega }_1-{\omega }_2)$, we obtain ${\omega }_1={\tau }_1+{\tau }_2$ and ${\omega }_2={\tau }_1-{\tau }_2$. We then define ${\omega }_{1d}={\tau }_{1d}+{\tau }_{2d},{\omega }_{2d}={\tau }_{1d}-{\tau }_{2d},{\tilde{\omega }}_1={\omega }_1-{\omega }_{1d}$, and ${\tilde{\omega }}_2={\omega }_2-{\omega }_{2d}$. Let $\tilde{\omega }=\omega -{\omega }_d$, where $\tilde{\omega }={[{\tilde{\omega }}_1,{\tilde{\omega }}_2]}^T$ and ${\omega }_d={[{\omega }_{1d},{\omega }_{2d}]}^T$. Moreover, from ${\tilde{\tau }}_1={\displaystyle \frac{1}{2}}({\tilde{\omega }}_1+{\tilde{\omega }}_2)$ and ${\tilde{\tau }}_2={\displaystyle \frac{1}{2}}\left({\tilde{\omega }}_1-{\tilde{\omega }}_2\right)$, applying (9) can obtain

$$M\dot{\tilde{\omega }}=\mathsf{\Pi }+\mathsf{\Phi }\mathsf{\Xi }+u$$

(38)

where matrices $\mathsf{\Pi }$, $\mathsf{\Phi }$, and $\mathsf{\Xi }$ are defined as

$$\mathsf{\Pi }=\left[\begin{array}{c}-d_1{\tilde{\omega }}_1-Cr{R}^{-1}{\omega }_2{\tau }_2\\ -d_2{\tilde{\omega }}_2+Cr{R}^{-1}{\omega }_1{\tau }_1\end{array}\right]$$

(39)

$$\mathsf{\Xi }={\left[\begin{array}{cccccc}{m}_1& m_2& m_{1}r& m_{2}r& m_{1}r{R}^{-1}& m_{2}r{R}^{-1}\end{array}\right]}^T$$

(40)

$$\mathsf{\Phi }=\left[\begin{array}{cccccc}{\mathsf{\Lambda }}_{11}& -{\mathsf{\Lambda }}_{12}& -{\mathsf{\Lambda }}_{21}& -{\mathsf{\Lambda }}_{22}& -{\mathsf{\Lambda }}_{31}& -{\mathsf{\Lambda }}_{32}\\ {\mathsf{\Lambda }}_{12}& -{\mathsf{\Lambda }}_{11}& -{\mathsf{\Lambda }}_{22}& -{\mathsf{\Lambda }}_{21}& -{\mathsf{\Lambda }}_{32}& -{\mathsf{\Lambda }}_{31}\end{array}\right]$$

(41)

with

$$\begin{array}{cc}\hfill {\mathsf{\Lambda }}_{1j}=& {\displaystyle \frac{\partial {\omega }_{jd}}{\partial {\eta }_z}}{\dot{\eta }}_z+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial {\eta }_{\theta }}}{\dot{\eta }}_{\theta }+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial z_e}}{s}_d\dot{s}+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial {\theta }_e}}({\displaystyle \frac{y_e}{z_e^2}}{x}_d^{\prime }\dot{s}-{\displaystyle \frac{x_e}{z_e^2}}{y}_d^{\prime }\dot{s})+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial x_d^{\prime }}}{\dot{x}}_d^{\prime }+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial y_d^{\prime }}}{\dot{y}}_d^{\prime }+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial {\mathsf{{\rm Y}}}_1}}{\dot{\mathsf{{\rm Y}}}}_1\hfill \\ \hfill & +{\displaystyle \frac{\partial {\omega }_{jd}}{\partial x_e}}{x}_d^{\prime }\dot{s}+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial y_e}}{y}_d^{\prime }\dot{s}+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial \dot{s}}}\ddot{s}+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial {\mathsf{{\rm Y}}}_2}}{\dot{\mathsf{{\rm Y}}}}_2+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial {\widehat{\xi }}_1}}{\dot{\widehat{\xi }}}_1+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial {\widehat{\xi }}_2}}{\dot{\widehat{\xi }}}_2+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial {\widehat{\xi }}_3}}{\dot{\widehat{\xi }}}_3\hfill \\ \hfill {\mathsf{\Lambda }}_{2j}& ={\displaystyle \frac{\partial {\omega }_{jd}}{\partial z_e}}(-cos({\theta }_e){\tau }_1)+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial {\theta }_e}}({\displaystyle \frac{y_e}{z_e^2}}cos(\theta )-{\displaystyle \frac{x_e}{z_e^2}}sin(\theta )){\tau }_1-{\displaystyle \frac{\partial {\omega }_{jd}}{\partial x_e}}{\tau }_{1}cos(\theta )-{\displaystyle \frac{\partial {\omega }_{jd}}{\partial y_e}}{\tau }_{1}sin(\theta )\hfill \\ \hfill {\mathsf{\Lambda }}_{3j}& ={\displaystyle \frac{\partial {\omega }_{jd}}{\partial {\theta }_e}}{\tau }_2+{\displaystyle \frac{\partial {\omega }_{jd}}{\partial \theta }}{\tau }_2\hfill \end{array}$$

(42)

where $j=1,2$ and $s_d={\displaystyle \frac{x_e}{z_e}}{x}_d^{\prime }+{\displaystyle \frac{y_e}{z_e}}{y}_d^{\prime }$. We design the virtual control input $\mu$ as

$$\mu =-K\tilde{\omega }-\mathsf{\Pi }-C\omega -\mathsf{\Phi }\widehat{\mathsf{\Xi }}-{\displaystyle \frac{1}{2}}\Delta$$

(43)

where $\mu ={[{\mu }_1,{\mu }_2]}^T$, $K>0$ is a positive definite constant matrix,

$$\Delta =\left[\begin{array}{c}-{\eta }_{\theta }{\widehat{\xi }}_3({\displaystyle \frac{y_e}{z_e^2}}cos(\theta )-{\displaystyle \frac{x_e}{z_e^2}}sin(\theta ))-{\eta }_{z}cos({\theta }_e)+{\eta }_{\theta }\\ -{\eta }_{\theta }{\widehat{\xi }}_3({\displaystyle \frac{y_e}{z_e^2}}cos(\theta )-{\displaystyle \frac{x_e}{z_e^2}}sin(\theta ))-{\eta }_{z}cos({\theta }_e)-{\eta }_{\theta }\end{array}\right].$$

(44)

The actual control input can then be obtained from (25)

$$u_i(t)={\displaystyle \frac{-K\tilde{\omega }-\mathsf{\Pi }-C\omega -\mathsf{\Phi }\widehat{\mathsf{\Xi }}-{\varrho }_{i,2}(t)n_i}{1+{\varrho }_{i,1}(t)m_i}}-{\displaystyle \frac{\Delta }{2+2{\varrho }_{i,1}(t)m_i}}.$$

(45)

$\widehat{\mathsf{\Xi }}$ is the estimate of $\mathsf{\Xi }$, which is updated by $\dot{\widehat{\mathsf{\Xi }}}=\mathsf{\Gamma }{\mathsf{\Phi }}^T\tilde{\omega }$, with $\mathsf{\Gamma }$ being a positive definite matrix.

The control structure diagram of this paper is shown in Figure 3. The main results of the paper are now formally stated in the following theorem.

Theorem 1.

For the nonholonomic mobile robot (8)–(9), applying the control law (45) with parameter update laws ensures that all the signals in the closed-loop system remain bounded, and the tracking errors $z_e$ and ${\theta }_e$ converge to the prescribed regions (21) within a bounded setting time. Moreover, the Zeno phenomenon can be avoided.

Proof. 

Define the Lyapunov function for the entire system as

$$V_2=V_1+{\displaystyle \frac{1}{2}}\left({\tilde{\omega }}^{T}M\tilde{\omega }+{\tilde{\mathsf{\Xi }}}^T{\mathsf{\Gamma }}^{-1}\tilde{\mathsf{\Xi }}\right)$$

(46)

where $V_1$ is defined in (33) and $\tilde{\mathsf{\Xi }}=\mathsf{\Xi }-\widehat{\mathsf{\Xi }}$. Differentiating $V_2$ yields

$${\dot{V}}_2=-{\dot{V}}_z-{\dot{V}}_{\theta }-{\tilde{\omega }}^T(K+D)\tilde{\omega }+\tilde{\omega }\vartheta n_i$$

(47)

where $\vartheta ={\displaystyle \frac{{\varrho }_{i,2}(t)}{1+{\varrho }_{i,1}(t)m_i}}$. Then, we can obtain

$$\begin{array}{cc}\hfill {\dot{V}}_2\le & -K_{1,1}{\left({\displaystyle \frac{1}{2}}{\eta }_z^2\right)}^{{\displaystyle \frac{3}{4}}}-K_{2,1}{\left({\displaystyle \frac{1}{2}}{z}_e^2\right)}^{{\displaystyle \frac{3}{4}}}-K_{1,2}{\left({\displaystyle \frac{1}{2}}{\eta }_z^2\right)}^2-K_{2,2}{\left({\displaystyle \frac{1}{2}}{z}_e^2\right)}^2\hfill \\ & -K_{1,1}{\left({\displaystyle \frac{1}{2}}{\eta }_{\theta }^2\right)}^{{\displaystyle \frac{3}{4}}}-K_{2,1}{\left({\displaystyle \frac{1}{2}}{\theta }_e^2\right)}^{{\displaystyle \frac{3}{4}}}-K_{1,2}{\left({\displaystyle \frac{1}{2}}{\eta }_{\theta }^2\right)}^2-K_{2,2}{\left({\displaystyle \frac{1}{2}}{\theta }_e^2\right)}^2\hfill \\ & -{\sigma }_1\sum _{j=1}^3{\left({\displaystyle \frac{1}{2n_{{\mathsf{\Theta }}_0}}}{\tilde{\xi }}_l^2\right)}^{{\displaystyle \frac{3}{4}}}-{\sigma }_2\sum _{j=1}^3\left(4-9{\epsilon }^{{\displaystyle \frac{4}{3}}}\right){\left({\displaystyle \frac{1}{2n_{{\mathsf{\Theta }}_0}}}{\tilde{\xi }}_l^2\right)}^2+\overline{C}\hfill \end{array}$$

(48)

where $\overline{C}={\displaystyle \frac{{\sigma }_1^4}{4{\sigma }^3}}+{\displaystyle \frac{1}{2}}({\displaystyle \frac{{\varrho }_{i,2}(t)n_i}{1+{\varrho }_{i,1}(t)m_i}}{)}^2$. Note that ${\sigma }_1,{\sigma }_2\ge 0$ are terms related to the fixed-time convergence rate analysis, not control design parameters. When a larger value of ${\sigma }_1$ is desirable, to prevent the residual term ${\displaystyle \frac{{\sigma }_1^4}{4{\sigma }^3}}$ from increasing excessively, a larger control design parameter $\sigma$. $l=1,2,3$, $K_{2,1}$, $K_{2,2}$ can be chosen.

From (47) and the definition of $V_2$ in (46) along with (33), the boundedness of ${\eta }_z$, ${\eta }_{\theta }$, $\tilde{\omega }$, ${\tilde{\xi }}_1$, ${\tilde{\xi }}_2$, ${\tilde{\xi }}_3$, and $\tilde{\mathsf{\Xi }}$ can be obtained. From (21), $z_e$ and ${\theta }_e$ are bounded. Thus, $x_e$ and $y_e$ are also bounded. From (36), we obtain that ${\tau }_{1d}$ and ${\tau }_{2d}$ are bounded. Because (21) is satisfied for all $t\ge 0$. Combining the definitions of ${\omega }_d$, ${\omega }_{1d}$, and ${\omega }_{2d}$ with the boundedness of ${\tau }_{1d}$ and ${\tau }_{2d}$, it follows that ${\omega }_d$ is bounded. Thus, $\omega ={\omega }_d+\tilde{\omega }$ is also bounded. The boundedness of ${\tau }_1$ and ${\tau }_2$ is then ensured. From (37), we conclude that ${\dot{\widehat{\xi }}}_1$, ${\dot{\widehat{\xi }}}_2$, and ${\dot{\widehat{\xi }}}_3$ are bounded. From (45) and (44), we obtain that control input $\mu$ and $\dot{\widehat{\mathsf{\Xi }}}$ are also bounded. Thus, $\dot{\omega }$ is bounded.

It can be concluded that the convergence time of the system (8) is fixed. According to Lemmas 1–3, to ensure fixed-time convergence of the system, when $V_2^2\ge {\displaystyle \frac{3\overline{C}}{\varrho {\gamma }_2}}$, the value range of the designed parameter $\varrho$ is (0,1). We have $\overline{C}\le {\displaystyle \frac{\varrho {\gamma }_2}{3}}{V}_2^2$. Then, it holds that

$$\begin{array}{cc}\hfill {\dot{V}}_2& \le -{\gamma }_1{V}_2^{{\displaystyle \frac{3}{4}}}-{\displaystyle \frac{{\gamma }_2}{3}}{V}_2^2+\overline{C}\hfill \\ & \le -{\gamma }_1{V}_2^{{\displaystyle \frac{3}{4}}}-(1-\varrho ){\displaystyle \frac{{\gamma }_2}{3}}{V}_2^2.\hfill \end{array}$$

(49)

By Lemma 1, $V_2$ will converge to the set $\left\{V_2:V_2\le \sqrt{{\displaystyle \frac{3\overline{C}}{\varrho {\gamma }_2}}}\right\}$ in fixed time, with the guaranteed convergence time estimated as

$$T_{\mathrm{fd}}\le T_{\max }={\displaystyle \frac{4}{{\gamma }_1}}+{\displaystyle \frac{3}{{\gamma }_2(1-\varrho )}}$$

(50)

where $\begin{array}{ccc}& {\gamma }_1\triangleq min\{K_{1,1},K_{1,2},{\sigma }_1\},\hfill & \hfill {\gamma }_2\triangleq min\{K_{2,1},K_{2,2},{\sigma }_2(4-9{\epsilon }^{{\displaystyle \frac{4}{3}}})\}\end{array}$, and $\epsilon >0$ are chosen to satisfy $\epsilon <{\left({\displaystyle \frac{2}{3}}\right)}^{{\displaystyle \frac{3}{2}}}$.

Furthermore, $\forall t\in [t_k,t_{k+1})$, one can obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}|{\mu }_i(t)-u_i(t)|& ={\displaystyle \frac{d}{dt}}{(({\mu }_i(t)-u_i(t))({\mu }_i(t)-u_i(t)))}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & =\mathrm{sign}({\mu }_i(t)-u_i(t))({\dot{\mu }}_i(t)-{\dot{u}}_i(t))\le |{\dot{\mu }}_i(t)|\hfill \end{array}$$

(51)

where $i=1,2$, and ${\mu }_i(t)$ is bounded and continuously differentiable with $|{\dot{\mu }}_i(t)|\le {\chi }_i$. Recalling ${\mu }_i(t_k)-u_i(t_k)=0$ and ${lim}_{t\to t_{k+1}}{\mu }_i(t)-u_i(t)={\mathcal{H}}_i,$ there exists a positive scalar $t^{★}$ such that $(t_{k+1}-t_k)\ge t^{★}\ge u_i/{\chi }_i$. This implies the existence of a minimum interval between triggering times, ensuring that Zeno behavior is effectively avoided in the proposed control scheme. This completes the proof. □

Remark 2.

This paper presents a control approach that aims to enable a mobile robot to follow a reference path, where the path is assumed to be generated by a path-planning algorithm. Various path-planning methods exist for mobile robots operating in environments with scattered obstacles, such as [42], where approaches are developed to find an optimal robust traversable path consistent with the physical constraints of the mobile robot. Combining our control method proposed with the path-planning strategies in [42] is a promising direction for future research.

Remark 3.

In recent work [43], a fixed-time adaptive guaranteed performance tracking control strategy was developed for nonholonomic mobile robots with asymmetric state constraints. The robot model considered in that study is analogous to (8) and (9). In addition, an event-triggered mechanism with a relative threshold was employed to conserve communication bandwidth between the controller and the actuator. Compared to [43], this paper introduces two key distinctions. First, while [43] addresses a trajectory tracking problem where the robot follows a desired trajectory generated by a virtual robot, this paper focuses on path following, where the robot adheres to a predefined path not necessarily derived from a virtual model. Second, instead of the asymmetric barrier Lyapunov function used in [43], this paper applies a novel universal asymmetric barrier function. This function has broader applicability and imposes lower demands on the control signal when the error is large, as discussed in Remarks 5 and 6 of [35]. Furthermore, differences in the controller structure arise from the distinct event-triggered mechanisms, barrier functions, and stability lemmas used in the two works.

Remark 4.

The method proposed in this paper shows potential for practical application. In scenarios where path-tracking accuracy is critical, such as in autonomous vehicle navigation or warehouse automation, ensuring performance within strict bounds can lead to safer and more reliable operations. In addition, by incorporating an event-triggered mechanism, control signals are only updated when the robot deviates from the path or is at risk of collision, thereby reducing unnecessary computation, conserving communication resources, and improving overall system efficiency. This is especially beneficial for logistics tasks, such as efficient warehouse retrieval and sorting. Moreover, by defining bounds for tracking error, the method improves the robustness of control response, which is particularly valuable in environments with varying or unknown dynamics.

Remark 5.

In [2], a framework was developed to address the control problem for nonholonomic mobile robots with constrained velocity and torque. The proposed controller ensures the convergence of tracking or stabilization errors and maintains the boundedness of closed-loop signals. However, the convergence rate is only “asymptotic”. In contrast, our proposed method enables the tracking error of the closed-loop system to converge to a small neighborhood around zero within a fixed time. Furthermore, [2] does not account for maximum overshoot, whereas our approach ensures predefined transient performance in the closed-loop system even in the presence of unknown system parameters.

Remark 6.

During the parameter selection process, if the goal is only to ensure system stability, most control parameters simply need to remain positive. For specific parameters, such as those defined in (20), the system’s desired output error performance—characteristics like maximum overshoot and steady-state error—can be achieved by appropriately tuning the performance function parameters. Specifically, reducing ${\phi }_{\infty }$ effectively minimizes the steady-state error. In addition, as demonstrated in the earlier analysis, the upper bound for the system’s convergence time can be determined a priori and is independent of the system’s initial conditions. By adjusting the values of ${\gamma }_1$, ${\gamma }_2$, and ϱ, different upper bounds for convergence time can be achieved. To accelerate the system’s convergence, increasing ${\gamma }_1$ and ${\gamma }_2$ while decreasing ϱ proves effective. However, excessively fast convergence may demand a higher control input. Consequently, finding an appropriate balance between convergence speed and control input magnitude is crucial for achieving the desired control performance.

4. Simulations

In this section, we present simulations to validate the effectiveness of the proposed control strategy. The mobile robot is modeled using (8) and (9). The physical parameters of the robot in (10) and (11) are provided in

Table 1. Robot physical parameters [36,43].

Parameter	R	$\overline{d}$	r	$m_c$	$m_{\omega }$
Value	0.75	0.3	0.15	30	1
Parameter	$l_c$	$l_w$	$l_m$	$d_1$	$d_2$
Value	15.625	0.005	0.0025	5	5

. Note that all these parameters are not utilized in the controller implementation, and the simulation does not include disturbances or measurement noise.

The reference trajectory is $q_d={[s,10cos(0.1s)]}^T$. The initial values of ${\widehat{\xi }}_1,{\widehat{\xi }}_2,{\widehat{\xi }}_3$, and $\widehat{\mathsf{\Xi }}$ are selected to be $0.6$ of their true values. Initial system states are set as $x(0)=0.3$, $y(0)=6.5$, and $\theta (0)=2.9$. We define the prescribed performance function as follows: ${\rho }_H(t)=0.2+3.8e^{-t}$, ${\rho }_L(t)=0.01+0.19e^{-t}$, ${\kappa }_H(t)=0.07+13.3e^{-t}$, and ${\kappa }_L(t)=0.07+13.3e^{-t}$. In this work, we set the intermediate variable $\alpha (t)=0.1+1.9e^{-t}$. Other design parameters are chosen as $n_{{\mathsf{\Theta }}_0}=2$, ${\sigma }_1={\sigma }_2=1$, $K_{1,1}=K_{1,2}=0.1$, $K_{2,1}=K_{2,2}=1$, $K=5I$, $p_1=1$, $p_2=0.2$, and $p_3=2$. The simulation results are presented in Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8. Despite the robot being subject to nonholonomic constraints and initially experiencing a relatively large tracking error, Figure 4 demonstrates that, under the proposed adaptive control torque, the robot successfully converges to the reference trajectory. Figure 5 and Figure 6 show the trajectories of the position error $z_e$ and the orientation error ${\theta }_e$, respectively. It is clear that both the position and orientation tracking errors rapidly converge and remain within their respective performance bounds, i.e., ${\rho }_L(t)<z_e(t)<{\rho }_H(t)$ and $-{\kappa }_L(t)<{\theta }_e(t)<$${\kappa }_H(t)$ for all $t\ge 0$. Figure 7 and Figure 8 depict the triggering moments and the release intervals of the controller under the event-triggered mechanism, effectively reducing communication resources.

In this example, we have selected five additional sets of event-triggered parameter values for the controller $u_1$ to evaluate the impact of the event-triggered mechanism. The event-triggered parameters are provided in

Table 2. Event-triggered parameter.

Parameters	$\mathit{p}1$	$\mathit{p}2$	$\mathit{p}3$	$\mathit{p}4$	$\mathit{p}5$
$m_i$	0.005	0.003	0.001	0.001	0.001
$n_i$	0.001	0.001	0.005	0.05	0.1

.

Figure 9 illustrates the relationship between the variables $m_i$ and $n_i$ and the trigger threshold. As $m_i$ and $n_i$ increase, the number of trigger events decreases, resulting in longer intervals between them. Conversely, triggering events more frequently over short intervals enhances the system’s control precision.

In addition, we select four different sets of initial values for comparison, as shown in

Table 3. Robot initial conditions.

Case	I	II	III	IV
$x(0)$	−1.8	0.6	−3.0	−2.3
$y(0)$	8.1	9.2	10.2	7.2
$\theta (0)$	−2.2	1.6	1.3	0.5

. The simulation results are illustrated in Figure 10.

It can be seen that, under different initial conditions, all the robots can track their reference trajectories effectively. This means that the system’s convergence is not affected by different initial conditions.

5. Conclusions and Future Work

In this paper, we proposed a fixed-time adaptive event-triggered control method with prescribed performance to address the asymmetric constraint problem of nonholonomic mobile robots. Theoretical analysis and simulations demonstrated that the proposed controller achieves accurate path-tracking while ensuring all closed-loop signals remain bounded. Additionally, the incorporation of an event-triggered control strategy significantly reduces the computational load and preserves communication bandwidth between the controller and actuator while eliminating Zeno behavior. This paper does not consider the physical constraints of the actuators in the controller design, which may degrade the performance of the closed-loop system in practical applications. Therefore, one direction for future work is to develop controllers that achieve fixed-time adaptive event-triggered control for nonholonomic mobile robots with input constraints. In addition, this paper only focuses on the control problem of a single robot, assuming direct access to the reference trajectory. Future research will explore distributed formation control for multiple mobile robots, where only a subset of the robots has access to the reference trajectory.