Exponential Trajectory Tracking Control of Nonholonomic Wheeled Mobile Robots

Abstract

Trajectory tracking control is important in order to realize autonomous driving of mobile robots. From a control standpoint, trajectory tracking can be stated as the problem of stabilizing a tracking error system that describes both position and orientation errors of the mobile robot with respect to a time-parameterized path. In this paper, we address the problem for the trajectory tracking of nonholonomic wheeled mobile robots, and an exponential trajectory tracking controller is designed. The stability analysis is concerned with studying the local exponential stability property of a cascade system, provided that two isolated subsystems are exponentially stable and under certain bound conditions for the interconnection term. A theoretical stability analysis of the dynamic behaviors of the closed-loop system is provided based on the Lyapunov stability theory, and an exponential stability result is proven. An explicit estimate of the set of feasible initial conditions for the error variables is determined. Simulation results for verification of the proposed tracking controller under different operating conditions are given. The obtained results show that the problem of trajectory tracking control of nonholonomic wheeled mobile robots is solved over a large class of reference trajectories with fast convergence and good transient performance.

1. Introduction

The control of wheeled mobile robots has been the subject of research interest for several decades [1,2,3]. The nonlinear behavior and the nonholonomic nature of these systems render its control a challenging task. Controlling a mobile robot can be broadly classified into three basic control problems: posture stabilization (parking problem), path tracking, and trajectory tracking [4]. In posture (position and orientation) stabilization, the mobile robot starting from an initial posture must reach a desired (reference) fixed posture. Stabilizing a mobile robot at a given posture leads to specific control problems. It is known that feedback point stabilization of nonholonomic systems like wheeled mobile robots cannot be achieved via smooth time-invariant control law due to the limitations imposed by Brockett’s necessary condition for feedback stabilization of such a system, and discontinuous or time-varying control strategies must be applied [5,6]. The path following consists of driving the mobile robot on a predefined path without time and speed requirements, i.e., independently from time and robot speed [7]. In contrast to path following, trajectory tracking consists of steering the mobile robot on a desired time-parameterized path. This is equivalent to tracking the position and orientation of a moving virtual reference mobile robot, which means that the robot must be at a specified position and orientation at each given moment. An important issue in controlling nonholonomic wheeled mobile robots arises from the fact that the number of configuration variables is greater than the number of control inputs.

The trajectory tracking control design for nonholonomic wheeled mobile robots has attracted considerable research attention, and various trajectory tracking controllers have been proposed in the literature using different control design techniques, with each of them having their strengths and weaknesses [8]. In [9,10,11,12], trajectory tracking controllers for wheeled mobile robots were designed using Lyapunov-based control techniques providing local asymptotic results. A hybrid control strategy combining mode-based control using Lyapunov design techniques and a deep reinforcement learning method for tracking control, which achieves local asymptotic stability of the error dynamics, was proposed in [13]. In order to improve the tracking accuracy, the kinematic control is serving as the ‘given control’ and the deep deterministic policy gradient method is used to learn an ‘acquired’ control to compensate for the existing errors. In [14], based on chain transformation of the kinematic equations of the nonholonomic mobile robots and the reference trajectory, a distributed control was constructed, in which components serving as dynamic oscillators were designed using the Lyapunov stability theory, providing local asymptotic convergence of the tracking errors. However, the controller relies on a persistent excitation condition upon the reference angular velocity, which limits the range of trajectories that can be tracked, such as straightline trajectories. In [15], a trajectory tracking controller was developed combining sliding mode and recursive backstepping techniques. Again, to achieve asymptotic tracking, this controller requires the persistent excitation of the reference angular velocity, and therefore, rectilinear reference trajectories are excluded. Sliding mode-based trajectory tracking controllers for unicycle mobile robots, ensuring asymptotic convergence in the presence of bounded disturbances at the kinematic level, were proposed in [16,17]. An adaptive neural controller based on a sliding mode control for robust mobile robot trajectory tracking with unknown dynamics, which achieves a global asymptotic result, was proposed in [18]. In [19], an approximated nonlinear model predictive control was proposed to solve the trajectory tracking problem for a differential drive mobile robot in a constrained region. In order to avoid the computational burden and to reduce the processing time of the nonlinear model predictive control, Euler approximation is used to transform the nonlinear optimization problem of nonlinear model predictive control into a constrained quadratic optimization problem. Local asymptotic stability was achieved with a good tracking performance. In [20], a trajectory tracking controls for nonholonomic mobile robots, where the position and orientation of the real and virtual mobile robots are measurable with respect to a global (fixed) coordinate system, which yields local asymptotic results, was proposed. A neural network-based adaptive controller for the trajectory tracking of wheeled mobile robots, where the controller gains are adaptively determined online using neural networks, was presented in [21]. In [22], a finite-time trajectory tracking controller was designed, where the finite-time stability theory and backstepping techniques were used for the control design. The desired trajectory is tracked within a finite time, providing globally asymptotical stability of the closed-loop tracking error system.

The cascade approach has been successfully applied in solving the trajectory tracking problem for nonholonomic mobile robots. Generally, it consists of a controller design, so that the resulting closed-loop system has a cascaded structure. Finite-time tracking controllers by using a cascaded control design that yields global asymptotic stability results were proposed in [23,24]. However, a specific feature of the control schemes is the nonzero (persistent) condition for the reference angular velocity, which excludes straight line trajectory tracking. In [25], a Lyapunov-based trajectory tracking controller, which achieves a global asymptotic result, was proposed. Appropriate Lyapunov functions were constructed for each of the subsystems of the cascaded error system and were firstly used for control design purposes and, after that, in the stability analysis stage. A tracking controller for a wheeled mobile robot using a cascaded control design, which guarantees global asymptotic tracking, was proposed in [26]. However, the persistent excitation of the reference angular velocity limits the tracking to curvilinear trajectories only. A cascaded-based trajectory tracking controller of wheeled mobile robots using only Cartesian position measurements, which yields local asymptotic results, was presented in [27]. A high-gain observer was designed to compensate for the lack of measurements for the reference linear velocity and orientation. Recently, in [28], based on a cascade system paradigm, a homogeneous controller was designed to solve the trajectory tracking problem for wheeled mobile robots, which achieves global asymptotic and finite-time stabilization of the tracking error dynamics in the absence of perturbations.

In this paper, we obtain an exponential solution to the trajectory tracking problem for nonholonomic unicycle-type mobile robots using a cascade-based control design. The motivation for using a cascaded system approach is determined by the specific structure of the system equations describing the tracking error dynamics, which could be broadly divided into position and orientation subsystems. The proposed controller uses relative position, orientation, and velocity information of the virtual robot, which generates the reference trajectory, for feedback control. It does not rely on the persistent excitation condition on the reference angular velocity, and a larger range of trajectories, including straight line trajectories, can also be tracked. The controller yields exponential tracking of a reference trajectory.

The main contributions of this paper are summarized as follows:

−

An exponential trajectory tracking controller for nonholonomic unicycle-type mobile robots is designed.

−

Theoretical stability analysis of the dynamic behaviors of the closed-loop system is provided, and an exponential stability result is proven.

−

An explicit estimate of the set of feasible initial conditions for the error variables is determined.

−

Verification of the proposed tracking controller under different operating conditions using MATLAB/Simulink is given.

The organization of the paper is as follows: Preliminaries and problem formulation is first given in Section 2. In Section 3, the main results: the tracking controller design and the stability analysis are given. In Section 4, the effectiveness of the proposed controller is demonstrated by simulations. Section 5 presents the conclusions.

2. Preliminaries and Problem Formulation

2.1. Preliminaries

In this section, we recall some lemmas that will be used to support the stability analysis of the proposed controller.

Lemma 1 

([29] Lemma 9.4). Consider the system

$$\dot{x}=f\left(t,x\right)+g\left(t,x\right),$$

(1)

where $f:[0,\infty )\times D\to R^n$ and g$:[0,\infty )\times D\to R^n$ are piecewise continuous in t and locally Lipshitz in x on $[0,\infty )\times D$, and $D\subset R^n$ is a domain that contains the origin $x=0$. The system (1) can be considered as a perturbation of the nominal system

$$\dot{x}=f\left(t,x\right).$$

(2)

Let x = 0 be an exponentially stable equilibrium point of the nominal system (2). Let V(t, x) be a Lyapunov function of the nominal system that satisfies

$$c_{1 }{‖x‖}^{2 }\le V(t,x)\le c_{1 }{‖x‖}^{2 },$$

(3)

$${\displaystyle \frac{\partial V}{\partial t}}+{\displaystyle \frac{\partial V}{\partial x}}f\left(t,x\right)\le -c_{3 }{‖x‖}^{2 }$$

(4)

$$‖{\displaystyle \frac{\partial V}{\partial x}}‖\le c_4 ‖x‖$$

(5)

for all $\left(t,x\right)\in [0, \infty )\times D$, where $D=\left\{x\in R^n |{‖x‖}_2<r\right\}$, for some positive constants $c_1$, $c_2$, $c_3$, and $c_4$. Suppose the perturbation term g(t, x) satisfies the bound

$$‖g(t,x)‖\le \gamma \left(t\right)‖x‖+\delta \left(t\right), \forall t\ge 0, \forall x\in D,$$

(6)

where $\gamma : R\to R$ is nonnegative and continuous for all $t\ge 0$, and satisfies

$$\underset{t_0}{\overset{t}{\int }}\gamma \left(\tau \right)d\tau \le \epsilon \left(t-t_0\right)+\eta ,$$

(7)

$$\epsilon <{\displaystyle \frac{c_1{c}_3}{c_2{c}_4}},$$

(8)

and $\delta : R\to R$ is nonnegative, continuous, and bounded for all $t\ge 0$. Then, provided that x(t₀) satisfies

$$‖x\left(t_0\right)‖<{\displaystyle \frac{r}{\rho }} \sqrt{{\displaystyle \frac{c_1}{c_2}}},$$

(9)

and $\underset{t\ge t_0}{\sup }\delta \left(t\right)$ satisfies

$$\underset{t\ge t_0}{\sup }\delta \left(t\right)<{\displaystyle \frac{{2c}_1\alpha r}{c_4\rho }},$$

(10)

the solution of the perturbed system (1) satisfies

$$‖x\left(t\right)‖\le \sqrt{{\displaystyle \frac{c_2}{c_1}}}{\rho ‖x\left(t_0\right)‖e}^{-\alpha (t-t_0)}+{\displaystyle \frac{c_1\rho }{2c_1}}\underset{t_o}{\overset{t}{\int }}{e}^{-\alpha \left(t-\tau \right)}\delta \left(\tau \right)d\tau ,$$

(11)

where

$$\begin{array}{ccc}\alpha ={\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{c_3}{c_2}}-\epsilon {\displaystyle \frac{c_4}{c_1}}\right]>0& & \rho =exp\left({\displaystyle \frac{c_4\eta }{2c_1}}\right)\end{array}\ge 1.$$

(12)

Lemma 2 

([29] Lemma 9.5). Case 1: If

$$\underset{0}{\overset{\infty }{\int }}\gamma \left(\tau \right)d\tau \le k,$$

(13)

then (7) is satisfied with ε = 0 and η = k.

Lemma 3 

([29] Lemma 9.6). Case 3: Suppose the conditions of Lemma 1 are satisfied and let x(t) denote the solutions of the perturbed system (1). If $\underset{t\to \infty }{\mathit{lim}}\delta \left(t\right)=0$, then $\underset{t\to \infty }{\mathit{lim}}x\left(t\right)=0$.

2.2. Problem Formulation

In this paper, we consider the problem of tracking a reference trajectory for nonholonomic unicycle-type wheeled mobile robots. This problem can also be formulated as tracking in position and orientation of a moving virtual reference mobile robot, which generates a desired trajectory for the actual robot to follow.

Consider the mobile robot shown in Figure 1. The kinematic scheme consists of a platform with two independently driving wheels. The wheels are assumed to roll without slipping and lateral sliding.

The following coordinate frames are assigned: an inertial (global) coordinate system Fxy in the plane of motion, a moving body-fixed coordinate system Pxy which has its origin placed at the mid-point P between the robot driving wheels, such that the x-axis is in the direction of the longitudinal robot base, and a moving reference coordinate system R$x_{R }{y}_R$ rigidly attached to the virtual reference robot (Figure 1). The coordinates of guide point P of the mobile robot with respect to the Fxy are denoted by (x, y). The angle θ is the orientation angle of the robot with respect to Fxy. The configuration of the system is described by the following vector of generalized coordinates $q={\left[\begin{array}{ccc}x& y& \theta \end{array}\right]}^T\in {\mathfrak{R}}^3$.

By the assumption of pure rolling without slipping and a lateral sliding condition of the wheels, the system is characterized by the following nonholonomic constraint on the generalized velocities $\dot{q }={\left[\begin{array}{ccc}\dot{x}& \dot{y}& \dot{\theta }\end{array}\right]}^T\in {\mathfrak{R}}^3$ as follows:

$$\left[\begin{array}{ccc}-sin\theta & cos\theta & 0\end{array}\right]\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta }\end{array}\right]=0.$$

(14)

The constraint Equation (14) can be converted into an affine driftless control system of the form

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta }\end{array}\right]=\left[\begin{array}{cc}cos\theta & 0\\ sin\theta & 0\\ 0& 1\end{array}\right] \left[\begin{array}{c}v\\ \omega \end{array}\right],$$

(15)

where v and ω are the mobile robot linear and angular velocities, respectively. Equation (15) describes the motion of the nonholonomic mobile robot in the plane.

The mobile robot cannot track an arbitrary trajectory, which is due to the nonholonomic constrain (14). For formulating the trajectory tracking problem, a feasible reference trajectory $q_r\left(t\right)={\left[\begin{array}{ccc}{x}_r\left(t\right)& y_r\left(t\right)& {\theta }_r\end{array}\left(t\right)\right]}^T$, which has to be tracked by the real mobile robot, is generated by a virtual nonholonomic mobile robot based on the same kinematic model (15) of the real robot in the form

$$\left[\begin{array}{c}{\dot{x}}_r\\ {\dot{y}}_r\\ {\dot{\theta }}_r\end{array}\right]=\left[\begin{array}{cc}cos{\theta }_r& 0\\ sin{\theta }_r& 0\\ 0& 1\end{array}\right] \left[\begin{array}{c}{v}_r\\ {\omega }_r\end{array}\right],$$

(16)

where $v_r$ and ${\omega }_r$ are the linear and angular velocities of the virtual reference mobile robot.

The coordinates and the orientation of the mobile robot with respect to the coordinate frame $R{x}_{R }{y}_R$ of the virtual reference robot define the tracking errors $\left(e_x\left(t\right), e_y\left(t\right),e_{\theta } \left(t\right)\right)$, i.e., its coordinates and orientation, with respect to a local moving coordinate system attached to the virtual mobile robot, which can be expressed as follows [30]:

$$\left[\begin{array}{c}{e}_x\\ e_y\\ e_{\theta }\end{array}\right]=\left[\begin{array}{ccc}cos{\theta }_r& sin{\theta }_r& 0\\ -sin{\theta }_r& cos{\theta }_r& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x-x_r\\ y-y_r\\ \theta -{\theta }_r\end{array}\right].$$

(17)

It is noted that the convergence of the tracking errors $\left(e_x\left(t\right), e_y\left(t\right),e_{\theta } \left(t\right)\right)$ to zero implies the same to $\left(\begin{array}{ccc}x\left(t\right)-x_r\left(t\right),& y\left(t\right)-x_r\left(t\right),& \theta \left(t\right)-{\theta }_r\left(t\right)\end{array}\right)$, since the transformation matrix is nonsingular for ${\theta }_r\in R$.

Differentiating (17) with respect to time and taking into account (15) and (16), after some work, we obtain the tracking error dynamics, described as

$$\begin{array}{c}{\dot{e}}_x=-v_r+{\omega }_r{e}_y+vcos{e}_{\theta }\hfill \\ {\dot{e}}_y=-{\omega }_r{e}_x+vsin{e}_{\theta }\hfill \\ {\dot{e}}_{\theta }=-{\omega }_r+\omega \hfill \end{array}.$$

(18)

The linear velocity v and angular velocity ω of the mobile robot are considered the control inputs of the system (18).

Assumption 1.

The reference linear and angular velocities of the virtual mobile robot are known, continuously differentiable and bounded, and satisfy ${0<v}_{rmin} \le v_r \le v_{rmax}$ and $-{\omega }_{rmin} \le {\omega }_r \le {\omega }_{rmax}$.

Assumption 2.

The tracking errors $\left(e_x\left(t\right), e_y\left(t\right),e_{\theta } \left(t\right)\right)$ are measurable.

The control objective can then be formulated to stabilize at zero the position and orientation tracking errors $\left(e_x\left(t\right), e_y\left(t\right),e_{\theta } \left(t\right)\right)$ with respect to the target position and orientation of the virtual reference robot.

3. Control Design and Stability Analysis

3.1. Control Design

In this section, we propose a nonlinear feedback trajectory tracking control for the linear and angular velocity (v, ω) of the mobile robot such that it can track a reference trajectory $\left(x_r\left(t\right),y_r\left(t\right), {\theta }_r\left(t\right)\right)$ generated by a virtual reference mobile robot.

The control design strategy that is used in this paper is to decompose the tracking error system (18) into two subsystems: the position subsystem and orientation subsystem and design the control laws for each of them separately.

For the position subsystem composed from the first and the second equation of (18), we design a control law for linear velocity v in the form

$$v^{\ast }=\sqrt{{\left({\nu }_1-a_1\right)}^2+{\left({\nu }_2-a_2\right)}^2},$$

(19)

and an auxiliary variable $\alpha$ is introduced, determined by

$$e_{\theta }^{\ast }=arctan\left({\displaystyle \frac{{\nu }_2-a_2}{{\nu }_1-a_1}}\right)≔\alpha ,$$

(20)

where ${\nu }_1=-k_e{e}_x$, ${\nu }_2=-k_e{e}_y$, $a_1≔-v_r+{\omega }_r{e}_y,$ and $a_2≔-{\omega }_r{e}_x$, and $k_e$ is a positive gain.

Remark 1.

It should be noted that the arctan function in (21) is bounded by $\mp \pi /2$. However, there will be a discontinuity in $e_{\theta }^{\ast }$ in the case when the denominator in (21) given by ${\nu }_1-a_1{=v}_r-k_e{e}_x-{\omega }_r{e}_{y }$ passes through zero. This discontinuity caused by division by zero is of type “jump”, but $e_{\theta }^{\ast }$ will stay finite everywhere. However, the controller to be designed will ensure that the denominator in (21) satisfies the condition ${\nu }_1-a_1>0$ for feasible initial conditions for ($e_{x }\left(t\right)$, ${ e}_{y }\left(t\right)$), i.e., initial conditions on a known bounded domain, for which this condition holds. Therefore, we first assume in the following that ${\nu }_1-a_1>0$. In Section 3.2.4, the nonzero lower bound of ${(\nu }_1-a_1)$, in order to prevent discontinuity of $e_{\theta }^{\ast }$, will be analyzed, and a bounded domain with initial conditions is obtained.

Using (19) and (20), the position error subsystem is obtained as follows:

$$\begin{array}{c}{\dot{e}}_x=-k_e{e}_x+v^{\ast }\left(\cos e_{\theta }-cos\alpha \right)\hfill \\ {\dot{e}}_y=-k_e{e}_y+v^{\ast }\left(sin{e}_{\theta }-sin\alpha \right).\hfill \end{array}$$

(21)

We introduce an error variable $\mu ={\displaystyle \frac{e_{\theta }-\alpha }{2}}$. Then, based on (19)–(21) and using standard trigonometric identities, the system (18) can be rewritten in the new coordinates ($e_x, e_y, \mu )$ in the form

$$\begin{array}{c}{\dot{e}}_x=-k_e{e}_x-2v^{\ast }sin\left(\mu +\alpha \right)sin\mu \hfill \\ {\dot{e}}_y=-k_e{e}_y+2v^{\ast }cos\left(\mu +\alpha \right)sin\mu \hfill \\ \dot{\mu }={\displaystyle \frac{1}{2}}\left[\left(\omega -{\omega }_r\right)-\dot{\alpha }\right].\hfill \end{array}$$

(22)

For the orientation subsystem given by the third equation of (22), the following feedback control law for ω is proposed:

$${\omega }^{\ast }={\omega }_r+\dot{\alpha }-2k_{\mu }\mu ,$$

(23)

where $k_{\mu }$ is a positive gain.

With the control (19) and (23), the resulting closed-loop system in the new coordinates is obtained as follows:

$$\begin{array}{c}{\dot{e}}_x=-k_e{e}_x-2v^{\ast }sin\left(\mu +\alpha \right)sin\mu \hfill \\ {\dot{e}}_y=-k_e{e}_y+2v^{\ast }cos\left(\mu +\alpha \right)sin\mu \hfill \\ \dot{\mu }=-k_{\mu }\mu .\hfill \end{array}$$

(24)

Denoting $e={\left[e_{x },e_y\right]}^T\in {\mathfrak{R}}^2$ and $A=diag\left({-k}_e,{-k}_e\right)\in {\mathfrak{R}}^{2\times 2}$, the cascaded system (24) can be presented in the form

$$\dot{e}=Ae+g\left(t,e,\mu \right)$$

(25)

$$\dot{\mu }=-k_{\mu }\mu ,$$

(26)

where

$$g\left(t,e,\mu \right)=\left[\begin{array}{c}{g}_1\\ g_2\end{array}\right]=\left[\begin{array}{c}-2v^{\ast }sin\left(\mu +\alpha \right)sin\mu \\ 2v^{\ast }cos\left(\mu +\alpha \right)sin\mu \end{array}\right]$$

(27)

is the interconnection term.

The point $x={\left[\begin{array}{cc}e& \mu \end{array}\right]}^T=0$ is an equilibrium point of the system (25) and (26).

It is to note that, if $g(t,e,\mu )=0$, (25) reduces to

$$\dot{e}=Ae.$$

(28)

Thus, we can view (25) as a nominal system in the form of (28), which is perturbed by the output of the system (26).

3.2. Stability Analysis

We now summarize our stability result in the following proposition:

Proposition 1.

Under Assumptions 1 and 2, the closed-loop system (25) and (26) is locally exponentially stable. The tracking errors (e, μ) converge asymptotically to zero.

Proof of Proposition 1.

Using the specific structure of the closed-loop system (25) and (26), we divide the proof into three parts corresponding to the cascade decomposition of the system, and we treat the stability property of each subsystem separately. In the first part, exponential stability of the μ-subsystem (26) is proven. In the second part, exponential stability of the e-subsystem (25) is proven, where it is considered as a perturbation of a nominal linear system perturbed by the output of the μ-subsystem. Finally, exponential stability of the overall system (25) and (26) is proven. □

3.2.1. μ-Subsystem

The μ-subsystem (26) has an equilibrium point at the origin $\mu =0$. The solution of the first-order linear differential equation (26), with the initial condition $\mu \left(t_0\right)$, is given by

$$\mu \left(t\right)={\mu (t}_0)e^{-k_{\mu }\left(t-t_0\right)}.$$

(29)

The system is exponentially stable, since $\mu \left(t\right)$ converges exponentially to the origin. Hence, we will consider $\mu \left(t\right)$ as an exponentially vanishing disturbance in the subsystem (25).

The solution (29) satisfies the bound

$$\begin{array}{ccc}\left|\mu \left(t\right)\right|& =& \left|{\mu (t}_0)\right|e^{-k_{\mu }(t-t_0)}\\ & \le & \left|{\mu (t}_0)\right|. \end{array}$$

(30)

3.2.2. e-Subsystem

In order to determine the stability property of the e-subsystem, we now view (25) as a form of system (1) in Lemma 1, where the output μ(t) of the μ-subsystem takes part of the perturbation term g(t, x) in place of the disturbance signals γ(t) and δ(t). In the following, we will show that the closed-loop system (25) satisfies the conditions of Lemmas 1–3.

Before proving the stability of the e-subsystem, let us derive some necessary bounds. First, using (20), an upper bound for the linear velocity $v^{\ast }$ of the mobile robot is obtained as follows:

$$\begin{array}{ccc}{v}^{\ast }& =& \sqrt{{\left({\nu }_1-a_1\right)}^2+{\left({\nu }_2-a_2\right)}^2}\hfill \\ & =& \sqrt{v_r^{2 }+\left(k_e^2+{\omega }_r^2\right)\left(e_x^2+e_y^2\right)-2v_r\left(k_e{e}_x+{\omega }_r{e}_y\right)}\hfill \\ & \le & \sqrt{v_{rmax}^{2 }+\left(k_e^2+{\omega }_{rmax}^2\right){‖e‖}^2+2v_r\sqrt{\left(k_e^2+{\omega }_{rmax}^2\right) }‖e‖}\hfill \\ & \le & v_{rmax}+\sqrt{k_e^2+{\omega }_{rmax}^2}‖e‖.\hfill \end{array}$$

(31)

Additionally, using (31) and (30), we obtain an upper bound for the perturbation term (27), given by

$$\begin{array}{ccc}‖g(t,e,\mu )‖& =& \sqrt{{\left[-2v^{\ast }sin\left(\mu +\alpha \right)sin\mu \right]}^2+{\left[2v^{\ast }cos\left(\mu +\alpha \right)sin\mu \right]}^2}\hfill \\ & =& 2v^{\ast }\left|sin\mu \right|\hfill \\ & \le & q_2\left|\mu \right|‖e‖+q_1\left|\mu \right|,\hfill \end{array}$$

(32)

where $q_1$ and $q_2$ are positive constants determined by

$$q_1=2v_{rmax}$$

(33)

$$q_2=2\sqrt{k_e^2+{\omega }_{rmax}^2},$$

(34)

and $\left|\mu \left(t\right)\right|$ is the nonnegative bounded disturbance for all $t\ge 0$. Hence, the interconnection term (27) satisfies the linear growth condition.

Remark 2.

We note that, setting $\gamma \left(t\right)=q_2\left|\mu \right|$ and $\delta \left(t\right)=q_1\left(t\right)$, the bound for the perturbation term can be written in the form of inequality (6) in Lemma 1. However, for clarity of exposition, we will keep the notation $\left|\mu \right|$ introduced in (32) for the disturbance, in order to emphasize that it is the output of the subsystem (26).

The e-subsystem (25) has an equilibrium point at the origin $e=0$. The nominal system $\dot{e}=Ae$ is a linear time-invariant system. The eigenvalues of the diagonal matrix $A=diag\left({-k}_e,{-k}_e\right)$ are equal to $-k_e$. Hence, the matrix A is Hurwitz, and the origin $e=0$ of the nominal system (28) is exponentially stable. For any given positive definite matrix Q, there exists a positive definite matrix P that satisfies the Lyapunov equation $PA+A^{T}P=-Q$, [29] (Theorem 4.6). With $Q=diag\left(\mathrm{1,1}\right){\in \mathfrak{R}}^{2\times 2}$, the matrix $P=diag\left({\displaystyle \frac{1}{2k_e}},{\displaystyle \frac{1}{2k_e}}\right)\in {\mathfrak{R}}^{2\times 2}$ is the unique solution of the Lyapunov equation.

The quadratic Lyapunov function in the form $V\left(e\right)=e^{T}Pe$ satisfies

$${\rho }_1{‖e‖}^2\le V\left(e\right)\le {\rho }_2{‖e‖}^2$$

(35)

$${\displaystyle \frac{\partial V}{\partial e}}Ae\le -{\rho }_3{‖e‖}^2$$

(36)

$$‖{\displaystyle \frac{\partial V}{\partial e}}‖\le {\rho }_4‖e‖,$$

(37)

where ${{\rho }_1={\rho }_2=\lambda }_{min}\left(P\right)={\lambda }_{max}\left(P\right)=1/2k_e$, ${{\rho }_3=\lambda }_{min}\left(Q\right)=1$, and ${\rho }_4={2\lambda }_{max}\left(P\right)=1/k_e$. It is noted that the conditions (35)–(37) are similar to (3)–(5) for the Lyapunov function for the nominal system (2) in Lemma 1.

Using (36) and (37), the derivative of V(e) along the trajectories of subsystem (25) satisfies

$$\begin{array}{ccc}\dot{V}& =& {\displaystyle \frac{\partial V }{\partial e}}\left(Ae+g(t,e,\mu )\right)\hfill \\ & =& {\displaystyle \frac{\partial V }{\partial e}}Ae+{\displaystyle \frac{\partial V }{\partial e}}g\left(t,e,\mu \right)\hfill \\ & \le & -{\rho }_3{‖e‖}^2+‖{\displaystyle \frac{\partial V}{\partial e}}‖‖g\left(t,e,\mu \right)‖\hfill \\ & \le & -{\rho }_3{‖e‖}^2+{\rho }_4‖e‖(q_2\left|\mu \left(t\right)\right|‖e‖+q_1\left|\mu \left(t\right)\right|)\hfill \\ & \le & -{\rho }_3{‖e‖}^2+{\rho }_4{q}_2\left|\mu \left(t\right)\right|{‖e‖}^2+{\rho }_4{q}_1\left|\mu \left(t\right)\right|‖e‖.\hfill \end{array}$$

(38)

When making use of (35), $\dot{V}$ can be majorized as

$$\begin{array}{ccc}\dot{V}& \le & -{\displaystyle \frac{{\rho }_3}{{\rho }_2}}V+{\displaystyle \frac{{\rho }_{4 }{q}_{2 }}{{\rho }_1}}\left|\mu \left(t\right)\right|V+{\rho }_4{q}_{1 }\left|\mu \left(t\right)\right|\sqrt{{\displaystyle \frac{V}{{\rho }_1}}}\\ & \le & -\left({\displaystyle \frac{{\rho }_3}{{\rho }_2}}-{\displaystyle \frac{{\rho }_{4 }{q}_{2 }}{{\rho }_1}}\left|\mu \left(t\right)\right|\right)V+{\displaystyle \frac{{\rho }_{4 }{q}_{1 }}{\sqrt{{\rho }_1}}}\left|\mu \left(t\right)\right|\sqrt{V}\end{array}.$$

(39)

By defining $U=\sqrt{V\left(e\right)}$, we can write (39) as

$$\begin{array}{ccc}\dot{U}& =& {\displaystyle \frac{\dot{V}}{2\sqrt{V}}} \\ & \le & -{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{\rho }_3}{{\rho }_2}}-{\displaystyle \frac{{\rho }_{4 }{q}_{2 }}{{\rho }_1}}\left|\mu \left(t\right)\right|\right)U+{\displaystyle \frac{{\rho }_{4 }{q}_{1 }}{\sqrt{{\rho }_1}}}\left|\mu \left(t\right)\right|\end{array}.$$

(40)

By integrating the differential inequality (40) over the time interval [$t_0$, $t$], we obtain

$$U\left(t\right)\le e^{\left[-{\displaystyle \frac{{\rho }_3}{2{\rho }_2}}(t-t_0)+{\displaystyle \frac{{{\rho }_{4}q}_2}{2{\rho }_1}}{\int }_{t_0}^t\left|\mu \left(\tau \right)\right|d\tau \right]}U\left(t_0\right)+{\displaystyle \frac{{\rho }_4{q}_1}{2{\rho }_1}}\underset{t_o}{\overset{t}{\int }}{e}^{\left[-{\displaystyle \frac{{\rho }_3}{2{\rho }_2}}(t-\tau )+{\displaystyle \frac{{\rho }_4{q}_2}{2{\rho }_1}}{\int }_{t_0}^t\left|\mu \left(\tau \right)\right|d\tau \right]}\left|\mu \left(\tau \right)\right|d\tau .$$

(41)

Using condition (35) in (41) yields

$$‖e\left(t\right)‖\le \sqrt{{\displaystyle \frac{{\rho }_2}{{\rho }_1}}}{e}^{\left[-{\displaystyle \frac{{\rho }_3}{2{\rho }_2}}(t-t_0)+{\displaystyle \frac{{{\rho }_{4}q}_2}{2{\rho }_1}}{\int }_{t_0}^t\left|\mu \left(\tau \right)\right|d\tau \right]}‖e\left(t_0\right)‖+{\displaystyle \frac{{\rho }_4{q}_1}{2{\rho }_1}}\underset{t_o}{\overset{t}{\int }}{e}^{\left[-{\displaystyle \frac{{\rho }_3}{2{\rho }_2}}(t-\tau )+{\displaystyle \frac{{\rho }_4{q}_2}{2{\rho }_1}}{\int }_{t_0}^t\left|\mu \left(\tau \right)\right|d\tau \right]}\left|\mu \left(\tau \right)\right|d\tau .$$

(42)

In order to obtain an estimate of the integral of $\left|\mu \left(\tau \right)\right|$ in (42), using (30), we obtain $\begin{array}{ccc}{\int }_{t_0}^t\left|\mu \left(\tau \right)\right|d\tau & =& {\displaystyle \frac{\left|\mu \left(t_0\right)\right|}{k_{\mu }}}\left(1-e^{-k_{\mu }(t-t_0}\right)\end{array} \begin{array}{cc}\le & {\displaystyle \frac{\left|\mu \left(t_0\right)\right|}{k_{\theta }\mu }}\end{array}$. In addition, since $\begin{array}{ccc}\underset{0}{\overset{\infty }{\int }}\left|\mu \left(\tau \right)\right|d\tau & =& {\displaystyle \frac{\left|\mu \left(0\right)\right|}{k_{\mu }}}\end{array}$, based on Lemma 2, the condition (7) from Lemma 1 can be applied with an upper bound for the integral as $\begin{array}{ccc}{\int }_{t_0}^t\left|\mu \left(\tau \right)\right|d\tau & \le & {\displaystyle \frac{\left|\mu \left(t_0\right)\right|}{k_{\mu }}}\end{array}$.

Denoting constants by β and δ as follows:

$$\beta ={\displaystyle \frac{{\rho }_3}{2{\rho }_2}}$$

(43)

$$\delta =e^{{\displaystyle \frac{{\rho }_4 q_2\left|\mu \left(t_0\right)\right|}{2{\rho }_1 k_{\mu }}}},$$

(44)

the solution e(t) of the perturbed subsystem (25) satisfies the inequality

$$‖e\left(t\right)‖\le \sqrt{{\displaystyle \frac{{\rho }_2}{{\rho }_1}}}{\delta ‖e\left(t_0\right)‖e}^{-\beta (t-t_0)}+{\displaystyle \frac{{\rho }_4{q}_1\delta }{2{\rho }_1}}\underset{t_0}{\overset{t}{\int }}{e}^{-\beta (t-\tau )} \left|\mu \left(\tau \right)\right|d\tau .$$

(45)

It is noted that all conditions of Lemma 1 are satisfied, and inequality (45) is obtained in the form of the inequality (11) from Lemma 1.

With $\mu \left(t\right)$ as an exponentially decaying disturbance, in the light of Lemma 3, we need to establish the exponential stability of the e-subsystem (25). For this end, substituting (30) into (45) yields

$$‖e\left(t\right)‖\le \sqrt{{\displaystyle \frac{{\rho }_2}{{\rho }_1}}}{\delta ‖e\left(t_0\right)‖e}^{-\beta (t-t_0)}+{\displaystyle \frac{{\rho }_4{q}_1\delta \left|{\mu (t}_0)\right| }{2{\rho }_1}}\underset{t_o}{\overset{t}{\int }}{e}^{-\beta (t-\tau )} e^{-k_{\mu }(\tau -t_0)}d\tau .$$

(46)

Letting ${\beta }_0=min\left\{k_{\mu },{\displaystyle \frac{\beta }{2}}\right\}={\displaystyle \frac{\beta }{2}}$, the integral in (46) can be bounded above as follows:

$$\begin{array}{ccc}\underset{t_o}{\overset{t}{\int }}{e}^{-\beta (t-\tau )} e^{-k_{\mu }(\tau -t_0)}d\tau & \le & \underset{t_o}{\overset{t}{\int }}{e}^{-2{\beta }_0(t-\tau )} e^{-{\beta }_0(\tau -t_0)}d\tau \\ & \le & {\displaystyle \frac{2}{\beta }}{e}^{-{\displaystyle \frac{\beta }{2}}(t-t_0)}. \end{array}$$

(47)

Using (47), we find that the solution e(t) of (25) satisfies

$$\begin{array}{ccc}‖e\left(t\right)‖& \le & \sqrt{{\displaystyle \frac{{\rho }_2}{{\rho }_1}}}{\delta ‖e\left(t_0\right)‖e}^{-\beta (t-t_0)}+{\displaystyle \frac{{\rho }_4{q}_1\delta \left|{\mu (t}_0)\right| }{2{\rho }_1}}{\displaystyle \frac{2}{\beta }}{e}^{-{\displaystyle \frac{\beta }{2}}(t-t_0)}\\ & \le & \left(\sqrt{{\displaystyle \frac{{\rho }_2}{{\rho }_1}}}\beta ‖e\left(t_0\right)‖+{\displaystyle \frac{{\rho }_4{q}_1\delta \left|{\mu (t}_0)\right| }{2{\rho }_1}}{\displaystyle \frac{2}{\beta }}\right)e^{-{\displaystyle \frac{\beta }{2}}(t-t_0)}, \end{array}$$

(48)

which proves the exponential stability of the e-subsystem. Hence, $e\left(t\right)={\left[e_x\left(t\right), e_y\left(t\right)\right]}^T$ converges to zero exponentially.

3.2.3. (e, $\mu$)-System

The solutions of (25) and (26) satisfy (48) and (49), respectively. Let $x={\left[\begin{array}{cc}e,& \mu \end{array}\right]}^T$. Using the inequality $‖x\left(t\right)‖\le ‖e\left(t\right)‖+‖\mu \left(t\right)‖$, and $k_{\mu }>{\displaystyle \frac{k_e}{2}}$ by design, yields

$$\begin{array}{ccc}‖x\left(t\right)‖& \le & \left(\sqrt{{\displaystyle \frac{{\rho }_2}{{\rho }_1}}}\beta ‖e\left(t_0\right)‖+{\displaystyle \frac{{\rho }_4{q}_1\delta \left|{\mu (t}_0)\right| }{2{\rho }_1}}{\displaystyle \frac{2}{\beta }}\right)e^{-{\displaystyle \frac{\beta }{2}}(t-t_0)} \\ & & + \left|{\mu (t}_0)\right|e^{-k_{\theta }(t-t_0)} \\ & \le & \left(\sqrt{{\displaystyle \frac{{\rho }_2}{{\rho }_1}}}\beta ‖e\left(t_0\right)‖+{\displaystyle \frac{{\rho }_4{q}_1\delta \left|{\mu (t}_0)\right| }{2{\rho }_1}}{\displaystyle \frac{2}{\beta }}+\left|{\mu (t}_0)\right|\right)e^{-{\displaystyle \frac{\beta }{2}}(t-t_0)}.\end{array}$$

(49)

Hence, the origin x = (e, μ) = 0 is an exponentially stable equilibrium point for the closed-loop system (25) and (26). The trajectory tracking errors ($e_x\left(t\right)$, $e_y\left(t\right)$, μ(t)) converge to zero exponentially. This completes the proof.

3.2.4. Initial Condition Discussion

We now point to how to obtain a bound for the initial conditions (e($t_0)$, μ($t_0)$) for (e$\left(t\right)$, μ(t)), such as to avoid discontinuity in (21) and in the coordinate transformation for μ, respectively.

The idea is to find constants $a_1$ and $a_2$, such that $‖e\left(t_0\right)‖<a_1$ and $\left|\mu \left(t_0\right)\right|<a_2$, provided that there exists known constants $s_1$ and $s_2$ that satisfy the inequalities $‖e\left(t\right)‖<s_1$ and $\left|\mu \left(t\right)\right|<s_2$, respectively.

First, let us find an upper bound for e(t). In order to avoid discontinuity caused by division by zero in (20), the denominator must satisfy the inequality ${\nu }_1-a_1>0$, which yields $k_e{e}_x+{\omega }_r{e}_y<v_r$. From the last inequality, we obtain a lower bound on the reference mobile robot velocity $v_r$, as follows:

$$\begin{array}{ccc}{k}_e{e}_x+{\omega }_r{e}_y& \le & \sqrt{\left(k_e^2+{\omega }_r^2\right)\left(e_x^2+e_y^2 \right)}\\ & \le & \sqrt{k_e^2+{\omega }_{rmax}^2} ‖e\left(t\right)‖\\ & <& v_{rmin} .\end{array}$$

(50)

where ${\omega }_{rmax}$ is the maximum value of the virtual reference robot angular velocity.

From inequality (50), the following upper bound for the solution e(t) of (25) is obtained:

$$‖e\left(t\right)‖<{\displaystyle \frac{v_{rmin}}{\sqrt{k_e^2+{\omega }_{rmax}^2}}}:=s.$$

(51)

Since, from (30), we have $\left|\mu \left(t\right)\right|\le \left|{\mu (t}_0)\right|$, from (45), the following bound for e(t) is obtained:

$$‖e\left(t\right)‖\le \sqrt{{\displaystyle \frac{{\rho }_2}{{\rho }_1}}}{\delta ‖e\left(t_0\right)‖e}^{-\beta (t-t_0)}+{\displaystyle \frac{{\rho }_4{q}_1\delta }{2{\rho }_1\beta }}\left|{\mu (t}_0)\right|\left(1-e^{-\beta (t-t_0}\right).$$

(52)

From (52), we have that

$$‖e\left(t\right)‖\le max\left\{\begin{array}{cc}\sqrt{{\displaystyle \frac{{\rho }_2}{{\rho }_1}}}\delta ‖e\left(t_0\right)‖,& {\displaystyle \frac{{\rho }_4{q}_1\delta }{2{\rho }_1\beta }}\left|{\mu (t}_0)\right|\end{array}\right\}.$$

(53)

From (53), it follows that the bound $‖e\left(t\right)‖<s$ obtained in (51) is satisfied if

$$‖e\left(t_0\right)‖<\sqrt{{\displaystyle \frac{{\rho }_1}{{\rho }_2}}}{\displaystyle \frac{s}{\delta }}$$

(54)

and

$$\left|{\mu (t}_0)\right|<{\displaystyle \frac{{2\rho }_1\beta s}{{\rho }_4{q}_{1 }\delta }}.$$

(55)

The conditions (54) and (55) are similar to conditions (9) and (10) in Lemma 1. In this way, in order to avoid discontinuity in the auxiliary variable α defined in (20) and, respectively, in the coordinate transformation for μ, the region for e(t) is limited, as determined in (51), and for this end, the initial conditions $e\left(t_0\right)$ and $\mu \left(t_0\right)$ must satisfy (54) and (55), respectively.

Remark 3.

It is noted that the exponential convergence to zero of ($e_x\left(t\right)$, $e_y\left(t\right)$, and μ(t)) implies that the orientation error $e_{\theta }$(t) converges also to zero as $t\to \infty$. Indeed, since $\mu ={\displaystyle \frac{e_{\theta }-\alpha }{2}}$, we obtain $e_{\theta }=2\mu +\alpha .$ From (21), we observe that α converges exponentially to zero when ${(e}_x\left(t\right)$, $e_y\left(t\right)$) converge to zero. Therefore, $e_{\theta }\left(t\right)$ converges exponentially to zero as $t\to \infty$.

Remark 4.

It is important to note that the bound s for the solution e(t) determined in (51) is crucial to avoid discontinuity in the auxiliary variable α defined in (20) and, subsequently, in the coordinate transformation for μ. However, different values of the magnitude of this bound result in different bounds for the initial values of the position error $‖e\left(t_0\right)‖$ and orientation error $\left|{\mu (t}_0)\right|$, defined in (54) and (55), respectively, with a proportional relationship. On the other hand, the controller gain k_e for the position–error dynamics largely influences the size of the bound s due to the fact that the gain k_e takes part in the denominator of the expression (51) for s, and different values of the controller gain k_e results in different values of this bound. From the specificity of the expression (51), it can be is seen that they are inversely related, and it follows that, the higher the value of k_e. the smaller the bound s. However, the controller gains determine the tracking performance for obtaining a faster or slower convergence rate of the closed-loop system dynamics, in the sense that larger controller gains can obtain faster convergence speed and vice versa. Hence, in order to increase the bounds for the initial conditions $e\left(t_0\right)$ and $\mu \left(t_0\right)$, the compromise is decreasing the values of the controller gain k_e, which leads to a decrease in the convergence speed of the closed-loop error dynamics.

4. Simulation Results

4.1. Tracking a Cuvilinear Trajectory

In this section, a number of simulations are performed in order to illustrate the effectiveness of the proposed feedback controller. In addition, the purpose of these simulations is also to verify the validity of the established bounds for the initial conditions e(t₀) and μ(t₀) given by (54) and (55), respectively, which guarantee continuity of the virtual control (21) and the state transformation for μ.

The motion of the mobile robot is simulated with the designed tracking controller (20)–(23).

4.1.1. Initial Conditions

The reference trajectory is generated by the virtual mobile robot, where the linear and angular velocities are set as

$$\begin{array}{ccc}{v}_r\left(t\right)=1.6-0.1sint,& \begin{array}{cc} & \end{array}& {\omega }_r\left(t\right)=-0.5sin0.25t,\end{array}$$

(56)

which implies that their minimum and maximum values are $v_{rmin}=1.5 \le v_r\left(t\right)\le 1.7=v_{rmax } \mathrm{m}/\mathrm{s}$ and ${\omega }_{rmin}=-0.5 \le {\omega }_r\left(t\right)\le 0.5={\omega }_{rmax } \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$. It is noted that the reference inputs $v_r$ and ${\omega }_r$ are bounded and continuously differentiable, and their derivatives are also bounded. The parameter gains of the controller (20)–(23) are taken as $k_{e }=0.5, k_{\mu }=3.3.$ The reference trajectory of the virtual mobile robot is generated with zero initial conditions: $x_{r }\left(0\right)=0$ m, $y_{r }\left(0\right)=0$ m, ${\theta }_{r }\left(0\right)=0 \mathrm{r}\mathrm{a}\mathrm{d}$.

Based on (51), the upper bound for $e\left(t\right)={\left[e_{x }\left(t\right),e_y\left(t\right)\right]}^T$ is obtained as

$$‖e\left(t\right)‖<{\displaystyle \frac{v_{rmin}}{\sqrt{k_e^2+{\omega }_{rmax}^2}}}:=s=2.12.$$

(57)

Regarding the initial conditions for the tracking errors ($e_{x }\left(t_0\right), e_y\left(t_0\right), \mu \left(t_0\right))$, using (30) and (51), as mentioned above, the initial conditions for $e\left(t_0\right)={\left[e_{x }\left(t_0\right),e_y\left(t_0\right)\right]}^T$ and $\mu \left(t_0\right)$ must satisfy the bounds (54) and (55), respectively, which ensures continuity in the auxiliary variable α and in the coordinate transformation for μ. Using (54) and (55), the bounds for the initial conditions $e\left(t_0\right)={\left[e_{x }\left(t_0\right),e_y\left(t_0\right)\right]}^T$ and $\mu \left(t_0\right)$ of the error variables are obtained as follows:

$$\begin{array}{ccc}‖e\left(t_0\right)‖<\sqrt{{\displaystyle \frac{{\rho }_1}{{\rho }_2}}}{\displaystyle \frac{s}{\delta }}=1.71 \mathrm{m},& \begin{array}{cc} & \end{array}& \left|{\mu (t}_0)\right|<{\displaystyle \frac{{2\rho }_1\beta s}{{\rho }_4{q}_{1 }\delta }}=0.5 \mathrm{r}\mathrm{a}\mathrm{d}.\end{array}$$

(58)

Since the two-norm of the position error is defined as $‖e‖=\sqrt{e_x^2+e_y^2}$, it is easy to obtain that the admissible initial values of the position errors are, for example, by taking

$$D=\left\{e\left(t_0\right)\in R^2 | \left|e_x\left(t_0\right)\right|<1.21 \mathrm{m}, \left|e_y\left(t_0\right)\right|<1. 21 \mathrm{m}\right\}.$$

(59)

Using the inverse coordinate transformation $e_{\theta }\left(t_0\right)=2\mu \left(t_0\right)+\alpha \left(t_0\right)$ with $\left|{\mu (t}_0)\right|<0.5 \mathrm{r}\mathrm{a}\mathrm{d}$ and $\left|{\alpha }_{max}\left(t_0\right)\right|<0.66 \mathrm{r}\mathrm{a}\mathrm{d}$, after a simple calculation, we obtain the bound for the admissible initial values of the orientation error as

$${|e}_{\theta }\left(t_0\right)|<0.34 \mathrm{r}\mathrm{a}\mathrm{d}.$$

(60)

Based on the established bounds of the initial conditions for the tracking errors, for the simulation tests, we chose $e_x\left(0\right)=1.2 \mathrm{m}$, $e_y\left(0\right)=-1.2 \mathrm{m}$, and $e_{\theta }\left(0\right)=-0.3 \mathrm{r}\mathrm{a}\mathrm{d}$, respectively, and $\mu \left(0\right)=-0.42 \mathrm{r}\mathrm{a}\mathrm{d}$. Thus, x$\left(0\right)=1.2 \mathrm{m}$, $y\left(0\right)=-1.2 \mathrm{m}$, and $\theta \left(0\right)=-0.3 \mathrm{r}\mathrm{a}\mathrm{d}$.

4.1.2. Results

The results of the simulation are depicted in Figure 2, Figure 3, Figure 4 and Figure 5. The control objective is to move the mobile robot along a prescribed path drawn by the virtual reference mobile robot, i.e., the mobile robot state $\left(\begin{array}{ccc}x\left(t\right),& y\left(t\right),& \theta \left(t\right)\end{array}\right)$ is able to track a reference trajectory $\left(x_r\left(t\right),{ y}_r\left(t\right), {\theta }_r\left(t\right)\right)$, generated by the virtual mobile robot. The tracking performance is shown in Figure 2a, where the moving paths in the xy-plane of the real mobile robot (green line) and the virtual reference mobile robot (red dashed line) are displayed. It can be observed that the path of the mobile robot approaches that of the virtual reference robot and successfully tracks it. The time responses of the tracking errors $(e_x, e_y, \mu )$ are shown in Figure 2b. It can be seen that the error coordinates converge to zero, which is in conformity with Proposition 1 in Section 2.

In Figure 3a are depicted the evolution in time of the velocity of the mobile robot (green line) and the virtual reference robot (red dashed line). The decrease in speed of the mobile robot in the first seconds is due to the initial conditions, in which it is situated ahead of the virtual robot. It takes about 8 s before the reference trajectory is successfully tracked and the velocity of the mobile robot converges to the velocity of the virtual reference mobile robot.

Figure 3b shows the evolution in time of the angular velocity ω of the mobile robot (green line) and the virtual reference robot ω_r (red dashed line). It can be observed that the reference velocity of the mobile robot converges to the reference angular velocity of the virtual robot.

As pointed out in Remark 3, the orientation error $e_{\theta }\left(t\right),$ which expresses the orientation of the real mobile robot with respect to the reference virtual robot, converges to zero as $t\to \infty$. In Figure 4 is depicted the evolution in time of the orientation error $e_{\theta }\left(t\right)$.

The two-norm of the position error $‖e‖=\sqrt{e_x^2+e_y^2}$ (blue line) and its upper bound obtained in (57) (red dashed line) are displayed in Figure 5a. From the simulation, it is clear that starting with the initial condition for the position errors $(e_x\left(0\right), e_y\left(0\right))$ from the domain determined in (59), the value of the position error two-norm $‖e‖$ is below the bound determined in (57), which guarantees continuity of the auxiliary variable (20) and, respectively, the state transformation for μ.

The norm of $\left|\mu \right|$ (blue line) and its upper bound obtained in (55) (red dashed line) are displayed in Figure 5b. Again, starting with the initial condition from the interval specified in (60) for admissible initial values for $\mu \left(0\right)$, $\left|\mu \right(t\left)\right|$ is below the specified bound.

4.2. Straight Line Trajectory Tracking

Two different tests for rectilinear trajectory tracking have been carried out in order to demonstrate the ability of the proposed controller to track straight line trajectory (Test 1), which is one of the basic tracking maneuvers. In Test 2, we demonstrate the trade-off between the following competing objectives: (i) larger domain of initial conditions for the tracking error errors $(e_x\left(0\right), e_y\left(0\right), \mu (0\left)\right)$ for which the continuity of the auxiliary variable (20) and, respectively, the state transformation for μ is guaranteed and (ii) faster convergence rate of the closed-loop error dynamics.

4.2.1. Initial Conditions

(a) Test 1. The straight line reference trajectory is generated by the virtual robot, where the linear and angular velocities are set as

$$\begin{array}{ccc}{v}_r\left(t\right)=1.6,& \begin{array}{cc} & \end{array}& {\omega }_r\left(t\right)=0.\end{array}$$

(61)

In this first test, the controller gains are taken as $k_{e }=0.5, k_{\mu }=3.3$, which are the same as in the simulation test for tracking a curvilinear trajectory. The reference trajectory of the virtual mobile robot is generated with zero initial conditions: $x_{r }\left(0\right)=0$ m, $y_{r }\left(0\right)=0$ m, ${\theta }_{r }\left(0\right)=0 \mathrm{r}\mathrm{a}\mathrm{d}$. Based on (51), the upper bound for $e\left(t\right)={\left[e_{x }\left(t\right),e_y\left(t\right)\right]}^T$ is obtained as

$$‖e\left(t\right)‖<3.2:=s.$$

(62)

Based on (54) and (55), and using the chosen values of the controller gains and the upper bound for $‖e\left(t\right)‖$ obtained in (62), the initial conditions for $e\left(t_0\right)={\left[e_{x }\left(t_0\right),e_y\left(t_0\right)\right]}^T$ and $\mu \left(t_0\right)$ have to satisfy the bounds

$$\begin{array}{ccc}‖e\left(t_0\right)‖<2.51 \mathrm{m},& \begin{array}{cc} & \end{array}& \left|{\mu (t}_0)\right|<0.80 \mathrm{r}\mathrm{a}\mathrm{d}.\end{array}$$

(63)

Using (63), the admissible initial values of the position and orientation errors are

$$D=\left\{e\left(t_0\right)\in R^2 | \left|e_x\left(t_0\right)\right|<1.78 \mathrm{m}, \left|e_y\left(t_0\right)\right|<1. 78 \mathrm{m}\right\}.$$

(64)

$${|e}_{\theta }\left(t_0\right)|<0.71 \mathrm{r}\mathrm{a}\mathrm{d}.$$

(65)

Based on the established bounds of the initial conditions for the tracking errors, for the simulation tests, we chose $e_x\left(0\right)=-1.7 \mathrm{m}$, $e_y\left(0\right)=-1.7 \mathrm{m}$, and $e_{\theta }\left(0\right)=0.6 \mathrm{r}\mathrm{a}\mathrm{d}$, respectively, and $\mu \left(0\right)=0.13 \mathrm{r}\mathrm{a}\mathrm{d}$. Thus, x$\left(0\right)=-1.7 \mathrm{m}$, $y\left(0\right)=-1.7 \mathrm{m}$, and $\theta \left(0\right)=0.6 \mathrm{r}\mathrm{a}\mathrm{d}$.

(b) Test 2. To demonstrate the trade-off between the larger domain of initial conditions for the tracking error errors $(e_x\left(0\right), e_y\left(0\right), \mu (0\left)\right)$, for which the continuity of the auxiliary variable (20) and the state transformation for μ is guaranteed, on the one hand, and faster convergence speed of the closed-loop error dynamics, on the other hand.

To this end, the values of the controller gains are chosen to be $k_{e }=0.2, k_{\mu }=0.7$, which are considerably smaller (more than two times smaller) compared to those in the simulation Test 1 ($k_{e }=0.5, k_{\mu }=3.3$) for tracking a straight line trajectory.

The straight line reference trajectory is generated by the virtual robot, where the linear and angular velocities are kept the same as in Test 1, as given in (61).

Based on (51), the upper bound for $e\left(t\right)={\left[e_{x }\left(t\right),e_y\left(t\right)\right]}^T$ is obtained as

$$‖e\left(t\right)‖<8.0:=s.$$

(66)

Using (66), the initial conditions for $e\left(t_0\right)={\left[e_{x }\left(t_0\right),e_y\left(t_0\right)\right]}^T$ and $\mu \left(t_0\right)$ have to satisfy the bounds

$$\begin{array}{ccc}‖e\left(t_0\right)‖<6.56 \mathrm{m},& \begin{array}{cc} & \end{array}& \left|{\mu (t}_0)\right|<0.98 \mathrm{r}\mathrm{a}\mathrm{d}.\end{array}$$

(67)

Using (67), the admissible initial values of the position and orientation errors are

$$D=\left\{e\left(t_0\right)\in R^2 | \left|e_x\left(t_0\right)\right|<4.64 \mathrm{m}, \left|e_y\left(t_0\right)\right|<4. 64 \mathrm{m}\right\}.$$

(68)

$${|e}_{\theta }\left(t_0\right)|<1.02 \mathrm{r}\mathrm{a}\mathrm{d}.$$

(69)

which is significantly larger than those determined in (64) and (65) for Test 1. Based on the established bounds for the initial conditions for the tracking errors, for the second simulation tests, we chose $e_x\left(0\right)=-4.5 \mathrm{m}$, $e_y\left(0\right)=-4.5 \mathrm{m}$, and $e_{\theta }\left(0\right)=0.6 \mathrm{r}\mathrm{a}\mathrm{d}$, respectively, and $\mu \left(0\right)=0.127 \mathrm{r}\mathrm{a}\mathrm{d}$. Thus, x$\left(0\right)=-4.5 \mathrm{m}$, $y\left(0\right)=-4.5 \mathrm{m}$, and $\theta \left(0\right)=0.6 \mathrm{r}\mathrm{a}\mathrm{d}$.

4.2.2. Results

(a) Test 1. For the first test of rectilinear trajectory tracking, the results of the simulation are depicted in Figure 6. The tracking performance is shown in Figure 6a, where the moving paths in the xy-plane of the real mobile robot (green line) and the virtual reference mobile robot (red dashed line) are displayed. It can be observed that the path of the mobile robot approaches that of the virtual reference robot and successfully tracks it, which demonstrates the ability of the mobile robot to track a straight line trajectory. The time responses of the tracking errors $(e_x, e_y, \mu )$ are shown in Figure 6b. It can be seen that the error coordinates converge to zero. Since the controller gains were chosen to be the same as in the simulation test of curvilinear trajectory tracking (Section 4.2.1), the rate of convergence is the same, and the mobile robot reaches the reference trajectory after approximately 8 s, which is the same as in the case of the curvilinear trajectory tracking (Figure 2b). From the simulation test, it can be seen that the straight line reference trajectory is successfully tracked, and from this point of view, it is an advantage over many other existing solutions that do not have this capability, including those proposed in [14,15,23,24,26].

(b) Test 2. For the second test of straight line trajectory tracking, the simulation results are depicted in Figure 7. Figure 7a shows that the trajectory of the mobile robot approaches and then tracks precisely the reference trajectory generated by the virtual robot. Figure 7b shows that the tracking errors tend asymptotically toward zero, which is consistent with the theoretical result in Proposition 1, but a decrease in the speed of convergence is observed.

Compared to the results obtained in Figure 6b from Test 1, in the second test, the convergence rate decreases by a little more than two times, which is approximately proportional to the decrease in the values of the controller gains. However, this is at the expense of the more than two times larger domain of the initial conditions compared to those obtained in Test 1. These results demonstrate the trade-off between a larger domain of initial conditions for the tracking error errors $(e_x\left(0\right), e_y\left(0\right), \mu \left(0\right))$ and a higher convergence rate of the closed-loop error dynamics.

4.3. Comparative Simulation

The performance of the mobile robot closed-loop system under the designed controller is evaluated via comparative simulation results from Reference [26] under the same initial conditions.

4.3.1. Initial Conditions

The reference velocities of the virtual reference robot are taken to be the same as those given in [26]: $v_r\left(t\right)=1.0-{\displaystyle \frac{t}{t+5}}$ m/s and ${\omega }_{r }=1.2+{\displaystyle \frac{1}{t+5}}$ rad/s. The initial conditions are also taken to be the same: $x_{r }\left(0\right)=0$ m, $y_{r }\left(0\right)=0$ m, and ${\theta }_{r }\left(0\right)=1 \mathrm{r}\mathrm{a}\mathrm{d}$, and x$\left(0\right)=0.4 \mathrm{m}$, $y\left(0\right)=0.3 \mathrm{m}$, and $\theta \left(0\right)=1 \mathrm{r}\mathrm{a}\mathrm{d}$. The parameter gains of the controller (20)–(23) are chosen as $k_{e }=1.4, k_{\mu }=3.3.$

4.3.2. Results

The results of the spiral motion simulation are depicted in Figure 8a,b. The tracking performance is shown in Figure 8a, where the moving paths in the xy-plane of the real mobile robot (green line) and the virtual reference mobile robot (red dashed line) are displayed. It can be observed that the path of the mobile robot approaches that of the virtual reference robot and successfully tracks it. The time responses of the tracking errors $(e_x, e_y, \mu )$ are shown in Figure 8b. It can be seen that the errors coordinates converge to zero asymptotically. Compared to the results presented in [26], the rate of convergence is comparable; even a little faster convergence is observed. In our case, the mobile robot reaches the reference trajectory after approximately 4 s compared to approximately 6 s in [26].

The simulation results confirm the validity of the proposed controller and the theoretical analysis from the previous section.

5. Conclusions

In this paper, a feedback controller for nonholonomic unicycle-type wheeled mobile robots was designed based on a cascade strategy, which ensures exponential tracking of the reference trajectory. The stability of the closed-loop system was analyzed using the Lyapunov stability theory, and local exponential stability property was established. An explicit estimate of the set of feasible initial conditions for the error variables was also determined. Simulation results were presented to demonstrate the validity, feasibility, and performance of the proposed exponential controller and show that the problem of trajectory tracking control of nonholonomic wheeled mobile robots was solved over a large class of reference trajectories with fast convergence and good transient performance.