Trajectory tracking control for wheeled mobile robot subject to generalized torque constraints

Abstract

The trajectory tracking control problem for wheeled mobile robot (WMR) subject to generalized torque constraints is studied in this paper. First, a more general model is obtained by taking the displacement between the centre of mass of the WMR and the midpoint of the driving wheels into consideration. Then, a virtual velocity controller and a generalized torque controller are designed based on a new Lyapunov function. Due to the advantages of the new Lyapunov function, the stability of the trajectory tracking error system can be derived directly and the constraint of the initial tracking errors is removed. Less conservative conditions for tuning the controller parameters to satisfy the saturation constraints are also derived. Finally, some numerical examples are given to show the effectiveness of the proposed controllers.

Keywords

Wheeled mobile robot trajectory tracking control generalized torque constraints double closed-loop control

Introduction

The wheeled mobile robot (WMR) has attracted many researchers’ attentions during the last several decades. As a typical nonholonomic constrained system, the WMR has the characteristics of nonlinearity, strong coupling, parameter uncertainties and so on, which make the control of the WMR more challenging (Brockett, 1983).

As a basic problem for the application of the WMR, trajectory tracking control has been widely investigated and many control methods have been applied in this area, such as the neural network (Asai et al., 2019; Chen et al., 2020; Yuan et al., 2014), adaptive control (Ayten et al., 2019; Boukens et al., 2017), active disturbance rejection control (Chen et al., 2018), sliding mode control (Mera et al., 2020; Yang et al., 2018), fuzzy logic control (Abdelwahab et al., 2020; Akka and Khaber, 2019), feedback linearization (Chwa, 2010; Montoya-Villegas et al., 2019), model predictive control (Wang et al., 2018; Yang et al., 2018) and so on.

A common characteristic of these results mentioned above is that the bounds of control inputs have not been taken into consideration. In recent years, saturation control has attracted considerable interest for control of nonlinear systems. In Chen et al. (2011), an auxiliary system was designed to tackle the nonsymmetric input constraints for the uncertain multi-input/multi-output (MIMO) nonlinear systems. In order to reduce the computational burden in backstepping design process (Wang et al., 2021), an adaptive radial basis function neural network–based dynamic surface control scheme was proposed for uncertain strict-feedback nonlinear systems under input saturation and external disturbance in Chen et al. (2014). In Miranda-Colorado (2022), an observer-based saturated proportional derivative controller was presented to solve the regulation problem of perturbed second-order nonlinear systems in the presence of input and state constraints.

As a class of practical nonlinear system, the saturation of the control input of the WMR is unavoidable due to the mechanical restriction. If the saturation is ignored, the tracking performance of the WMR will be degraded; even worse, the WMR may not be able to track the desired trajectory (Ren and Beard, 2004). So it is necessary to take the input saturation into consideration when designing the trajectory tracking controllers. Regarding the saturation constraints at the kinematic level, the velocity saturation has been widely considered. In Lee et al. (2001), based on the backstepping technique and LaSalle’s Invariance principle, the tracking and stabilization problems of the mobile robot under velocity constraints were solved simultaneously. In Jiang et al. (2001), a kind of kinematic controllers were designed via the application of passivity theory and normalization method, which could solve the stabilization and tracking problem under velocity constraints, respectively. In Wang (2008), the semiglobal stabilizability with saturated inputs was investigated based on a kinematic state transformation. The global asymptotic stabilization and tracking control of the WMR were studied in Su and Zheng (2010). In Huang (2009), sinusoidal functions were used to reduce the surges and fluctuations of the outputs of the controllers in polar coordinates; however, the bounds of the saturation were not considered explicitly. A novel trajectory tracking controller was designed via the constrained directions method in Zeiaee et al. (2016). In Thomas et al. (2021), a discrete-time sliding mode control scheme was presented based on the equivalent control approach for the unicycle mobile robot with bounded inputs.

In practice, the mobile robot is ultimately driven by motors mounted on the driving wheels, and it is more meaningful to consider the input constraints at the dynamic level. In Chen et al. (2009), a moving horizon $H_{\infty}$ tracking control scheme was presented in the presence of both actuators saturations and external disturbances. In Chen (2014), based on the finite-time stability theory, a tracking control scheme was proposed considering the saturation constraints on the velocities and torques. Nevertheless, the controller parameters were hard to be determined. In Chen and Jia (2014), a simple tracking controller combining with two first-order filters was presented, which simplified the procedure of tuning controller parameters to satisfy the input constraints. A robust neural network-based control scheme was proposed in Huang et al. (2019), which realized trajectory tracking and stabilization simultaneously in the presence of uncertainties, external disturbances and input saturation. In Wang et al. (2021), both the velocities and accelerations saturation constraints were considered, and an adaptive visual tracking controller was designed for the WMR with an onboard camera. Although some results have been obtained concerning the saturation of the WMR, there are still problems that have not been fully addressed. First, in order to guarantee the stability of the system and avoid the equilibrium point $\pm π$ of tracking error of the heading angle (Fukao et al., 2000), the WMR’s initial states should be located in a specific region. Second, the conditions for tuning controller parameters are always too conservative as some terms are relaxed too much to obtain the upper bound, which may result in very slow convergence of the tracking errors. All the concerns mentioned above motivate the study of this paper.

Different from Bai et al. (2020) which considers the velocity constraints of the WMR, we focus on the trajectory tracking control problem under generalized torque constraints in this paper. Our main contributions can be summarized as follows: (a) A more general model of the WMR is considered, which takes the displacement of the centre of mass of the WMR and the midpoint of the axis of the driving wheels into consideration. (b) A new Lyapunov function is proposed, based on which the stability of the trajectory tracking error system can be derived directly and the constraint of the initial tracking error of the WMR is removed. (c) Less conservative conditions for tuning the controller parameters are obtained, which is verified by the simulations.

The rest of the paper is organized as follows: the system model of the WMR and problem statement are given in the ‘Problem statement’ section. The main results are obtained in the ‘Main results’ section. Some numerical simulations are given in the ‘Simulation results’ section, and the conclusion is given in the ‘Conclusion’ section.

Notations: The symbols ∥·∥ and |·| represent the Euclidean norm of a vector and the absolute value of an element, respectively. The symbol $R$ denotes the set of all real numbers. The symbol $max {x_{1}, x_{2}, \dots, x_{n}}$ represents the maximum value of the elements from $x_{1}$ to $x_{n}$ . The function $\arctan (\cdot)$ denotes inverse tangent function.

Problem statement

The WMR considered in this paper is shown in Figure 1. The latter two wheels are driving wheels. XOY is the global coordinate frame and xay is the local coordinate frame. c is the centre of mass of the WMR, with its coordinates $(x_{c}, y_{c})$ in the global frame. $θ_{c}$ is the WMR’s heading angle. Then, the posture of the WMR can be denoted as $q_{c} = [x_{c}, y_{c}, θ_{c}]^{T}$ . a is the midpoint of the axis of the driving wheels. d is the displacement between a and c.

Figure 1.

Wheeled mobile robot.

Let $ν$ and $ω$ represent the linear velocity and angular velocity of the midpoint of the axis of the driving wheels, respectively. Assume that the wheels of the WMR are pure rolling and there is no skidding when the WMR is moving; that is, the WMR can only move in the direction of axis of symmetry of the driving wheels and the wheels are non-slipping. Then, the following nonholonomic constraint should be satisfied (Fierro and Lewis, 1997, 1998)

{\overset{\cdot}{y}}_{c} \cos θ_{c} - {\overset{\cdot}{x}}_{c} \sin θ_{c} - d {\overset{\cdot}{θ}}_{c} = 0

(1)

The kinematic model of the WMR under the nonholonomic constraint can be obtained as

{\begin{matrix} {\overset{\cdot}{x}}_{c} = ν \cos θ_{c} - d ω \sin θ_{c} \\ {\overset{\cdot}{y}}_{c} = ν \sin θ_{c} + d ω \cos θ_{c} \\ {\overset{\cdot}{θ}}_{c} = ω \end{matrix}

(2)

The reference trajectory is denoted as $q_{r} = [x_{r}, y_{r}, θ_{r}]^{T}$ , which is smooth and known priori. The reference linear velocity $ν_{r}$ and angular velocity $ω_{r}$ can be chosen as (Klancar et al., 2011)

ν_{r} = \sqrt{{\overset{\cdot}{x}}_{r}^{2} + {\overset{\cdot}{y}}_{r}^{2}}, ω_{r} = \frac{{\overset{\cdot}{x}}_{r} {\overset{\cdot\cdot}{y}}_{r} - {\overset{\cdot\cdot}{x}}_{r} {\overset{\cdot}{y}}_{r}}{{\overset{\cdot}{x}}_{r}^{2} + {\overset{\cdot}{y}}_{r}^{2}}

(3)

Then, the reference trajectory is generated by the following equation

{\begin{matrix} {\overset{\cdot}{x}}_{r} = ν_{r} \cos θ_{r} - d ω_{r} \sin θ_{r} \\ {\overset{\cdot}{y}}_{r} = ν_{r} \sin θ_{r} + d ω_{r} \cos θ_{r} \\ {\overset{\cdot}{θ}}_{r} = ω_{r} \end{matrix}

(4)

The dynamic model of WMR can be described as (Chen et al., 2012)

{\begin{matrix} \overset{\cdot}{ν} = τ_{1} \\ \overset{\cdot}{ω} = τ_{2} \end{matrix}

(5)

where $τ_{1}$ and $τ_{2}$ are generalized torque inputs.

The trajectory tracking error of the WMR is defined as (Kanayama et al., 1990)

q_{e} = [\begin{matrix} x_{e} \\ y_{e} \\ θ_{e} \end{matrix}] = [\begin{matrix} \cos θ_{c} & \sin θ_{c} & 0 \\ - \sin θ_{c} & \cos θ_{c} & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} x_{r} - x_{c} \\ y_{r} - y_{c} \\ θ_{r} - θ_{c} \end{matrix}]

(6)

Differentiating equation (6) along the trajectory of equation (2) and combining with equation (5), the dynamics of the trajectory tracking error can be described as

{\begin{matrix} {\overset{\cdot}{x}}_{e} = ω y_{e} + ν_{r} \cos θ_{e} - ν - d ω_{r} \sin θ_{e} \\ {\overset{\cdot}{y}}_{e} = - ω x_{e} + ν_{r} \sin θ_{e} + d ω_{r} \cos θ_{e} - d ω \\ {\overset{\cdot}{θ}}_{e} = ω_{r} - ω \\ \overset{\cdot}{ν} = τ_{1} \\ \overset{\cdot}{ω} = τ_{2} \end{matrix}

(7)

Assumption 1. The generalized torque inputs $τ_{1}$ and $τ_{2}$ are subject to the following constraints

| τ_{1} | \leq τ_{1 \max}, | τ_{2} | \leq τ_{2 \max}

(8)

where $τ_{1 \max}$ and $τ_{2 \max}$ are known positive constants.

Assumption 2. The desired velocities $ν_{r}$ and $ω_{r}$ and their derivatives ${\overset{\cdot}{ν}}_{r}$ and ${\overset{\cdot}{ω}}_{r}$ are bounded and satisfy the following constraints

\begin{matrix} 0 < ν_{r} \leq γ_{ν}, | ω_{r} | \leq γ_{ω}, \\ | {\overset{\cdot}{ν}}_{r} | \leq μ_{ν} \leq τ_{1 \max}, | {\overset{\cdot}{ω}}_{r} | \leq μ_{ω} \leq τ_{2 \max} \end{matrix}

(9)

where $γ_{ν}$ , $γ_{ω}$ , $μ_{ν}$ and $μ_{ω}$ are known positive constants.

Remark 1. The reference velocities $ν_{r}$ and $ω_{r}$ are obtained by trajectory planning. $ν_{r} > 0$ means that the WMR moves in the forward direction, which can be easily satisfied by a proper trajectory planner (Klancar et al., 2011).

Our goal in this paper is to design the torque controllers $τ_{1}$ and $τ_{2}$ in the presence of the constraints given in equation (8) such that the WMR can track the reference trajectory asymptotically, that is

lim_{t \to \infty} ‖ q_{e} ‖ = 0

(10)

Lemma 1. If $f : [0, \infty) \to R$ is uniformly continuous and $lim_{t \to \infty} \int_{0}^{t} f (τ) d τ$ exists and is finite, then $lim_{t \to \infty} f (t) = 0$ (Khalil, 2002).

Main results

The control schematic of the WMR is shown in Figure 2. First, the kinematic model of the WMR is considered, and a virtual kinematic controller $[ν_{c}, ω_{c}]^{T}$ is designed to guarantee that $lim_{t \to \infty} ∥ q_{e} ∥ = 0$ . Then, the dynamic model of the WMR is taken into consideration, and a dynamic controller is designed to make the real velocity of the WMR, that is, $[ν, ω]^{T}$ , follow the output of the virtual kinematic controller $[ν_{c}, ω_{c}]^{T}$ . Finally, the asymptotical stability of the whole trajectory tracking system is realized.

Figure 2.

Trajectory tracking control system structure.

Kinematic controller design

In this subsection, a new Lyapunov function is proposed, based on which a new virtual kinematic controller is derived and the asymptotical stability of the tracking error system is achieved based on the kinemics of the WMR.

Only considering the kinemics of the WMR, the tracking error system can be described as

{\begin{matrix} {\overset{\cdot}{x}}_{e} = ω y_{e} + ν_{r} \cos θ_{e} - ν - d ω_{r} \sin θ_{e} \\ {\overset{\cdot}{y}}_{e} = - ω x_{e} + ν_{r} \sin θ_{e} + d ω_{r} \cos θ_{e} - d ω \\ {\overset{\cdot}{θ}}_{e} = ω_{r} - ω \end{matrix}

(11)

Theorem 1. The equilibrium point of equation (11) $q_{e} = [0, 0, 2 i π]^{T}, (i = 0, 1, 2, \dots)$ is asymptotically stable by the virtual velocity controller given in equation (12)

{\begin{array}{l} ν_{c} = ν_{r} \cos θ_{e} - \frac{2 ω_{c}}{k_{θ}} \sin \frac{θ_{e}}{2} + k_{x} \frac{X_{e}}{ρ} \\ ω_{c} = ω_{r} + ν_{r} (Y_{e} \frac{β k_{y} \cos \frac{θ_{e}}{2}}{ρ} + α k_{θ} \sin \frac{θ_{e}}{2}) \end{array}

(12)

where

\begin{matrix} X_{e} = x_{e} + d (1 - \cos θ_{e}) \\ Y_{e} = y_{e} - d \sin θ_{e} + \frac{2}{k_{θ}} \sin \frac{θ_{e}}{2} \\ ρ = \sqrt{1 + {X_{e}}^{2} + {Y_{e}}^{2}} \end{matrix}

(13)

and $k_{x}$ , $k_{y}$ , $k_{θ}$ , $α$ and $β$ are real positive constants, with $α + β = 2$ .

Proof: Choose the Lyapunov function as follows

V_{1} = \sqrt{1 + {X_{e}}^{2} + {Y_{e}}^{2}} + \frac{2}{k_{y}} (1 - \cos \frac{θ_{e}}{2}) - 1

(14)

Differentiating equation (14), we obtain

\begin{matrix} {\overset{\cdot}{V}}_{1} = \frac{1}{ρ} (X_{e} {\overset{\cdot}{X}}_{e} + Y_{e} {\overset{\cdot}{Y}}_{e}) + \frac{{\overset{\cdot}{θ}}_{e}}{k_{y}} \sin \frac{θ_{e}}{2} \\ = \frac{1}{ρ} X_{e} ({\overset{\cdot}{x}}_{e} + d {\overset{\cdot}{θ}}_{e} \sin θ_{e}) + \frac{1}{ρ} Y_{e} (\begin{matrix} {\overset{\cdot}{y}}_{e} - d {\overset{\cdot}{θ}}_{e} \cos θ_{e} + \\ \frac{{\overset{\cdot}{θ}}_{e}}{k_{θ}} \cos \frac{θ_{e}}{2} \end{matrix}) \\ + \frac{{\overset{\cdot}{θ}}_{e}}{k_{y}} \sin \frac{θ_{e}}{2} \\ = \frac{1}{ρ} X_{e} (y_{e} ω + ν_{r} \cos θ_{e} - ν - d ω \sin θ_{e}) \\ + \frac{1}{ρ} Y_{e} (\begin{matrix} - x_{e} ω + ν_{r} \sin θ_{e} + d ω (\cos θ_{e} - 1) \\ + \frac{(ω_{r} - ω)}{k_{θ}} \cos \frac{θ_{e}}{2} \end{matrix}) \\ + \frac{(ω_{r} - ω)}{k_{y}} \sin \frac{θ_{e}}{2} \\ = \frac{1}{ρ} X_{e} (ν_{r} \cos θ_{e} - ν - \frac{2 ω}{k_{θ}} \sin \frac{θ_{e}}{2}) + \frac{1}{ρ} ν_{r} \sin θ_{e} Y_{e} + \\ (ω_{r} - ω) (\frac{1}{ρ k_{θ}} \cos \frac{θ_{e}}{2} Y_{e} + \frac{1}{k_{y}} \sin \frac{θ_{e}}{2}) \end{matrix}

(15)

If the linear velocity and angular velocity, $ν$ and $ω$ , respectively, are chosen as $ν_{c}$ and $ω_{c}$ given in equation (12), it follows

{\overset{\cdot}{V}}_{1} = - \frac{k_{x}}{ρ^{2}} X_{e}^{2} - \frac{β k_{y}}{ρ^{2} k_{θ}} ν_{r} \overset{2}{\cos} \frac{θ_{e}}{2} Y_{e}^{2} - \frac{α k_{θ}}{k_{y}} ν_{r} \overset{2}{\sin} \frac{θ_{e}}{2}

(16)

As $α$ , $β$ , $k_{x}$ , $k_{y}$ , $k_{θ}$ and $ν_{r}$ are positive, ${\overset{\cdot}{V}}_{1} \leq 0$ . Differentiating equation (16) with the controller given in equation (12), we have

\begin{array}{l} {\overset{\cdot\cdot}{V}}_{1} = - 2 k_{x} \frac{X_{e}}{ρ} \cdot \frac{d (\frac{X_{e}}{ρ})}{d t} - \frac{β k_{y}}{k_{θ}} (\begin{array}{l} {\overset{\cdot}{ν}}_{r} \frac{\cos^{2} \frac{θ_{e}}{2} {Y_{e}}^{2}}{ρ^{2}} \\ + 2 ν_{r} \frac{\cos \frac{θ_{e}}{2} Y_{e}}{ρ} \cdot \frac{d (\frac{\cos \frac{θ_{e}}{2} Y_{e}}{ρ})}{d t} \end{array}) - \frac{α k_{θ}}{k_{y}} (\begin{array}{l} {\overset{\cdot}{ν}}_{r} \sin^{2} \frac{θ_{e}}{2} \\ + \frac{1}{2} ν_{r} {\overset{\cdot}{θ}}_{e} \sin θ_{e} \end{array}) \\ = - 2 k_{x} \frac{X_{e}}{ρ} (\begin{array}{l} \frac{ω Y_{e} + ν_{r} \cos θ_{e} - ν - \frac{2 ω}{k_{θ}} \sin \frac{θ_{e}}{2}}{\sqrt{1 + X_{e}^{2} + Y_{e}^{2}}} \\ - \frac{(X_{e} (\begin{array}{l} X_{e} (ν_{r} \cos θ_{e} - ν - \frac{2 ω}{k_{θ}} \sin \frac{θ_{e}}{2}) \\ + Y_{e} (ν_{r} \sin θ_{e} + \frac{{\overset{\cdot}{θ}}_{e}}{k_{θ}} \cos \frac{θ_{e}}{2}) \end{array}))}{{(1 + {X_{e}}^{2} + {Y_{e}}^{2})}^{\frac{3}{2}}} \end{array}) \\ - \frac{β k_{y}}{k_{θ}} (\begin{array}{l} {\overset{\cdot}{ν}}_{r} \frac{\cos^{2} \frac{θ_{e}}{2} {Y_{e}}^{2}}{ρ^{2}} + 2 ν_{r} \frac{\cos \frac{θ_{e}}{2} Y_{e}}{ρ} \cdot \\ (\begin{array}{l} \frac{\cos \frac{θ_{e}}{2} (- ω X_{e} + ν_{r} \sin θ_{e} + \frac{{\overset{\cdot}{θ}}_{e}}{k_{θ}} \cos \frac{θ_{e}}{2}) - Y_{e} \frac{{\overset{\cdot}{θ}}_{e}}{2} \sin \frac{θ_{e}}{2}}{\sqrt{1 + X_{e}^{2} + Y_{e}^{2}}} \\ - \frac{(\cos \frac{θ_{e}}{2} Y_{e} (\begin{array}{l} X_{e} (ν_{r} \cos θ_{e} - ν - \frac{2 ω}{k_{θ}} \sin \frac{θ_{e}}{2}) \\ + Y_{e} (ν_{r} \sin θ_{e} + \frac{{\overset{\cdot}{θ}}_{e}}{k_{θ}} \cos \frac{θ_{e}}{2}) \end{array}))}{{(1 + {X_{e}}^{2} + {Y_{e}}^{2})}^{\frac{3}{2}}} \end{array}) \end{array}) \end{array}

(17)

As $V_{1} \geq 0$ and ${\overset{\cdot}{V}}_{1} \leq 0$ , $V_{1}$ is bounded. Hence, $\int_{0}^{\infty} {\overset{\cdot}{V}}_{1} dt = V_{1} (\infty) - V_{1} (0)$ is finite. Thus, $X_{e}$ and $Y_{e}$ are bounded. It is obtained from equations (9) and (12) that $ν_{c}$ and $ω_{c}$ are bounded. The linear and angular velocity $ν$ and $ω$ are chosen as $ν_{c}$ and $ω_{c}$ ; it follows from equation (11) that ${\overset{\cdot}{x}}_{e}$ , ${\overset{\cdot}{y}}_{e}$ and ${\overset{\cdot}{θ}}_{e}$ are also bounded. From equation (17), we can obtain that ${\overset{\cdot\cdot}{V}}_{1}$ is bounded. Thus, ${\overset{\cdot}{V}}_{1}$ is uniformly continuous. According to Lemma 1, it is concluded that $lim_{t \to \infty} {\overset{\cdot}{V}}_{1} = 0$ , that is, $\sin (θ_{e} / 2), X_{e}, Y_{e} \to 0$ , as $t \to \infty$ ; then, we can obtain that $x_{e}, y_{e} \to 0$ and $θ_{e} \to 2 i π$ , $i = 0, 1, 2, \dots$ . Thus, the equilibrium point of equation (11) $q_{e} = [0, 0, 2 i π]^{T}, (i = 0, 1, 2, \dots)$ is asymptotically stable. This completes the proof.

Remark 2. A new Lyapunov function is proposed and the derivative of the new Lyapunov function contains all terms of $x_{e}$ , $y_{e}$ and $θ_{e}$ , from which it is easier to obtain the stability of the tracking error system than the traditional methods.

Remark 3. The attraction region of the $θ_{e}$ is $(- π, π)$ by the traditional methods (Kanayama et al., 1990) and the initial tracking error should not be too large to avoid the equilibrium of $π$ . In this paper, the attraction region of the $θ_{e}$ is extended to $(- 2 π, 2 π)$ and the restriction of the initial tracking error is removed.

Dynamic controller design

In this subsection, a generalized torque controller is designed to make the velocities of the WMR, that is, $ν$ and $ω$ , follow the virtual kinematic controllers $ν_{c}$ and $ω_{c}$ without violating the constraints given in equation (8). And the asymptotical stability of the whole trajectory tracking system is achieved.

The tracking errors of the linear velocity and angular velocity are defined as

{\begin{matrix} ν_{e} = ν - ν_{c} \\ ω_{e} = ω - ω_{c} \end{matrix}

(18)

Theorem 2. Applying the virtual kinematic controller (equation (12)) and the torque controller (equation (19)) to the dynamics of the trajectory tracking error (equation (7)), the equilibrium point of the whole trajectory tracking system $[x_{e}, y_{e}, θ_{e}, ν_{e}, ω_{e}]^{T} = [0, 0, 2 i π, 0, 0]^{T}, (i = 0, 1, 2, \dots)$ is asymptotically stable.

{\begin{matrix} τ_{1} = {\overset{\cdot}{ν}}_{c} + \frac{b_{1}}{2} (- \frac{ν_{e}}{a_{1}} + \frac{X_{e}}{ρ}) \\ τ_{2} = {\overset{\cdot}{ω}}_{c} + \frac{b_{2}}{2} (- \frac{ω_{e}}{a_{2}} + \frac{2 \sin \frac{θ_{e}}{2} X_{e} + \cos \frac{θ_{e}}{2} Y_{e}}{ρ k_{θ}} + \frac{1}{k_{y}} \sin \frac{θ_{e}}{2}) \end{matrix}

(19)

where $a_{1}$ , $a_{2}$ , $b_{1}$ , $b_{2}$ , $k_{y}$ and $k_{θ}$ are positive. And the generalized torque constraints (equation (8)) are satisfied if the parameters $α$ , $β$ , $k_{x}$ , $k_{y}$ , $k_{θ}$ , $a_{1}$ , $a_{2}$ , $b_{1}$ and $b_{2}$ are chosen under the following constraints

a_{1} \leq | ν_{c} |_{\max}

(20a)

a_{2} (\frac{3}{k_{θ}} + \frac{1}{k_{y}}) \leq | ω_{c} |_{\max}

(20b)

\begin{matrix} μ_{ω} + μ_{ν} (β k_{y} + α k_{θ}) + β k_{y} γ_{ν} (\begin{matrix} (2 + \frac{2}{k_{θ}}) | ω_{c} |_{\max} + \frac{5}{2} γ_{ν} \\ + | ν_{c} |_{\max} \end{matrix}) \\ + \frac{β k_{y} (4 + k_{θ}) + α {k_{θ}}^{2}}{2 k_{θ}} γ_{ν} (\begin{matrix} γ_{ν} (β k_{y} + α k_{θ}) \\ + | ω_{c} |_{\max} \end{matrix}) \\ + \frac{b_{2}}{2} (\frac{1}{a_{2}} | ω_{c} |_{\max} + \frac{3}{k_{θ}} + \frac{1}{k_{y}}) \leq τ_{2 \max} \end{matrix}

(20c)

\begin{matrix} μ_{ν} + (\frac{2 | ω_{c} |_{\max} + k_{x}}{2 k_{θ}} + γ_{ν}) (γ_{ν} (β k_{y} + α k_{θ}) + | ω_{c} |_{\max}) \\ + \frac{2}{k_{θ}} (τ_{2 \max} - \frac{b_{2}}{2} (\frac{1}{a_{2}} | ω_{c} |_{\max} + \frac{3}{k_{θ}} + \frac{1}{k_{y}})) \\ + k_{x} ((2 + \frac{8}{k_{θ}}) | ω_{c} |_{\max} + \frac{5}{2} γ_{ν} + 4 | ν_{c} |_{\max}) \\ + \frac{b_{1}}{2} (\frac{1}{a_{1}} | ν_{c} |_{\max} + 1) \leq τ_{1 \max} \end{matrix}

(20d)

where

| ω_{c} |_{\max} = γ_{ω} + γ_{ν} (β k_{y} + α k_{θ})

(21a)

| ν_{c} |_{\max} = γ_{ν} + \frac{2}{k_{θ}} (γ_{ω} + γ_{ν} (β k_{y} + α k_{θ})) + k_{x}

(21b)

Proof: Taking the absolute value of the controller (equation (12)), we can get

\begin{matrix} | ω_{c} | \leq | ω_{r} | + | ν_{r} | (β k_{y} + α k_{θ}) \\ \leq γ_{ω} + γ_{ν} (β k_{y} + α k_{θ}) \\ \overset{Δ}{=} | ω_{c} |_{\max} \\ | ν_{c} | \leq | ν_{r} | + \frac{2}{k_{θ}} | ω_{c} |_{\max} + k_{x} \\ \leq γ_{ν} + \frac{2}{k_{θ}} (γ_{ω} + γ_{ν} (β k_{y} + α k_{θ})) + k_{x} \\ \overset{Δ}{=} | ν_{c} |_{\max} \end{matrix}

(22)

From equations (7), (18) and (19), we can obtain

{\overset{\cdot}{ν}}_{e} = \frac{b_{1}}{2} (- \frac{ν_{e}}{a_{1}} + \frac{X_{e}}{ρ})

(23a)

{\overset{\cdot}{ω}}_{e} = \frac{b_{2}}{2} (- \frac{ω_{e}}{a_{2}} + \frac{2 \sin \frac{θ_{e}}{2} X_{e} + \cos \frac{θ_{e}}{2} Y_{e}}{ρ k_{θ}} + \frac{1}{k_{y}} \sin \frac{θ_{e}}{2})

(23b)

It is obtained from equation (23a) that

| ν_{e} | \leq max {| ν_{e} (0) |, a_{1}}

(24)

Let

\begin{matrix} a_{1} \leq | ν_{e} (0) | \\ \leq | ν (0) - ν_{c} (0) | \\ \leq | ν_{c} (0) | \leq {| ν_{c} |}_{\max} \end{matrix}

(25)

Then

| ν_{e} | \leq | ν_{e} (0) | \leq {| ν_{c} |}_{\max}

(26)

Similarly, it can be obtained from equation (23b) that

| ω_{e} | \leq max {| ω_{e} (0) |, a_{2} (\frac{3}{k_{θ}} + \frac{1}{k_{y}})}

(27)

Let

a_{2} (\frac{3}{k_{θ}} + \frac{1}{k_{y}}) \leq {| ω_{c} |}_{\max}

(28)

Then

| ω_{e} | \leq | ω_{e} (0) | \leq {| ω_{c} |}_{\max}

(29)

According to equation (18), it follows

{\begin{matrix} ν = ν_{e} + ν_{c} \\ ω = ω_{e} + ω_{c} \end{matrix}

(30)

From equations (26), (29) and (30), we can obtain

\begin{matrix} | ν | \leq | ν_{e} | + | ν_{c} | \\ \leq | ν_{e} | + {| ν_{c} |}_{\max} \\ \leq 2 {| ν_{c} |}_{\max} \end{matrix}

(31)

\begin{matrix} | ω | \leq | ω_{e} | + | ω_{c} | \\ \leq | ω_{e} | + {| ω_{c} |}_{\max} \\ \leq 2 {| ω_{c} |}_{\max} \end{matrix}

(32)

Taking the derivative of the virtual velocities $ν_{c}$ and $ω_{c}$ , we can get

\begin{matrix} {\overset{\cdot}{ω}}_{c} = {\overset{\cdot}{ω}}_{r} + {\overset{\cdot}{ν}}_{r} (Y_{e} \frac{β k_{y} \cos \frac{θ_{e}}{2}}{ρ} + α k_{θ} \sin \frac{θ_{e}}{2}) \\ + ν_{r} (\frac{d (Y_{e} \frac{β k_{y} \cos \frac{θ_{e}}{2}}{ρ} + α k_{θ} \sin \frac{θ_{e}}{2})}{dt}) \end{matrix}

(33)

\begin{matrix} \frac{d (Y_{e} \frac{β k_{y} \cos \frac{θ_{e}}{2}}{ρ} + α k_{θ} \sin \frac{θ_{e}}{2})}{dt} \\ = β k_{y} \frac{{\overset{\cdot}{Y}}_{e} \cos \frac{θ_{e}}{2} - Y_{e} \frac{{\overset{\cdot}{θ}}_{e}}{2} \sin \frac{θ_{e}}{2}}{\sqrt{1 + {X_{e}}^{2} + {Y_{e}}^{2}}} \\ - β k_{y} \frac{Y_{e} \cos \frac{θ_{e}}{2} (X_{e} {\overset{\cdot}{X}}_{e} + Y_{e} {\overset{\cdot}{Y}}_{e})}{{(1 + {X_{e}}^{2} + {Y_{e}}^{2})}^{\frac{3}{2}}} + α k_{θ} \frac{{\overset{\cdot}{θ}}_{e}}{2} \cos \frac{θ_{e}}{2} \\ = \frac{β k_{y} ((- ω X_{e} + ν_{r} \sin θ_{e} + \frac{{\overset{\cdot}{θ}}_{e}}{k_{θ}} \cos \frac{θ_{e}}{2}) \cos \frac{θ_{e}}{2} - Y_{e} \frac{{\overset{\cdot}{θ}}_{e}}{2} \sin \frac{θ_{e}}{2})}{\sqrt{1 + {X_{e}}^{2} + {Y_{e}}^{2}}} \\ - \frac{β k_{y} Y_{e} \cos \frac{θ_{e}}{2} (\begin{matrix} X_{e} (ν_{r} \cos θ_{e} - ν - \frac{2 ω}{k_{θ}} \sin \frac{θ_{e}}{2}) \\ + Y_{e} (ν_{r} \sin θ_{e} + \frac{{\overset{\cdot}{θ}}_{e}}{k_{θ}} \cos \frac{θ_{e}}{2}) \end{matrix})}{{(1 + {X_{e}}^{2} + {Y_{e}}^{2})}^{\frac{3}{2}}} \\ + α k_{θ} \frac{{\overset{\cdot}{θ}}_{e}}{2} \cos \frac{θ_{e}}{2} \end{matrix}

(34)

From equations (33) and (34), we can obtain

\begin{matrix} | {\overset{\cdot}{ω}}_{c} | \leq | {\overset{\cdot}{ω}}_{r} | + | {\overset{\cdot}{ν}}_{r} | (β k_{y} + α k_{θ}) + | ν_{r} | | \frac{d (Y_{e} \frac{β k_{y} \cos \frac{θ_{e}}{2}}{ρ} + α k_{θ} \sin \frac{θ_{e}}{2})}{dt} | \\ \leq | {\overset{\cdot}{ω}}_{r} | + | {\overset{\cdot}{ν}}_{r} | (β k_{y} + α k_{θ}) + β k_{y} | ν_{r} | \\ (| ω | + | ν_{r} | + \frac{1}{k_{θ}} | {\overset{\cdot}{θ}}_{e} | + \frac{1}{2} | {\overset{\cdot}{θ}}_{e} |) + \end{matrix}

\begin{matrix} β k_{y} | ν_{r} | (\frac{1}{2} (| ν_{r} | + | ν | + \frac{2}{k_{θ}} | ω |) + | ν_{r} | + \frac{1}{k_{θ}} | {\overset{\cdot}{θ}}_{e} |) \\ + \frac{α k_{θ}}{2} | ν_{r} | | {\overset{\cdot}{θ}}_{e} | \\ \leq | {\overset{\cdot}{ω}}_{r} | + | {\overset{\cdot}{ν}}_{r} | (β k_{y} + α k_{θ}) + β k_{y} | ν_{r} | \\ (\begin{matrix} (2 + \frac{2}{k_{θ}}) {| ω_{c} |}_{\max} \\ + \frac{5}{2} | ν_{r} | + {| ν_{c} |}_{\max} \end{matrix}) \\ + \frac{β k_{y} (4 + k_{θ}) + α {k_{θ}}^{2}}{2 k_{θ}} | ν_{r} | | {\overset{\cdot}{θ}}_{e} | \end{matrix}

(35)

According to equations (7), (12) and (30), it follows

\begin{matrix} | {\overset{\cdot}{θ}}_{e} | = | ω_{r} - ω | \\ = | ω_{r} - ω_{e} - ω_{c} | \\ \leq | ω_{r} - ω_{c} | + | ω_{e} | \\ \leq | - ν_{r} (Y_{e} \frac{β k_{y} \cos \frac{θ_{e}}{2}}{ρ} + α k_{θ} \sin \frac{θ_{e}}{2}) | + | ω_{e} | \\ \leq | ν_{r} | (β k_{y} + α k_{θ}) + {| ω_{c} |}_{\max} \end{matrix}

(36)

Substituting equation (36) into equation (35), we can obtain

\begin{matrix} | {\overset{\cdot}{ω}}_{c} | \leq | {\overset{\cdot}{ω}}_{r} | + | {\overset{\cdot}{ν}}_{r} | (β k_{y} + α k_{θ}) + β k_{y} | ν_{r} | \\ (\begin{matrix} (2 + \frac{2}{k_{θ}}) {| ω_{c} |}_{\max} \\ + \frac{5}{2} | ν_{r} | + {| ν_{c} |}_{\max} \end{matrix}) + \\ \frac{β k_{y} (4 + k_{θ}) + α {k_{θ}}^{2}}{2 k_{θ}} | ν_{r} | \cdot (| ν_{r} | (β k_{y} + α k_{θ}) + {| ω_{c} |}_{\max}) \\ \leq μ_{ω} + μ_{ν} (β k_{y} + α k_{θ}) + β k_{y} γ_{ν} \\ (\begin{matrix} (2 + \frac{2}{k_{θ}}) {| ω_{c} |}_{\max} \\ + \frac{5}{2} γ_{ν} + {| ν_{c} |}_{\max} \end{matrix}) + \\ \frac{β k_{y} (4 + k_{θ}) + α {k_{θ}}^{2}}{2 k_{θ}} γ_{ν} \cdot (γ_{ν} (β k_{y} + α k_{θ}) + {| ω_{c} |}_{\max}) \end{matrix}

(37)

\begin{matrix} {\overset{\cdot}{ν}}_{c} = {\overset{\cdot}{ν}}_{r} \cos θ_{e} - ν_{r} {\overset{\cdot}{θ}}_{e} \sin θ_{e} - \frac{2}{k_{θ}} ({\overset{\cdot}{ω}}_{c} \sin \frac{θ_{e}}{2} + ω_{c} \frac{{\overset{\cdot}{θ}}_{e}}{2} \cos \frac{θ_{e}}{2}) \\ + k_{x} (\frac{d (\frac{X_{e}}{ρ})}{dt}) \end{matrix}

(38)

\begin{matrix} \frac{d (\frac{X_{e}}{ρ})}{dt} \\ = \frac{{\overset{\cdot}{X}}_{e} \sqrt{1 + {X_{e}}^{2} + {Y_{e}}^{2}} - X_{e} \frac{X_{e} {\overset{\cdot}{X}}_{e} + Y_{e} {\overset{\cdot}{Y}}_{e}}{\sqrt{1 + {X_{e}}^{2} + {Y_{e}}^{2}}}}{1 + {X_{e}}^{2} + {Y_{e}}^{2}} \\ = \frac{ω Y_{e} + ν_{r} \cos θ_{e} - ν - \frac{2 ω}{k_{θ}} \sin \frac{θ_{e}}{2}}{\sqrt{1 + X_{e}^{2} + Y_{e}^{2}}} \\ - \frac{(X_{e} (\begin{matrix} X_{e} (ν_{r} \cos θ_{e} - ν - \frac{2 ω}{k_{θ}} \sin \frac{θ_{e}}{2}) \\ + Y_{e} (ν_{r} \sin θ_{e} + \frac{{\overset{\cdot}{θ}}_{e}}{k_{θ}} \cos \frac{θ_{e}}{2}) \end{matrix}))}{{(1 + {X_{e}}^{2} + {Y_{e}}^{2})}^{\frac{3}{2}}} \end{matrix}

(39)

\begin{matrix} | {\overset{\cdot}{ν}}_{c} | \leq | {\overset{\cdot}{ν}}_{r} | + | ν_{r} | | {\overset{\cdot}{θ}}_{e} | + \frac{2}{k_{θ}} (| {\overset{\cdot}{ω}}_{c} | + \frac{1}{2} | ω_{c} | | {\overset{\cdot}{θ}}_{e} |) + k_{x} | \frac{d (\frac{X_{e}}{ρ})}{dt} | \\ \leq | {\overset{\cdot}{ν}}_{r} | + | ν_{r} | | {\overset{\cdot}{θ}}_{e} | + \frac{2}{k_{θ}} (| {\overset{\cdot}{ω}}_{c} | + \frac{1}{2} | ω_{c} | | {\overset{\cdot}{θ}}_{e} |) \\ + k_{x} (| ω | + | ν_{r} | + | ν | + \frac{2}{k_{θ}} | ω |) \\ + k_{x} (| ν_{r} | + | ν | + \frac{2}{k_{θ}} | ω | + \frac{1}{2} (| ν_{r} | + \frac{1}{k_{θ}} | {\overset{\cdot}{θ}}_{e} |)) \\ \leq μ_{ν} + (\frac{2 {| ω_{c} |}_{\max} + k_{x}}{2 k_{θ}} + γ_{ν}) (γ_{ν} (β k_{y} + α k_{θ}) + {| ω_{c} |}_{\max}) \\ + \frac{2}{k_{θ}} | {\overset{\cdot}{ω}}_{c} | + k_{x} (\begin{matrix} (2 + \frac{8}{k_{θ}}) {| ω_{c} |}_{\max} \\ + \frac{5}{2} γ_{ν} + 4 {| ν_{c} |}_{\max} \end{matrix}) \end{matrix}

(40)

According to equation (19), we can get

\begin{matrix} | τ_{2} | \leq | {\overset{\cdot}{ω}}_{c} | + \frac{b_{2}}{2} (\frac{1}{a_{2}} | ω_{e} | + \frac{3}{k_{θ}} + \frac{1}{k_{y}}) \\ \leq | {\overset{\cdot}{ω}}_{c} | + \frac{b_{2}}{2} (\frac{1}{a_{2}} {| ω_{c} |}_{\max} + \frac{3}{k_{θ}} + \frac{1}{k_{y}}) \end{matrix}

(41)

Substituting equations (20c) and (37), we can obtain

\begin{matrix} | τ_{2} | \leq μ_{ω} + μ_{ν} (β k_{y} + α k_{θ}) + β k_{y} γ_{ν} (\begin{matrix} (2 + \frac{2}{k_{θ}}) {| ω_{c} |}_{\max} \\ + \frac{5}{2} γ_{ν} + {| ν_{c} |}_{\max} \end{matrix}) \\ + \frac{β k_{y} (4 + k_{θ}) + α {k_{θ}}^{2}}{2 k_{θ}} γ_{ν} \cdot \\ (γ_{ν} (β k_{y} + α k_{θ}) + {| ω_{c} |}_{\max}) + \frac{b_{2}}{2} (\frac{1}{a_{2}} {| ω_{c} |}_{\max} + \frac{3}{k_{θ}} + \frac{1}{k_{y}}) \\ \leq τ_{2 \max} \end{matrix}

(42)

We can get from equations (41) and (42) that

| {\overset{\cdot}{ω}}_{c} | \leq τ_{2 \max} - \frac{b_{2}}{2} (\frac{1}{a_{2}} {| ω_{c} |}_{\max} + \frac{3}{k_{θ}} + \frac{1}{k_{y}})

(43)

According to equation (19), we can obtain

\begin{matrix} | τ_{1} | \leq | {\overset{\cdot}{ν}}_{c} | + \frac{b_{1}}{2} (\frac{1}{a_{1}} | ν_{e} | + 1) \\ \leq | {\overset{\cdot}{ν}}_{c} | + \frac{b_{1}}{2} (\frac{1}{a_{1}} {| ν_{c} |}_{\max} + 1) \end{matrix}

(44)

Substituting equations (20d), (40) and (43), it follows

\begin{matrix} | τ_{1} | \leq μ_{ν} + (\frac{2 {| ω_{c} |}_{\max} + k_{x}}{2 k_{θ}} + γ_{ν}) (γ_{ν} (β k_{y} + α k_{θ}) + {| ω_{c} |}_{\max}) \\ + \frac{2}{k_{θ}} | {\overset{\cdot}{ω}}_{c} | \\ + k_{x} ((2 + \frac{8}{k_{θ}}) {| ω_{c} |}_{\max} + \frac{5}{2} γ_{ν} + 4 {| ν_{c} |}_{\max}) \\ + \frac{b_{1}}{2} (\frac{1}{a_{1}} {| ν_{c} |}_{\max} + 1) \end{matrix}

\begin{matrix} \leq μ_{ν} + (\frac{2 {| ω_{c} |}_{\max} + k_{x}}{2 k_{θ}} + γ_{ν}) (γ_{ν} (β k_{y} + α k_{θ}) + {| ω_{c} |}_{\max}) \\ + \frac{2}{k_{θ}} (τ_{2 \max} - \frac{b_{2}}{2} (\frac{1}{a_{2}} {| ω_{c} |}_{\max} + \frac{3}{k_{θ}} + \frac{1}{k_{y}})) \\ + k_{x} ((2 + \frac{8}{k_{θ}}) {| ω_{c} |}_{\max} + \frac{5}{2} γ_{ν} + 4 {| ν_{c} |}_{\max}) \\ + \frac{b_{1}}{2} (\frac{1}{a_{1}} {| ν_{c} |}_{\max} + 1) \leq τ_{1 \max} \end{matrix}

(45)

Next, we will analyze the stability of the whole trajectory tracking system. Choose the Lyapunov function as follows

V_{2} = V_{1} + \frac{1}{b_{1}} ν_{e}^{2} + \frac{1}{b_{2}} ω_{e}^{2}

(46)

Differentiating equation (46), we can get

\begin{matrix} {\overset{\cdot}{V}}_{2} = \frac{1}{ρ} X_{e} (ν_{r} \cos θ_{e} - ν - \frac{2 ω}{k_{θ}} \sin \frac{θ_{e}}{2}) + \frac{1}{ρ} ν_{r} \sin θ_{e} Y_{e} \\ + (ω_{r} - ω) (\frac{1}{ρ k_{θ}} \cos \frac{θ_{e}}{2} Y_{e} + \frac{1}{k_{y}} \sin \frac{θ_{e}}{2}) \\ + \frac{2}{b_{1}} ν_{e} {\overset{\cdot}{ν}}_{e} + \frac{2}{b_{2}} ω_{e} {\overset{\cdot}{ω}}_{e} \end{matrix}

(47)

Substituting equation (30) into equation (47), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{2} = \frac{1}{ρ} X_{e} (ν_{r} \cos θ_{e} - ν_{e} - ν_{c} - \frac{2 (ω_{e} + ω_{c})}{k_{θ}} \sin \frac{θ_{e}}{2}) \\ + \frac{1}{ρ} ν_{r} \sin θ_{e} Y_{e} + (ω_{r} - ω_{e} - ω_{c}) \\ (\frac{1}{ρ k_{θ}} \cos \frac{θ_{e}}{2} Y_{e} + \frac{1}{k_{y}} \sin \frac{θ_{e}}{2}) + \frac{2}{b_{1}} ν_{e} {\overset{\cdot}{ν}}_{e} + \frac{2}{b_{2}} ω_{e} {\overset{\cdot}{ω}}_{e} \\ = \frac{1}{ρ} X_{e} (ν_{r} \cos θ_{e} - ν_{c} - \frac{2 ω_{c}}{k_{θ}} \sin \frac{θ_{e}}{2}) + \frac{1}{ρ} ν_{r} \sin θ_{e} Y_{e} \\ + (ω_{r} - ω_{c}) (\frac{1}{ρ k_{θ}} \cos \frac{θ_{e}}{2} Y_{e} + \frac{1}{k_{y}} \sin \frac{θ_{e}}{2}) - \frac{X_{e} ν_{e}}{ρ} \\ - \frac{2 X_{e} ω_{e} \sin \frac{θ_{e}}{2}}{ρ k_{θ}} - ω_{e} (\frac{1}{ρ k_{θ}} \cos \frac{θ_{e}}{2} Y_{e} + \frac{1}{k_{y}} \sin \frac{θ_{e}}{2}) \\ + \frac{2}{b_{1}} ν_{e} (τ_{1} - {\overset{\cdot}{ν}}_{c}) + \frac{2}{b_{2}} ω_{e} (τ_{2} - {\overset{\cdot}{ω}}_{c}) \end{matrix}

(48)

Substituting equations (12) and (19) into equation (47), it follows

\begin{matrix} {\overset{\cdot}{V}}_{2} = - \frac{k_{x}}{ρ^{2}} X_{e}^{2} - \frac{β k_{y}}{ρ^{2} k_{θ}} ν_{r} \overset{2}{\cos} \frac{θ_{e}}{2} Y_{e}^{2} - \frac{α k_{θ}}{k_{y}} ν_{r} \overset{2}{\sin} \frac{θ_{e}}{2} \\ - \frac{1}{a_{1}} {ν_{e}}^{2} - \frac{1}{a_{2}} {ω_{e}}^{2} \\ = {\overset{\cdot}{V}}_{1} - \frac{1}{a_{1}} {ν_{e}}^{2} - \frac{1}{a_{2}} {ω_{e}}^{2} \end{matrix}

(49)

As $α$ , $β$ , $k_{x}$ , $k_{y}$ , $k_{θ}$ , $a_{1}$ , $a_{2}$ and $ν_{r}$ are positive, ${\overset{\cdot}{V}}_{2} \leq 0$ .

It is obtained from equations (23a) and (23b) that

\begin{matrix} | {\overset{\cdot}{ν}}_{e} | \leq \frac{b_{1}}{2} (\frac{| ν_{e} |}{a_{1}} + 1) \\ \leq \frac{b_{1}}{2} (\frac{{| ν_{c} |}_{\max}}{a_{1}} + 1) \end{matrix}

(50)

\begin{matrix} | {\overset{\cdot}{ω}}_{e} | \leq \frac{b_{2}}{2} (\frac{| ω_{e} |}{a_{2}} + \frac{3}{k_{θ}} + \frac{1}{k_{y}}) \\ \leq \frac{b_{2}}{2} (\frac{{| ω_{c} |}_{\max}}{a_{2}} + \frac{3}{k_{θ}} + \frac{1}{k_{y}}) \end{matrix}

(51)

It follows from equation (22) that $| {\overset{\cdot}{ν}}_{e} |$ and $| {\overset{\cdot}{ω}}_{e} |$ are bounded, that is, ${\overset{\cdot}{ν}}_{e}$ and ${\overset{\cdot}{ω}}_{e}$ are bounded. Thus, $ν_{e}$ and $ω_{e}$ are uniformly continuous. It can be obtained from the proof of Theorem 1 that ${\overset{\cdot}{V}}_{1}$ is uniformly continuous. Thus, ${\overset{\cdot}{V}}_{2}$ is uniformly continuous. As $V_{2} \geq 0$ and ${\overset{\cdot}{V}}_{2} \leq 0$ , $\int_{0}^{\infty} {\overset{\cdot}{V}}_{2} dt = V_{2} (\infty) - V_{2} (0)$ is finite. According to Lemma 1, it is easy to obtain that $lim_{t \to \infty} {\overset{\cdot}{V}}_{2} = 0$ . Following the similar procedure of the proof of Theorem 1, we can get that $q_{e}$ is asymptotically stable and the velocity tracking errors $ν_{e}$ and $ω_{e}$ asymptotically converge to zero, that is, $ν$ and $ω$ can follow the virtual kinematic controllers $ν_{c}$ and $ω_{c}$ . Thus, the equilibrium point of the whole trajectory tracking system $[x_{e}, y_{e}, θ_{e}, ν_{e}, ω_{e}]^{T} = [0, 0, 2 i π, 0, 0]^{T}, (i = 0, 1, 2, \dots)$ is asymptotically stable. This completes the proof.

Remark 4. Some tips are given for tuning the parameters in equation (20). As $k_{θ}$ appears in the denominator, $k_{θ}$ can be relatively large, then small $α$ and $k_{y}$ can guarantee $| ω_{c} |_{\max}$ relatively small. Then, $k_{x}$ , $a_{1}$ , $a_{2}$ , $b_{1}$ and $b_{2}$ can be tuned to satisfy the constraints of equation (20).

Simulation results

In this section, some examples are provided to verify the effectiveness of the proposed approach in this paper.

Example 1. In this example, we consider the case that the centre of the mass of the WMR coincides with the midpoint of the axis of the driving wheels, that is, $d = 0$ . The same problem is also studied in Chen et al. (2012). The linear velocity and angular velocity of the reference trajectory are $ν_{r} = 1.5 m / s$ and $ω_{r} = 0.8 rad / s$ , respectively. The initial value of the reference trajectory is $[0, 0, 0]^{T}$ , and the initial value of the WMR is $[0.5, - 0.5, 0.2 π]^{T}$ . The constraints are given as follows: $γ_{ν} = 1.5$ , $γ_{ω} = 1.2$ , $μ_{ν} = 2.0$ , $μ_{ω} = 1.8$ , $τ_{1 \max} = 12.5$ and $τ_{2 \max} = 12.2$ . The parameters of the controller in this paper are chosen as follows: $k_{x} = 0.1$ , $k_{y} = 0.2$ , $k_{θ} = 10$ , $α = 0.02$ , $β = 1.98$ , $a_{1} = 0.1$ , $a_{2} = 0.13$ , $b_{1} = 0.28$ and $b_{2} = 0.21$ .

The quantitative analysis is performed by considering the following performance indexes (Miranda-Colorado, 2021)

ISV = \int_{0}^{T} {τ_{1}}^{2} + {τ_{2}}^{2} dt

(52)

ITAE = \int_{0}^{T} t | q_{e} (i) | dt, i = 1, 2, 3

(53)

where T denotes the simulation time, $q_{e} (1) = x_{e}$ , $q_{e} (2) = y_{e}$ and $q_{e} (3) = θ_{e}$ , which are defined in equation (6). The performance index ISV offers quantitative information concerning large control input values, and ITAE provides quantitative information about large deviations in the tracking error signal. The quantitative data are shown in Table 1. According to the performance index ISV, it is easy to see that the approach in this paper can guarantee the control input values relatively larger. As a result, the WMR can track the reference trajectory in less time without violating the input constraints. $ITAE s$ show the smaller deviations obtained by the controller proposed in this paper.

Table 1.

Performance indexes ISV and ITAE for the WMR in Example 1.

	ISV	ITAE
		$x_{e}$	$y_{e}$	$θ_{e}$
This paper	2.2392	12.8526	3.0796	7.0991
Chen et al.	1.4472	71.2974	85.4782	18.1937

The simulation results by the controller in this paper and the controller in Chen et al. (2012) are shown in Figures 3 to 6. Figure 3 shows the tracking trajectories by the controllers proposed in this paper and in Chen et al. (2012), where the circle line in red is the reference trajectory and the blue dotted line is the actual trajectory of the WMR. From Figure 3, it is easy to see that the WMR can track the reference trajectory in a short time by the controller proposed in this paper, while the tracking task takes more time by the controller in Chen et al. (2012). The tracking errors are also shown in Figure 6. The torques of the controllers are shown in Figure 4, from which we can see that the generalized torque constraints are satisfied by both controllers. However, the maximum torque of the controller in Chen et al. (2012) is less than $1.5 N . m$ , while the maximum torque of the controller in this paper is more than $2 N . m$ , which shows the less conservativeness of the controller proposed in this paper with the quantitative analysis of the performance index ISV. The tracking errors of the velocity are also shown in Figure 5, which verifies the convergence of the velocity tracking errors.

Example 2. In this example, the reference trajectory is described as

x_{r} = 3 \sin (\frac{2 π t}{50}), y_{r} = 2 \sin (\frac{4 π t}{50})

(54)

The initial value of the reference trajectory is $[0, 0, \arctan (4 / 3)]^{T}$ and the initial value of the WMR is $[- 1, 0, 0.2 π]^{T}$ . Other simulation configurations are the same as that in Example 1.

Figure 3.

Tracking trajectory comparison in Example 1.

Figure 4.

Generalized torques comparison in Example 1.

Figure 5.

Velocity tracking errors by the controller in this paper in Example 1.

Figure 6.

Tracking errors comparison in Example 1.

The quantitative data are shown in Table 2. The performance index ISV shows that the approach in this paper can guarantee the control input values relatively larger, which results in faster convergence rates of tracking errors without violating the input constraints. $ITAE s$ show the smaller deviations obtained by the controller proposed in this paper.

Table 2.

Performance indexes ISV and ITAE for the WMR in Example 2.

	ISV	ITAE
	ISV	$x_{e}$	$y_{e}$	$θ_{e}$
This paper	1.4796	37.4977	35.0532	18.5177
Chen et al.	1.3260	189.1530	269.7274	33.0785

The simulation results by the controller in this paper and the controller in Chen et al. (2012) are shown in Figures 7 to 10. Figure 7 shows the tracking trajectories by the controllers proposed in this paper and in Chen et al. (2012). The legend description of Figure 7 is the same as that of Figure 3. From Figure 7, it is easy to see that the WMR can track the reference trajectory in a short time by the controller proposed in this paper, while the tracking task takes more time by the controller in Chen et al. (2012). The tracking errors are also shown in Figure 9. The torques of the controllers are shown in Figure 8, frow which we can see that the generalized torque constraints are satisfied by both controllers. However, the maximum torque of the controller in Chen et al. (2012) is less than $0.5 N . m$ , while the maximum torque of the controller in this paper is more than $1 N . m$ , which shows the less conservativeness of the controller proposed in this paper in combination with the analysis of the performance index ISV. The tracking errors of the velocity are also shown in Figure 10, which verifies the convergence of the velocity tracking errors.

Figure 7.

Tracking trajectory comparison in Example 2.

Figure 8.

Generalized torques comparison in Example 2.

Figure 9.

Tracking errors comparison in Example 2.

Figure 10.

Velocity tracking errors by the controller in this paper in Example 2.

Example 3. In this example, the case when $d > 0$ is considered. All the parameters are the same as in Example 1 except that the initial value is $[1, 0, 0.2 π]^{T}$ and $d = 0.25 m$ . The tracking trajectory is shown in Figure 11, the tracking errors are shown in Figure 12, the control torques are shown in Figure 13 and the velocity tracking errors are shown in Figure 14. From the simulation, we can see that the WMR can track the reference trajectory and generalized torque constraints are satisfied, which illustrates the effectiveness of the proposed controller.

Figure 11.

Tracking trajectory when $d = 0.25 m$ .

Figure 12.

Tracking errors when $d = 0.25 m$ .

Figure 13.

Control torques when $d = 0.25 m$ .

Figure 14.

Velocity tracking errors when $d = 0.25 m$ .

Conclusion

In this paper, the tracking control problem for the WMR under generalized torque constraints is investigated. A more general model is derived by taking the displacement between the centre of mass of the WMR and the midpoint of the axis of the driving wheels into consideration. A new Lyapunov function is proposed, based on which a virtual velocity controller and a generalized torque controller are designed to guarantee the stability of the tracking error systems. Less conservative conditions for tuning the controller parameters to satisfy the saturation constraints are also derived. Finally, some numerical examples illustrate the effectiveness of the proposed controller. In further research, we will consider more specific dynamic model of the WMR under disturbances and the experimental evaluation of the saturated controller on the real WMR platform.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under Grants (61773146, 61973102), by Guangdong Provincial Key Laboratory of Intelligent Decision and Cooperative Control and by the China Scholarship Council.

ORCID iD

Chaojie Shen

References

Abdelwahab

Parque

Elbab

AMF

, et al. (2020) Trajectory tracking of wheeled mobile robots using z-number based fuzzy logic. IEEE Access 8: 18426–18441.

Akka

Khaber

(2019) Optimal fuzzy tracking control with obstacles avoidance for a mobile robot based on Takagi-Sugeno fuzzy model. Transactions of the Institute of Measurement and Control 41(10): 2772–2781.

Asai

Chen

Takami

(2019) Neural network trajectory tracking of tracked mobile robot. In: Proceedings of the 2019 16th international multi-conference on systems, signals & devices (SSD), Istanbul, 21–24 March, pp. 225–230. New York: IEEE.

Ayten

Çiplak

Dumlu

(2019) Implementation a fractional-order adaptive model-based PID-type sliding mode speed control for wheeled mobile robot. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 233(8): 1067–1084.

Bai

Sun

Chen

(2020) Trajectory tracking control for wheeled mobile robots with input saturation. In: Proceedings of the 2020 7th international conference on information, cybernetics, and computational social systems (ICCSS), Guangzhou, China, 13–15 November, pp. 537–540. New York: IEEE.

Boukens

Boukabou

Chadli

(2017) Robust adaptive neural network-based trajectory tracking control approach for nonholonomic electrically driven mobile robots. Robotics and Autonomous Systems 92: 30–40.

Brockett

(1983) Asymptotic stability and feedback stabilization. Differential Geometric Control Theory 27(1): 181–191.

Chen

Gao

Ding

, et al. (2018) Trajectory tracking control of WMRs with lateral and longitudinal slippage based on active disturbance rejection control. Robotics and Autonomous Systems 107: 236–245.

Chen

(2014) Robust stabilization for a class of dynamic feedback uncertain nonholonomic mobile robots with input saturation. International Journal of Control, Automation and Systems 12(6): 1216–1224.

10.

Chen

Wang

, et al. (2009) Moving horizon H_∞ tracking control of wheeled mobile robots with actuator saturation. IEEE Transactions on Control Systems Technology 17(2): 449–457.

11.

Chen

Wang

Zhang

, et al. (2012) Saturated tracking control for nonholonomic mobile robots with dynamic feedback. Transactions of the Institute of Measurement and Control 35(2): 105–116.

12.

Chen

Ren

(2011) Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints. Automatica 47(3): 452–465.

13.

Chen

Tao

Jiang

(2014) Dynamic surface control using neural networks for a class of uncertain nonlinear systems with input saturation. IEEE Transactions on Neural Networks and Learning Systems 26(9): 2086–2097.

14.

Chen

Jia

(2014) Simple tracking controller for unicycle-type mobile robots with velocity and torque constraints. Transactions of the Institute of Measurement & Control 37(2): 211–218.

15.

Chen

Liu

, et al. (2020) Adaptive-neural-network-based trajectory tracking control for a nonholonomic wheeled mobile robot with velocity constraints. IEEE Transactions on Industrial Electronics 68(6): 5057–5067.

16.

Chwa

(2010) Tracking control of differential-drive wheeled mobile robots using a backstepping-like feedback linearization. IEEE Transactions on Systems, Man, and Cybernetics – Part A: Systems and Humans 40(6): 1285–1295.

17.

Fierro

Lewis

(1997) Control of a nonholomic mobile robot: Backstepping kinematics into dynamics. Journal of Robotic Systems 14(3): 149–163.

18.

Fierro

Lewis

(1998) Control of a nonholonomic mobile robot using neural networks. IEEE Transactions on Neural Networks 9(4): 589–600.

19.

Fukao

Nakagawa

Adachi

(2000) Adaptive tracking control of a nonholonomic mobile robot. IEEE Transactions on Robotics and Automation 16(5): 609–615.

20.

Huang

Zhou

, et al. (2019) Robust neural network–based tracking control and stabilization of a wheeled mobile robot with input saturation. International Journal of Robust and Nonlinear Control 29(2): 375–392.

21.

Huang

(2009) Control approach for tracking a moving target by a wheeled mobile robot with limited velocities. IET Control Theory & Applications 3(12): 1565–1577.

22.

Jiang

Lefeber

Nijmeijer

(2001) Saturated stabilization and tracking of a nonholonomic mobile robot. Systems & Control Letters 42(5): 327–332.

23.

Kanayama

Kimura

Miyazaki

, et al. (1990) A stable tracking control method for an autonomous mobile robot. In: Proceedings of the IEEE international conference on robotics and automation, Cincinnati, OH, 13–18 May, pp. 384–389. New York: IEEE.

24.

Khalil

(2002) Nonlinear Systems. Upper Saddle River, NJ: Prentice Hall.

25.

Klancar

Matko

Blazic

(2011) A control strategy for platoons of differential drive wheeled mobile robot. Robotics and Autonomous Systems 59(2): 57–64.

26.

Lee

Song

Lee

, et al. (2001) Tracking control of unicycle-modeled mobile robots using a saturation feedback controller. IEEE Transactions on Control Systems Technology 9(2): 305–318.

27.

Mera

Ríos

Martínez

(2020) A sliding-mode based controller for trajectory tracking of perturbed unicycle mobile robots. Control Engineering Practice 102: 104548.

28.

Miranda-Colorado

(2021) Robust observer-based anti-swing control of 2D-crane systems with load hoisting-lowering. Nonlinear Dynamics 104(4): 3581–3596.

29.

Miranda-Colorado

(2022) Observer-based saturated proportional derivative control of perturbed second-order systems: Prescribed input and velocity constraints. ISA Transactions 122: 336–345.

30.

Montoya-Villegas

Moreno-Valenzuela

Pérez-Alcocer

(2019) A feedback linearization-based motion controller for a UWMR with experimental evaluations. Robotica 37(6): 1073–1089.

31.

Ren

Beard

(2004) Trajectory tracking for unmanned air vehicles with velocity and heading rate constraints. IEEE Transactions on Control Systems Technology 12(5): 706–716.

32.

Zheng

(2010) Global asymptotic stabilization and tracking of wheeled mobile robots with actuator saturation. In: Proceedings of the 2010 IEEE international conference on robotics and biomimetics, Tianjin, China, 14–18 December, pp. 345–350. New York: IEEE.

33.

Thomas

Bandyopadhyay

Vachhani

(2021) Discrete-time sliding mode control design for unicycle robot with bounded inputs. IEEE Transactions on Circuits and Systems II: Express Briefs 68(8): 2912–2916.

34.

Wang

Iwasaki

(2021) Command filtered adaptive backstepping control for dual-motor servo systems with torque disturbance and uncertainties. IEEE Transactions on Industrial Electronics 69(2): 1773–1781.

35.

Wang

(2008) Semiglobal practical stabilization of nonholonomic wheeled mobile robots with saturated inputs. Automatica 44(3): 816–822.

36.

Wang

Liu

Yang

, et al. (2018) Trajectory tracking of an omni-directional wheeled mobile robot using a model predictive control strategy. Applied Sciences 8(2): 231.

37.

Wang

Zhang

Fang

(2021) Visual tracking of mobile robots with both velocity and acceleration saturation constraints. Mechanical Systems and Signal Processing 150: 107274.

38.

Yang

Wang

, et al. (2018) Adaptive and sliding mode tracking control for wheeled mobile robots with unknown visual parameters. Transactions of the Institute of Measurement and Control 40(1): 269–278.

39.

Yuan

Cheng

Jiang

, et al. (2014) A novel self-tuning proportional–integral–derivative controller based on a radial basis function network for trajectory tracking of service robots. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 228(3): 167–181.

40.

Zeiaee

Soltani-Zarrin

Langari

(2016) A novel approach for tracking control of differential drive robots subject to hard input constraints. In: Proceedings of the 2016 American control conference (ACC), Boston, MA, 6–8 July, pp. 2098–2103. New York: IEEE.