Saturated tracking control for nonholonomic mobile robots with dynamic feedback

Abstract

The saturated tracking control problem is addressed for nonholonomic mobile robots with dynamic feedback in this paper. A finite-time control technique and the virtual-controller-tracked method are adopted in this paper. The main contribution and innovation can be summarized as follows. First, the smooth kinematic tracking controller of Jiang et al. is taken as a virtual control law for the dynamic feedback model. Second, a continuous and bounded dynamic feedback controller is proposed to make the generalized velocity converge to the kinematic (virtual) controller in a finite time for any initial values of tracking errors in the specified attraction region. Third, all of the states of the tracking error system are proved to go to zero as time goes to infinity. In the mean time, the control inputs are bounded by the prespecified bounds at any time. Finally, the simulation results show the effectiveness of the proposed control design approach.

Keywords

Attraction region dynamic feedback inputs saturation nonholonomic mobile robots tracking control

Introduction

Owing to Brockett’s theorem (Brockett 1983), it is well known that nonholonomic systems cannot be stabilized to a point with pure smooth (or even continuous) state feedback control, although it is controllable. In order to overcome this difficulty, lots of feedback stabilization approaches have been proposed, which basically can be classified into four cases: continuous time-varying feedback control laws (Murray and Sastry 1993; Teel et al. 1992; Tian and Li 2002); discontinuous feedback control laws (Astolfi 1996; Bloch and Drakunov 1995a; Canudas de Wit and Sordalen 1992); hybrid feedback control laws (Sordalen and Egeland 1995); and optimal control laws (Hussein and Bloch 2008; Qu et al. 2006; Soueres et al. 2001). Recently, adaptive output feedback stabilization for nonholonomic systems with uncertainties or with strong nonlinear drifts were considered by Ju et al. (2009) and Zheng and Wu (2009), respectively, and adaptive stabilization of high-order nonholonomic systems with strong nonlinear drifts has been also studied in Gao et al. (2011).

Another control problem of nonholonomic systems is the trajectory tracking problem. It is not clear that the stabilization methodologies available now may be applied directly to tracking problems for nonholonomic systems, which has aroused the attention from more and more researchers. For example, based on sliding mode control, a discontinuous feedback tracking controller is given for nonholonomic dynamic systems by Bloch and Drakunov (1995b). And a new sliding-mode control method (Chwa 2004) has been obtained for tracking control of nonholonomic wheeled mobile robots in polar coordinates. As for a class of nonholonomic chained form control systems, the tracking problem is addressed by Jiang and Nijmeijer (1999) with a recursive technique. Park et al. (2009) have studied the trajectory tracking problem of nonholonomic wheeled mobile robots with model uncertainties and external disturbances by using adaptive neural sliding mode control, and in Park et al. (2010), the same authors have presented a simple adaptive control approach for trajectory tracking of electrically driven nonholonomic mobile robots with dynamic surface control methodology.

However, neither stabilization problems nor trajectory tracking problems of nonholonomic systems above have a common characteristic: the bounds of control inputs were not considered. From a practical point of view, it is important to design saturated controllers for some nonholonomic mechanical systems. That is because any actuator always has a limitation of the physical inputs. However, until now, only a few researchers have focused on the design of controllers with saturated inputs for such systems (Chen et al. 2009; Jiang et al. 2001; Luo and Tsiotras 2000; Wang 2008), etc., the control strategies of which are based on a kinematic model or switching technique (hence, discontinuous control law). Although there have also been many results for other systems with constrained inputs (for example Ailon 2010; Chen et al. 2011; He et al. 2007; Lan et al. 2006; Yu et al. 2010; Zhou et al. 2010), it is not easy to extend these results to the trajectory tracking control for the nonholonomic mobile robots based on the dynamic feedback model. Indeed, addressing the nonholonomic control problems with bounded inputs at dynamic levels is more realistic in engineering practice, where the practical torque and force are chosen as direct control inputs.

This article considers the saturated tracking control problem of nonholonomic mobile robots based on dynamic feedback. First of all, a specified attraction region $D$ is presented in advance, the size of which depends on the saturated levels of control inputs. Then, thanks to Jiang et al. (2001), its kinematic tracking controller is adopted as a virtual controller for the dynamic feedback system. Our main contributions can be summarized as the following several respects:

The dynamic feedback tracking controller is designed to make the generalized velocity converge to the virtual velocity controller in a finite time for any initial values of tracking errors in the specified attraction region.

All of the states of tracking errors are proved to go to zero as time goes to infinity.

Furthermore, the control inputs are continuous and bounded at any time by the saturated levels given in advance.

The structure of the paper is as follows: Section 2 gives a formalization of the problem considered in the paper and some proper assumptions. Section 3 states our main results including three preliminary lemmas, controller design and stability analysis. Section 4 provides an illustrative numerical example and the corresponding simulation results of the proposed methodology. Finally, a conclusion is shown in Section 5.

Problem statement

As shown in Figure 1, the two fixed rear wheels of the robot are controlled independently by motors, and a front castor wheel prevents the robot from tipping over as it moves on a plane. Assuming that the radii $r$ are identical for all of the wheels and the distance $2 R$ between the fixed wheels is a known positive constant.

Figure 1

Nonholonomic wheeled mobile robot.

Such kinds of nonholonomic wheeled mobile robots have been studied by many researchers (see Canudas de Wit et al. 1996; Jiang et al. 2001; Kolmanovsky and McClamvoch 1995, and references therein). Using the same method of Campion et al. (1996), we can obtain its kinematic model described by the following equations:

{\begin{matrix} \overset{\cdot}{x} = v \cos θ \\ \overset{\cdot}{y} = v \sin θ \\ \overset{\cdot}{θ} = ω \end{matrix}

(1)

where $(x, y)$ is the position of the mass center $P$ of the robot moving in the plane $X$ – $Y$ , $v$ is the forward linear velocity, $ω$ is the steering angular velocity and $θ$ denotes its heading angle from the horizontal axis $X$ . We usually regard $v$ and $ω$ as control inputs in this kinematic model.

The dynamic feedback model corresponding to system (1) is

{\begin{matrix} \overset{\cdot}{x} = v \cos θ \\ \overset{\cdot}{y} = v \sin θ \\ \overset{\cdot}{θ} = ω \\ \overset{\cdot}{v} = τ_{1} \\ \overset{\cdot}{ω} = τ_{2} \end{matrix}

(2)

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

As shown in Figure 2, when considering the trajectory tracking problem of nonholonomic wheeled mobile robots, we can suppose that the desired reference trajectory $(x_{r}, y_{r}, θ_{r})$ satisfies the same motion equations as system (1):

Figure 2

Trajectory tracking of nonholonomic wheeled mobile robots.

{\begin{matrix} {\overset{\cdot}{x}}_{r} = v_{r} \cos θ_{r} \\ {\overset{\cdot}{y}}_{r} = v_{r} \sin θ_{r} \\ {\overset{\cdot}{θ}}_{r} = ω_{r} \end{matrix}

where $v_{r}$ and $ω_{r}$ denote the reference line velocity and angular velocity of the reference robot. Jiang et al. (2001) addressed saturated tracking control for the kinematic model (1), while, in this paper, our control task is to design a bounded, continuous, dynamic feedback tracking controller $(τ_{1}, τ_{2})$ for (2) such that

lim_{t \to \infty} (q (t) - q_{r} (t)) = 0,

(3)

where

q (t) = (x (t), y (t), θ (t)), q_{r} (t) = (x_{r} (t), y_{r} (t), θ_{r} (t)) .

Let

q_{e} (t) = (x_{e} (t), y_{e} (t), θ_{e} (t)),

where $x_{e} (t)$ , $y_{e} (t)$ and $θ_{e} (t)$ denote the tracking errors defined by

(\begin{matrix} x_{e} \\ y_{e} \\ θ_{e} \end{matrix}) = (\begin{matrix} \cos θ & \sin θ & 0 \\ - \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix} x_{r} - x \\ y_{r} - y \\ θ_{r} - θ \end{matrix}) .

(4)

Obviously, from (4), for any value of $θ$ , formula (3) holds if and only if

lim_{t \to \infty} q_{e} (t) = 0 .

(5)

The derivative of $q_{e} (t)$ can be calculated as

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

We consider the saturated output tracking control for the dynamic system (2) in this paper (i.e. the pose and angle $q (t)$ of this robot is seen as an output vector). Hence, the stabilization problem of the tracking-error system above with dynamic feedback as follows will be discussed:

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

(6)

where the generalized torque inputs $τ_{1}$ and $τ_{2}$ are subject to the bounded constraints:

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

(7)

where $τ_{1 \max}$ and $τ_{2 \max}$ are two positive constants given in advance.

Assumption 1. The reference velocities $v_{r} (t)$ , $ω_{r} (t)$ and their derivatives ${\overset{\cdot}{v}}_{r} (t)$ , ${\overset{\cdot}{ω}}_{r} (t)$ are bounded at any time $t \in [0, \infty],$

| v_{r} (t) | \leq v_{rmax}, | ω_{r} (t) | \leq ω_{rmax}, | {\overset{\cdot}{v}}_{r} (t) | \leq {\bar{v}}_{rmax}, | {\overset{\cdot}{ω}}_{r} (t) | \leq {\bar{ω}}_{rmax},

(1a)

where $v_{rmax}$ , $ω_{rmax}$ , ${\bar{v}}_{rmax}$ , and ${\bar{ω}}_{rmax}$ are known positive constants given arbitrarily in advance. Either $v_{r} (t)$ or $ω_{r} (t)$ does not converge to zero.

Assumption 2. The saturated levels of inputs satisfy that

τ_{1 \max} > {\bar{v}}_{rmax} + q (v_{rmax} + μ_{1})^{λ_{1}} + (1 + δ) v_{rmax} (ω_{rmax} + μ_{2})^{λ_{2}}

(2a)

τ_{2 \max} > {\bar{ω}}_{rmax} + q (v_{rmax} + μ_{1})^{λ_{1}} + (1 + δ) v_{rmax} (ω_{rmax} + μ_{2})^{λ_{2}}

(3a)

where

q, δ > 0, max {1, 2 v_{rmax}} < μ_{1} < + \infty, 1 \leq μ_{2} < + \infty, 1 < λ_{1}, λ_{2} < 2

(4a)

are also known constants given arbitrarily in advance.

Remark 1. Assumption 1 is reasonable because the velocities and accelerators of practical vehicles are always bounded.

Remark 2. The right-hand sides of the inequalities of Assumption 2 are equivalent to the constraints of the upper bounds of reference velocities and accelerators allowed. However, the left-hand sides of these inequalities imply that the real torques or forces must be required to exceed these bounds. In practice, just as noted by Ailon (2010), it is significant to estimate how much lower bounds of the inputs needed in order to drive a mechanical system to track a desired trajectory with bounded force and torque inputs. For the saturated tracking control problem, as pointed out by Jiang et al. (2001) and Chen et al. (2011), it is a quite standard assumption to suppose that $τ_{1 \max} > {\bar{v}}_{rmax}$ and $τ_{2 \max} > {\bar{ω}}_{rmax}$ . Moreover, noting that the positive constant $q$ can be chosen sufficiently small $(q \to 0^{+})$ , and in the next term, $δ$ , $μ_{2}$ and $λ_{2}$ can also be selected as small as possible (simultaneously satisfying (4a)). In fact, the second term $(1 + δ) v_{rmax} (ω_{rmax} + μ_{2})^{λ_{2}}$ depends on the initial condition (i.e. the domain of attraction given in advance). Therefore, this assumption is not rigorous.

Then, the control objective is to design continuous and bounded inputs $τ_{1}$ and $τ_{2}$ such that system (6) can be asymptotically stabilized to an equilibrium point $(0, 0, 0, v_{r}, ω_{r})$ . Simultaneously, the saturated constraints (7) holds for any time $t \geq 0$ .

Main results

In this section, the main results are presented. First, the following three lemmas are given, which are needed for the proof of our conclusion later.

Lemma 1. Let

f (t_{1}) = \int_{0}^{1} \cos (s t_{1}) ds, \forall t_{1} \in R^{1}

then $f (t_{1})$ is a smooth function in $t_{1}$ and satisfies that

| f (t_{1}) | \leq 1, | f' (t_{1}) | \leq 2,

where $f' (t_{1})$ denotes the derivative of $f (t_{1})$ with respect to $t_{1}$ .

Proof. See Appendix A. □

Remark 3. The expression of function $f (t_{1})$ was first proposed by Jiang et al. (2001). This lemma gives the bounds of $f (t_{1})$ and $f' (t_{1})$ .

Lemma 2. Consider a first-order system $\overset{\cdot}{η} = u$ , where $η$ and $u \in R^{1}$ denote its state and input, respectively. Take a continuous control law

u = - ksgn (η) | η |^{α},

where $k > 0, α \in (0, 1)$ are two design parameters. Then, the closed-loop system can be described as

\overset{\cdot}{η} = - ksgn (η) | η |^{α}

(8)

The solution of system (8) converges to zero as $t \to T_{0}$ and $η (t) \equiv 0$ for any $t \geq T_{0}$ , where

T_{0} = \frac{V^{β} (0)}{k β 2^{1 - β}}, β = \frac{1 - α}{2}, V (0) = \frac{η^{2} (0)}{2} .

Proof. See Appendix B. □

Remark 4. As illustrated by the system (8) of Bhat and Bernstein (1998), the right-hand side of (8) is continuous everywhere and locally Lipschitz everywhere except the origin. Therefore, every initial condition in $ℝ \ {0}$ has a unique solution in forward time.

Lemma 3. Let $z (t) \in R^{1}$ be a continuous function, for any given positive constants $C_{1}, C_{2}$ and $\tilde{T}$ , if

z^{2} (t) \leq C_{1} + C_{2} \int_{0}^{\tilde{T}} | z (ζ) | d ζ, \forall t \geq 0

(9)

then, an upper bound of $| z (t) |$ can be obtained as follows:

| z (t) | < C_{2} \tilde{T} + \sqrt{C_{1}}, \forall t \geq 0 .

(10)

Proof. See Appendix C. □

Next, we give our main results.

Theorem 1. For arbitrarily given positive constants $v_{rmax}$ , $ω_{rmax}$ , ${\bar{v}}_{rmax}$ , ${\bar{ω}}_{rmax}$ satisfying (1a) in Assumption 1, choose $q$ , $δ$ , $μ_{1}$ , $μ_{2}$ , $λ_{1}$ , $λ_{2}$ , $τ_{1 \max}$ and $τ_{2 \max}$ to satisfy (2a)–(4a) in Assumption 2. Again, for any given sufficiently small positive numbers $p_{1}$ , $p_{2}$ and $p_{3} (< q)$ , choose $M$ and positive design parameters $k_{j} (j = 1, 2, \dots, 5)$ , $l_{1}$ , $l_{2}$ , $α_{1}$ , $α_{2}$ , $β_{1}$ , $β_{2}$ such that

{\begin{matrix} q {(v_{rmax + μ_{1}})}^{λ_{1}} + (1 + δ) v_{rmax} {(ω_{rmax} + μ_{2})}^{λ_{2}} < M < min {τ_{1 \max} - {\bar{v}}_{rmax}, τ_{2 \max} - {\bar{ω}}_{rmax}} \\ 2 - λ_{i} \leq α_{i} < 1, β_{i} = \frac{1 - α_{i}}{2}, i = 1, 2 \\ 2 v_{rmax} + k_{1} \leq μ_{1} \\ \frac{1}{2} k_{2} v_{rmax} + k_{3} \leq μ_{2} \\ k_{2} \leq min {p_{1}, p_{2}, p_{1}^{2}, p_{2}^{2}, \frac{2 p_{3}}{v_{rmax}}} \\ k_{2} {(1 + \frac{1}{2 k_{4} β_{1}})}^{2} \leq 2 p_{3}^{2} \\ \frac{1}{k_{5} β_{2}} \leq min {(1 + δ) v_{rmax}, \frac{2 (1 + δ) v_{rmax}}{M}} \\ \frac{k_{2}}{4 k_{4} β_{1}} \leq p_{3} \\ \frac{1}{2} k_{1} l_{1} e^{\frac{2}{k_{2}} M} \leq δ v_{rmax} \\ k_{1} l_{1} + k_{4} \leq p_{3} \\ \frac{k_{2}}{2} v_{rmax}^{2} + k_{3} v_{rmax} + \frac{1}{2} k_{1} l_{1} e^{\frac{2}{k_{2}} M} (ω_{rmax} + \frac{k_{2}}{2} v_{rmax} + k_{3}) + k_{1}^{2} l_{1} \leq τ_{1 \max} - {\bar{v}}_{rmax} - M \\ \frac{3}{2} k_{2} v_{rmax} + k_{3} l_{2} + k_{5} \leq (1 + δ) v_{rmax} \\ \frac{1}{2} k_{2} {\bar{v}}_{rmax} + k_{2} v_{rmax} (\frac{1}{2} ω_{rmax} + \frac{1}{4} k_{2} v_{rmax} + \frac{1}{2} k_{3} + v_{rmax} + \frac{k_{2} v_{rmax}}{2} + k_{3} + \frac{k_{1} + v_{rmax}}{2}) \\ + k_{3} l_{2} (\frac{1}{2} k_{2} v_{rmax} + k_{3}) \leq τ_{2 \max} - {\bar{ω}}_{rmax} - M . \end{matrix}

(11)

If the initial tracking error of system (6) belongs to a regional $D$ defined by

D \overset{Δ}{=} {(x_{e} (0), y_{e} (0), θ_{e} (0), v_{e} (0), ω_{e} (0)) \in R^{5} | p_{1} | x_{e} (0) | + p_{2} | y_{e} (0) | + | θ_{e} (0) | + \frac{1}{2} θ_{e}^{2} (0)

+ p_{3} (| v_{e} (0) | + μ_{1})^{λ_{1}} + (1 + δ) v_{rmax} (| ω_{e} (0) | + μ_{2})^{λ_{2}} \leq M},

where $v_{e} (0)$ and $ω_{e} (0)$ are the initial velocity errors,

v_{e} (0) = v_{r} (0) - v (0), ω_{e} (0) = ω_{r} (0) - ω (0) .

Suppose that $v_{d}$ and $ω_{d}$ are two smooth kinematic reference velocities, $ξ_{1}$ and $ξ_{2}$ are two auxiliary state variables:

{\begin{matrix} v_{d} = v_{r} \cos θ_{e} + k_{1} \tanh (l_{1} x_{e}) \\ ω_{d} = ω_{r} + \frac{k_{2} v_{r} y_{e}}{1 + x_{e}^{2} + y_{e}^{2}} f (θ_{e}) + k_{3} \tanh (l_{2} θ_{e}) \\ ξ_{1} = v - v_{d} \\ ξ_{2} = ω - ω_{d} \end{matrix}

(12)

where $f (θ_{e}) = \int_{0}^{1} \cos (s θ_{e}) ds$ is a smooth function with respect to $θ_{e}$ . Take the following continuous state feedback laws:

{\begin{matrix} τ_{1} = {\overset{\cdot}{v}}_{d} - k_{4} sgn (ξ_{1}) | ξ_{1} |^{α_{1}} \\ τ_{2} = {\overset{\cdot}{ω}}_{d} - k_{5} sgn (ξ_{2}) | ξ_{2} |^{α_{2}} . \end{matrix}

(13)

Then, the closed-loop system of the tracking errors composed by (6) and (13) can be asymptotically stabilized to the equilibrium point $(0, 0, 0, v_{r}, ω_{r})$ . In the meantime, the input signals are saturated in any time, i.e. $| τ_{1} (t) | \leq τ_{1 \max}$ , $| τ_{2} (t) | \leq τ_{2 \max}$ , $\forall t \geq 0$ .

Proof. From (6), (12) and (13), it yields

{\begin{matrix} {\overset{\cdot}{x}}_{e} = (ξ_{2} + ω_{d}) y_{e} - (ξ_{1} + v_{d}) + v_{r} \cos θ_{e} \\ {\overset{\cdot}{y}}_{e} = - (ξ_{2} + ω_{d}) x_{e} + v_{r} \sin θ_{e} \\ {\overset{\cdot}{θ}}_{e} = ω_{r} - (ξ_{2} + ω_{d}) \\ {\overset{\cdot}{ξ}}_{1} = - k_{4} sgn (ξ_{1}) | ξ_{1} |^{α_{1}} \\ {\overset{\cdot}{ξ}}_{2} = - k_{5} sgn (ξ_{2}) | ξ_{2} |^{α_{2}} \end{matrix}

(14)

According to Lemma 2, there exist two positive constants

T_{1} = \frac{{[\frac{1}{2} ξ_{1}^{2} (0)]}^{β_{1}}}{k_{4} β_{1} 2^{1 - β_{1}}} = \frac{| ξ_{1} {(0) |}^{2 β_{1}}}{2 k_{4} β_{1}}, T_{2} = \frac{{[\frac{1}{2} ξ_{2}^{2} (0)]}^{β_{2}}}{k_{5} β_{2} 2^{1 - β_{2}}} = \frac{| ξ_{2} {(0) |}^{2 β_{2}}}{2 k_{5} β_{2}}

such that

ξ_{1} (t) \equiv 0, t \geq T_{1}, and ξ_{2} (t) \equiv 0, t \geq T_{2} .

(15)

Then $q_{e} (t) \to 0$ as $t \to \infty$ (see Jiang et al. 2001, for details). From (12) we have $v_{d} (t) \to v_{r} (t)$ , $ω_{d} (t) \to ω_{r} (t)$ as $t \to \infty$ . Hence, $v (t) \to v_{r} (t)$ , $ω (t) \to ω_{r} (t)$ as $t \to \infty$ .

By Lemmas 1 and 2, it is clear that the state feedback law (13) is continuous. Next, we prove that the control inputs (13) satisfy the saturation constraints (7) in any time. Consider an auxiliary function that is positive definite and radially unbounded with respect to $(x_{e}, y_{e}, θ_{e})$ ,

V_{1} (t) = \frac{k_{2}}{2} \ln (1 + x_{e}^{2} (t) + y_{e}^{2} (t)) + \frac{1}{2} θ_{e}^{2} (t) .

(16)

Taking the time derivative of $V_{1} (t)$ along (14), we have

{\overset{\cdot}{V}}_{1} (t) = k_{2} \frac{x_{e} {\overset{\cdot}{x}}_{e} + y_{e} {\overset{\cdot}{y}}_{e}}{1 + x_{e}^{2} + y_{e}^{2}} + θ_{e} {\overset{\cdot}{θ}}_{e}

= k_{2} \frac{x_{e} (ξ_{2} + ω_{d}) y_{e} - x_{e} (ξ_{1} + v_{d}) + x_{e} v_{r} \cos θ_{e} - y_{e} (ξ_{2} + ω_{d}) x_{e} + y_{e} v_{r} \sin θ_{e}}{1 + x_{e}^{2} + y_{e}^{2}} + θ_{e} (ω_{r} - (ξ_{2} + ω_{d})) .

Substituting the virtual control laws (12) into the formula above, yields

{\overset{\cdot}{V}}_{1} (t) = k_{2} \frac{- x_{e} (ξ_{1} + v_{r} \cos θ_{e} + k_{1} \tanh (l_{1} x_{e})) + x_{e} v_{r} \cos θ_{e} + y_{e} v_{r} \sin θ_{e}}{1 + x_{e}^{2} + y_{e}^{2}}

+ θ_{e} [ω_{r} - (ξ_{2} + ω_{r} + \frac{k_{2} v_{r} y_{e} f (θ_{e})}{1 + x_{e}^{2} + y_{e}^{2}} + k_{3} \tanh (l_{2} θ_{e}))] .

Noting that $θ_{e} f (θ_{e}) = \sin θ_{e}$ , the formula above can be simplified as

{\overset{\cdot}{V}}_{1} (t) = - k_{1} k_{2} \frac{x_{e} \tanh (l_{1} x_{e})}{1 + x_{e}^{2} + y_{e}^{2}} - k_{3} θ_{e} \tanh (l_{2} θ_{e}) - k_{2} \frac{x_{e} ξ_{1}}{1 + x_{e}^{2} + y_{e}^{2}} - θ_{e} ξ_{2} .

Because

x_{e} \tanh (l_{1} x_{e}) \geq 0, θ_{e} \tanh (l_{2} θ_{e}) \geq 0, \frac{| x_{e} |}{1 + x_{e}^{2} + y_{e}^{2}} \leq \frac{1}{2},

we have

{\overset{\cdot}{V}}_{1} (t) \leq \frac{k_{2}}{2} | ξ_{1} (t) | + | θ_{e} (t) | \cdot | ξ_{2} (t) | .

Integrating it from 0 to $t$ , it has

V_{1} (t) \leq V_{1} (0) + \int_{0}^{t} (\frac{k_{2}}{2} | ξ_{1} (ρ) | + | θ_{e} (ρ) | \cdot | ξ_{2} (ρ) |) d ρ \leq V_{1} (0) + \frac{k_{2}}{2} \int_{0}^{T_{1}} | ξ_{1} (ρ) | d ρ + \int_{0}^{T_{2}} | θ_{e} (ρ) | \cdot | ξ_{2} (ρ) | d ρ, \forall t \geq 0 .

(17)

The reason is explained below. If $t \leq T_{i}$ ( $i = 1, 2$ ), then

\int_{0}^{t} | ξ_{i} (ρ) | d ρ \leq \int_{0}^{T_{i}} | ξ_{i} (ρ) | d ρ .

(18)

On the other hand, if $t > T_{i}$ , according to (15), we have $ξ_{i} (ρ) \equiv 0$ as $ρ \geq T_{i}$ , so (18) is also valid. Thus, (17) holds clearly.

By using (16), we have

\frac{θ_{e}^{2} (t)}{2} \leq V_{1} (t) \leq V_{1} (0) + \frac{k_{2}}{2} \int_{0}^{T_{1}} | ξ_{1} (ρ) | d ρ + \int_{0}^{T_{2}} | θ_{e} (ρ) | \cdot | ξ_{2} (ρ) | d ρ, \forall t \geq 0 .

(19)

From (14), it can be seen that $| ξ_{i} (t) |$ ( $i = 1, 2$ ) is a monotonically decreasing function. Hence, (19) can be rewritten as

θ_{e}^{2} (t) \leq 2 V_{1} (0) + k_{2} | ξ_{1} (0) | T_{1} + 2 | ξ_{2} (0) | \int_{0}^{T_{2}} | θ_{e} (ρ) | d ρ, \forall t \geq 0 .

Let

C_{1} = 2 V_{1} (0) + k_{2} | ξ_{1} (0) | T_{1}, C_{2} = 2 | ξ_{2} (0) |, \tilde{T} = T_{2} .

From Lemma 3, we can obtain

| θ_{e} (t) | < 2 | ξ_{2} (0) | T_{2} + \sqrt{2 V_{1} (0) + k_{2} | ξ_{1} (0) | T_{1}}

\leq 2 | ξ_{2} (0) | T_{2} + \sqrt{k_{2} (x_{e}^{2} (0) + y_{e}^{2} (0)) + θ_{e}^{2} (0) + \frac{k_{2}}{2} (ξ_{1}^{2} (0) + T_{1}^{2})} .

Applying a simple inequality described by

\sqrt{\sum_{i = 1}^{n} a_{i}} \leq \sum_{i = 1}^{n} \sqrt{a_{i}}, a_{i} \geq 0, i = 1, 2, \dots, n,

we have

| θ_{e} (t) | < 2 | ξ_{2} (0) | T_{2} + \sqrt{k_{2}} (| x_{e} (0) | + | y_{e} (0) | + \frac{| ξ_{1} (0) | + T_{1}}{\sqrt{2}}) + | θ_{e} (0) | .

(20)

Substituting the expressions of $T_{1}$ and $T_{2}$ into (20),

| θ_{e} (t) | < \frac{| ξ_{2} {(0) |}^{2 β_{2} + 1}}{k_{5} β_{2}} + \sqrt{k_{2}} (| x_{e} (0) | + | y_{e} (0) | + \frac{| ξ_{1} (0) |}{\sqrt{2}} + \frac{| ξ_{1} {(0) |}^{2 β_{1}}}{2 \sqrt{2} k_{4} β_{1}}) + | θ_{e} (0) |

(21)

Noting that,

| ξ_{1} (0) | = | v (0) - v_{d} (0) | \leq | v_{e} (0) | + | v_{r} (0) | + (| v_{r} (0) | + k_{1}) \leq | v_{e} (0) | + 2 v_{rmax} + k_{1} .

By the third inequality in (11), we can obtain

| ξ_{1} (0) | \leq | v_{e} (0) | + μ_{1} .

(22)

Similarly, we have

| ξ_{2} (0) | = | ω (0) - ω_{d} (0) | = | ω (0) - ω_{r} (0) - \frac{k_{2} v_{r} (0) y_{e} (0)}{1 + x_{e}^{2} (0) + y_{e}^{2} (0)} f (θ_{e} (0)) - k_{3} \tanh (l_{2} θ_{e} (0)) |

Because

\frac{| y_{e} (0) |}{1 + x_{e}^{2} (0) + y_{e}^{2} (0)} \leq \frac{1}{2}, | f (θ_{e} (0)) | \leq 1, | \tanh (l_{2} θ_{e} (0)) | \leq 1,

(23)

and from the fourth inequality in (11), we have

| ξ_{2} (0) | \leq | ω_{e} (0) | + \frac{1}{2} k_{2} v_{rmax} + k_{3} \leq | ω_{e} (0) | + μ_{2} .

(24)

Then according to Assumption 2 and the second inequality in (11), from (22) and (24), we can obtain

{\begin{matrix} | ξ_{1} {(0) |}^{2 β_{1} + 1} \leq {(| v_{e} (0) | + μ_{1})}^{2 β_{1} + 1} \leq {(| v_{e} (0) | + μ_{1})}^{λ_{1}} \\ | ξ_{2} {(0) |}^{2 β_{2} + 1} \leq {(| ω_{e} (0) | + μ_{2})}^{2 β_{2} + 1} \leq {(| ω_{e} (0) | + μ_{2})}^{λ_{2}} . \end{matrix}

(25)

So, from (21), we have

| θ_{e} (t) | < \frac{{(ω_{e} (0) + μ_{2})}^{λ_{2}}}{k_{5} β_{2}} + \sqrt{k_{2}} [| x_{e} (0) | + | y_{e} (0) | + \frac{1}{\sqrt{2}} (1 + \frac{1}{2 k_{4} β_{1}}) {(| v_{e} (0) | + μ_{1})}^{λ_{1}}] + | θ_{e} (0) | .

From the fifth to the seventh inequality in (11), we have

\sqrt{k_{2}} \leq min {p_{1}, p_{2}}, \sqrt{\frac{k_{2}}{2}} (1 + \frac{1}{2 k_{4} β_{1}}) \leq p_{3}, \frac{1}{k_{5} β_{2}} \leq (1 + δ) v_{rmax}

it implies that $| θ_{e} (t) | \leq M$ . According to the inequalities (fifth, seventh and eighth) in (11), we have

k_{2} \leq min {p_{1}, p_{2}}, \frac{k_{2}}{4 k_{4} β_{1}} \leq p_{3}, \frac{M}{2 k_{5} β_{2}} \leq (1 + δ) v_{rmax} .

By (17), (25) and the three inequalities above, we can obtain

V_{1} (t) \leq V_{1} (0) + \frac{1}{2} k_{2} | ξ_{1} (0) | T_{1} + | ξ_{2} (0) | M T_{2} = k_{2} \ln \sqrt{1 + x_{e}^{2} (0) + y_{e}^{2} (0)} + \frac{1}{2} θ_{e}^{2} (0) + \frac{k_{2} | ξ_{1} {(0) |}^{2 β_{1} + 1}}{4 k_{4} β_{1}} + M \frac{| ξ_{2} {(0) |}^{2 β_{2} + 1}}{2 k_{5} β_{2}} \leq k_{2} (| x_{e} (0) | + | y_{e} (0) |) + \frac{1}{2} θ_{e}^{2} (0) + \frac{k_{2} {(| v_{e} (0) + μ_{1} |)}^{λ_{1}}}{4 k_{4} β_{1}} + M \frac{{(ω_{e} (0) + μ_{2})}^{λ_{2}}}{2 k_{5} β_{2}} \leq M

Thus,

\frac{k_{2}}{2} \ln (1 + x_{e}^{2} (t) + y_{e}^{2} (t)) \leq V_{1} (t) \leq M, i . e . 1 + x_{e}^{2} (t) + y_{e}^{2} (t) \leq e^{\frac{2 M}{k_{2}}},

the bounds of $| x_{e} (t) |$ and $| y_{e} (t) |$ are given by

| x_{e} (t) | \leq \frac{1 + x_{e}^{2} (t)}{2} \leq \frac{1}{2} e^{\frac{2 M}{k_{2}}} \overset{Δ}{=} x_{emax}, | y_{e} (t) | \leq \frac{1 + y_{e}^{2} (t)}{2} \leq \frac{1}{2} e^{\frac{2 M}{k_{2}}} \overset{Δ}{=} y_{emax} .

(26)

Next, we consider the boundedness of ${\overset{\cdot}{v}}_{d} (t)$ and ${\overset{\cdot}{ω}}_{d} (t)$ .

| {\overset{\cdot}{v}}_{d} | = | {\overset{\cdot}{v}}_{r} \cos θ_{e} - v_{r} \sin θ_{e} {\overset{\cdot}{θ}}_{e} + k_{1} l_{1} {\overset{\cdot}{x}}_{e} \sec h^{2} (l_{1} x_{e}) |

= | {\overset{\cdot}{v}}_{r} \cos θ_{e} - v_{r} \sin θ_{e} [ω_{r} - (ξ_{2} + ω_{r} + \frac{k_{2} v_{r} y_{e}}{1 + x_{e}^{2} + y_{e}^{2}} f (θ_{e}) + k_{3} \tanh (l_{2} θ_{e}))] + k_{1} l_{1} [(ξ_{2} + ω_{r} + \frac{k_{2} v_{r} y_{e}}{1 + x_{e}^{2} + y_{e}^{2}} f (θ_{e}) + k_{3} \tanh (l_{2} θ_{e})) y_{e} - (ξ_{1} + v_{r} \cos θ_{e} + k_{1} \tanh (l_{1} x_{e})) + v_{r} \cos θ_{e}] \sec h^{2} (l_{1} x_{e}) |

where $\sec h^{2} (s) = \frac{d}{ds} (\tanh (s))$ for any $s \in R^{1}$ . Noting that

| f (θ_{e}) | \leq 1, | f' (θ_{e}) | \leq 2, | \tanh (s) | \leq 1, | \sec h^{2} (s) | \leq 1,

(27)

we have

| {\overset{\cdot}{v}}_{d} | \leq {\bar{v}}_{rmax} + v_{rmax} (| ξ_{2} (0) | + \frac{k_{2}}{2} v_{rmax} + k_{3}) + k_{1} l_{1} [(| ξ_{2} (0) | + ω_{rmax} + \frac{k_{2}}{2} v_{rmax} + k_{3}) y_{emax} + | ξ_{1} (0) | + k_{1}]

\leq {\bar{v}}_{rmax} + v_{rmax} (| ω_{e} (0) | + μ_{2} + \frac{k_{2}}{2} v_{rmax} + k_{3}) + k_{1} l_{1} [(| ω_{e} (0) | + μ_{2} + ω_{rmax} + \frac{k_{2}}{2} v_{rmax} + k_{3}) y_{emax}

+ | v_{e} (0) | + μ_{1} + k_{1}] .

Similarly, we have

| {\overset{\cdot}{ω}}_{d} | = | {\overset{\cdot}{ω}}_{r} + k_{2} \frac{d}{dt} (\frac{v_{r} y_{e} f (θ_{e})}{1 + x_{e}^{2} + y_{e}^{2}}) + k_{3} l_{2} {\overset{\cdot}{θ}}_{e} \sec h^{2} (l_{2} θ_{e}) |

= | {\overset{\cdot}{ω}}_{r} + k_{2} \frac{{\overset{\cdot}{v}}_{r} y_{e} f (θ_{e}) + v_{r} {\overset{\cdot}{y}}_{e} f (θ_{e}) + v_{r} y_{e} {\overset{\cdot}{θ}}_{e} f' (θ_{e})}{1 + x_{e}^{2} + y_{e}^{2}} - 2 k_{2} \frac{v_{r} y_{e} f (θ_{e}) (x_{e} {\overset{\cdot}{x}}_{e} + y_{e} {\overset{\cdot}{y}}_{e})}{{(1 + x_{e}^{2} + y_{e}^{2})}^{2}} + k_{3} l_{2} {\overset{\cdot}{θ}}_{e} \sec h^{2} (l_{2} θ_{e}) |

= | {\overset{\cdot}{ω}}_{r} + \frac{k_{2}}{1 + x_{e}^{2} + y_{e}^{2}} {{\overset{\cdot}{v}}_{r} y_{e} f (θ_{e}) + v_{r} [- (ξ_{2} + ω_{r} + \frac{k_{2} v_{r} y_{e}}{1 + x_{e}^{2} + y_{e}^{2}} f (θ_{e}) + k_{3} \tanh (l_{2} θ_{e})) x_{e}

+ v_{r} \sin θ_{e}] f (θ_{e}) + v_{r} y_{e} [ω_{r} - (ξ_{2} + ω_{r} + \frac{k_{2} v_{r} y_{e}}{1 + x_{e}^{2} + y_{e}^{2}} f (θ_{e}) + k_{3} \tanh (l_{2} θ_{e}))] f' (θ_{e})}

- 2 k_{2} \frac{v_{r} y_{e} f (θ_{e}) [- x_{e} (ξ_{1} + v_{r} \cos θ_{e} + k_{1} \tanh (l_{1} x_{e})) + x_{e} v_{r} \cos θ_{e} + y_{e} v_{r} \sin θ_{e}]}{{(1 + x_{e}^{2} + y_{e}^{2})}^{2}}

+ k_{3} l_{2} [ω_{r} - (ξ_{2} + ω_{r} + \frac{k_{2} v_{r} y_{e}}{1 + x_{e}^{2} + y_{e}^{2}} f (θ_{e}) + k_{3} \tanh (l_{2} θ_{e}))] \sec h^{2} (l_{2} θ_{e}) | .

Once again applying (23), (27) and from Assumption 1, we can obtain

| {\overset{\cdot}{ω}}_{d} | \leq {\bar{ω}}_{rmax} + k_{2} [\frac{1}{2} {\bar{v}}_{rmax} + v_{rmax} (\frac{| ξ_{2} (0) | + ω_{rmax}}{2} + \frac{k_{2} v_{rmax}}{4} + \frac{1}{2} k_{3} + v_{rmax})]

+ k_{2} v_{rmax} (| ξ_{2} (0) | + \frac{1}{2} k_{2} v_{rmax} + k_{3}) + \frac{k_{2}}{2} v_{rmax} (| ξ_{1} (0) | + k_{1} + v_{rmax})

+ k_{3} l_{2} (| ξ_{2} (0) | + \frac{1}{2} k_{2} v_{rmax} + k_{3})

Substituting (22) and (24) into the formula above, it becomes

| {\overset{\cdot}{ω}}_{d} | \leq {\bar{ω}}_{rmax} + k_{2} [\frac{1}{2} {\bar{v}}_{rmax} + v_{rmax} (\frac{| ω_{e} (0) | + μ_{2} + ω_{rmax}}{2} + \frac{k_{2}}{4} v_{rmax} + \frac{k_{3}}{2} + v_{rmax})]

+ k_{2} v_{rmax} (| ω_{e} (0) | + μ_{2} + \frac{k_{2}}{2} v_{rmax} + k_{3}) + \frac{k_{2}}{2} v_{rmax} (| v_{e} (0) | + μ_{1} + k_{1} + v_{rmax})

+ k_{3} l_{2} (| ω_{e} (0) | + μ_{2} + \frac{k_{2}}{2} v_{rmax} + k_{3})

Finally, we obtain the saturability of $τ_{1}$ and $τ_{2}$ :

| τ_{1} | \leq | {\overset{\cdot}{v}}_{d} | + k_{4} | ξ_{1} |^{α_{1}}

\leq {\bar{v}}_{rmax} + v_{rmax} (| ω_{e} (0) | + μ_{2} + \frac{k_{2}}{2} v_{rmax} + k_{3}) + k_{1} l_{1} [(| ω_{e} (0) | + μ_{2} + ω_{rmax} + \frac{k_{2}}{2} v_{rmax} + k_{3}) y_{emax}

+ | v_{e} (0) | + μ_{1} + k_{1}] + k_{4} (| v_{e} (0) | + μ_{1})^{λ_{1}}

From (22), (24) and (26), it follows that

| τ_{1} | \leq (k_{1} l_{1} + k_{4}) (| v_{e} (0) | + μ_{1})^{λ_{1}} + (v_{rmax} + \frac{1}{2} k_{1} l_{1} e^{\frac{2 M}{k_{2}}}) (| ω_{e} (0) | + μ_{2})^{λ_{2}} + {\bar{v}}_{rmax}

+ \frac{k_{2}}{2} v_{rmax}^{2} + k_{3} v_{rmax} + \frac{1}{2} k_{1} l_{1} e^{\frac{2 M}{k_{2}}} (ω_{rmax} + \frac{k_{2}}{2} v_{rmax} + k_{3}) + k_{1}^{2} l_{1} .

From these inequalities (9th to 11th) in (11), it can be seen that $| τ_{1} (t) | \leq τ_{1 \max}$ for all $t \geq 0$ . In the same way, we can obtain that

| τ_{2} | \leq | {\overset{\cdot}{ω}}_{d} | + k_{5} | ξ_{2} |^{α_{2}}

\leq {\bar{ω}}_{rmax} + k_{2} [\frac{1}{2} {\bar{v}}_{rmax} + v_{rmax} (\frac{| ω_{e} (0) | + μ_{2} + ω_{rmax}}{2} + \frac{k_{2}}{4} v_{rmax} + \frac{k_{3}}{2} + v_{rmax})]

+ k_{2} v_{rmax} (| ω_{e} (0) | + μ_{2} + \frac{k_{2}}{2} v_{rmax} + k_{3}) + \frac{k_{2}}{2} v_{rmax} (| v_{e} (0) | + μ_{1} + k_{1} + v_{rmax})

+ k_{3} l_{2} (| ω_{e} (0) | + μ_{2} + \frac{k_{2}}{2} v_{rmax} + k_{3}) + k_{5} (| ω_{e} (0) | + μ_{2})^{λ_{2}} .

Taking into account the range of $μ_{i}$ and $λ_{i}$ ( $i = 1, 2$ ), it has

| τ_{2} | \leq \frac{k_{2}}{2} v_{rmax} (| v_{e} (0) | + μ_{1})^{λ_{1}} + (\frac{3}{2} k_{2} v_{rmax} + k_{3} l_{2} + k_{5}) (| ω_{e} (0) | + μ_{2})^{λ_{2}} + {\bar{ω}}_{rmax} + \frac{1}{2} k_{2} {\bar{v}}_{rmax} + k_{2} v_{rmax}

(\frac{1}{2} ω_{rmax} + \frac{k_{2}}{4} v_{rmax} + \frac{k_{3}}{2} + v_{rmax} + \frac{k_{2} v_{rmax}}{2} + k_{3} + \frac{k_{1} + v_{rmax}}{2}) + k_{3} l_{2} (\frac{k_{2}}{2} v_{rmax} + k_{3}) .

Hence, according to the 5th, 12th and 13th inequality in (11), we also have $| τ_{2} (t) | \leq τ_{2 \max}$ for any $t \geq 0$ . □

Remark 5. The special attraction region $D$ is a well-defined, nonempty convex–compact set, the size of which depends on the saturated bounds of control inputs. It can be seen from the first inequality in (11) that $M$ can be chosen as large as possible if the bounds of the control inputs ( $τ_{1 \max}, τ_{2 \max}$ ) could be allowed to be selected very large, which means that the covered radius of domain $D$ may be very large. For example, when the initial velocities are equal to zero, i.e. $v (0) = 0$ and $ω (0) = 0$ , the special attraction region can be simplified as

D \overset{Δ}{=} {(x_{e} (0), y_{e} (0), θ_{e} (0)) \in R^{3} | p_{1} | x_{e} (0) | + p_{2} | y_{e} (0) | + | θ_{e} (0) | + \frac{1}{2} θ_{e}^{2} (0) \leq M - m}

where

m = q (v_{rmax + μ_{1}})^{λ_{1}} + (1 + δ) v_{rmax} (ω_{rmax} + μ_{2})^{λ_{2}} .

According to the first inequality in (11), it is obvious that $M > m$ . For simplicity, take $p_{1} = p_{2} = 0.1$ , $M - m = 1$ . In the case of $v (0) = 0, ω (0) = 0$ , Figure 3 shows that the special attraction region $D$ is a well-defined, nonempty convex–compact set.

Figure 3

The special attraction region $D$ with $v (0) = 0$ , $ω (0) = 0$ .

Remark 6. Although it is difficult to find the set of all solutions for the inequalities (11), a group of appropriate parameters can be obtained one by one according to the following method. First, for arbitrarily given $1 < λ_{1}, λ_{2} < 2$ , we can select $α_{i}$ and $β_{i}$ $(i = 1, 2)$ such that the second inequality in (11) holds. Then, for arbitrarily given positive constants $v_{rmax}$ , $ω_{rmax}$ , ${\bar{v}}_{rmax}$ , ${\bar{ω}}_{rmax}$ satisfying (1a) in Assumption 1, choose $q$ , $δ$ , $μ_{1}$ , $μ_{2}$ , $λ_{1}$ , $λ_{2}$ , $τ_{1 \max}$ and $τ_{2 \max}$ to satisfy (2a)–(4a) in Assumption 2. Also, $δ$ satisfies the following condition,

(1 + δ)^{2} > \frac{1}{β_{2} v_{rmax}^{2} min {1, \frac{2}{min {τ_{1 \max} - {\bar{v}}_{rmax}, τ_{2 \max} - {\bar{ω}}_{rmax}}}}} .

(r1)

Choosing $M$ to satisfy the first inequality in (11). For any given sufficiently small positive numbers $p_{1}$ , $p_{2}$ and $p_{3} (< q)$ , other parameters can be taken by the following steps.

Step 1. From the first inequality in (11), we have

min {1, \frac{2}{M}} \geq min {1, \frac{2}{min {τ_{1 \max} - {\bar{v}}_{rmax}, τ_{2 \max} - {\bar{ω}}_{rmax}}}}

Then, the formula (r1) means that

(1 + δ)^{2} > \frac{1}{β_{2} v_{rmax}^{2} min {1, \frac{2}{M}}} .

It can be rewritten as

\frac{1}{β_{2} min {(1 + δ) v_{rmax}, \frac{2 (1 + δ) v_{rmax}}{M}}} < (1 + δ) v_{rmax} .

From the 7th and the 12th inequalities in (11), $k_{5}$ is taken such that

\frac{1}{β_{2} min {(1 + δ) v_{rmax}, \frac{2 (1 + δ) v_{rmax}}{M}}} \leq k_{5} < (1 + δ) v_{rmax} .

Step 2. From the 10th inequality in (11), we choose $k_{4}$ satisfying that $k_{4} < p_{3}$ .

Step 3. From the 4th and the 11th inequalities in (11), we choose $k_{3}$ satisfying that

k_{3} < min {μ_{2}, \frac{τ_{1 \max} - {\bar{v}}_{rmax} - M}{v_{rmax}}} .

Step 4. According to the 12th and the 13th inequalities in (11), $l_{2}$ can be selected by

k_{3} l_{2} < (1 + δ) v_{rmax} - k_{5}, k_{3}^{2} l_{2} < τ_{2 \max} - {\bar{ω}}_{rmax} - M .

Step 5. In this step, $k_{2}$ will be chosen. For the 11th inequality in (11), let

k_{2, 11} \overset{Δ}{=} \frac{2 (τ_{1 \max} - {\bar{v}}_{rmax} - M - k_{3} v_{rmax})}{v_{rmax}^{2}} .

As for the 12th inequality in (11), let

k_{2, 12} \overset{Δ}{=} \frac{2 ((1 + δ) v_{rmax} - k_{5} - k_{3} l_{2})}{3 v_{rmax}} .

Suppose for a moment that $k_{2} \leq 1$ . For the 13th inequality in (11), let

k_{2, 13} \overset{Δ}{=} \frac{τ_{2 \max} - {\bar{ω}}_{rmax} - M - k_{3}^{2} l_{2}}{\frac{{\bar{v}}_{rmax}}{2} + v_{rmax} (\frac{ω_{rmax}}{2} + \frac{v_{rmax}}{4} + \frac{k_{3}}{2} + v_{rmax} + \frac{v_{rmax}}{2} + k_{3} + \frac{v_{rmax}}{2}) + \frac{k_{3} l_{2} v_{rmax}}{2}} .

Then, by the 4th, 5th, 6th, 8th, 11th, 12th and the 13th inequalities in (11), $k_{2}$ can be chosen to satisfy that

k_{2} < min {1, \frac{2 (μ_{2} - k_{3})}{v_{rmax}}, p_{1}, p_{2}, p_{1}^{2}, p_{2}^{2}, \frac{2 p_{3}}{v_{rmax}}, \frac{2 p_{3}^{2}}{{(1 + \frac{1}{2 k_{4} β_{1}})}^{2}}, 4 k_{4} β_{1} p_{3}, k_{2, 11}, k_{2, 12}, k_{2, 13}} .

Step 6. From the 13th inequality in (11), let

χ \overset{Δ}{=} \frac{k_{2} {\bar{v}}_{rmax}}{2} + k_{2} v_{rmax} (\frac{ω_{rmax}}{2} + \frac{k_{2} v_{rmax}}{4} + \frac{k_{3}}{2} + v_{rmax} + \frac{k_{2} v_{rmax}}{2} + k_{3} + \frac{v_{rmax}}{2}) + k_{3} l_{2} (\frac{k_{2} v_{rmax}}{2} + k_{3}) .

Combining the 3rd and the 13th inequalities in (11), we select $k_{1}$ satisfying that

k_{1} < min {μ_{1} - 2 v_{rmax}, \frac{2 (τ_{2 \max} - {\bar{ω}}_{rmax} - M - χ)}{k_{2} v_{rmax}}} .

Step 7. In the last step, by the 9th, 10th and the 11th inequalities in (11), we can choose $l_{1}$ such that

l_{1} \leq min {\frac{2 δ v_{rmax}}{k_{1} e^{\frac{2 M}{k_{2}}}}, \frac{p_{3} - k_{4}}{k_{1}}, \frac{τ_{1 \max} - {\bar{v}}_{rmax} - M - k_{3} v_{rmax} - \frac{k_{2} v_{rmax}^{2}}{2}}{\frac{1}{2} k_{1} e^{\frac{2 M}{k_{2}}} (ω_{rmax} + \frac{k_{2} v_{rmax}}{2} + k_{3}) + k_{1}^{2}}} .

Thus, it is easy to obtain a feasible solution of (11) in this way. In order to show this method more clearly, the following example is given to explain how we picked the parameters for (11) one-by-one:

abc + ab + ad + bc < m_{1},

(r2)

\frac{d}{bc} < m_{2},

(r3)

bcd + c + d < m_{3} .

(r4)

Assume that $a$ , $b$ , $c$ and $d$ are positive design parameters decided by (r2)–(r4), where $m_{1}$ , $m_{2}$ and $m_{3}$ are known positive constants. In the first place, according to the particularity of these inequalities, $c$ and $b$ can be chosen first.

Step 1. By formula (r4), selecting $c$ satisfying that $c < m_{3}$ .

Step 2. From (r2), we can chose $b$ such that $bc < m_{1}$ . Then, $c$ and $b$ can be seen as known constants, and the next two steps are used to select $d$ and $a$ .

Step 3. According to (r3) and (r4), $d$ can be chosen such that $d < min {bc m_{2}, \frac{m_{3} - c}{bc + 1}}$ .

Step 4. From (r2), choosing $a$ satisfying that $a < \frac{m_{1} - bc}{bc + b + d}$ .

Remark 7. In engineering systems, if the practical inputs of energy are bounded and the operating area is given in advance with special shape and size, then the feedback gain coefficients should be subject to certain constraints. Equations (11) just reflect such constraints information.

Simulations

In this section, the controller proposed is used to show how the saturated stabilization of the closed-loop system can be achieved. We demonstrate the effectiveness of our methods by an example.

In the following simulation, we assume that: $v_{r} = v_{rmax} = 1.5$ , $ω_{r} = ω_{rmax} = 1.2$ , ${\bar{v}}_{rmax} = 2.0$ , ${\bar{ω}}_{rmax} = 1.8$ , $q = 0.5$ , $δ = 0.1$ , $μ_{1} = 3.1$ , $μ_{2} = 1$ , $λ_{1} = λ_{2} = 1.5$ , $τ_{1 \max} = 12.5$ , $τ_{2 \max} = 12.2$ . According to Theorem 1, we select design parameters: $α_{1} = α_{2} = 0.5$ , $β_{1} = β_{2} = 0.25$ , $M = 10.35$ , $p_{1} = p_{2} = p_{3} = 1.0$ . Suppose that the initial tracking errors are chosen as: $(x_{e} (0), y_{e} (0), θ_{e} (0), v_{e} (0), ω_{e} (0)) = (- 0.2, 0.3, 0, 0.5, 0) \in D$ . From Remark 6, the feedback coefficients $k_{i}$ and $l_{j}$ ( $i = 1, 2, 3, 4, 5$ ; $j = 1, 2$ ) can be taken as $k_{1} = 0.1$ , $k_{2} = 0.2$ , $k_{3} = 0.6$ , $k_{4} = 0.5$ , $k_{5} = 0.8$ , and $l_{1} = l_{2} = 1.0$ .

Figures 4 –7 show some simulation results with MATLAB. From Figure 4, it can be seen that all of the tracking error states of the closed system asymptotically converge to the origin equilibrium point, i.e. $x_{e} (t) \to 0$ , $y_{e} (t) \to 0$ , $θ_{e} (t) \to 0$ as $t \to \infty$ .

Figure 4

Tracking error variables.

Figure 5

Auxiliary state variables.

Figure 6

Bounded inputs.

Figure 7

Trajectory tracking.

Figure 5 shows that the auxiliary state variables converge to zero in a finite time, after this, $ξ_{1} (t) \equiv 0$ , $ξ_{2} (t) \equiv 0$ . This means that the dynamic feedback controller converge to the virtual control law in a finite time.

By Figure 6, it shows that the inputs keep continuity and boundedness by the saturated levels $τ_{1 \max} = 12.5$ and $τ_{2 \max} = 12.2$ in any time. Furthermore, the bounded inputs are also convergent.

Figure 7 demonstrates the good tracking performance of actual robot with respect to the desired trajectory.

Conclusion

The tracking problem based on dynamic feedback for the nonholonomic wheeled mobile robot with inputs saturation has been discussed in this paper. The contributions of this paper include having an applied finite-time control technique and back-stepping procedure such that the continuous, saturated tracking controller can be obtained. The closed-loop system of tracking errors can be asymptotically stabilized to zero equilibrium point. We work on extending the results to the saturated robust tracking problem of nonholonomic systems with uncertain parameters or disturbance in the coming time.

Footnotes

Appendix A. Proof of Lemma 1

The function $f (t_{1})$ can be rewritten as

f (t_{1}) = {\begin{matrix} \frac{\sin t_{1}}{t_{1}} & t_{1} \neq 0, \\ 1 & t_{1} \neq 0 . \end{matrix}

It follows that $| f (t_{1}) | \leq 1$ , and that

f^{'} (t_{1}) = {\begin{matrix} \frac{t_{1} \cos t_{1} - \sin t_{1}}{t_{1}^{2}} & t_{1} \neq 0, \\ 0 & t_{1} \neq 0 . \end{matrix}

Applying L’Hospitals rule, we have

lim_{t_{1} \to 0} \frac{t_{1} \cos t_{1} - \sin t_{1}}{t_{1}^{2}} = lim_{t_{1} \to 0} \frac{\cos t_{1} - t_{1} \sin t_{1} - \cos t_{1}}{2 t_{1}} = 0 .

Thus, $f' (t_{1}) \in C^{0}$ , i.e. $f (t_{1}) \in C^{1}$ . For all positive integers $n > 1$ , the higher-order derivative of $f (t_{1})$ can be calculated with a recurrence method:

f^{(n)} (t_{1}) = {\begin{matrix} \frac{{(- 1)}^{n - 1} n! (t_{1} \cos t_{1} - \sin t_{1}) + {(- 1)}^{n - 1} \frac{n!}{2} t_{1}^{2} \sin t_{1} + {(- 1)}^{n} \frac{n!}{3} t_{1}^{3} \cos t_{1} + \dots + {(- 1)}^{n - 1} t_{1}^{n} [\frac{1 + {(- 1)}^{n}}{2} \sin t_{1} + \frac{1 - {(- 1)}^{n}}{2} \cos t_{1}]}{t_{1}^{n + 1}} & t_{1} \neq 0, \\ {(- 1)}^{[\frac{n}{2}]} \cdot \frac{1 + {(- 1)}^{n}}{2 (n + 1)} & t_{1} = 0, \end{matrix}

where $[x]$ denotes the largest integer that is less than or equal to $x$ . Similarly, we can prove that

lim_{t_{1} \to 0} f^{(n)} (t_{1}) = f^{(n)} (0),

which means that $f (t_{1}) \in C^{\infty}$ , since any number of derivatives of $f (t_{1})$ are continuous as $t_{1} \neq 0$ .

Next, we prove that $| f' (t_{1}) | \leq 2$ . First, if $| t_{1} | > 1$ , then we have

| f' (t_{1}) | \leq \frac{| t_{1} | + 1}{t_{1}^{2}} \leq \frac{| t_{1} | + 1}{| t_{1} |} \leq 2 .

Second, if $t_{1} = 0$ , it is obvious that $| f^{'} (0) | = 0 \leq 2$ . Finally, if $0 < | t_{1} | \leq 1$ , we have

f' (t_{1}) t_{1}^{2} = t_{1} \cos t_{1} - \sin t_{1} .

Note that $f' (t_{1})$ is differentiable with respect to $t_{1}$ as $t_{1} \neq 0$ . Taking the derivative of the formula above

(28)

f'' (t_{1}) t_{1}^{2} + 2 f' (t_{1}) t_{1} = - t_{1} \sin t_{1}

Substituting the expression of $f' (t_{1})$ into (28), we have

(29)

f'' (t_{1}) t_{1}^{3} = - t_{1}^{2} \sin t_{1} - 2 t_{1} \cos t_{1} + 2 \sin t_{1} \overset{Δ}{=} h (t_{1})

Then,

h' (t_{1}) = - 2 t_{1} \sin t_{1} - t_{1}^{2} \cos t_{1} - 2 \cos t_{1} + 2 t_{1} \sin t_{1} + 2 \cos t_{1} = - t_{1}^{2} \cos t_{1} < 0, | t_{1} | \in (0, 1] .

Hence, $| f' (t_{1}) | \leq 2$ for all $t_{1}$ .

Appendix B. Proof of Lemma 2

Take a Lyapunov function candidate,

V = \frac{1}{2} η^{2}

its derivative is

\overset{\cdot}{V} = - k η sgn (η) | η |^{α} = - k (\sqrt{2})^{1 + α} V^{\frac{1 + α}{2}}

according to Bhat and Bernstein (1998) and Hong et al. (2001), we have

lim_{t \to T_{0}} η (t) = 0 and η (t) \equiv 0, as t \geq T_{0} .

This completes the proof of Lemma 2.

Appendix C. Proof of Lemma 3

Because $| z (t) |$ is continuous, it may be assumed that there exist some $\tilde{t} \in [0, \tilde{T}]$ such that

| z (\tilde{t}) | = \tilde{M} \overset{Δ}{=} max_{t \in [0, \tilde{T}]} {| z (t) |} .

We give our claim by contradiction. Suppose that (10) does not hold and for some $t_{1} \in [0, \infty)$ ,

(30)

| z (t_{1}) | \geq C_{2} \tilde{T} + \sqrt{C_{1}} .

This problem will be discussed in two cases:

which contradicts the assumption (30). This completes the proof of Lemma 3.

Funding

This work was supported by the National Science Foundation (grant number 60874002), the Key Project of Shanghai Education Committee (grant number 09ZZ158), the Key Discipline of Shanghai (grant number S30501), the Scientific Research Foundation for Returned Scholars, Ministry of education of China (grant number BZX/11H002), the Youth Fund of Hohai University (grant number 2010B23514) and the Natural Science Foundation of Hebei Education Committee (grant number Z2011119).

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