Fixed-time integral sliding mode tracking control of a wheeled mobile robot

Ling Ma; Chenghu Wang; Cheng Ge; Hui Liu; Bo Li

doi:10.20517/ces.2023.14

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Research Article | Open Access | 28 Jun 2023

Fixed-time integral sliding mode tracking control of a wheeled mobile robot

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Ling Ma

,

Chenghu Wang

, ...

Bo Li

Complex Eng Syst 2023;3:10.

10.20517/ces.2023.14 | © The Author(s) 2023.

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Abstract

This paper presents a fixed-time integral sliding mode control scheme for a nonholonomic wheeled mobile robot (WMR). To achieve the trajectory tracking mission, the dynamic model of a WMR is first transformed into a second-order attitude subsystem and a third-order position subsystem. Two novel continuous fixed-time disturbance observers are proposed to estimate the external disturbances of the two subsystems, respectively. Then, trajectory tracking controllers are designed for two subsystems by utilizing the reconstructed information obtained from the disturbance observers. Additionally, an auxiliary variable that incorporates the Gaussian error function is introduced to address the chattering problem of the control system. Finally, the proposed control scheme is validated by a wheeled mobile robotic experimental platform.

Keywords

Wheeled mobile robot, trajectory tracking, disturbance observer, fixed-time stability, integral sliding mode control

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1. INTRODUCTION

During the past decades, the wheeled mobile robot (WMR) has attracted extensive attention as it is widely used in various fields. The research on the WMR mainly includes robot positioning, motion planning, and motion control, among which the motion control is a fundamental problem. There are three main parts of the motion control, including point stabilization, path planning, and trajectory tracking ^[1]. The trajectory tracking control is a significant field in motion control, which has been studied extensively in recent years ^[2]. In practical engineering applications, a WMR is a highly coupled system with nonholonomic constraints and external disturbances. Hence, it is significant to design an anti-interference trajectory tracking control scheme with superior performance. At present, the design of the tracking controller of a WMR is mainly based on two types: one is to consider only the kinematic model ^[3], while the other is to design on the basis of kinematic and dynamic models ^[4]. The kinematic model-based control only considers the linear velocity and angular velocity as the control inputs. Compared with the kinematic model, the introduction of dynamic models can solve the external disturbance problem and the crucial nonholonomic constraint problem ^[5].

In ^[6], the system with nonhonolomic constraints was transformed into an extended chain system by coordinate transformation. On this basis, some scholars have designed the trajectory tracking control schemes by transforming the kinematic model of a WMR into a chain structure ^[7]. In practice, there is a problem called "excellent velocity tracking" ^[8] when designing a trajectory tracking controller only based on a kinematic system. Thus, it is more reasonable to take the force or torque as inputs of the control system instead of the speed. Meanwhile, external disturbances can be further taken into account. Nevertheless, the design process of the controller that simultaneously incorporates both the kinematic and dynamic models is complicated. The work of Zhai and Song ^[9] transformed the dynamic error system into second-order and third-order subsystems. And an intermediate variable related to the position error is introduced to tackle the problem of constructing a control method for a third-order system using the terminal sliding mode control. However, the aforementioned control schemes can only achieve finite time stability. It is noteworthy that the upper limit of the convergence time is unknown and dependent on the initial states of the control system. To overcome this problem, fixed-time stable control methods are proposed ^[10]. In reference ^[11], a new integral sliding mode-based control (ISMC) scheme was developed and applied on the dynamic model of the WMR to enable the WMR to track the desired trajectory in a fixed time. However, there exists the singularity problem, making the WMR unable to track the arbitrary trajectories and limiting its practical application when the desired angular velocity is zero.

In the practical motion environment, there are external disturbances and uncertainties that can deteriorate the performance of the control system. To cope with the problem, an observer-based control scheme is an efficient method with disturbance-rejection performance ^[12]. The traditional observers can only achieve asymptotic stability of the observation errors, whereas the finite time disturbance observers were designed to improve the performance of the observer ^[13]. On this basis, the fault-tolerant attitude control problem of spacecraft under external disturbances was solved by the introduction of a continuous finite-time observer ^[14], which also restrains the chattering phenomenon. Zhang et al. put forward a novel continuous practical fixed-time disturbance observer and applied it on a WMR, which can not only avoid the chattering problem but also improve the ability to attenuate disturbance ^[15]. Different from the work of Zhang, the Gaussian error function, which is sometimes called probability integral ^[16], can also be used to develop a control scheme that improves the chattering problem ^[17].

Motivated by the above discussions, an integral sliding mode-based fixed-time trajectory tracking control scheme is proposed by combining the kinematic model with the dynamic model of a WMR in this paper. (1) A continuous fixed-time disturbance observer using the Gaussian error function is proposed, which avoids the chattering problem and estimates the external disturbance of a WMR accurately. (2) An auxiliary variable incorporating variable exponential coefficients is introduced to simplify the design process of the controller for the third-order subsystem and avoid the singularity problem simultaneously. (3) The reliability and effectiveness of the designed control scheme are verified by a comparative experiment conducted on a wheeled mobile experimental platform.

2. PRELIMINARIES AND PROBLEM STATEMENT

2.1. Preliminaries

Lemma 1^[18] Consider the following system as

(1)

\dot{x} = f (x), x (0) = x_{0}, x \in R

If there exists a positive definite Lyapunov function $V (x)$ , which satisfies $\dot{V} (x) \leq - m_{1} V^{a} (x) - n_{1} V^{b} (x) + ϱ$ , where $m_{1}, n_{1}$ , and $ϱ$ are all positive constants. $0 < a < 1, b > 1$ are real numbers. Then the origin of the system (2) is fixed-time stable, and the settling time is bounded by $t_{1} \leq \frac{1}{m_{1} ϑ (1 - a)} + \frac{1}{n_{1} ϑ (b - 1)}$ with $0 < ϑ < 1$ .

Lemma 2^[16] The Gaussian error function is defined as follows:

(2)

\erf (x) = \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- 2 t^{2}} d t

where $e$ is the natural constant. If $0 \leq x < 1$ , then the Gaussian error function will satisfy $\frac{1}{2} x \leq \erf (x) \leq 2 x$ .

Lemma 3^[19] For $x \in R$ and $μ > 0$ , one gets the following chain of inequalities: $x \tanh (\frac{x}{μ}) < x \erf (\frac{x}{μ}) < | x |$ .

Lemma 4^[20] The following inequality will hold $| x | - \frac{ε}{κ} \leq x \tanh (κ x)$ for any $κ > 0$ and for any $ε \in R$ , where $ε = e^{- (ε + 1)}$ . Then, $ε = 0.2785$ can be obtained.

2.2. Dynamic model of WMR

A nonholonomic WMR system is shown in Figure 1. It consists of two balance wheels and two driving wheels, and the line between the balance wheels is perpendicular to the line between the driving wheels. The distance between the driving wheel and the barycentric coordinate is $R$ , and $r$ is the radius of the driving wheel. The position and attitude control is achieved by independent direct current motors, which provide the appropriate torques to the driving wheels. One assumes that the center of mass of the WMR coincides with the geometric center. Then, the dynamic model of the WMR is expressed in the form of ^[21]

(3)

{\begin{matrix} \begin{aligned} \dot{x} = v \cos θ \\ \dot{y} = v \sin θ \\ \dot{θ} = ω \\ J \dot{ω} = u_{1} + d_{1} \\ m \dot{v} = u_{2} + d_{2} \end{aligned} \end{matrix}

Fixed-time integral sliding mode tracking control of a wheeled mobile robot

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Figure 1. Physical model of a WMR. WMR: wheeled mobile robot.

with $u_{1} = \frac{R}{r} (τ_{1} - τ_{2})$ , and $u_{2} = \frac{1}{r} (τ_{1} + τ_{2})$ . $τ_{1}$ and $τ_{2}$ present the control torques. $v$ and $ω$ are the linear and angular velocities of the WMR, respectively. $m$ denotes the mass, and $J$ the moment of inertia. $(x, y)$ is the actual coordinates. $θ$ is the orientation of the vehicle counterclockwise from the positive direction of the $X$ axis. $(\dot{x}, \dot{y}, \dot{θ})$ denotes the motion of the WMR. $d_{1}$ and $d_{2}$ represent the external disturbances.

The reference trajectory is defined as

(4)

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

where $x_{r}$ , $y_{r}$ , and $θ_{r}$ denote the position and attitude states of the virtual WMR, respectively.

Assumption 1: Suppose $ω_{r}$ , ${\dot{ω}}_{r}$ , $v_{r}$ , and ${\dot{v}}_{r}$ are satisfied with $| ω_{r} | \leq ω_{r_{m a x}}$ , $| {\dot{ω}}_{r} | \leq ω_{1_{m a x}}$ , $| v_{r} | \leq v_{r_{m a x}}$ , And $| {\dot{v}}_{r} | \leq v_{1_{m a x}}$ , where $ω_{r_{m a x}}$ , $ω_{1_{m a x}}$ , $v_{r_{m a x}}$ , and $v_{1_{m a x}}$ are positive constants.

Assumption 2: Suppose the $d_{1}$ , $d_{2}$ , and their derivatives exist with bounds, which is given by $| d_{1} | \leq k_{1 m}, | d_{2} | \leq k_{2 m}$ , where $k_{1 m}$ and $k_{2 m}$ are all positive constants.

Then, the tracking errors of the WMR are expressed as

(5)

[\begin{matrix} x_{e} \\ y_{e} \\ θ_{e} \end{matrix}] = [\begin{array}{ccc} \cos θ & \sin θ & 0 \\ - \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{array}] [\begin{matrix} x - x_{r} \\ y - y_{r} \\ θ - θ_{r} \end{matrix}] .

Furthermore, the error dynamics system could be transformed in the form of

(6)

{\begin{matrix} \begin{aligned} {\dot{x}}_{e} = ω y_{e} - v + v_{r} \cos θ_{e} \\ {\dot{y}}_{e} = v_{r} \sin θ_{e} - ω x_{e} \\ {\dot{θ}}_{e} = ω - ω_{r} \\ J \dot{ω} = u_{1} + d_{1} \\ m \dot{v} = u_{2} + d_{2} \end{aligned} \end{matrix}

To simplify the whole design process, the system (6) can be divided into two subsystems, which contain a second-order subsystem:

(7)

{\begin{matrix} \begin{aligned} {\dot{θ}}_{e} = ω - ω_{r} \\ J \dot{ω} = u_{1} + d_{1} \end{aligned} \end{matrix}

and a third-order subsystem:

(8)

{\begin{matrix} \begin{aligned} {\dot{x}}_{e} = ω y_{e} - v + v_{r} \cos θ_{e} \\ {\dot{y}}_{e} = v_{r} \sin θ_{e} - ω x_{e} \\ m \dot{v} = u_{2} + d_{2} \end{aligned} \end{matrix}

3. FIXED-TIME TRAJECTORY CONTROL

In this section, a fixed-time sliding mode control scheme is developed to realize the fast and high-accuracy trajectory tracking control of a WMR under external disturbances. Firstly, a fixed-time disturbance observer and a new fixed-time sliding mode surface are proposed for the second-order subsystem (7). On this basis, a fixed-time controller is constructed to make the error variables, $θ_{e}$ and ${\dot{θ}}_{e}$ , converge into a small region around the origin. Then, a fixed-time controller is developed for the third-order subsystem (8), which guarantees that the system state variables, $x_{e}, y_{e},$ and $v$ , are all uniformly ultimately bounded, and the tracking errors, $x_{e}$ and $y_{e}$ , can converge into a small region around the origin in a fixed time.

3.1. Tracking control laws design for the second-order subsystem

Fixed-time disturbance observer

Firstly, for the attitude error subsystem (7), define an auxiliary variable as

(9)

ς_{1} = ω - ϖ_{1}

where $ϖ_{1}$ satisfies

(10)

{\dot{ϖ}}_{1} = \frac{1}{J} u_{1} + l_{11} \erf (ς_{1}) + l_{12} | ς_{1} |^{γ_{1}} \erf (ς_{1})

The parameters $l_{11}$ and $l_{12}$ are positive constants with $l_{11} > k_{1 m} / J$ . Let variable exponential coefficient $γ_{1} = \frac{λ_{0} ς_{1}^{2}}{1 + μ_{0} ς_{1}^{2}}$ with $λ_{0}$ and $μ_{0}$ satisfying $0 < μ_{0} < 1$ and $1 + μ_{0} < λ_{0}$ .

Theorem 1For the second-order subsystem (7), if the disturbance observer is constructed as

(11)

{\hat{d}}_{1} = J (l_{11} \erf (ς_{1}) + l_{12} | ς_{1} |^{γ_{1}} \erf (ς_{1}))

then it can estimate $d_{1}$ accurately in a fixed time. That is to say, the observation error ${\tilde{d}}_{1} = d_{1} - {\hat{d}}_{1}$ can converge into a small region within a fixed time.

Proof of Theorem 1 Select a Lyapunov function as $V_{2} = ς_{1}^{2}$ , differentiating it, one has

(12)

\begin{aligned} {\dot{V}}_{2} & = 2 ς_{1} (\dot{ω} - {\dot{ϖ}}_{1}) \\ = 2 ς_{1} (\frac{1}{J} (u_{1} + d_{1}) - (\frac{1}{J} u_{1} + l_{11} \erf (ς_{1}) + l_{12} | ς_{1} |^{γ_{1}} \erf (ς_{1}))) \\ = 2 ς_{1} (- l_{11} \erf (ς_{1}) - l_{12} | ς_{1} |^{γ_{1}} \erf (ς_{1}) + \frac{1}{J} d_{1}) \\ = - 2 (l_{11} ς_{1} \erf (ς_{1}) - \frac{1}{J} ς_{1} d_{1} + l_{12} ς_{1} | ς_{1} |^{γ_{1}} \erf (ς_{1})) \\ \leq - 2 (l_{11} ς_{1} t a n h (ς_{1}) - \frac{1}{J} ς_{1} d_{1} + l_{12} ς_{1} | ς_{1} |^{γ_{1}} \erf (ς_{1})) \\ \leq - 2 (l_{11} | ς_{1} | - l_{11} ϵ_{1} - \frac{k_{1 m}}{J} | ς_{1} | + l_{12} ς_{1} | ς_{1} |^{γ_{1}} \erf (ς_{1})) \\ \leq - 2 l_{12} | ς_{1} | | ς_{1} |^{γ_{1}} \erf (| ς_{1} |) + 2 l_{11} ϵ_{1} \\ = - 2 l_{12} | ς_{1} |^{γ_{1} + 1} \erf (| ς_{1} |) + 2 l_{11} ϵ_{1} \end{aligned}

where $ϵ_{1}$ is a positive constant.

Case 1 When $V_{2} > 1$ and $| ς_{1} | > 1$ , one has $\erf (| ς_{1} |) > \erf (1)$ and $\frac{λ_{0} ς_{1}^{2}}{1 + μ_{0} ς_{1}^{2}} \geq \frac{λ_{0}}{1 + μ_{0}} > 1$ . Then (12) can be rewritten as

(13)

\begin{aligned} {\dot{V}}_{2} & \leq - 2 (l_{12} \erf (1) - l_{11} ϵ_{1}) | ς_{1} |^{\frac{λ_{0}}{1 + μ_{0}} + 1} \\ \leq - 2 (l_{12} \erf (1) - l_{11} ϵ_{1}) V_{2}^{\frac{λ_{0} + μ_{0} + 1}{2 (1 + μ_{0})}} \end{aligned}

As $l_{12} \erf (1) - l_{11} ϵ_{1} > 0$ and ${\bar{γ}}_{1} = \frac{λ_{0} + μ_{0} + 1}{2 (1 + μ_{0})} > 1$ , then all the solutions of ${V_{2} > 1}$ will reach the set ${V_{2} \leq 1}$ within a fixed time $t_{f 1} \leq \frac{1}{2 (l_{12} \erf (1) - l_{11} ϵ_{1}) ({\bar{γ}}_{1} - 1)}$ .

Case 2 In the converse case $V_{2} \leq 1$ , one has

(14)

\begin{aligned} {\dot{V}}_{2} & \leq - 2 l_{12} | ς_{1} | | ς_{1} |^{γ_{1}} \erf (| ς_{1} |) + 2 l_{11} ϵ_{1} \end{aligned}

As $1 + μ_{0} ς_{1}^{2} \geq 1$ and $| ς_{1} | \leq 1$ , it can be obtained that $min (| ς_{1} |^{γ_{1}}) \geq min (| ς_{1} |^{λ_{0} ς_{1}^{2}}) = e^{\frac{- λ_{0}}{2 e}}$ . Considering the Lemma Lemma3, then (14) is converted into the following form

(15)

\begin{aligned} {\dot{V}}_{2} & \leq - 2 l_{12} | ς_{1} | | ς_{1} |^{γ_{1}} \tanh (| ς_{1} |) + 2 l_{11} ϵ_{1} \\ \leq 2 l_{12} | ς_{1} | | ς_{1} |^{γ_{1}} + 2 l_{12} | ς_{1} |^{γ_{1}} ϵ_{2} + 2 l_{11} ϵ_{1} \\ \leq - 2 l_{12} | ς_{1} | | ς_{1} |^{γ_{1}} + 2 l_{12} ϵ_{2} + 2 l_{11} ϵ_{1} \\ \leq - 2 l_{12} | ς_{1} |^{γ_{1} + 1} + 2 l_{11} ϵ_{1} + 2 l_{12} ϵ_{2} \\ \leq - 2 l_{12} e^{\frac{- λ_{0}}{2 e}} | ς_{1} | + 2 l_{11} ϵ_{1} + l_{12} ϵ_{2} \\ \leq - b_{1} V_{2}^{\frac{1}{2}} + \tilde{ϵ} \\ \leq - l_{13} b_{1} V_{2}^{\frac{1}{2}} - (1 - l_{13}) b_{1} V_{2}^{\frac{1}{2}} + \tilde{ϵ} \end{aligned}

with $b_{1} = 2 l_{12} e^{\frac{- λ_{0}}{2 e}}$ , and $\tilde{ϵ} = 2 l_{11} ϵ_{1} + 2 l_{12} ϵ_{2}$ . When $0 < l_{13} < 1$ , and $\tilde{ϵ} - (1 - l_{13}) b_{1} V_{2}^{\frac{1}{2}} \geq 0$ , (15) can be simplified as ${\dot{V}}_{2} \leq - l_{13} b_{1} V_{2}^{\frac{1}{2}}$ . Then, the solution of $V_{2}$ will reach a small set $Δ_{1}$ , which is defined as $Δ_{1} = {ς_{1} | V_{1} (ς_{1}) \leq (\frac{\tilde{ϵ}}{b_{1} (1 - l_{13})})^{2}}$ within a settling time $t_{f 2} \leq \frac{2}{b_{1} l_{13}}$ .

In view of the above two cases, the auxiliary variable $ς_{1}$ will converge into a small set $Δ_{1} = {ς_{1} | V_{1} (ς_{1}) \leq (\frac{\tilde{ϵ}}{b_{1} (1 - l_{13})})^{2}}$ within settling time $t_{f} = t_{f 1} + t_{f 2}$ .

Then, the disturbance observation error

(16)

\begin{aligned} {\tilde{d}}_{1} & = d_{1} - {\hat{d}}_{1} \\ = d_{1} - J (l_{11} \erf (ς_{1}) + l_{12} | ς_{1} |^{γ_{1}} \erf (ς_{1})) \end{aligned}

The disturbance $d_{1}$ is bounded according to Assumption 1. Thus, the disturbance observer (11) can estimate $d_{1}$ accurately, and the observation error ${\tilde{d}}_{1}$ can remain in a small set

Δ_{2} = {ς_{1} | | ς_{1} | \leq k_{1 m} + J (l_{11} \erf (Δ_{1}) + l_{12} | Δ_{1} |^{{\bar{γ}}_{1}} \erf (Δ_{1}))}

after a fixed time, where

{\bar{γ}}_{1} = \frac{λ_{0} Δ_{1}^{2}}{1 + μ_{0} Δ_{1}^{2}}

.

3.1.2. Fixed-time sliding mode controller

For the subsystem (7), define $ω_{e} = ω - ω_{r}$ . A fixed-time integral sliding mode surface is introduced as follows ^[22]

(17)

s_{1} = ω_{e} + \int_{0}^{t} (k_{11} (⌈ θ_{e} ⌋^{p_{1}} + ⌈ θ_{e} ⌋^{q_{1}}) + k_{12} (⌈ ω_{e} ⌋^{p_{2}} + ⌈ ω_{e} ⌋^{q_{2}})) d τ

with $0 < p_{i} < 1$ , $q_{i} > 1$ , and $(i = 1, 2)$ . For any $x \in R$ , $α \in R^{+}$ , the notation is defined as $⌈ x ⌋^{α} = | x |^{α} s i g n (x)$ . Based on the sliding mode surface as (17), the fixed-time controller is designed as follows:

(18)

u_{1} = - J (k_{11} (⌈ θ_{e} ⌋^{p_{1}} + ⌈ θ_{e} ⌋^{q_{1}}) + k_{12} (⌈ ω_{e} ⌋^{p_{2}} + ⌈ ω_{e} ⌋^{q_{2}}) + α_{1} ⌈ s_{1} ⌋^{p_{3}} + α_{2} ⌈ s_{1} ⌋^{q_{3}} + α_{3} \erf (s_{1}) - {\dot{ω}}_{r}) - {\hat{d}}_{1}

where $α_{i}, k_{1 i}, (i = 1, 2)$ are positive constants, $α_{3}$ satisfies $α_{3} \geq \frac{k_{1 m}}{J}$ . In addition, $p_{i}, q_{i}, (i = 1, 2, 3)$ are all positive odd integers with $0 < p_{i} < 1, q_{i} > 1$ .

Theorem 2For the second-order system (7), if the fixed-time controller is constructed in the form of (18), then the real sliding mode variable will converge into a small set within a fixed time.

Proof of Theorem 2 Choose a Lyapunov function as $V_{3} = \frac{1}{2} s_{1}^{2}$ and refer to Lemma 2 to Lemma 4, the time derivative of $V_{3}$ is

(19)

\begin{aligned} {\dot{V}}_{3} & = s_{1} ({\dot{ω}}_{e} + k_{11} (⌈ θ_{e} ⌋^{p_{1}} + ⌈ θ_{e} ⌋^{q_{1}}) + k_{12} (⌈ ω_{e} ⌋^{p_{2}} + ⌈ ω_{e} ⌋^{q_{2}}) \\ = s_{1} (\frac{1}{J} (u_{1} + d_{1}) - {\dot{ω}}_{r} + k_{11} (⌈ θ_{e} ⌋^{p_{1}} + ⌈ θ_{e} ⌋^{q_{1}}) + k_{12} (⌈ ω_{e} ⌋^{p_{2}} + ⌈ ω_{e} ⌋^{q_{2}})) \\ = s_{1} (- α_{1} ⌈ s_{1} ⌋^{p_{3}} - α_{2} ⌈ s_{1} ⌋^{q_{3}} - α_{3} \erf (s_{1}) + \frac{1}{J} {\tilde{d}}_{1}) \\ \leq - α_{1} | s_{1} |^{p_{3} + 1} - α_{2} | s_{1} |^{q_{3} + 1} - α_{3} s_{1} \tanh (s_{1}) + \frac{1}{J} | s_{1} | | {\tilde{d}}_{1} | \\ \leq - 2^{{\bar{p}}_{3}} α_{1} V_{2}^{{\bar{p}}_{3}} - 2^{{\bar{q}}_{3}} α_{2} V_{2}^{{\bar{q}}_{3}} + {\bar{ϑ}}_{1} \end{aligned}

where ${\bar{p}}_{3} = \frac{p_{3} + 1}{2}, {\bar{q}}_{3} = \frac{q_{3} + 1}{2}, {\bar{ϑ}}_{1} = α_{3} ϑ_{1}$ with $ϑ_{1}$ being a positive constant. By using Lemma Lemma4, the second-order system (7) is fixed-time stable. The sliding mode surface $s_{1}$ will converge into a small region $Δ_{3} = {s_{1} | V (s_{1}) \leq m i n {(\frac{c_{2}}{α_{1} 2^{{\bar{p}}_{3}}})^{\frac{1}{{\bar{p}}_{3}}}, (\frac{c_{2}}{α_{2} 2^{{\bar{q}}_{3}}})^{\frac{1}{{\bar{p}}_{3}}}}}$ around the origin in a fixed time $t_{s_{1}}$ , which is determined by $t_{s_{1}} \leq \frac{1}{α_{1} 2^{{\bar{p}}_{3}} ϕ_{1} (1 - {\bar{p}}_{3})} + \frac{1}{α_{2} 2^{{\bar{q}}_{3}} ϕ_{1} ({\bar{q}}_{3} - 1)}$ . Then, one can obtain that variables $θ_{e}$ and $ω_{e}$ converge to zero along the real sliding mode in a fixed time^[23].

3.2. Tracking control laws design for the third-order subsystem

After the angular error $θ_{e}$ converges to zero according to Theorem 2, one can obtain that $\sin θ_{e}$ equals zero, and $\cos θ_{e}$ equals 1. The system (8) can be simplified as

(20)

{\begin{matrix} \begin{aligned} {\dot{x}}_{e} = ω_{r} y_{e} - v + v_{r} \\ {\dot{y}}_{e} = - ω_{r} x_{e} \\ m \dot{v} = u_{2} + d_{2} \end{aligned} \end{matrix}

3.2.1. Fixed-time disturbance observer

Introduce the following auxiliary variable for the simplified third-order subsystem (20)

(21)

ς_{2} = v - ϖ_{2}

where $ϖ_{2}$ satisfies

(22)

{\dot{ϖ}}_{2} = \frac{1}{m} u_{2} + l_{21} \erf (ς_{2}) + l_{22} | ς_{2} |^{γ_{2}} \erf (ς_{2})

where $γ_{2} = \frac{λ_{3} ς_{1}^{2}}{1 + μ_{3} ς_{1}^{2}}$ , and $λ_{3}$ and $μ_{3}$ are integers satisfying the constraints: $0 < μ_{3} < 1$ , $1 + μ_{3} < λ_{3}$ . The parameters $l_{21}$ and $l_{22}$ are positive constants with $l_{21} > k_{2 m}$ and $l_{22} > 0$ .

Theorem 3For the simplified third-order subsystem (20), a fixed-time disturbance observer is developed in the form of

(23)

{\hat{d}}_{2} = m (l_{21} \erf (ς_{2}) + l_{22} | ς_{2} |^{γ_{2}} \erf (ς_{2}))

then it can estimate $d_{2}$ in a fixed time, and the observation error ${\tilde{d}}_{2} = d_{2} - {\hat{d}}_{2}$ can converge into a small region around the origin within a fixed time $t_{d_{2}}$ .

Proof of Theorem 3 Similar to the proof of Theorem 1.

3.2.2. Fixed-time sliding mode controller

For the third-order subsystem (20), introduce the following auxiliary variable:

(24)

ξ = x_{e} + \int_{0}^{t} (λ_{1} \erf (x_{e}) - λ_{2} \erf (y_{e}) + λ_{3} x_{e} y_{e} \erf (y_{e}) + k_{1} {| x_{e} |}^{γ_{3}} \erf (\frac{x_{e}}{ϵ_{3}})) d τ

where $λ_{1}$ $λ_{2}$ , $k_{1}$ , and $ϵ_{3}$ are positive constants, $λ_{2} \leq λ_{1} \erf (1)$ . Let $γ_{3} = \frac{λ_{4} {x_{e}}^{2}}{1 + μ_{4} {x_{e}}^{2}}$ , with $λ_{4}$ and $μ_{4}$ being integers and $0 < μ_{4} < 1$ , $1 + μ_{4} < λ_{4}$ .

Select a fixed-time sliding mode surface as:

(25)

s_{2} = \dot{ξ} + \int_{0}^{t} (k_{21} (⌈ ξ ⌋^{p_{4}} + ⌈ ξ ⌋^{q_{4}}) + k_{22} (⌈ \dot{ξ} ⌋^{p_{5}} + ⌈ \dot{ξ} ⌋^{q_{5}})) d τ

where $k_{21}$ and $k_{22}$ are positive constants, $0 < p_{i} < 1$ and $q_{i} > 1$ , $i = 4, 5$ are positive odd integers.

Theorem 4For the third-order subsystem (20), if the fixed-time sliding mode surface is chosen as (25) and the fixed-time controller is designed as (26),

(26)

\begin{aligned} u_{2} = & m ({\dot{v}}_{r} + {\dot{ω}}_{r} y_{e} + ω_{r} {\dot{y}}_{e} + \dot{h} (x_{e}, y_{e}) + k_{21} (⌈ ξ ⌋^{p_{4}} + ⌈ ξ ⌋^{q_{4}}) \\ + k_{22} (⌈ \dot{ξ} ⌋^{p_{5}} + ⌈ \dot{ξ} ⌋^{q_{5}}) + β_{1} ⌈ s_{2} ⌋^{p_{6}} + β_{2} ⌈ s_{2} ⌋^{q_{6}} + β_{3} \erf (s_{2})) + {\hat{d}}_{2} \end{aligned}

then the sliding mode surface $s_{2}$ is fixed-time stable, which will converge into a small region of origin within settling time $t_{s_{2}} \leq \frac{1}{α_{4} 2^{{\bar{p}}_{6}} ϕ_{2} (1 - {\bar{p}}_{6})} + \frac{1}{α_{5} 2^{{\bar{q}}_{6}} ϕ_{2} ({\bar{q}}_{6} - 1)}$ .

$h (x_{e}, y_{e}) = λ_{1} \erf (x_{e}) - λ_{2} \erf (y_{e}) + λ_{3} x_{e} y_{e} \erf (y_{e}) + k_{1} {| x_{e} |}^{γ_{3}} \erf (\frac{x_{e}}{ϵ_{2}})$

, in which $β_{1}, β_{2}$ are positive constants, and $p_{6}, q_{6}$ are positive odd integers satisfying $0 < p_{6} < 1$ , $q_{6} > 1$ .

Proof of Theorem 4 The proof process will be conducted in 3 steps: (1) After the angular error $θ_{e}$ converges to zero according to Theorem 2, $s_{2}$ and the auxiliary variable $ξ$ can converge into a small region around the origin within a fixed time; (2) The error variables $x_{e}, y_{e}$ can converge into a small region around the origin within a fixed time; (3) It should be proved that $x_{e}$ and $y_{e}$ do not escape to infinity before the angular error $θ_{e}$ converges to zero.

Step 1 Select a positive Lyapunov function $V_{4} = \frac{1}{2} s_{2}^{2}$ , differentiating it and substituting (24)-(26) yields to

(27)

\begin{aligned} {\dot{V}}_{4} & = s_{2} {\dot{s}}_{2} \\ = s_{2} (- β_{1} ⌈ s_{2} ⌋^{p_{6}} - β_{2} ⌈ s_{2} ⌋^{q_{6}} - β_{3} \erf (s_{2}) + \frac{1}{m} (\hat{d_{2}} - d_{2})) \\ \leq - β_{1} | s_{2} |^{p_{6} + 1} - β_{2} | s_{2} |^{q_{6} + 1} - β_{3} s_{2} \erf (s_{2}) + \frac{1}{m} s_{2} {\tilde{d}}_{2} \\ \leq - β_{1} | s_{2} |^{p_{6} + 1} - β_{2} | s_{2} |^{q_{6} + 1} - β_{3} s_{2} \tanh (s_{2}) + \frac{1}{m} s_{2} {\tilde{d}}_{2} \\ \leq - β_{1} | s_{2} |^{p_{6} + 1} - β_{2} | s_{2} |^{q_{6} + 1} - β_{3} | s_{2} | + \frac{1}{m} | s_{2} | | {\tilde{d}}_{2} | \\ \leq - 2^{{\bar{p}}_{6}} β_{1} V_{3}^{{\bar{p}}_{6}} - 2^{{\bar{q}}_{6}} β_{2} V_{3}^{{\bar{q}}_{6}} + {\bar{ϑ}}_{2} \end{aligned}

where ${\bar{p}}_{6} = \frac{p_{6} + 1}{2}, {\bar{q}}_{6} = \frac{q_{6} + 1}{2}$ , ${\bar{ϑ}}_{2} = β_{3} ϑ_{2}$ with $ϑ_{2}$ being a positive constant. Using the Lemma Lemma4, the third-order subsystem (14) is fixed-time stable, and $s_{2}$ will converge into a small set $Δ_{4} = {s_{2} | V (s_{2}) \leq m i n {(\frac{c_{3}}{α_{1} 2^{{\bar{p}}_{6}}})^{\frac{1}{{\bar{p}}_{6}}}, (\frac{c_{3}}{α_{2} 2^{{\bar{q}}_{6}}})^{\frac{1}{{\bar{p}}_{6}}}}}$ around zero in the fixed time $t_{s_{2}}$ , which is determined by

(28)

t_{s_{2}} \leq \frac{1}{α_{4} 2^{{\bar{p}}_{6}} ϕ_{2} (1 - {\bar{p}}_{6})} + \frac{1}{α_{5} 2^{{\bar{q}}_{6}} ϕ_{2} ({\bar{q}}_{6} - 1)}

Then, $s_{2}$ will hold in a small region of origin, which guarantees a real sliding mode surface ^[23]. Therefore, the auxiliary variable $ξ$ and its derivative $\dot{ξ}$ will also converge into the origin along the sliding mode surface ^[24].

Step 2 According to (25), when the auxiliary $\dot{ξ} = 0$ , one has

(29)

{\dot{x}}_{e} = - λ_{1} \erf (x_{e}) + λ_{2} \erf (y_{e}) - λ_{3} x_{e} y_{e} \erf (y_{e}) - k_{1} {| x_{e} |}^{γ_{3}} \erf (\frac{x_{e}}{ϵ_{3}})

Choose a Lyapunov function as $V_{5} = x_{e}^{2}$ , the time derivative of $V_{5}$ is

(30)

\begin{aligned} {\dot{V}}_{5} & = 2 x_{e} {\dot{x}}_{e} \\ = - 2 (λ_{1} x_{e} \erf (x_{e}) - λ_{2} x_{e} \erf (y_{e}) + λ_{3} x_{e}^{2} y_{e} \erf (y_{e}) + k_{1} {| x_{e} |}^{γ_{3}} \erf (\frac{x_{e}}{ϵ_{3}})) \\ \leq - 2 (λ_{1} | x_{e} | - λ_{2} | x_{e} | - λ_{1} ϵ_{4} + k_{1} x_{e} | x_{e} |^{γ_{3}} \erf (\frac{x_{e}}{ϵ_{3}})) \\ \leq - 2 (k_{1} ϵ_{3} | x_{e} | | x_{e} |^{γ_{3}} \erf (| \frac{x_{e}}{ϵ_{3}} |) - λ_{1} ϵ_{4}) \end{aligned}

where $ϵ_{4}$ is a positive constant. The rest of the proof is similar to the proof of Theorem 2. There exists a constant $0 < ϑ_{3} < 1$ such that the variable $x_{e}$ will reach and keep in a small region $Δ_{5}$ around the origin within a fixed time $T_{2}$ :

(31)

T_{2} \leq \frac{1}{k_{1} ϑ_{3} e^{- \frac{λ_{4}}{2 e}}} + \frac{1}{2 k_{1} ϵ_{3} (\frac{λ_{4}}{1 + μ_{4}} - 1)}

Then, it can be obtained that the ${\dot{x}}_{e}$ is a uniformly continuous form (29). Employ Barbalat Lemma ^[25] to prove ${\dot{x}}_{e} \to 0$ as $t \to \infty$ , then ${\dot{x}}_{e}$ is bounded after the variable $x_{e}$ converges. Hence, there exists a small region $Δ_{5}$ around the origin that $y_{e}$ can converge into $Δ_{5}$ .

Step 3 Before the angular error $θ_{e}$ converges to zero, $θ_{e} \neq 0$ , such that subsystem (13) cannot be simplified as (19). It should be proved that system state variables $x_{e}, y_{e},$ and $v$ are bounded before the angular error $θ_{e}$ converges to zero.

Consider the following bounded function:

(32)

V_{6} = \frac{1}{2} x_{e}^{2} + \frac{1}{2} y_{e}^{2} + | v |

The time derivative of $V_{6}$ is

(33)

\begin{aligned} {\dot{V}}_{6} & \leq | x_{e} | | {\dot{x}}_{e} | + | y_{e} | | {\dot{y}}_{e} | + | \dot{v} | \\ \leq | x_{e} | | {\dot{x}}_{e} | + | y_{e} | | {\dot{y}}_{e} | + m (ω_{r} {\dot{y}}_{e} + | \dot{h} (x_{e}, y_{e}) | + k_{21} (| ξ |^{p_{4}} + | ξ |^{q_{4}}) + k_{22} (| \dot{ξ} |^{p_{5}} + | \dot{ξ} |^{q_{5}}) \\ + β_{1} | s |_{2}^{p_{6}} + β_{2} | s |_{2}^{q_{6}}) + \frac{| {\tilde{d}}_{2} |}{m} \\ \leq | x_{e} | | {\dot{x}}_{e} | + | y_{e} | | {\dot{y}}_{e} | + m (| {\dot{y}}_{e} | + k_{21} (| ξ |^{p_{4}} + | ξ |^{q_{4}}) + k_{22} (| \dot{ξ} |^{p_{5}} + | \dot{ξ} |^{q_{5}}) + β_{1} | s |_{2}^{p_{6}} + β_{2} | s |_{2}^{q_{6}} \\ + λ_{3} (| {\dot{x}}_{e} | | y_{e} | + | x_{e} | | {\dot{y}}_{e} | + | x_{e} | | y_{e} | | {\dot{y}}_{e} |) + \frac{k_{1} λ_{4} | x_{e} | | {\dot{x}}_{e} |}{1 + μ_{0} x_{e}^{2}} | x_{e} |^{γ_{3}} (1 + \frac{2 | x_{e} |}{1 + μ_{4} x_{e}^{2}}) \\ + \frac{2 k_{1}}{ϵ_{2} \sqrt{π}} | x_{e} |^{γ_{3}} | {\dot{x}}_{e} |) + \frac{| {\tilde{d}}_{2} |}{m} \\ \leq | x_{e} | | {\dot{x}}_{e} | + | y_{e} | | {\dot{y}}_{e} | + m (| {\dot{y}}_{e} | + k_{21} (| ξ |^{p_{4}} + | ξ |^{q_{4}}) + k_{22} (| \dot{ξ} |^{p_{5}} + | \dot{ξ} |^{q_{5}}) + β_{1} | s |_{2}^{p_{6}} + β_{2} | s |_{2}^{q_{6}} \\ + λ_{3} (| {\dot{x}}_{e} | | y_{e} | + | x_{e} | | {\dot{y}}_{e} | + | x_{e} | | y_{e} | | {\dot{y}}_{e} |) + \frac{k_{1} λ_{4}}{μ_{4}} (1 + 2 | x_{e} |) | x_{e} |^{\frac{λ_{4}}{μ_{4}} - 1} | x_{e} | \\ + \frac{2 k_{1}}{ϵ_{2} \sqrt{π}} | x_{e} |^{\frac{λ_{4}}{μ_{4}}} | x_{e} |) + \frac{| {\tilde{d}}_{2} |}{m} \end{aligned}

Let $η_{1} = \sqrt{x_{e}^{2} + y_{e}^{2} + | v |} \geq η > 1$ , then one has the following inequalities: $| x_{e} | \leq η_{1}, | y_{e} | \leq η_{1}, | v | \leq η_{1}$ . Furthermore, there exist positive constants $a_{i} (i = 3, 4), b_{l}, c_{l}, (l = 4, 5, . . ., 9)$ , which satisfy $| x_{e} |^{\frac{λ_{4}}{μ_{4}}} \leq a_{3} η_{1}$ , $| x_{e} |^{\frac{λ_{4}}{μ_{4}} - 1} \leq a_{4} η_{1}$ , $| s_{2} |^{p_{6}} < b_{4} + c_{4} η_{1}$ , $| s_{2} |^{q_{6}} < b_{5} + c_{5} η_{1}$ , $| ξ |^{p_{4}} < b_{6} + c_{6} η_{1}$ , $| ξ |^{q_{4}} < b_{7} + c_{7} η_{1}$ , $| \dot{ξ} |^{p_{5}} < b_{8} + c_{8} η_{1}$ , $| \dot{ξ} |^{q_{5}} < b_{9} + c_{9} η_{1}$ . According to Theorem 3, the state variable $θ_{e}, ω$ will converge into the origin within a fixed time $t_{s_{1}}$ , then one has $θ_{e} \leq θ_{m}$ , $| ω | \leq ω_{m}$ , $| {\tilde{d}}_{2} | < k_{d}$ . Further $| {\dot{y}}_{e} | < | {\dot{x}}_{e} | \leq v_{1_{m a x}} + ω_{m} η_{1}$ can be obtained. Then, (33) can be simplified as

(34)

\begin{aligned} {\dot{V}}_{6} & \leq (2 η_{1} + 2 m λ_{3} + η_{1}^{2} + m) (v_{1_{m a x}} + ω_{m} η_{1}) + \frac{m k_{1} λ_{4}}{μ_{4}} (v_{1_{m a x}} + ω_{m} η_{1}) a_{4} η_{1} \\ + \frac{2 m k_{1}}{ϵ_{2} \sqrt{π}} a_{3} η_{1} (v_{1_{m a x}} + ω_{m} η_{1}) + \frac{k_{d}}{m} \\ \leq \frac{η_{1}^{2}}{2} 2 (2 v_{1_{m a x}} + 3 ω_{m} + 2 m λ_{3} + m + \frac{2 m k_{1} a_{4} λ_{4}}{μ_{4}} (v_{1_{m a x}} + ω_{m}) + \frac{2 m a_{3} k_{1}}{ϵ_{2} \sqrt{π}} (v_{1_{m a x}} + ω_{m})) \\ + 2 m λ_{3} v_{1_{m a x}} + \frac{k_{d}}{m} \\ \leq K V_{6} + ϱ_{1} \end{aligned}

where $K$ and $ϱ_{1}$ satisfy the following constraints:

(35)

\begin{aligned} K & = 2 (2 v_{1_{m a x}} + 3 ω_{m} + 2 m λ_{3} + m + \frac{2 m k_{1} a_{4} λ_{4}}{μ_{4}} (v_{1_{m a x}} + ω_{m}) + \frac{2 m a_{3} k_{1}}{ϵ_{2} \sqrt{π}} (v_{1_{m a x}} + ω_{m})) \end{aligned}

(36)

\begin{aligned} ϱ_{1} & = 2 m λ_{3} v_{1_{m a x}} + \frac{k_{d}}{m} \end{aligned}

On the contrary, if $η_{1} > 1$ , there exists a positive constant $ϱ_{2}$ , which satisfies ${\dot{V}}_{6} \leq ϱ_{2}$ . One has ${\dot{V}}_{6} \leq K V_{6} + ϱ_{3}$ for the state variable $x_{e}, y_{e}, v$ . Further, before the angular error $θ_{e}$ converges to zero, one can obtain

(37)

V_{6} \leq (V_{6} (0) + \frac{ϱ_{3}}{K}) e^{K t} - \frac{ϱ_{3}}{K}

Remark 1 The auxiliary variable $ξ$ in (24) can reduce the order of the third-order subsystem, which simplifies the process of the controller design. In addition, the controller developed in this paper can guarantee that the system state variables converge in a fixed time and the chattering problem is solved by using the error function erf( $\cdot$ ). Furthermore, utilizing the variable exponent coefficient in (24) avoids the common singularity problem.

4. EXPERIMENT RESULTS

To verify the effectiveness of the proposed control scheme, the trajectory tracking experiment is implemented on a Quanser QBot 2e mobile robot platform composed of a QBot 2e mobile robot, an OptiTrack system with 12 infrared cameras, and a computer. The experimental platform is presented in Figure 2. The whole closed-loop experiment structure is as follows: The simulation diagram is compiled on the host computer equipped with MATLAB/Simulink to transform the simulation into an executable file. And the control scheme is written to the Gumstix computer embedded in the QBot 2e through wireless communication protocol. The real-time position information of the QBot 2e is obtained by the OptiTrack positioning system. Then the host computer calculates the information and transmits them to the embedded computer of a WMR for the input of real-time calculation of executable files. So as to complete the trajectory experiment of the mobile robot.

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Figure 2. The Quanser QBot 2e Mobile Robot Platform.

In the experiment, the physical parameters of the QBot 2e are chosen as follows: $m = 4 k g$ , $J = 2.5$ $k g \cdot m^{2}$ . The desired reference trajectory is set as $x_{r} = \cos (0.2 t) m, y_{r} = \sin (0.2 t) m$ . The initial values of the reference and practical trajectories are $[x_{r} (0), y_{r} (0), θ_{r} (0)]^{T} = [1, 0, π / 2]^{T}$ , $[x (0), y (0), θ (0)]^{T} = [0.7, - 0.02, π / 6]^{T}$ , respectively. The main relevant parameters of the proposed control scheme are as follows: $k_{1} = 0.001$ , $k_{11} = k_{12} = 0.9$ , $k_{21} = 0.05$ , $k_{22} = 0.06$ , $ϵ_{2} = 0.00001$ . Choose the parameters $α_{1} = 2$ , $α_{2} = 0.5$ , $β_{1} = β_{2} = 1$ for the sliding mode surface $s_{1}$ in (17) and $s_{2}$ in (25), respectively.

It is obvious that the WMR trajectory tracking mission can be achieved by the designed control method as plotted in the red track in Figure 3. The time response curves of the sliding mode surfaces, $s_{1}$ and $s_{2}$ , are shown in Figure 4, which converge very quickly. To illustrate the excellence of the proposed control method, a comparative experiment on the trajectory tracking of WMRs is conducted between this work and reference ^[26]. The control inputs of the designed control scheme and reference are shown in Figure 5, which are nonsingular and continuous. Figure 6 illustrates the tracking errors in this experiment, which have a big fluctuation due to the influence of external disturbances. In the experiment, the external disturbance is from the experimental environment, such as uneven ground. The observed disturbance values are shown in Figure 7, which indicates the effectiveness of the proposed disturbance observer in this work. From the experimental results, it can be concluded that the designed control scheme has the robustness against the external disturbance and high tracking accuracy.

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Figure 3. The comparative trajectory tracking experiment results of the WMR. WMR: wheeled mobile robot.

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Figure 4. Comparative results of sliding mode surfaces in the experiment.

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Figure 5. Comparative results of control torques in the experiment.

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Figure 6. Comparative results of tracking errors $x_{e}$ , $y_{e}$ , $θ_{e}$ in the experiment.

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Figure 7. Disturbance estimation $\hat{d_{1}}$ and $\hat{d_{2}}$ in the experiment.

5. DISCUSSION

In this paper, a universal control scheme for fixed-time trajectory tracking based on ISMC is put forward. The dynamic model of the WMR has been transformed into two error subsystems. Then utilizing the fixed-time technology and ISMC, a new fixed-time disturbance observer has been proposed and applied on the two error subsystems. Furthermore, an observer-based tracking control method has been proposed to achieve a trajectory tracking mission for the WMR, and guarantee the tracking error converges within a fixed time. Finally, the proposed control approach has been verified by a mobile robotic platform, and the experimental results show fine control performances. Our future work will focus on how to realize the formation tracking control of multi-wheeled mobile robots in both theory and experiment.

DECLARATIONS

Acknowledgments

The authors would like to thank the editors and reviewers for their valuable comments dedicated to this article.

Authors' contributions

Made significant contributions in writing, methodology, review: Li B

Executed writing-original draft, experiment verification: Ma L

Methodology validation: Wang C

Data result analysis: Ge C

Supervision and modification: Liu H

Availability of data and materials

Not applicable.

Financial support and sponsorship

This work was supported in part by the National Natural Science Foundation of China (62073212), Natural Science Foundation of Shanghai (23ZR1426600), and Innovation Fund of Chinese Universities Industry-University-Research (2021ZYB05004).

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Copyright

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Cite This Article

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Fixed-time integral sliding mode tracking control of a wheeled mobile robot

Ling Ma, ... Bo Li

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Ma, L.; Wang, C.; Ge, C.; Liu, H.; Li, B. Fixed-time integral sliding mode tracking control of a wheeled mobile robot. Complex Eng. Syst. 2023, 3, 10. http://dx.doi.org/10.20517/ces.2023.14

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Contents

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Keywords

1. INTRODUCTION

2. PRELIMINARIES AND PROBLEM STATEMENT

3. FIXED-TIME TRAJECTORY CONTROL

4. EXPERIMENT RESULTS

5. DISCUSSION

DECLARATIONS

REFERENCES

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Fixed-time integral sliding mode tracking control of a wheeled mobile robot

Abstract

Keywords

1. INTRODUCTION

2. PRELIMINARIES AND PROBLEM STATEMENT

2.1. Preliminaries

2.2. Dynamic model of WMR

3. FIXED-TIME TRAJECTORY CONTROL

3.1. Tracking control laws design for the second-order subsystem

Fixed-time disturbance observer

3.1.2. Fixed-time sliding mode controller

3.2. Tracking control laws design for the third-order subsystem

3.2.1. Fixed-time disturbance observer

3.2.2. Fixed-time sliding mode controller

4. EXPERIMENT RESULTS

5. DISCUSSION

DECLARATIONS

Acknowledgments

Authors' contributions

Availability of data and materials

Financial support and sponsorship

Conflicts of interest

Ethical approval and consent to participate

Consent for publication

Copyright

REFERENCES

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