Tracking Control of a Two-Wheeled Mobile Robot Using Integral Sliding Mode Control and a Linear Quadratic Regulator

Abstract

In this work, an effective control framework is proposed for a two-wheeled mobile robot (TWMR) operating under time-varying disturbances and uncertain system parameters. To enhance robustness against these uncertainties, an integral sliding mode control (ISMC) method is adopted. A mathematical model of the TWMR is obtained in the state form, and an ISMC law is designed. The proposed control law comprises two terms: a nominal term and a discontinuous term. The nominal term is designed based on the linearized model and optimal control to eliminate any steady-state error, while the discontinuous term is designed based on the sliding surface and the reaching law to force the system state onto the sliding surface under changing disturbances and parameter variations. These two terms are combined to constitute the overall control law. The performance and robustness of the proposed method are assessed through simulation under different uncertainty conditions of the TWMR.

1. Introduction

A two-wheeled mobile robot (TWMR) is a type of robot with three degrees of freedom (3-DOF) that uses two wheels on the right and left sides. TWMRs are underactuated and highly nonlinear systems that cannot maintain self-balancing and drive without constant control of the two wheels’ speed and direction against gravity and uncertainties. TWMRs have gained increasing popularity in a variety of consumer and industrial applications due to their relatively simple structure, versatility, and ability to access narrow workspaces. These systems have been the subject of extensive research interest worldwide for several decades, dating back to William G. Walter’s development of the first electronic autonomous robot, known as the tortoise, in 1948 [1]. Recent research studies have focused on improving the performance, perception, and autonomy of TWMRs. One area of active research is the improvement of control algorithms to ones that can better handle disturbances and uncertainties. The existing methods to stabilize and drive TWMRs include system modeling [2,3,4], PID control [5,6,7], optimal control [8,9,10], H∞ control [11], model predictive control (MPC) [12,13], intelligent control [14,15,16,17], sliding mode control (SMC) [18,19,20,21,22,23], and feedback linearization [24].

Among the studies related to the modeling of TWMRs, the authors of [2] obtained a 1-DOF model of a monoaxial bicycle while dealing with the postural control problem of its tilt angle. In [3], the authors investigated the dynamics of a two-wheeled inverted pendulum robot and examined the robot’s behavior on an inclined surface, as well as its turning effects and stability. The authors of [4] analyzed the kinematics of a robot’s wheels and body separately and then established the whole kinematics model of the robot system.

Early control studies on TWMRs predominantly employed PID control strategies. In [5], the authors developed a wheeled mobile robot equipped with a pair of DC motors, a driver board, an STM32 microcontroller, and two Hall sensors, and applied a PID algorithm to regulate the rotational speeds of the DC motors. In [6], a single tethered robot was modeled as a MIMO system, and both a conventional PID controller and a fuzzy PID controller with self-tuning scaling factors were designed to achieve the desired control performance. In [7], a two-wheeled self-balancing robot model subject to road disturbance effects was established and simulated using MATLAB (2021a) Simscape Multibody, where robot balancing and stability were achieved through PID control and linear quadratic regulator (LQR) approaches tailored to specific control objectives.

The work presented in [8] developed a nonlinear optimal control strategy for a two-wheeled inverted pendulum (TWIP) mobile platform subject to parameter uncertainties, utilizing the state-dependent Riccati equation approach. They developed a state-dependent coefficient matrix for the control law based on the dynamic characteristics of the TWIP mobile platform. The authors of [9] proposed an adaptive LQR control method for a TWMR with unknown system dynamics and disturbances to track the desired trajectory, wherein an adaptive observer was adopted to estimate the unknown system states and disturbances in real time. The authors of [10] proposed a model-free LQR control method for a TWMR using a deep neural network and reinforcement learning to approximate the Q-function of the LQR controller. In [11], a robust LQR controller combined with an H∞ control scheme was developed for a TWMR operating under uncertainties and external disturbances to ensure satisfactory closed-loop stability and performance.

In [12], a hierarchical cascade control scheme was developed for a two-wheeled robot manipulator with varying mass. An adaptive MPC was implemented in the inner loop to regulate the fast-varying center-of-gravity angle, while an outer-loop linear quadratic Gaussian (LQG) controller generated the reference set-point. The authors of [13] proposed an MPC optimized via a genetic algorithm (GA) to improve the trajectory tracking effect of Mecanum-wheeled mobile robots (MWMRs), where the MPC considered the mechanical and speed constraints of the MWMRs, as well as the changes in the curvature of the tracking trajectory.

In [14], a dual-loop adaptive type-II fuzzy control scheme was proposed for wheeled mobile robots (WMRs) operating under wheel slip and external disturbances. The outer loop addresses position tracking and the inner loop regulates velocity tracking, with each loop combining a stabilizing feedback controller and an adaptive type-II fuzzy controller to compensate for unknown dynamics. Another fuzzy control study for a differential-drive wheeled mobile robot performing path-following was reported in [15], where the parameters of all fuzzy logic controller membership functions were automatically optimized using a GA. In [16], a fuzzy controller incorporating an LQR was designed for a TWMR to direct the orientation of the chassis. The parameters of the LQR were computed using a GA. The authors of [17] developed an intelligent system consisting of two neural networks for the control of a wheeled robot. The architecture employs two networks: one to identify the obstacle’s position and size, and the other to generate a continuous trajectory toward the destination while accounting for the acquired information, destination coordinates, and terminal orientation.

Recently, the application of nonlinear control techniques for TWMRs has attracted the attention of researchers [18,19,20,21,22,23,24]. In [18], a nonlinear sliding mode controller for a TWMR was proposed to track a desired trajectory in the presence of disturbances and uncertainties. In [19], an adaptive sliding mode-based fault-tolerant controller for WMRs under actuator faults, disturbances, and unknown system parameters was proposed, where the adaptive law adjusts the parameters of the SMC in real time. The study in [20] proposed a robust adaptive output-feedback trajectory tracking controller for a wheeled mobile robot (WMR) based on the sliding mode super-twisting algorithm. The study in [21] proposed a robust control framework that integrates adaptive observer design with sliding mode control to address terrain inclination and disturbances in a two-wheeled self-balancing robot, enabling estimation of unmeasured states and the inclination angle. In [22], a control scheme combining sliding mode control with modern control techniques was presented for a two-wheeled inverted pendulum. A robust integral sliding mode controller was designed to address short- and long-term disturbances, constant uncertainties, and parameter variations. In [23], a backstepping control algorithm with speed compensation control was proposed to solve the path tracking control problem of mobile robots under nonholonomic constraints and external interference factors. While these control methods provide satisfactory performance, there remains significant room for improvement, particularly concerning computational efficiency and implementation complexity. Specifically, advanced techniques like adaptive SMC and high-order SMC require complex, resource-intensive computations for both implementation and stability proofs. Likewise, the recursive nature of the backstepping algorithm causes it to suffer from an explosion of complexity, where the resulting control law becomes computationally burden as the system’s order increases.

Therefore, we present a design for a TWMR tracking controller combining an LQR and SMC. The objective of this study is to utilize the benefits offered by both the optimal nature and robustness of SMC. First, a nominal term is designed based on the linearized model of the TWMR and optimal control theory to eliminate any steady-state error. Subsequently, a discontinuous control term is designed based on the sliding surface and reaching law to drive the system states onto the sliding surface in the presence of time-varying disturbances and parameter variations. Then, the two terms are combined to constitute the overall control law. The effectiveness of the proposed method is illustrated through a set of simulation works. The organization of this paper is as follows: Section 1 provides an introduction. Section 2 describes the mathematical modeling of a TWMR system. Section 3 describes the integral sliding mode control (ISMC) design based on SMC and an LQR. Section 4 simulates the closed-loop system and compares its performance with that of the SMC.

2. Mathematical Modeling of a TWMR System

2.1. Nonlinear Model

The first step in designing a control law that properly guides a TWMR to a desired signal is to construct its mathematical model. A TWMR can rotate around its body’s vertical axis and can also move backward and forward along the central axis between its two wheels. Figure 1 shows the coordinate system of the TWMR.

Recently, a variety of mathematical descriptions of TWMRs have been presented in the literature [2,3,4]. By applying the Newton–Euler or Lagrangian method, a nonlinear dynamic model of a TWMR can be derived in the following form:

$$M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)=Nu$$

(1)

where $\mathit{M}(\mathit{q})\in R^{3\times 3}$ is the inertia matrix, $\mathit{C}(\mathit{q},\dot{\mathit{q}})\in R^{3\times 3}$ is the centrifugal and Coriolis force matrix, $\mathit{G}(\mathit{q})\in R^3$ is the gravity vector, $\mathit{N}\in R^{3\times 2}$ is the input matrix, $\mathit{u}\in R^2$ is the input vector, and $\mathit{q}={\left[x \theta \delta \right]}^T\in R^3$. $x$, $\theta$, and $\delta$ are the position of the wheel [m], the pitch angle [rad], and the yaw angle [rad] of the TWMR, respectively.

By taking the inverse of $\mathit{M}(\mathit{q})$ and defining the state vector $\mathit{x}={\left[x_1 x_2 x_3 x_4 x_5 x_6\right]}^T\in R^6$, the control input vector $\mathit{u}={\left[C_L{C}_R\right]}^T\in R^2$, and the output vector $\mathit{y}={\left[y_1{y}_2\right]}^T\in R^2$, Equation (1) can be rewritten in the nonlinear state space form

$$\dot{x}=f\left(x\right)+g\left(x\right)u, y=h(x),$$

(2)

with

$$\mathit{f}\left(\mathit{x}\right)=\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 0& 0& f_2\left(x\right)& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& f_4\left(x\right)& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& f_6\left(x\right)& 0& 0& 0\end{array}\right], \mathit{g}\left(\mathit{x}\right)=\left[\begin{array}{cc}0& 0\\ g_2\left(x\right)& g_2\left(x\right)\\ 0& 0\\ g_4\left(x\right)& g_4\left(x\right)\\ 0& 0\\ g_6\left(x\right)& -g_6\left(x\right)\end{array}\right], \mathit{h}(\mathit{x})=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0\end{array}\right],$$

(3)

where $x_1=x$, $x_3=\theta$, $x_5=\delta$, and $x_2$, $x_4$, and $x_6$ are the corresponding time derivatives.

The functions in Equation (2) are given by

$$f_2(x)={\displaystyle \frac{1}{{\eta }_1(x_3)}}{\displaystyle \frac{{\mathrm{s}\mathrm{i}\mathrm{n}x}_3}{x_3}}[-m_B^2{L}^{2}g{ \cos x}_3+q_1{m}_{B}L{ \cos }^2{x}_3+m_{B}L(q_1+q_2{ \cos }^2{x}_3)x_6^2],$$

(4)

$$f_4(x)={\displaystyle \frac{1}{{\eta }_1(x_3)}}{\displaystyle \frac{{\mathrm{s}\mathrm{i}\mathrm{n}x}_3}{x_3}}[q_3{m}_{B}Lg-m_B^2{L}^2{x}_4^2{\mathrm{c}\mathrm{o}\mathrm{s}x}_3-(m_B^2{L}^2+q_2{q}_3)x_6^2{ \cos x}_3],$$

(5)

$$f_6(x)={\displaystyle \frac{1}{{\eta }_2(x_3)}}{\displaystyle \frac{{\mathrm{s}\mathrm{i}\mathrm{n}x}_3}{x_3}}(2q_2{x}_4{x}_6{ \cos x}_3-m_{B}L{x}_2{x}_6),$$

(6)

$$g_2(x)={\displaystyle \frac{1}{{\eta }_1(x_3)}}(q_1/r+m_{B}L{ \cos x}_3),$$

(7)

$$g_4(x)=-{\displaystyle \frac{1}{{\eta }_1(x_3)}}[(m_{B}L{ \cos x}_3)/r+q_3],$$

(8)

$$g_6(x)=-{\displaystyle \frac{1}{{\eta }_2(x_3)}}(d/r),$$

(9)

where ${\eta }_1(x_3)=m_B^2{L}^2{\sin }^2{x}_3+m_B{I}_2+2(m_W+J/r^2)q_1$ and ${\eta }_2(x_3)=I_3+2J_v+2(m_W+J/r^2)d^2-q_3{\sin }^2{x}_3$ with constant terms $q_1=I_2+m_B{L}^2$, $q_2=I_3-I_1-m_B{L}^2$, and $q_3=m_B+2m_W+2J/r^2$. The parameters employed in Equation (2) are defined in

Table 1. Physical parameter values of a magnetic levitation train system.

Parameters	Value	Unit
Mass of the wheel ($m_W$)	2	kg
Mass of the body ($m_B$)	45	kg
Radius of the wheel ($r$)	0.2032	m
Distance between the center of the wheels and the TWMR’s center of gravity ($L$)	0.135	m
Distance between the contact patches of the wheels ($d$)	0.6	m
Gravity constant ($g$)	9.81	m/s2
Moment of inertia of the pendulum body ($I_1$, $I_2$, $I_3$)	1.9, 2.1, 1.6	kg·m2
Moment of inertia of the wheel for the wheel axis ($J$)	0.02	kg·m2
Moment of inertia of the wheel for the vertical axis ($J_v$)	0.04	kg·m2

.

2.2. Linearized Model

By choosing the origin as an equilibrium point of the system—that is, setting ${\mathit{x}}_0=\mathbf{0}$ and ${\mathit{u}}_0=\mathbf{0}$—and applying the Taylor series approximation, Equation (2) can be represented in state space form as

$$\dot{\mathit{x}}(t)=\mathit{A}\mathit{x}(t)+\mathit{B}\mathit{u}(t), \mathit{y}(t)=\mathit{C}\mathit{x}(t),$$

(10)

where $\mathit{A}\in R^{n\times n}$, $\mathit{B}\in R^{n\times m}$, $\mathit{C}\in R^{l\times n}$, and $n$, $m$, and $l$ are 6, 2, and 2, respectively. The Jacobian matrices of partial derivatives are given by

$$\mathit{A}={\left.{\displaystyle \frac{\partial \mathit{f}(\mathit{x})}{\partial \mathit{x}}}\right|}_{\begin{array}{c}{\mathit{x}}_0=\mathbf{0}\\ {\mathit{u}}_0=\mathbf{0}\end{array}}=\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 0& 0& a_{23}& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& a_{43}& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right],$$

(11)

$$\mathit{B}={\left.{\displaystyle \frac{\partial \mathit{g}(\mathit{x})}{\partial \mathit{u}}}\right|}_{\begin{array}{c}{\mathit{x}}_0=\mathbf{0}\\ {\mathit{u}}_0=\mathbf{0}\end{array}}=\left[\begin{array}{cc}0& 0\\ b_{21}& b_{22}\\ 0& 0\\ b_{41}& b_{42}\\ 0& 0\\ b_{61}& b_{62}\end{array}\right],$$

(12)

$$\mathit{C}=\mathit{h}\left(\mathit{x}\right),$$

(13)

with coefficients

$$a_{23}=-{\displaystyle \frac{1}{q_4}}{m}_B^{2}g{L}^2,$$

(14)

$$a_{43}={\displaystyle \frac{1}{q_4}}{q}_3{m}_{B}gL,$$

(15)

$$b_{21}=b_{22}={\displaystyle \frac{1}{q_4}}(q_1/r+m_{B}L),$$

(16)

$$b_{41}=b_{42}=-{\displaystyle \frac{1}{q_4}}[(m_{B}L)/r+q_3],$$

(17)

$$b_{61}=b_{62}=-{\displaystyle \frac{1}{q_5}}(d/r),$$

(18)

$$q_4=q_1{q}_3-m_B^2{L}^2,$$

(19)

$$q_5=I_3+2J_v+2(m_W+J/r^2)d^2,$$

(20)

It can be verified that the pair ($\mathit{A}$, $\mathit{B}$) in Equation (10) is controllable with the parameters in Table 1.

2.3. Model with Uncertainties

As expected, the linearized model is valid in a small region around the equilibrium point. In order to express Equation (10) more closely to the nonlinear model in Equation (2), we further consider uncertainties in the following form:

$$\dot{\mathit{x}}(t)=\mathit{A}\mathit{x}(t)+\mathit{B}\mathit{u}(t)+\mathit{M}\mathit{\xi }(\mathit{x},t),$$

(21)

where $\mathit{\xi }(\mathit{x},t)\in R^l$ is the uncertainty due to model dynamics and/or external disturbances, and $\mathit{M}$ is a matrix with appropriate dimensions. Some assumptions are considered:

1.

$rank\left(\mathit{B}\right)=m$, and the pair ($\mathit{A}$, $\mathit{B}$) is controllable.

2.

$\mathit{\xi }(\mathit{x},t)$ is unknown but upper-bounded for all $t$.

3.

$\mathit{M}$ is known and lies in the range space of matrix $\mathit{B}$; therefore, it is possible to write $\mathit{M}=\mathit{B}\mathit{D}$ for some $\mathit{D}\in R^{m+1}$.

Then, Equation (21) can be rewritten as

$$\dot{\mathit{x}}(t)=\mathit{A}\mathit{x}(t)+\mathit{B}\left[\mathit{u}(t)+\mathit{D}\mathit{\xi }(\mathit{x},t)\right],$$

(22)

In order to design a tracking controller for the TWMR, a new state vector is defined as follows:

$$\mathit{v}=\int (\mathit{y}-{\mathit{y}}_r)dt,$$

(23)

where ${\mathit{y}}_r\in R^l$ is the desired signal, and $\mathit{v}\in R^l$. Differentiating both sides of Equation (23) and combining it with Equation (22) yields the following augmented system:

$$\dot{\tilde{\mathit{x}}}(t)=\tilde{\mathit{A}}\tilde{\mathit{x}}(t)+\tilde{\mathit{B}}[\mathit{u}(t)+\mathit{D}\mathit{\xi }(\tilde{\mathit{x}},t)]+{\mathit{f}}_{{\mathit{y}}_r}(t),$$

(24)

where

$$\tilde{\mathit{x}}=\left[\begin{array}{c}\mathit{x}\\ \mathit{v}\end{array}\right], \tilde{\mathit{A}}=\left[\begin{array}{cc}\mathit{A}& \mathbf{0}\\ \mathit{C}& \mathbf{0}\end{array}\right], \tilde{\mathit{B}}=\left[\begin{array}{c}\mathit{B}\\ \mathbf{0}\end{array}\right] \mathrm{and} {\mathit{f}}_{{\mathit{y}}_r}(t)=\left[\begin{array}{c}0\\ -1\end{array}\right]{\mathit{y}}_{\mathit{r}},$$

(25)

Note that the augmented system in Equation (24) is controllable if the pair ($\mathit{A}$, $\mathit{B}$) of the original system has rank $n$ and the following condition is satisfied [24]:

$$rank\left[\begin{array}{cc}\mathit{B}& \mathit{A}\\ \mathbf{0}& -\mathit{C}\end{array}\right]=n+l,$$

(26)

3. Design of an Integral Sliding Mode Control Law

Sliding mode control (SMC) forces the system states to converge to a pre-defined sliding surface and slide along it until they enter the reaching phase. SMC typically consists of two components: a nominal (equivalent) control and a discontinuous (switching) control [25,26]. The nominal control is responsible for forcing the system state to move towards the sliding surface, while the discontinuous control is responsible for positioning the system state on the sliding surface and keeping it there to achieve the desired performance characteristics. Since the objective of this work is to design a robust control law for the system in Equation (21) that is insensitive to uncertainties and external disturbances, we assume that the integral sliding mode control (ISMC) law $\mathit{u}$ also has two components, nominal control ${\mathit{u}}_e$ and discontinuous control ${\mathit{u}}_n$, defined as follows:

$$\mathit{u}(t)={\mathit{u}}_e(t)+{\mathit{u}}_n(t),$$

(27)

3.1. Nominal Control Law

A nominal controller is derived for the nominal system by neglecting the uncertain components in Equation (24).

$$\dot{\tilde{\mathit{x}}}(t)=\tilde{\mathit{A}}\tilde{\mathit{x}}(t)+\tilde{\mathit{B}}{\mathit{u}}_e(t),$$

(28)

Then, the feedback control law is obtained as

$${\mathit{u}}_e(t)=-\mathit{K}\tilde{\mathit{x}}(t),$$

(29)

where the gain matrix $\mathit{K}$ can be obtained in several ways, but we apply optimal control theory so that the quadratic cost function is minimized.

$$\min {\int }_0^{\mathrm{\infty }}({\tilde{\mathit{x}}}^T\mathit{Q}\tilde{\mathit{x}}+{\mathit{u}}_e^T\mathit{R}{\mathit{u}}_e)dt,$$

(30)

Here, $\mathit{Q}$ is a positive semi-definite matrix and $\mathit{R}$ is a positive definite matrix. They penalize deviations from the desired state and control effort. $\mathit{K}$ is obtained as

$$\mathit{K}=-{\mathit{R}}^{-1}{\tilde{\mathit{B}}}^T\mathit{P},$$

(31)

where $\mathit{P}$ is the solution of the Riccati equation

$$\mathit{P}\tilde{\mathit{A}}+{\tilde{\mathit{A}}}^T\mathit{P}-\mathit{P}\tilde{\mathit{B}}{\mathit{R}}^{-1}{\tilde{\mathit{B}}}^T\mathit{P}+\mathit{Q}=\mathbf{0},$$

(32)

Rewriting Equation (29) together with Equation (23) results in

$${\mathit{u}}_e(t)=-{\mathit{K}}_{1}x(t)+{\mathit{K}}_2\int ({\mathit{y}}_r-\mathit{y})dt,$$

(33)

where $\mathit{K}=\left[{\mathit{K}}_1 {\mathit{K}}_2\right]$.

3.2. Discontinuous Control Law

The ISMC is developed using a sliding surface and a reaching law, with the design process divided into two parts: defining an integral sliding surface on which the sliding motion occurs and designing a control law that ensures the system states converge to and remain on the specified sliding surface.

For ISMC, the following sliding surface is defined:

$$\mathit{s}(t)=\mathit{G}\tilde{\mathit{x}}(t)+\mathit{z}(t),$$

(34)

where $\mathit{G}\in R^{l\times \left(n+l\right)}$ denotes the design freedom and $\mathit{z}\in R^l$ induces the integral term. $\mathit{G}$ can be selected as the left pseudo-inverse of $\tilde{\mathit{B}}$, such that $\mathit{G}\tilde{\mathit{B}}$ is invertible:

$$\mathit{G}=({\tilde{\mathit{B}}}^T\tilde{\mathit{B}}{)}^{-1}{\tilde{\mathit{B}}}^T,$$

(35)

During the sliding mode, $\mathit{s}=\dot{\mathit{s}}\left(\mathit{t}\right)=0$. Differentiation of the sliding surface yields

$$\dot{\mathit{s}}(t)=\mathit{G}\dot{\tilde{\mathit{x}}}(t)+\dot{\mathit{z}}(t)=\mathbf{0},$$

(36)

Substituting Equations (24) and (27) into Equation (36) gives

$$\dot{\mathit{s}}(t)=\mathit{G}\tilde{\mathit{A}}\tilde{\mathit{x}}(t)+\mathit{G}\tilde{\mathit{B}}{\mathit{u}}_e(t)+\mathit{G}{\mathit{f}}_{y_r}(t)+\dot{\mathit{z}}(t)+\mathit{G}\tilde{\mathit{B}}{\mathit{u}}_n(t)+\mathit{G}\tilde{\mathit{B}}\mathit{D}\mathit{\xi }(\mathit{x},t)=\mathbf{0},$$

(37)

From the condition that the 4 terms related to the nominal control must be 0,

$$\dot{\mathit{z}}(t)=-\mathit{G}[\tilde{\mathit{A}}\tilde{\mathit{x}}(t)+\tilde{\mathit{B}}{\mathit{u}}_e(t)+{\mathit{f}}_{y_r}(t)],$$

(38)

Integrating Equation (38) with the initial condition $\mathit{z}\left(0\right)=-\mathit{G}\tilde{\mathit{x}}(0)$ and substituting this into Equation (34) gives the sliding surface

$$\mathit{s}(t)=\mathit{G}\tilde{\mathit{x}}(t)-\mathit{G}\tilde{\mathit{x}}(0)-\mathit{G}{\int }_0^t[\tilde{\mathit{A}}\tilde{\mathit{x}}(\tau )+\tilde{\mathit{B}}{\mathit{u}}_e(\tau )+{\mathit{f}}_{y_r}(\tau )]d\tau ,$$

(39)

where $\mathit{z}\left(0\right)=-\mathit{G}\tilde{\mathit{x}}(0)$ guarantees $\mathit{s}\left(0\right)=\mathbf{0}$ to eliminate the reaching phase.

From the condition that the remaining two terms related to the discontinuous control in Equation (37) must be 0, the discontinuous control ${\mathit{u}}_n$ is designed as

$${\mathit{u}}_n(t)=-(\mathit{G}\tilde{\mathit{B}}{)}^{-1}\cdot \eta {\displaystyle \frac{\mathit{s}(t)}{‖\mathit{s}(t)‖}} \mathrm{for} \mathit{s}(t)\ne \mathbf{0},$$

(40)

where ${\displaystyle \frac{\mathit{s}(t)}{‖\mathit{s}(t)‖}}$ is the unit vector component, and $\eta$ is a positive constant.

3.3. Overall Control Law

The nominal control and the discontinuous control are combined to form a single control input. Thus, the overall control law in Equation (27) becomes

$$\mathit{u}(t)=-\mathit{K}\tilde{\mathit{x}}(t)-(\mathit{G}\tilde{\mathit{B}}{)}^{-1}\cdot \eta {\displaystyle \frac{\mathit{s}(t)}{‖\mathit{s}(t)‖}}=-{\mathit{K}}_{1}x+{\mathit{K}}_2\int ({\mathit{y}}_r-\mathit{y})dt-(\mathit{G}\tilde{\mathit{B}}{)}^{-1}\cdot \eta {\displaystyle \frac{\mathit{s}(t)}{‖\mathit{s}(t)‖}}.$$

(41)

If we want to further mitigate the chattering effect, ${\displaystyle \frac{\mathit{s}(t)}{‖\mathit{s}(t)‖}}$ can be replaced by one of its continuous approximations [26].

The finite time convergence of the feedback system is proved by the Lyapunov approach.

Theorem 1. 

The uncertain system in Equation (24) can be asymptotically stabilized with the integral sliding surface in Equation (34) and the control law in Equation (41), if$\eta$is chosen such that the condition$\eta \ge \left|\right|\mathit{G}\tilde{\mathit{B}}\mathit{D}\mathit{\xi }(\tilde{\mathit{x}},t)\left|\right|+\rho$is satisfied.

Proof. 

Choose a Lyapunov function as follows:

$$\mathit{V}(t)={\displaystyle \frac{1}{2}}{\mathit{s}}^T(t)\mathit{s}(t),$$

(42)

The sufficient condition for the existence of a sliding mode is that [25,26]

$$\dot{\mathit{V}}(t)={\mathit{s}}^T(t)\dot{\mathit{s}}(t)<\mathbf{0},$$

(43)

In order to prove the condition in Equation (43), we first differentiate Equation (34) and then substitute Equation (24) into it.

$$\dot{\mathit{s}}(t)=\mathit{G}\dot{\tilde{\mathit{x}}}(t)+\dot{\mathit{z}}(t)=\mathit{G}\tilde{\mathit{A}}\tilde{\mathit{x}}(t)+\mathit{G}\tilde{\mathit{B}}[\mathit{u}(t)+\mathit{D}\mathit{\xi }(\tilde{\mathit{x}},t)]+\mathit{G}{\mathit{f}}_{y_r}(t)+\dot{\mathit{z}}(t),$$

(44)

Substituting Equations (28), (38), and (41) into Equation (44) and rearranging yields

$$\begin{array}{c}\dot{\mathit{s}}(t)=\mathit{G}\tilde{\mathit{A}}\tilde{\mathit{x}}(t)+\mathit{G}\tilde{\mathit{B}}[-\mathit{K}\tilde{\mathit{x}}(t)-(\mathit{G}\tilde{\mathit{B}}{)}^{-1}\cdot \eta {\displaystyle \frac{\mathit{s}\left(t\right)}{‖\mathit{s}\left(t\right)‖}}+\mathit{D}\mathit{\xi }(\tilde{\mathit{x}},t)]+\mathit{G}{\mathit{f}}_{y_r}\left(t\right)-\mathit{G}\left[\tilde{\mathit{A}}\tilde{\mathit{x}}\left(t\right)-\tilde{\mathit{B}}\mathit{K}\tilde{\mathit{x}}\left(t\right)\right]-\mathit{G}{\mathit{f}}_{y_r}\left(t\right),\\ =\mathit{G}\left(\tilde{\mathit{A}}-\tilde{\mathit{B}}\mathit{K}\right)\tilde{\mathit{x}}\left(t\right)-\eta {\displaystyle \frac{\mathit{s}\left(t\right)}{‖\mathit{s}\left(t\right)‖}}+\mathit{G}\tilde{\mathit{B}}\mathit{D}\mathit{\xi }\left(\tilde{\mathit{x}},t\right)+\mathit{G}{\mathit{f}}_{y_r}\left(t\right)-\mathit{G}\left(\tilde{\mathit{A}}-\tilde{\mathit{B}}\mathit{K}\right)\tilde{\mathit{x}}\left(t\right)-\mathit{G}{\mathit{f}}_{y_r}\left(t\right),\\ =-\eta {\displaystyle \frac{\mathit{s}(t)}{‖\mathit{s}(t)‖}}+\mathit{G}\tilde{\mathit{B}}\mathit{D}\mathit{\xi }(\tilde{\mathit{x}},t),\end{array}$$

(45)

Pre-multiplying both sides of Equation (45) by ${\mathit{s}}^T(t)$ yields

$${\mathit{s}}^T(t)\dot{\mathit{s}}(t)=-\eta {\displaystyle \frac{{\mathit{s}}^T(t)\mathit{s}(t)}{\left|\right|\mathit{s}(t)\left|\right|}}+{\mathit{s}}^T(t)\mathit{G}\tilde{\mathit{B}}\mathit{D}\mathit{\xi }(\tilde{\mathit{x}},t),$$

(46)

and through the property ${\mathit{s}}^T\left(t\right)\mathit{s}\left(t\right)={‖\mathit{s}(t)‖}^2$, Equation (46) becomes

$$\begin{array}{c}{\mathit{s}}^T\left(t\right)\dot{\mathit{s}}\left(t\right)=-\eta ‖\mathit{s}\left(t\right)‖+{\mathit{s}}^T\left(t\right)\mathit{G}\tilde{\mathit{B}}\mathit{D}\mathit{\xi }\left(\tilde{\mathit{x}},t\right),\\ \le -\eta ‖\mathit{s}\left(t\right)‖+‖{\mathit{s}}^T\left(t\right)\mathit{G}\tilde{\mathit{B}}\mathit{D}\mathit{\xi }\left(\tilde{\mathit{x}},t\right)‖,\\ \le -\eta ‖\mathit{s}\left(t\right)‖+‖{\mathit{s}}^T(t)‖\cdot ‖\mathit{G}\tilde{\mathit{B}}\mathit{D}\mathit{\xi }(\tilde{\mathit{x}},t)‖,\end{array}$$

(47)

Applying the property $‖{\mathit{s}}^T\left(t\right)‖=‖\mathit{s}\left(t\right)‖$ again to Equation (29) gives

$${\mathit{s}}^T(t)\dot{\mathit{s}}(t)\le ‖\mathit{s}(t)‖\left(-\eta +‖\mathit{G}\tilde{\mathit{B}}\mathit{D}\mathit{\xi }(\tilde{\mathit{x}},t)‖\right),$$

(48)

If we choose a positive scalar $\eta$ such that

$$\eta \ge ‖\mathit{G}\tilde{\mathit{B}}\mathit{D}\mathit{\xi }(\tilde{\mathit{x}},t)‖+\rho ,$$

(49)

where $\rho$ is a small positive constant, the condition in Equation (43) is satisfied. As can be seen from Equation (49), if the value of $\eta$ is greater than upper bound of uncertainty, the Lyapunov function is gradually decreasing and the sliding surface $\mathit{s}(t)$ tends to zero in the given time. □

4. Simulation Results and Discussion

In this section, we describe a set of simulation works carried out on the nonlinear model in Equation (2) to demonstrate the efficiency of the proposed method. The results were compared with those of sliding mode control (SMC). The SMC uses Equation (33) as nominal control, but removes the integral element from the sliding plane in Equation (34). The tracking performance was quantitatively assessed using standard metrics: overshoot ($M_p$), rise time ($t_r$), settling time ($t_s$), and the integral absolute error ($IAE$), where $IAE=\int \left|e(t)\right|dt$. To measure robustness during disturbance rejection and parameter variation, we calculated the $IAE$ and the peak perturbation ($M_{peak}$). $M_{peak}$ is defined as the maximum absolute deviation from the output.

4.1. System Parameters and Controller Settings

The TMWR parameters and values in Table 1 were used for the simulation [8,22].

The weighing matrices $\mathit{Q}$ and $\mathit{R}$ are generally chosen to achieve a trade-off between tracking performance and control effort. In this study, they were chosen through trial and error to ensure an acceptable transient response while the system is less affected by measurement noise. With the choice of

$$Q=diag\left(15, 1, 15, 1, 15, 1, 350, 30\right),$$

(50)

$$R=diag\left(0.1, 0.1\right),$$

(51)

the feedback gain matrices ${\mathit{K}}_1$ and ${\mathit{K}}_2$ were obtained as

$${\mathit{K}}_1=\left[\begin{array}{ccc}-61.47& -44.27& -91.85\\ -61.47& -44.27& -91.85\end{array}\begin{array}{ccc}-21.78& -13.69& -4.59\\ -21.78& 13.69& 4.59\end{array}\right],$$

(52)

$${\mathit{K}}_2=\left[\begin{array}{cc}-41.83& -12.25\\ -41.83& 12.25\end{array}\right],$$

(53)

For the discontinuous control ${\mathit{u}}_n$, $\mathit{G}$ was chosen to be $‖\mathit{G}\tilde{\mathit{B}}\mathit{D}‖=1$ and $\eta$ was set to 2.5 by considering the upper bound of $\mathit{\xi }(\tilde{\mathit{x}},t)$.

4.2. Performance Under Setpoint Changes

To illustrate the tracking performance of the proposed method, a closed-loop TWMR system without uncertainties was evaluated for step-type reference changes. The following desired signals and initial conditions were configured for two different simulation cases:

• Case 1: Reference ${\mathit{y}}_r={[1,\mathsf{\pi }/6]}^{ T}$ and initial condition $\mathit{x}(0)=\mathbf{0}$;
• Case 2: Reference ${\mathit{y}}_r={\left[2, \mathsf{\pi }/2\right]}^{ T}$ and initial condition $\mathit{x}(0)={\left[0,0,-\mathsf{\pi }/6,0,-2\mathsf{\pi }/3,0\right]}^{ T}$.

Figure 2 and Figure 3 show the responses of the position, pitch angle, and yaw angle for both Case 1 and Case 2. It can be seen from Figure 2 and Figure 3 that the position and yaw angle converged to each desired value within 4 s, whereas the pitch angle maintained self-balancing within 4 s.

We calculated $M_p$, $t_r$, $t_s$, and $IAE$ of the outputs to quantitatively measure the performance of the proposed method.

Table 2. Tracking performance under reference input changes.

Case	Output	Controller	Performance Indices
			${\mathit{M}}_{\mathit{p}}$	${\mathit{t}}_{\mathit{r}}$	${\mathit{t}}_{\mathit{s}}$	$\mathit{I}\mathit{A}\mathit{E}$
1	$y_1$(m)	ISMC	4.705	1.187	2.238	1.557
		SMC	5.508	1.135	2.952	1.615
	$y_2$(deg)	ISMC	0.000	1.640	2.483	33.389
		SMC	1.452	1.544	2.480	35.315
2	$y_1$(m)	ISMC	1.070	1.387	2.373	2.965
		SMC	0.881	1.388	2.379	2.993
	$y_2$(deg)	ISMC	0.512	0.747	1.010	102.72
		SMC	0.934	0.796	1.091	106.41

summarizes these results.

Comparing the results in Table 2, the ISMC’s overshoot observed in Case 2 was marginally larger than that of Case 1. In both cases, the $IAE$ of ISMC is shown to be less than that of SMC.

4.3. Performance Under External Disturbance

A second simulation was conducted to test the proposed method’s robustness to external disturbances, such as stones or irregularities in the terrain, which can transmit external torque to the wheels when the TWMR travels on the terrain. Disturbance ${\left[-2\mathrm{s}\mathrm{i}\mathrm{n}(t), -\mathrm{c}\mathrm{o}\mathrm{s}(t)\right]}^T$ in the form of a trigonometric function was applied to the left and right wheels at 2 s after the outputs approached the desired value ${\mathit{y}}_r={\left[1, \mathsf{\pi }/6\right]}^{ T}$ from the initial state $\mathit{x}\left(0\right)=\mathbf{0}$. Figure 4 shows the responses of the position, pitch angle and yaw angle. It can be clearly seen that the proposed ISMC maintained all states near steady-state values with minimal oscillation, whereas the SMC exhibited more oscillatory responses due to input chattering.

To quantitatively demonstrate the disturbance rejection performance, we calculated $M_{peak}$ and $IAE$ and summarized them in

Table 3. Regulating performance under disturbance.

Output	Controller	Performance Indices
		${\mathit{M}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$	$\mathit{I}\mathit{A}\mathit{E}$
${\mathrm{y}}_1$ (m)	ISMC	0.001	0.005
	SMC	0.026	0.189
${\mathrm{y}}_2$ (deg)	ISMC	0.238	1.183
	SMC	4.439	34.962

. Table 3 confirms the superiority of our method.

4.4. Performance Under Disturbance Plus Parameter Changes

A third simulation was conducted to confirm the robustness of the proposed method to both external disturbances and parameter changes. Considering that the mass of the wheels slightly changes due to mud when the TWMR drives on a muddy road, four parameters were changed by adding about 40% of their nominal values: $m_W=2.0+0.8\mathrm{s}\mathrm{i}\mathrm{n}(t)$, $I_1=1.9+0.76\mathrm{s}\mathrm{i}\mathrm{n}(t)$, $I_2= 2.1+0.84\mathrm{s}\mathrm{i}\mathrm{n}(t)$, and $I_3=1.6+0.64\mathrm{s}\mathrm{i}\mathrm{n}(t)$. We applied the disturbance ${\left[-2\mathrm{s}\mathrm{i}\mathrm{n}(t), -\mathrm{c}\mathrm{o}\mathrm{s}(t)\right]}^T$ at 2 s while tracking the setpoint ${\mathit{y}}_r={\left[2,\mathsf{\pi }/2\right]}^{ T}$ from the initial state $\mathit{x}(0)={\left[0,0,-\mathsf{\pi }/9,0,0,0\right]}^{ T}$. Figure 5 shows the responses of the closed-loop system.

The figures clearly demonstrate that the proposed ISMC completely eliminated uncertainties, resulting in stable responses, whereas the SMC continued to exhibit oscillations.

Table 4. Regulating performance under disturbance and parameter variations.

Output	Controller	Performance Indices
		${\mathit{M}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$	$\mathit{I}\mathit{A}\mathit{E}$
${\mathrm{y}}_1$ (m)	ISMC	0.001	0.005
	SMC	0.028	0.186
${\mathrm{y}}_2$ (deg)	ISMC	0.234	1.158
	SMC	4.429	35.311

summarizes the results of $M_{peak}$ and $IAE$. This evidence highlights the superior robustness of the proposed ISMC.

5. Conclusions

In this paper, we proposed an integral sliding mode control (ISMC) strategy for controlling the position and yaw angle and self-stabilizing the pitch angle of a TWMR. In order to maintain control under time-varying disturbances and parameter variations in the TWMR, a nonlinear mathematical model using Newtonian law was obtained, and a control law consisting of nominal and discontinuous control was derived. The nominal control was designed based on the linearized model and optimal control theory. In order to design a tracking controller, an augmented model was used. The discontinuous control was designed such that it successfully rejects disturbances and parameter variations. Finally, a set of simulation studies were performed to evaluate the performance and robustness of the proposed method under different uncertainty conditions. The TWMR control using ISMC with optimal features was more robust and more effective than SMC. The transition of the proposed control law to real-world systems necessitates future investigations into special areas including nonlinear observer design for estimating unmeasurable variables and IIR filter design for mitigating sensor noise.