PSO-Based Optimal Tracking Control of Mobile Robots with Unknown Wheel Slipping

Abstract

Wheel slipping during trajectory tracking presents significant challenges for wheeled mobile robots (WMRs), degrading accuracy and stability on low-friction or dynamic terrain. Effective control requires addressing unknown slipping parameters while balancing tracking precision and energy efficiency. To address this challenge, a control framework integrating a sliding mode observer (SMO), an improved particle swarm optimization (PSO) algorithm, and a linear quadratic regulator (LQR) is proposed. First, a dynamic model incorporating longitudinal slipping is established. Second, an SMO is designed to estimate the slipping ratio in real-time, with chattering suppressed using a low-pass filter. Finally, an improved PSO algorithm featuring a nonlinear cosine-decreasing inertia weight strategy optimizes the LQR weighting matrices (Q/R) online to both minimize tracking errors and control energy consumption. Simulations including both circular and sine wave trajectories demonstrate that the SMO achieves rapid and accurate slipping ratio estimation, while the PSO-optimized LQR significantly enhances tracking accuracy, achieves smoother control inputs, and maintains stability under varying slipping conditions.

1. Introduction

Wheeled Mobile Robots (WMRs) have emerged as a transformative technology in modern automation, driving innovation in advanced manufacturing, logistics, agriculture, and service sectors. Their ability to navigate dynamic environments flexibly and efficiently makes them indispensable in scenarios ranging from precision material handling to autonomous field operations [1,2,3]. However, their practical deployment is constrained by two critical challenges: inherent nonholonomic motion constraints, which complicate control design, and unavoidable wheel slipping—caused by low-friction surfaces, abrupt acceleration, or uneven terrain—which introduces nonlinearity and uncertainty, degrading tracking accuracy and threatening stability.

Existing trajectory tracking methods for WMRs, such as sliding mode control [4,5,6], backstepping control [7,8,9], fuzzy logic control [10,11,12], adaptive control [13,14,15], neural network-based control [16,17,18], active disturbance rejection control [19,20,21], and pure pursuit control [22,23], have achieved significant progress but often rely on idealized assumptions of zero wheel slipping. In reality, slipping disrupts the kinematic relationship between wheel motion and robot pose, leading to accumulated errors and potential instability. This gap between theory and practice underscores the need for robust control strategies that explicitly address slipping.

Recent efforts to mitigate slipping effects have followed two main methods. One approach treats slipping as a lumped disturbance, using observers to compensate for its impact. For example, Kang et al. [24] employed a generalized extended state observer (GESO) to estimate lumped disturbances, including unknown slipping and sliding effects, subsequently proposing a GESO-based robust tracking controller. Li et al. [25] proposed a novel fixed-time control scheme. This method constructs a fixed-time velocity control law using intermediate variables to avoid singularity issues and develops a fast fixed-time disturbance observer to estimate slipping and external disturbances. Experiments have verified that it can achieve high-precision tracking within a preset time. Chen et al. [26] obtained the virtual velocity control law via a desired disturbance observer, later formulating a robust tracking control method aligned with the actual system’s trajectory goal. Nie et al. [27] proposed a low-complexity safety control method. This method overcomes the limitations of traditional model control point selection by introducing a novel linearization model combined with virtual points and directly constrains the tracking error using a preset performance function. It achieves the goal of strictly limiting the robot trajectory tracking error within a preset safety range without the need to obtain or estimate complex slipping parameters. In reference [28], autoregressive wavelet neural networks were used to approximate model uncertainties and slipping limitations. This enabled the development of a controller that can track a trajectory in an adaptive manner and avoid obstacles. Ryu et al. [29] designed a robust controller treating wheel slipping as lumped disturbance, validating its effectiveness via simulation and experiment. FEI et al. [30] presented a nonlinear extended state observer (NESO) leveraging fuzzy neural networks to estimate system states and total disturbances, thereby enabling active compensation for unmodeled dynamics. In addition, data-driven control strategies, such as the γ-DDPC framework proposed by Breschi et al. [31], provide a new way of thinking about wheeled robot slip as system uncertainty by dealing with system uncertainty and noise through regularization techniques. The studies mentioned above all treat slipping as a black-box disturbance, failing to leverage the internal parameter information of slipping. Controller parameters are not dynamically adjusted based on the slipping magnitude, which may restrict performance under varying slipping conditions.

The second path involves slipping parameter estimation. Wang et al. [32] established a mathematical model incorporating both skidding and sliding dynamics. Cui et al. [33] proposed an adaptive tracking control scheme for wheeled mobile robots with unknown longitudinal and lateral slipping by integrating a sliding mode observer for real-time slipping parameter estimation and a backstepping controller, guaranteeing global stability via the Lyapunov theory and demonstrating robust performance under large initial errors. Song et al. [34] proposed a nonlinear sliding mode observer to estimate the slipping parameters of unmanned wheeled vehicles with high accuracy, verifying its effectiveness using dedicated linear and 2D test benches. However, these methods [32,33,34] typically estimate the slipping ratio as a system parameter but lack adaptation of the controller parameters based on the estimated slipping conditions. Consequently, the controller struggles to maintain optimal performance across varying slipping conditions.

Ni et al. [35] designed an improved LQR controller, integrating feedforward control and an adaptive gain strategy for real-time QR matrix adjustment based on tracking error. The above literature is able to adjust the LQR controller parameters in real time according to the error, but not according to the difference in the system slipping ratio.

In summary, existing methods have not yet solved the problem of real-time weight optimization updates under different slipping parameters. In this paper, we estimate the slipping ratio parameters in real time and use the improved PSO algorithm to combine with the LQR controller to adjust the parameter sizes of the controller to dynamically adapt the controller to varying slipping conditions and operational scenarios. The main contributions of this article are as follows:

(1)

To address the model mismatch problem introduced by unknown slipping parameters, a kinematic model incorporating longitudinal slipping dynamics is derived to characterize the system behavior, a sliding mode observer is used to observe the slipping parameter, and a first-order low-pass filter is used to reduce the chattering phenomenon.

(2)

To reduce reliance on empirical selection of LQR weights, a PSO algorithm with nonlinear inertia weights is proposed to optimize the Q/R matrix in real time to improve the tracking performance of the controller.

(3)

A closed-loop framework integrating slipping estimation and optimization control is constructed, significantly enhancing the tracking accuracy, control smoothness, and robust stability of WMRs with unknown slipping condition.

2. Kinematic Modeling with Slipping

There are many types of wheeled mobile robots, among which two-wheeled differential drive robots are favored due to their advantages of exhibiting a simple structure and a flexible control system. Figure 1 shows a classic kinematic model of a two-wheel differentially driven wheeled mobile robot. The structure includes a frame of coordinates $F_1(X,Y)$, centered at the coordinate origin, and a frame of move coordinates $F_2(x,y)$, with the geometric center O as the origin ω, denotes the angular velocity of the mobile robot, which is rotating around the origin O of the coordinate system $F_2(x,y)$. Instead of a steering mechanism, the robot achieves turning by adjusting the speed difference between the two wheels to realize the sliding of the robot body, then completing the steering action of different radii. To simplify the model analysis, it is assumed that the body is perfectly symmetrical and that the center of mass of the robot and the center of its shape are not offset from one point. In this study, we focused on the longitudinal slipping phenomenon. In the ideal case of a wheel without slipping, its kinematics can be expressed as

$$\begin{array}{l}{v}_R=r{\omega }_R\\ v_L=r{\omega }_L\end{array}$$

(1)

where r denotes the wheel’s rolling radius, the linear and angular velocities of the left wheel are denoted by $v_L$ and ${\omega }_L$, and the linear and angular velocities of the right wheel are denoted by $v_R$ and ${\omega }_R$.

As Figure 1 shows, the forward velocity v and the steering angular velocity ω can be characterized by the following equation:

$$\begin{array}{c}v={\displaystyle \frac{v_{\mathrm{L}}+v_{\mathrm{R}}}{2}}={\displaystyle \frac{r{\omega }_{\mathrm{L}}+r{\omega }_{\mathrm{R}}}{2}}\\ \dot{\theta }=\omega ={\displaystyle \frac{v_{\mathrm{L}}-v_{\mathrm{R}}}{b}}={\displaystyle \frac{r{\omega }_{\mathrm{L}}-r{\omega }_{\mathrm{R}}}{b}}\end{array}$$

(2)

where b denotes the wheelbase.

To describe the phenomenon of longitudinal slipping more accurately, the concept of a longitudinal wheel slipping ratio is introduced. The introduction of the longitudinal wheel slipping ratio means that the linear velocities of the left and right wheels become

$$\begin{array}{l}{v}_R^{\prime }=r{\omega }_R(1-i_R)\\ v_L^{\prime }=r{\omega }_L(1-i_L)\end{array}$$

(3)

where $i_L$ is the left wheel slipping ratio, and $i_R$ is the right wheeled slipping ratio.

In coordinate system $F_1(X,Y)$, without wheel slipping, the position and attitude of WMR can be described as follows:

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

(4)

With the introduction of the longitudinal slipping ratio in the standard model, the model with longitudinal slipping included in the coordinate system $F_2(x,y)$ can be defined as described below:

$$\begin{array}{c}\dot{x}={\displaystyle \frac{r{\omega }_{\mathrm{L}}(1-i_{\mathrm{L}})+r{\omega }_{\mathrm{R}}(1-i_{\mathrm{R}})}{2}}\\ \dot{y}={\displaystyle \frac{r{\omega }_{\mathrm{L}}(1-i_{\mathrm{L}})+r{\omega }_{\mathrm{R}}(1-i_{\mathrm{R}})}{2}}\\ \dot{\theta }={\displaystyle \frac{r{\omega }_{\mathrm{L}}(1-i_{\mathrm{L}})-r{\omega }_{\mathrm{R}}(1-i_{\mathrm{R}})}{b}}\end{array}$$

(5)

As shown in Figure 1, there is a transformation relationship between coordinate system $F_1(X,Y)$ and coordinate system $F_2(x,y)$, and the transformation equation between them is shown below:

$$\left[\begin{array}{c}X\\ Y\end{array}\right]=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

(6)

Therefore, the following equation can be applied in order to represent the WMR’s kinematic model within the global reference system $F_1(X,Y)$, considering the longitudinal slipping case:

$$\begin{array}{c}\dot{X}={\displaystyle \frac{r{\omega }_{\mathrm{L}}(1-i_{\mathrm{L}})+r{\omega }_{\mathrm{R}}(1-i_{\mathrm{R}})}{2}}\cos \theta \\ \dot{Y}={\displaystyle \frac{r{\omega }_{\mathrm{L}}(1-i_{\mathrm{L}})+r{\omega }_{\mathrm{R}}(1-i_{\mathrm{R}})}{2}}\sin \theta \\ \dot{\theta }={\displaystyle \frac{r{\omega }_{\mathrm{L}}(1-i_{\mathrm{L}})-r{\omega }_{\mathrm{R}}(1-i_{\mathrm{R}})}{b}}\end{array}$$

(7)

3. Mobile Robot Trajectory Tracking Model

During the trajectory tracking process, the mobile robot must determine the error between its actual pose and the desired pose in real time in order to promptly adjust its control strategy. When tracking a given reference trajectory, the trajectory tracking diagram can be simplified to that shown in Figure 2. The variables ${(X,Y,\theta )}^{\mathrm{T}}$ and ${(X^r,Y^r,{\theta }^r)}^{\mathrm{T}}$ denote the true and desired positions of the mobile robot, respectively, and the relationship between these positions is given by the following kinematic equations:

$$\left[\begin{array}{c}{e}_x\\ e_y\\ e_{\theta }\end{array}\right]=\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0\\ -\sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{X}^r-X\\ Y^r-Y\\ {\theta }^r-\theta \end{array}\right]$$

(8)

where $e_x$ denotes the transverse error, $e_y$ denotes the longitudinal error, and $e_{\theta }$ denotes the directional error.

Applying the derivative to Equation (8) yields

$$\begin{array}{c}{\dot{e}}_x=\omega e_y+v^r\cos e_{\theta }-v\\ {\dot{e}}_y=-\omega e_x+v^r\sin e_{\theta }\\ {\dot{e}}_{\theta }=\omega -{\omega }^r\end{array}$$

(9)

where $\omega$ represents the angular velocity control input, v denotes the linear velocity control input, ${\omega }^r$ is the reference angular velocity, and $v^r$ denotes the reference linear velocity.

This is obtained by linearizing the system equation at the tracking trajectory.

$$\left[\begin{array}{c}{\dot{e}}_x\\ {\dot{e}}_y\\ {\dot{e}}_{\theta }\end{array}\right]=\left[\begin{array}{ccc}0& {\omega }^r& 0\\ -{\omega }^r& 0& v^r\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{e}_x\\ e_y\\ e_{\theta }\end{array}\right]+\left[\begin{array}{cc}-1& 0\\ 0& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}v-v^r\\ \omega -{\omega }^r\end{array}\right]$$

(10)

4. Slipping Estimation Using Sliding Mode Observer

4.1. Sliding Mode Observer Design

The measurement of the longitudinal slipping ratio of the wheel is challenging due to the complexity of direct measurement and the presence of noise in the measurement data. The sliding mode observer exhibits strong robustness to model uncertainty and noise. Therefore, a sliding mode observer is employed to estimate the longitudinal slipping ratio of each wheel. Here, $\widehat{X}$, $\widehat{Y}$, and $\widehat{\theta }$ are used to denote the estimation of the position and angle of the WMR. $\widehat{v}$ represents the estimated linear velocity of the WMR, and $\widehat{\omega }$ represents its estimated angular velocity.

The above analysis leads to the design of a sliding mode observer with respect to position, angle, and linear velocities.

$$\begin{array}{c}\dot{\widehat{X}}=\widehat{v}\cos \widehat{\theta }+k_x\mathrm{sgn}(X-\widehat{X})\\ \dot{\widehat{Y}}=\widehat{v}\sin \widehat{\theta }+k_y\mathrm{sgn}(Y-\widehat{Y})\\ \dot{\widehat{\theta }}=\widehat{\omega }+k_{\theta }\mathrm{sgn}(\theta -\widehat{\theta })\\ \widehat{v}=k_v\left(\mathrm{sgn}(X-\widehat{X})\cos \widehat{\theta }+\mathrm{sgn}(Y-\widehat{Y})\sin \widehat{\theta }\right)\\ \dot{\widehat{\omega }}=k_{\omega }\mathrm{sgn}(\theta -\widehat{\theta })\end{array}$$

(11)

in the equation, $k_x$, $k_y$, $k_{\theta }$, $k_v$, and $k_{\omega }$ represent the sliding mode gain.

Calculate the wheel speed using the wheel speed formula:

$$\begin{array}{l}{\widehat{v}}_R=\widehat{v}+{\displaystyle \frac{b}{2}}\widehat{\omega }\\ {\widehat{v}}_L=\widehat{v}-{\displaystyle \frac{b}{2}}\widehat{\omega }\end{array}$$

(12)

where b is the wheelbase, the estimated linear velocities for the left and right wheels are ${\widehat{v}}_L$ and ${\widehat{v}}_R$, respectively.

Therefore, the estimated slipping ratio of the wheel is calculated by the following equation:

$$\begin{array}{l}{\widehat{i}}_R=\left\{\begin{array}{c}1-{\displaystyle \frac{{\widehat{v}}_R}{r{\omega }_R}}\mathrm{if}{\displaystyle |{\omega }_R|}>\epsilon \\ 0\mathrm{otherwise}\end{array}\right.\\ {\widehat{i}}_L=\left\{\begin{array}{c}1-{\displaystyle \frac{{\widehat{v}}_L}{r{\omega }_L}}\mathrm{if}{\displaystyle |{\omega }_R|}>\epsilon \\ 0\mathrm{otherwise}\end{array}\right.\end{array}$$

(13)

where r is the wheel radius, ${\widehat{i}}_L$ indicates an estimate of the left-hand wheel slipping ratio, and ${\widehat{i}}_R$ indicates an estimate of the right-hand wheel slipping ratio. ${\omega }_R$ and ${\omega }_L$ are the commanded angular velocities of wheels, and $\epsilon =0.05$ denotes the threshold to avoid dividing by zero.

Additionally, a low-pass filter (LPF) is employed to mitigate the chattering phenomenon inherent in the sliding mode observer (SMO). It is designed as shown in Figure 3, and the relationship between input u and output y of LPF is given as

$$A\dot{y}+y=u,y(0)=u(0)$$

(14)

where $u=[{\widehat{i}}_R,{\widehat{i}}_L]$, and $A=diag[a_1,a_2]$, $a_1$, $a_2$ are the filter parameters, and u is the input.

The relationship between the cutoff frequency and filter parameters is as follows:

$$f_c={\displaystyle \frac{1}{2\pi a}}$$

(15)

With a cut-off frequency of $f_c=20 \mathrm{Hz}$, the LPF effectively attenuates high-frequency oscillations while preserving the essential dynamic characteristics of the slip ratio signal. Furthermore, owing to its structural simplicity and computational efficiency, the filter is well-suited for real-time implementation in embedded systems.

4.2. Proof of Sliding Mode Observer

In this section, the stability of the sliding mode observer is proven using the Lyapunov theory.

The switching function of the SMO is defined by the following equation:

$$\begin{array}{l}{s}_1=X-\widehat{X}\\ s_2=Y-\widehat{Y}\\ s_3=\theta -\widehat{\theta }\end{array}$$

(16)

As noted in reference [36], a sliding mode observer achieves stability only if 1. the error state converges to the sliding mode surface from any initial error in finite time; 2. the system exhibits stability within a neighborhood of this surface. Select the appropriate sliding mode gain so that $s_i\cdot {\dot{s}}_i<0$, i = 1, 2, 3.

Define the positive definite Lyapunov function:

$$\mathrm{V}={\displaystyle \frac{1}{2}}{s}^2={\displaystyle \frac{1}{2}}(s_1^2+s_2^2+s_3^2)$$

(17)

Since the slipping ratio is physically bounded, $i_L<1$ and $i_R<1$.

It is evident that there is an inherent upper limit that exists between the actual velocity difference and the estimated velocity difference.

$$\begin{array}{l}\left|v-\widehat{v}\right|<D_v\\ \left|\omega -\widehat{\omega }\right|<D_{\omega }\end{array}$$

(18)

where $D_v$ and $D_{\omega }$ are constants greater than zero.

The estimated speed is usually constrained by the control inputs:

$$\left|\widehat{v}\right|<v_{\max }$$

(19)

where $v_{\max }$ is a constant greater than zero.

Angular observation error $s_3=\theta -\widehat{\theta }$ is bounded in system operation:

$$\left|\theta -\widehat{\theta }\right|<M$$

(20)

where M is a constant greater than zero.

Furthermore, the trigonometric functions satisfy Lipschitz continuity:

$$\begin{array}{l}\left|\cos \theta -\cos \widehat{\theta }\right|\le \left|\theta -\widehat{\theta }\right|\\ \left|\sin \theta -\sin \widehat{\theta }\right|\le \left|\theta -\widehat{\theta }\right|\end{array}$$

(21)

Thus,

$$\begin{array}{l}\left|v\cos \theta -\widehat{v}\cos \widehat{\theta }\right|\le \left|v-\widehat{v}\right|+\left|\widehat{v}\right|\cdot \left|\cos \theta -\cos \widehat{\theta }\right|\le D_v+v_{\max }\left|\theta -\widehat{\theta }\right|\\ \le D_v+v_{\max }M\end{array}$$

(22)

identically obtained.

$$\begin{array}{l}\left|v\sin \theta -\widehat{v}\sin \widehat{\theta }\right|\le D_v+v_{\max }M\\ \left|\omega -\widehat{\omega }\right|\le D_{\omega }\end{array}$$

(23)

Define the upper bound:

$$\begin{array}{l}{\Delta }_1=D_v+v_{\max }M\\ {\Delta }_2=D_v+v_{\max }M\\ {\Delta }_3=D_{\omega }\end{array}$$

(24)

Taking the time derivative of Equation (17) yields

$$\begin{array}{l}\dot{\mathrm{V}}={\mathrm{s}}_1{\dot{s}}_1+{\mathrm{s}}_2{\dot{s}}_2+{\mathrm{s}}_3{\dot{s}}_3=s_1(v\cos \theta -\widehat{v}\cos \widehat{\theta }-k_x\mathrm{sgn}(s_1))\\ +s_2(v\sin \theta -\widehat{v}\sin \widehat{\theta }-\mathrm{sgn}(s_2))+s_3(\omega -\widehat{\omega }-\mathrm{sgn}(s_3))\end{array}$$

(25)

From Equations (22)–(24), we derive

$$\dot{\mathrm{V}}\le \left|s_1\right|{\Delta }_1+\left|s_2\right|{\Delta }_2+\left|s_3\right|{\Delta }_3-s_1{k}_x\mathrm{sgn}(s_1)-s_2{k}_y\mathrm{sgn}(s_2)-s_3{k}_{\theta }\mathrm{sgn}(s_3)$$

(26)

Since the sign function satisfies $s_i\mathrm{sgn}(s_i)=\left|s_i\right|$.

Thus,

$$\dot{\mathrm{V}}\le \left|s_1\right|({\Delta }_1-k_x)+\left|s_2\right|({\Delta }_2-k_y)+\left|s_3\right|({\Delta }_3-k_{\theta })$$

(27)

Select sliding mode gain $k_x$, $k_y$, and $k_{\theta }$ so that the following satisfies the requirement.

$$\begin{array}{l}{k}_x>D_v+v_{\max }M\\ k_y>D_v+v_{\max }M\\ k_{\theta }>D_{\omega }.\end{array}$$

(28)

Consequently, we have

$$\dot{V}\le \left|s_1\right|({\mathsf{\Delta }}_1-k_x)+\left|s_2\right|({\mathsf{\Delta }}_2-k_y)+\left|s_3\right|({\mathsf{\Delta }}_3-k_{\theta })<0$$

(29)

As can be seen from the above analysis, when sliding mode gains of $k_x$, $k_y$, and $k_{\theta }$ satisfy Equation (28), the observation error of the sliding mode observer converges to zero within a finite time.

5. LQR Control Design with PSO-Based Optimization

5.1. Linearization of System Equations

Applying the linear quadratic regulator (LQR) to nonlinear systems offers significant advantages in solving control problems. However, wheeled mobile robot systems are nonlinear. Therefore, to apply the LQR controller, the system equations need to be linearized near the operating point. The linearized system matrix is as follows:

$$A=\left[\begin{array}{ccc}0& {\omega }^r& 0\\ -{\omega }^r& 0& v^r\\ 0& 0& 0\end{array}\right]B=\left[\begin{array}{cc}-1& 0\\ 0& 0\\ 0& -1\end{array}\right]$$

(30)

The cost function [37] is given by the following equation:

$$J={\displaystyle {\int }_0^{\infty }[x^{\mathrm{T}}(t)Qx(t)+u^{\mathrm{T}}(t)Ru(t)]dt}$$

(31)

where $Q=diag[\begin{array}{ccc}{q}_1& q_2& q_3\end{array}]$, $q_1$, $q_2$, $q_3$ represent the weight coefficient of $e_x$, $e_y$, $e_{\theta }$. $R=diag\left[\begin{array}{cc}{r}_1& r_2\end{array}\right]$, $r_1$ and $r_2$ denote the weight coefficients of the control inputs v and w.

The matrix K is given by the following equation:

$$K=R^{-1}{B}^{\mathrm{T}}P$$

(32)

The P matrix is a constant matrix whose exact value can be derived by solving the Riccati equation, whose mathematical expression is shown below:

$$A^{\mathrm{T}}P+PA-PB{R}^{-1}{B}^{\mathrm{T}}P+Q=0$$

(33)

Therefore, the control law $u=-KX$ is applied to the system, which can minimize the system’s performance index $K=\left[\begin{array}{cc}{K}_1& K_2\end{array}\right]$, and $K_1$, $K_2$ are the feedback coefficients of LQR control.

As shown above, the system’s optimal control performance is primarily affected by the weight matrices Q and R. However, there is no fixed pattern for the selection of these two matrices, and it is usually chosen empirically by the designer. Not only is it difficult to ensure that the final output of the optimal feedback control coefficients will lead to a truly optimal system, but the whole process is time-consuming. The adverse effect of subjectivity that arises from relying on experience to determine the optimality of the system can be mitigated by using the PSO algorithm to optimize the weight matrices.

5.2. Enhanced PSO with Nonlinear Inertia Weight Strategy

Particle swarm optimization (PSO) was proposed in 1995 [38], and the idea behind its design is inspired by the cooperative behavior of groups of organisms, such as birds and fish. In the algorithm, each candidate solution is treated as a particle, which dynamically adjusts its flight direction and velocity in the solution space by tracking its individual optimal and population optimal positions. This process gradually approaches the optimal solution. The following equations are utilized in order to effect changes in the velocity and position of each individual particle:

$$\begin{array}{c}{v}_{i+1}=\omega v_i+c_1\times r_1(p_{best}-x_i)+c_2\times r_2(g_{best}-x_i)\\ x_{i+1}=x_i+v_i\end{array}$$

(34)

where $\omega$ is the non-negative inertia weight; $p_{best}$ and $g_{best}$ are the best past positions of the individual and the population; $v_i$ is the velocity of the ith particle; $c_1$ and $c_2$ represent the individual learning factor and group learning coefficients, respectively; $r_1$ and $r_2$ denote two random numbers uniformly distributed in the range [0, 1], which serve to introduce randomness and make the search process randomly exploratory; $x_i$ denotes the current position of the i-th particle.

The inertia weight $\omega$ exerts significant impacts on the global optimization performance, and compared to fixed inertia weights, the linear decreasing inertia weight strategy [39] improves the performance, i.e., the inertia weight decreases linearly, leading to faster fitness function convergence compared to that of fixed weights.

The decreasing inertia weight factor of the PSO algorithm optimized using the strategy of linear decreasing inertia weight is as follows:

$$w^T=w_{\max }-(w_{\max }-w_{\min })(T/N)$$

(35)

the inertia weight coefficient for the Tth iteration is represented as $w^T$, the maximum value is $w_{\max }$, and $w_{\min }$ denotes the minimum value of the inertia weights. T denotes the number of iterations currently in progress. It is important to note that the maximum number of iterations that can be performed is denoted by N.

Although the linear decreasing inertia weight strategy outperforms that of fixed weights, it still exhibits limitations; the global search duration is short in the early iterations, the local search ability is weak, and it can easily miss the global optimum. Therefore, this paper proposes a nonlinear decreasing weight update strategy:

$$w^T=w_{\min }+{\displaystyle \frac{(w_{\max }-w_{\min })}{2}}\left(1+\cos \left({\displaystyle \frac{\pi T}{N}}\right)\right)$$

(36)

the inertia weight coefficient for the Tth iteration is represented as $w^T$, the maximum value is $w_{\max }$, $w_{\min }$ denotes the minimum value of the inertia weights, and T denotes the number of iterations currently in progress. It is important to note that the maximum number of iterations that can be performed is denoted by N.

With this weight adjustment strategy, the inertia weight value at the beginning of the iteration is large and decreases slowly, enhancing exploration of the solution space, and the inertia weights decrease rapidly as the number of iterations approaches the maximum number of iterations, enhancing focusing on the optimal solution and preventing convergence to local optimality. The comparison between the linear decreasing inertia weighting strategy and the cosine decreasing inertia weighting strategy is shown in Figure 4.

The fitness function employs a weighted performance index to satisfy both the tracking performance and energy efficiency, and the fitness function is as follows:

$$F=0.7ISE+0.2E+0.1e_{end}$$

(37)

where ISE is the integrated squared error, E is the energy consumption of the control input, and $e_{end}$ is the terminal position error. Each component is defined as

$$\mathrm{ISE}={\displaystyle {\int }_0^{\mathrm{T}}\left[e_x^2(t)+e_y^2(t)+e_{\theta }^2(t)\right]}dt$$

(38)

where $e_x$ denotes the transverse error, the longitudinal error is represented by $e_y$ and $e_{\theta }$ denotes the angular error. T denotes the simulation time

$$E={\displaystyle {\int }_0^{\mathrm{T}}{‖u(t)‖}^2}dt$$

(39)

where $u(t)={[v,w]}^{\mathrm{T}}$ denotes control input.

$$e_{end}=\sqrt{{\left(x(T_{end})-x^r(T_{end})\right)}^2+{\left(y(T_{end})-y^r(T_{end})\right)}^2}$$

(40)

where $x(T_{end})$ and $y(T_{end})$ denote the position of the wheeled mobile robot at the final moment, and $x^r(T_{end})$ and $y^r(T_{end})$ indicate the expected position at the final moment.

The flow diagram of the algorithm using the cosine weight update strategy is shown in Figure 5.

6. Simulation Results

In this section, we will apply the formulas derived in prior sections to simulate the WMR with longitudinal slipping. To rigorously evaluate the tracking performance of the controller under varying operating conditions, two reference trajectories are selected for simulation: a circular trajectory and a sine wave trajectory. The relevant parameters of wheeled mobile robots are shown in

Table 1. Parameters of WMR.

Parameters	b	r
Value	1 m	0.5 m

. The five observation gains of the sliding mode observer are selected as shown in

Table 2. Parameters of SMO.

Parameters	kx	ky	kθ	kv	kw
Value	6.0	6.0	4.0	3.2	3.0

. The robot’s position is directly measurable via sensors. If the population size is too small, it tends to fall into local optimization, and if the population size is too large, the growth is no longer significant; the maximum number of iterations, as the termination condition of the optimization, takes too long to optimize if it is too large, and optimization is terminated before reaching the optimal result if it is too small. A good balance between the algorithm’s solution speed and the quality of the optimal solution can be achieved by using the following particle swarm optimization (PSO) parameter configurations. The parameter configurations of the improved particle swarm optimization algorithm are shown in

Table 3. Parameters of PSO.

Parameters	Particle number	Number of iterations	c1	c2
Value	50	100	2.5	1.5
Parameters	Range of Q matrix	Range of R matrix	wmax	wmin
Value	[0–50]	[0.01–10]	0.7	0.1

. We analyzed the computational load of the PSO algorithm. The optimization process was performed using MATLAB 2024a on a desktop platform (Intel i7-12700 K DELL, Xiamen, China), with each run taking approximately 120 s.

To compare the effectiveness of the three weighted updating methods, we choose a circular reference trajectory for simulation, whose specific parameters are given in Section 6.1, and the fitness convergence curve is shown in Figure 6.

As shown in the results, the cosine weights updating method proposed in this paper exhibits the fastest decreasing adaptation and the smallest final adaptation. The PSO-optimized LQR algorithm proposed in this paper displays a large searchable space and a high probability of finding the optimal solution in the early iteration period due to the slow reduction of inertia weights and avoids falling into a local optimum in the later stage due to the fast decrease in inertia weights, finding better approximate solutions.

6.1. Circular Trajectory Simulation

The equation for a given circular reference trajectory is

$$\begin{array}{l}{X}^r=5\cos t\\ Y^r=5\sin t\end{array}$$

(41)

where t denotes the simulation time, 0 < t < 50 s.

The linear velocity of the WMR is expected to be $v_r=2.5 \mathrm{m}/\mathrm{s}$; the expected angular velocity is ${\omega }_r=0.5 \mathrm{rad}/\mathrm{s}$. The parameters of the Q, R matrix optimized by the PSO are as follows: Q = diag [11.760, 29.763, 12.219], R = diag [0.010, 0.010], and the initial position and orientation of the WMR are set as ${\left[\begin{array}{ccc}X(0)& Y(0)& \theta (0)\end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{ccc}6\mathrm{m}& 0& 0\end{array}\right]}^{\mathrm{T}}$. The initial desired pose is given by the trajectory equation ${\left[\begin{array}{ccc}{X}^r(0)& Y^r(0)& {\theta }^r(0)\end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{ccc}5\mathrm{m}& 0& 0\end{array}\right]}^{\mathrm{T}}$. The verification of tracking performance in the slipping case will be performed according to the following parameters, i.e., the abrupt change of $i_L$ and $i_R$ parameters is simulated, and it is assumed that $i_L=0.15\sin 0.2(t-15)$, and $i_R=-0.15\sin 0.2(t-15)$ at t = 15 s. For comparison, unoptimized Q and R matrices are selected, i.e., Q = diag [10, 10, 8]; R = diag [1, 1]. The tracking results are shown in Figure 7, Figure 8, Figure 9 and Figure 10. Two performance indicators were selected: maximum deviation and convergence time. The maximum deviation value was $\sqrt{e_x^2+e_y^2}$ deviation. The convergence time is the time required for the error to enter and remain within the steady-state band (5%). The two performance indicators of the circular reference trajectory are shown in

Table 4. Circular reference trajectory performance indicators.

Performance Indicators	Conventional LQR	PSO-Optimized LQR	Reference [33] Method
Maximum deviation	0.35 m	0.09 m	0.15 m
Convergence time	1.23 s	1.19 s	0.85 s

.

As illustrated in Figure 7 and Figure 8, a comparison of the steady-state error between the optimized model and the baseline model demonstrates a significant reduction in error magnitude following optimization. Figure 8 further reveals that the conventional LQR controller experiences substantial tracking deviations once wheel slipping occurs. Conversely, Figure 10 indicates that the SMO converges to the true slipping ratio within 3 s, thereby delivering precise parameter updates to the PSO-LQR controller. This integrated “estimation-optimization” closed-loop architecture sustains system stability under conditions of unknown slipping and effectively mitigates the tracking divergence problem inherent in traditional LQR controllers when confronted with model uncertainties.

A quantitative analysis of the performance indicators in Table 4 further supports these observations. The PSO-optimized LQR achieves a maximum deviation of only 0.09 m, significantly lower than the 0.35 m of the conventional LQR and also superior to the 0.15 m of the reference [33] method. Regarding convergence time, the PSO-LQR (1.19 s) shows a slight improvement over the conventional LQR (1.23 s), although it is slightly slower than the reference [33] method (0.85 s). This indicates that while the PSO-LQR may not be the fastest in terms of initial convergence, it offers a superior balance by achieving significantly higher steady-state accuracy, which is often more critical for precise trajectory tracking.

6.2. Sine Wave Trajectory Simulation

The sine wave reference trajectory is defined as

$$\begin{array}{c}{X}^r=0.5t\\ Y^r=5\sin 0.05t\end{array}$$

(42)

where t denotes the simulation time, 0 < t < 50 s. The original coordinates of the wheeled mobile robot are defined as follows: ${\left[\begin{array}{ccc}X(0)& Y(0)& \theta (0)\end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{ccc}1\mathrm{m}& 0\mathrm{m}& 0\mathrm{m}\end{array}\right]}^{\mathrm{T}}$. The initial desired pose is assigned a value of ${\left[\begin{array}{ccc}{X}^r(0)& Y^r(0)& {\theta }^r(0)\end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{ccc}0\mathrm{m}& 0\mathrm{m}& 0\mathrm{m}\end{array}\right]}^{\mathrm{T}}$. The verification of tracking performance in the slipping case will be performed according to the following parameters, i.e., the abrupt change of $i_L$$i_R$ parameter is simulated, and it is assumed that $i_L$ becomes 0.1 and $i_R$ becomes 0.15 at t = 10 s. The parameters of the Q and R matrix optimized by the PSO algorithm are as follows: Q = diag [8.916, 19.920, 14.607], and R = diag [0.026, 0.050]. At the same time, SMO can accurately track changes in slipping parameters. Unoptimized Q and R matrices are selected, i.e., Q = diag [10, 10, 5], and R = diag [8, 8]. The simulation results are shown below in Figure 11, Figure 12, Figure 13 and Figure 14. The two performance indicators of the sine wave reference trajectory are shown in

Table 5. Sine wave reference trajectory performance indicators.

Performance Indicators	Conventional LQR	PSO-Optimized LQR	Reference [33] Method
Maximum deviation	0.15 m	0.06 m	0.06 m
Convergence time	3.21 s	2.77 s	2.84 s

.

The sine wave reference trajectory exhibits significant periodic curvature fluctuations, which poses a serious challenge to the trajectory tracking control due to its stringent dynamic requirements. As shown in Figure 11a and Figure 12a, the LQR controller with unoptimized parameters exhibits significant limitations when tracking such complex paths. Large tracking errors are observed, particularly in intervals where curvature is maximal.

The performance metrics in Table 5 provide a clear quantitative comparison. The maximum deviation for the conventional LQR is 0.15 m, which is reduced to 0.06 m by the PSO-LQR, matching the performance of the reference [33] method. For convergence time, the PSO-LQR (2.77 s) shows a notable improvement over that in the conventional LQR (3.21 s), although it remains slightly slower than in the reference [33] method (2.26 s). These results demonstrate that the proposed PSO-LQR controller effectively handles the challenging sine wave trajectory, achieving high tracking accuracy and improved convergence compared to those of the conventional approach, while generating smoother control inputs, as seen in Figure 13b.

In contrast, the control input transitions smoothly between acceleration and deceleration phases, directly enhancing system durability. These improvements confirm the controller’s effectiveness in complex navigation scenarios with continuously changing curvature.

7. Conclusions

In this study, the key challenge of the trajectory tracking control of wheeled mobile robots (WMRs) with wheel slipping is addressed. To improve tracking accuracy and robustness, an integrated control framework is proposed, combining three key components: First, a dynamic model incorporating longitudinal slipping parameters is developed. Second, a sliding mode observer (SMO) with an LPF is designed. The SMO provides a fast and accurate estimation of the slipping ratio, while the low-pass filter effectively suppresses chattering of the observation. Finally, the LQR controller parameters are optimized by PSO. The improved PSO algorithm with nonlinear cosine decreasing inertia weights optimizes the Q and R matrices in the LQR cost function to minimize the tracking error and control energy consumption. This is validated by extensive simulations of circular and sine wave trajectories. Compared to a manually adjusted LQR, the PSO-LQR controller significantly reduces steady-state tracking errors and produces smoother control inputs. Smooth control inputs extend the service life of control motors in real-world operation. The composite framework maintains stability under complex slipping conditions, even during high curvature maneuvers. However, this study still exhibits some limitations. First, the dynamic model assumes that the robot’s mass is symmetrical and that there is no lateral slip, which may differ from certain real-world scenarios. Second, the simulation was conducted in an ideal environment and did not consider actual factors such as sensor noise and communication delays. Future work will focus on verifying this algorithm using a physical experimental platform and further studying the incorporation of lateral slip and more complex dynamic models into the control framework.