An Adaptive Control Based on Improved Gray Wolf Algorithm for Mobile Robots

Abstract

In this paper, a novel intelligent controller for the trajectory tracking control of a nonholonomic mobile robot with time-varying parameter uncertainty and external disturbances in the case of tire hysteresis loss is proposed. Based on tire dynamics principles, a dynamic and kinematic model of a nonholonomic mobile robot is established, and the neural network approximation model of the system’s nonlinear term caused by many coupling factors when the robot enters a roll is given. Then, in order to adaptively estimate the unknown upper bounds on the uncertainties and perturbations for each subsystem in real time, a novel adaptive law employed online as a gain parameter is designed to solve the problem of inter-system coupling and reduce the transient response time of the system with lower uncertainties. Additionally, based on improved gray wolf optimizer and fuzzy system techniques, an adaptive algorithm using the gray wolf optimizer study space as the output variable of the fuzzy system to expand the search area of the gray wolves is developed to optimize the controller parameters online. Finally, the efficacy of the proposed intelligent control scheme and the feasibility of the proposed algorithm are verified by the 2023a version of MATLAB/Simulink platform.

1. Introduction

With the characteristics of moving ability, high efficiency, and flexible control, mobile robots, which can assist or take the place of people to complete some repetitive and dangerous work and improve the production efficiency and safety of the working environment [1], have been deeply researched and widely applied in the service industry, industry, defense, agriculture, and other fields [2,3,4]. An intelligent, accurate, safe, and reliable control system, as the key to the execution of tasks, is crucial to promote the development and application of mobile robot technology.

The objective of the robot movement control system is to obtain the information of the robot’s traveling speed and angle in real time, and constantly reduce the distance and angle deviation between the robot and the target point, while guiding the robot to move accurately, smoothly, and quickly to the stationary or moving target point. Lots of research on the motion control of mobile robots has been carried out, and various motion control techniques and solutions for nonholonomic mobile robots have been proposed. In [5], a robust output feedback controller, which can ensure the system status asymptotically converges to the origin by treating the disturbance as an extended state and constructing an extended state observer (ESO), is proposed for mobile robotic systems with parameter uncertainty, measurement angle deviation, and non-zero external disturbances. In [6], the authors focus on designing an action tracking control method for nonholonomic mobile robots, taking into account the actuator dynamics, and the feasibility of the nonholonomic mobile robot following the specified trajectory is verified by designing a trajectory tracking controller at the actuator aspect. In [7], a robot kinematics controller based on Lyapunov theory and a PID controller for DC motors are discussed. And, by estimating localization through integrating the robot movement into a fixed sampling frequency, the method of precise control of the mobile robots is given. In [8], a robust synchronization controller that can be simply tuned and adapted to various control scenarios is developed. Then, an algorithm for recognizing the system parameters for both online and offline scenarios using particle swarm optimization is proposed and evaluated. The method, which can successfully improve its convergence speed, accuracy, and stability without prior knowledge, is implemented on the Ballbot robot. In [9], the robust adaptive neural network tracking controller is developed and adaptive laws are introduced to estimate a local upper bound of the mechanical subsystem and the electrical subsystem of the nonholonomic mobile robot. In [10], a robust adaptive controller that addresses the parameter uncertainty of a wheeled mobile robot (WMR) is proposed. And, utilizing a smooth finite-time control technique, a trajectory following controller is proposed. Additionally, in order to deal with the system uncertainties of the nonholonomic WMR, the method of combining an observer and compensator is adopted in the design of the tracking controller. In [11], for the sake of improving the motion control accuracy and robustness of the system, a hierarchical improved fuzzy dynamic sliding mode control method, which can incorporate a tuning mechanism and operate in a similar way to the online processing of uncertainties without imposing a significant computational burden, is proposed for automated guided vehicles (AGVs) to deal with the uncertainty of the system.

To significantly enhance the convergence speed and robustness of the controller, computational intelligence technology has been favored by many scholars; especially, heuristic algorithms have been well applied, such as particle swarm optimization (PSO) [8,12,13], genetic algorithm (GA) [14,15], bee colony optimization [16,17,18], ant colony optimization (ACO) [19,20,21], gray wolf optimizer (GWO) [22,23], etc. And the GA and PSO, which can be particularly used for self-tuning in controlling movement laws, are the most widely used algorithms in research. The GWO, which is a heuristic algorithm with a clear structure, fewer configuration parameters, and less likelihood of falling into local extremes, can offer superior performance advantages with regard to optimal control precision and convergence speed. However, due to its late introduction, few studies on methods based on the GWO have been implemented as adaptive processes, especially for the motion control of mobile robots.

With the development of robotics technology and the increase in social demand, the performance requirements for mobile robot systems are becoming higher and higher. For example, electrical power substation inspection robots are required to be capable of smooth movement at high speeds in all weather conditions [24]. The weight of the heliostat cleaning robot [25,26] is relatively large, and the robot is required to clean the solar mirrors stably on off-road surfaces such as sand and hard soil. The logistics transportation robot has a maximum payload capacity of up to 200 tons [27], and the Handle handling robot can achieve a speed of 15 km/h [28]. Some mobile robots with larger loads or faster speeds may be prone to tilting and rolling when operating in large-angle turns or complex environments where the upper bounds of the speed and load cannot be well estimated. Therefore, there are still some challenges and problems to be solved in the motion control of mobile robots under tilting and slipping conditions [29]. On the other hand, in order to solve the design problem of mobile robot controllers, which is exceptionally challenging due to the influence of highly time-varying uncertainties in the parameters of DC motors caused by strong coupling terms between electrical and mechanical systems, numerous classes of adaptive control methods have been proposed [30,31]. However, unfortunately, only the time-varying uncertain parameter scalar terms of electromechanical coupling were considered. In this paper, using the techniques of a neural network and robust control, an intelligent controller for the motion control of a nonholonomic mobile robot with time-varying parameter uncertainty and external interference under the condition of tire hysteresis loss is proposed. At first, aimed at the motion control problem of a mobile robot with time-varying parameter uncertainty and external interference under the condition of tire hysteresis loss, based on the tire dynamics principle, dynamic and kinematic models of the nonholonomic mobile robot were established. The neural network approximation model of the system’s nonlinear term, which is caused by many coupling factors when the robot rolls, is given. And, by estimating the local undetermined upper bounds that correspond to each of the subsystem perturbations as well as uncertainty terms such as the reducer gear clearance, changes in road friction coefficients, etc., a novel adaptive law for the real-time adaptive estimation of uncertainties and unknown upper bounds of perturbations of each subsystem is designed, while ensuring the system is able to stably track any desired trajectory in real time without requiring any previous understanding of its model. Furthermore, based on fuzzy system technology and the improved GWO, a self-tuning online optimization algorithm, whose learning space is used as the output variable by the fuzzy system to expand the search area of the gray wolf, is given. And the output of the algorithm can serve as the online input for the designed control law, which realizes the real-time self-tuning of any parameter and adapts to any desired trajectory. Finally, simulations are employed to confirm the suggested intelligent controller’s efficacy.

The remainder of this article is structured as follows. Description of the nonholonomic mobile robot system and some relevant characteristics and parameters is given in Section 2. The proposed controller is designed in Section 3. Section 4 provides a detailed procedure of the improvement process of the GWO. The resilience of the designed controller and the effectiveness of the optimization algorithm are verified through simulations in Section 5. Lastly, the conclusions are presented in Section 6.

2. Problem Formulation

In order to comprehensively describe the lateral tilt problem of mobile robots during rapid turns, loading, or driving on complex road surfaces, one can establish a global coordinate system for the motion process of mobile robots, as illustrated in Figure 1a [30]. The local coordinate system, which can contain the lateral tilting motion of the system, is shown in Figure 1b,c.

For the mobile robot model shown above, consider a kinematic system for a mobile robot as follows [32]

$$m{v}_l\left(\dot{\gamma }+\omega \right)-m_a{l}_x\ddot{\psi }+{\tau }_o\left(t\right)=F_{y}G\left(\theta \left(t\right)\right)$$

(1)

$$I_z\ddot{\theta }=a\left(F_{y1}+F_{y2}\right)\cos \theta -b(F_{y3}+F_{y4})$$

(2)

$$I_x\ddot{\psi }+D_s\dot{\psi }+K_s\psi =m_a{v}_l\left(\dot{\gamma }+\omega \right)l_x+m_{a}g{l}_x\psi$$

(3)

where m is the mass of the entire mobile robot system, $m_a$ represents the sprung weight of the system, $\theta (t)$ represents the front wheels’ angle of the system, $v_l$ is the longitudinal velocity, $\omega$ is the yaw velocity, $\gamma$ is the side slip angle, $l_x$ is the distance from the center of the sprung mass to the x-axis, $\psi$ is the roll angle of the system, $I_z$ is the moment of inertia about the z-axis, $I_x$ represents the moment of inertia about the x-axis, $l_f$ and $l_r$ are the distances from the system’s center of mass to the front and rear axes, respectively. ${\tau }_o\left(t\right)$ represents the external and unknown disturbances, $F_y=\left[F_{yi}\right]\in R^{1\times 4}$, where $F_{yi}$ is the cornering force of the i-th tire, $G\left(\theta \left(t\right)\right)={\left[\cos \theta \left(t\right),\cos \theta \left(t\right),\mathrm{1,1}\right]}^T$ is the steering coefficient matrix, g is the gravitational acceleration, $D_s$ is the roll damping of the suspension, $K_s$ is the suspension roll stiffness.

The cornering force of tires $F_{yi}$ in the above system (1)–(3) exhibits strong nonlinearity, especially under high-speed turning angles and heavy load conditions. The performance of the mobile robot varies significantly across both the longitudinal and lateral ranges. To guarantee the stability of the mobile robot, one can consider the force condition of the system tires, as shown in Figure 2, and utilize the H.B. Pacejke tire model [33] to establish the dynamic model of the tire. The tire model under nonlinear operating conditions can be formulated as (4).

$$F_{yi}=D_i\sin \left\{C{\tan }^{-1}\left[B_i{\alpha }_i-E_i\left(B_i{\alpha }_i-{\tan }^{-1}\left(B_i{\alpha }_i\right)\right)\right]\right\}$$

(4)

where $B_i=b_3\sin \left(b_4{\tan }^{-1}\left(b_5{F}_{zi}\right)\right)/\left(C{D}_i\right)$ is the stiffness factor, the shape factor C is constant, $D_i=b_1{F}_{zi}^2+b_2{F}_{zi}$ and $E_i=b_6{F}_{zi}^2+b_7{F}_{zi}+b_8$ represent the peak and curvature factors, respectively, $b_1~b_8$ are all constants, $F_{zi}$ is the vertical force applied to the ith tire.

Assuming that the four tires of the system have identical physical properties, the front wheels’ steering angle is the same, and the side deflections of the tires on both sides are equal, then the side deflections of the front and rear tires are as follows

$${\alpha }_f=\left(\gamma +{\displaystyle \frac{l_f\omega }{v_l}}\right)-\theta$$

(5)

$${\alpha }_r=\gamma -{\displaystyle \frac{l_r\omega }{v_l}}$$

(6)

The vertical forces on each tire for the system during normal driving without roll are as follows

$$F_{z1}=F_{z2}={\displaystyle \frac{mg{l}_r}{2L}}$$

(7)

$$F_{z3}=F_{z4}={\displaystyle \frac{mg{l}_f}{2L}}$$

(8)

where L is the distance between the centers of the front and rear wheels.

When a roll occurs, the vertical forces on each tire change due to load transfer as follows

$$\begin{gathered} F_{z1}={\displaystyle \frac{mg{l}_r}{2L}}-{\displaystyle \frac{m_s{v}_l\omega h{l}_r}{dL}}, \\ {F}_{z2}={\displaystyle \frac{mg{l}_r}{2L}}+{\displaystyle \frac{m_s{v}_l\omega h{l}_r}{dL}}, \\ {F}_{z3}={\displaystyle \frac{mg{l}_f}{2L}}-{\displaystyle \frac{m_s{v}_l\omega h{l}_f}{dL}}, \\ {F}_{z4}={\displaystyle \frac{mg{l}_f}{2L}}+{\displaystyle \frac{m_s{v}_l\omega h{l}_f}{dL}}, \end{gathered}$$

(9)

where h is the system center of mass height and d is the wheel base of the system.

Substituting the nonlinear tire model into the dynamic Equations (1) and (2) of the system, we obtain

$$m{v}_l\left(\dot{\gamma }+\omega \right)-m_a{l}_x\ddot{\psi }+{\tau }_o={\displaystyle \frac{mg}{L}}\left(l_r\cos \theta +l_f\right)$$

(10)

$$I_z\ddot{\theta }={\displaystyle \frac{mg{l}_f}{L}}\left(\cos \theta -1\right)$$

(11)

Define $x_1=\gamma ,x_2=\theta ,x_3={\dot{x}}_2,x_4=\psi ,x_5={\dot{x}}_4$, then the dynamical model of the whole system can be written as

$$\left\{\begin{array}{c}{\dot{x}}_1={\displaystyle \frac{1}{m{v}_l}}\left[{\displaystyle \frac{mg}{L}}\left(l_r\cos \theta +l_f\right)+m_a{l}_x{\dot{x}}_5-{\tau }_o\right]-\omega \hfill \\ {\dot{x}}_2=x_3\hfill \\ {\dot{x}}_3={\displaystyle \frac{mg{l}_f}{I_{z}L}}\left(\cos x_2-1\right)\hfill \\ {\dot{x}}_4=x_5\hfill \\ {\dot{x}}_5={\displaystyle \frac{1}{I_x}}[m_a{v}_l\left({\dot{x}}_1+\omega \right)l_x-(m_{a}g{l}_x-K_s)x_4-D_s{x}_5]\hfill \end{array}\right.$$

(12)

or with a short notation as follows

$$\left\{\begin{array}{c}{\dot{x}}_1=j_1\left(x\right)+x_5\hfill \\ {\dot{x}}_2=x_3\hfill \\ {\dot{x}}_3=j_2\left(x\right)\hfill \\ {\dot{x}}_4=x_5\hfill \\ {\dot{x}}_5=j_3\left(x\right)+f{v}_l\hfill \end{array}\right.$$

(13)

where $x={\left[x_1,x_2,x_3,x_4\right]}^T$, $j_1\left(x\right)={\displaystyle \frac{1}{m{v}_l}}\left[{\displaystyle \frac{mg}{L}}\left(l_r\cos \theta +l_f\right)+m_a{l}_x{\dot{x}}_5-{\tau }_o\right]-\omega -x_5$, $j_2\left(x\right)={\displaystyle \frac{mg{l}_f}{I_{z}L}}\left(\cos x_2-1\right),j_3\left(x\right)={\displaystyle \frac{1}{I_x}}[m_a{v}_l{\dot{x}}_1{l}_x-(m_{a}g{l}_x-K_s)x_4-D_s{x}_5],f={\displaystyle \frac{m_a{l}_x\omega }{I_x}}$. Assume that all functions $j_1\left(x\right),j_2\left(x\right),j_3\left(x\right)$ in $\mathsf{\Omega }\in R$ are unknown.

Based on this, considering that mobile robots in practical applications are affected by rolling and external unknown disturbances, without loss of generality, one can obtain generalized system (10), (11), and (3) to a general mobile robot. As a result, one can have derived the generalized nonholonomic constraint equations and dynamic equations for mobile robots under non-ideal conditions as (14) and (15), respectively.

$$\left\{\begin{array}{c}\dot{y}\cos \theta -\dot{x}\sin \theta -d_0\dot{\theta }=\varsigma \\ \dot{x}\cos \theta +\dot{y}\sin \theta +b_0\dot{\theta }=r{\zeta }_r\\ \dot{x}\cos \theta +\dot{y}\sin \theta -b_0\dot{\theta }=r{\zeta }_l\end{array}\right.$$

(14)

$$\dot{q}=\mathrm{z}\left(q\right)\vartheta +\mathsf{\Phi }\left(q,\mu \right)$$

(15)

where $\varsigma$ is the lateral skidding velocity of the mobile robot, $\zeta ={\left[{\zeta }_r,{\zeta }_l\right]}^T$ is the disturbance of the angular velocity vector due to the two drive wheels, (x, y) is the inertial coordinate of the center of mass, $d_0$ represents the distance from the geometric center of mass to the actual center of mass, r denotes the actuated wheel’s radius, q$={\left[x,y,\theta \right]}^T$, $\mathsf{\Phi }\left(q,\varsigma \right)$ is the disturbance vector caused by the nonholonomic constraints of the system, $\vartheta ={[\nu \omega ]}^T$.

The dynamics of the nonholonomic mobile robot in the non-ideal case can be described by the Lagrange equation as

$$\begin{array}{c}\mathrm{M}\left(\mathrm{q}\right)\ddot{q}+O\left(q,\dot{q}\right)\dot{q}+G\left(q\right)\\ =E\left(q\right)\tau \left(t\right)+K^T\left(q\right)\lambda +{\tau }_0\left(t\right)\end{array}$$

(16)

where M(q) is the bounded inertia matrix, $O\left(q,\dot{q}\right)$ denotes a Coriolis and centripetal torque vector, G(q) denotes the gravitational vector, E(q) is the input transformation matrix, $\tau \left(t\right)$ is the torque term, $K^T\left(q\right)\lambda$ denotes the nonholonomic constraint force, where K(q) is the full rank matrix relevant to the nonholonomic constraints and $\lambda$ is a Lagrange multiplier, and ${\tau }_0\left(t\right)$ denotes the unknown and external applied perturbations.

Here, one can construct a suitable matrix $S\left(q\right)=\left[-J_1^{-1}\left(q\right)J_2\left(q\right)-J_2^{-1}\left(q\right)J_3\left(q\right) I_2\right],$ which divides linearly independent vectors in a generalized coordinate space, satisfying J(q)S(q) = 0, substituting into (16) yields $\overline{M}\left(q\right)=S^T\left(q\right)M\left(q\right)S\left(q\right)$, $\overline{E}\left(q\right)=S^T\left(q\right)E\left(q\right)$.

We assume the trajectory tracking error as r$\left(t\right)=q_d\left(t\right)-q_{d,a}\left(t\right)$ where $q_d\left(t\right)$ and $q_{d,a}\left(t\right)$ are the desired trajectories and actual trajectories of the robot system, respectively. What is more, the tracking error of the filter is defined as $f\left(t\right)=\dot{r}\left(t\right)+Ar\left(t\right)$, where A is a positive definite matrix. Then, we obtain the error states as $\tilde{z}\left(t\right)={\left[r^T\left(t\right)f^T\left(t\right)\right]}^T$. The complete error state dynamic equation for the kinematic force of the robot system is provided by [34]

$$\dot{\tilde{z}}\left(t\right)=A\left(q,\dot{q}\right)\tilde{z}\left(t\right)+u\left(t\right)$$

(17)

where a control input $u\left(t\right)=H\left(f_e\right)-\overline{E}\left(q\right)\tau$ with $f_e={\left[q_{d,a}^T,{\dot{q}}_{d,a}^T,q_d^T,{\dot{q}}_d^T,{\ddot{q}}_d^T\right]}^T$.

For the control problem, it can be reduced to a motion control system design issue, where the adjustable controller parameters can be self-tuned in real time to effectively handle various unknown external disturbances ${\tau }_0$. This ensures that the robot system accurately tracks the desired trajectory $q_d\left(t\right)$, even when the kinematic matrices of the mobile robot system are all entirely unknown.

The nonholonomic robotic intelligent control system proposed in the paper is shown in Figure 3, and the equation is as follows

$$\overline{E}(q)\tau \left(t\right)=\widehat{H}-u_o\left(t\right)+u_r\left(t\right)$$

(18)

where $\overline{E}(q)$ denotes the transformation matrix with known inputs, $u_o\left(t\right)$ denotes the linear quadratic regulator (LQR) term, $\widehat{H}$ denotes the neural network term, and $u_r\left(t\right)$ denotes the robust term.

It should be emphasized that the traditional LQR controllers’ parameters are calculated with the Riccati equation [35], which brings a challenge to the real-time self-tuning of some parameters. Then, a new adaptive algorithm to adjust the controller parameters in real time will be given.

Substituting (18) into (17) yields the following equation of the augmented error dynamic

$$\dot{\tilde{z}}\left(t\right)=A\tilde{z}\left(t\right)+B[u_o\left(t\right)+\tilde{H}-u_r\left(t\right)]$$

(19)

where $B=\left[\begin{array}{c}0\\ {\overline{M}}^{-1}\left(q\right)\end{array}\right]$ represents the matrix of coefficient and $\tilde{H}=H-\widehat{H}$ denotes an approximation error of the neural network.

The optimal control term ensures that the following quadratic function can be minimized [36]

$$J_0={\int }_{t_0}^{\infty }L\left(\tilde{z},u\right)dt$$

(20)

with the Lagrange function

$$L\left(\tilde{z},u\right)={\displaystyle \frac{1}{2}}{\tilde{z}}^T\left(t\right)Q\tilde{x}\left(t\right)+{\displaystyle \frac{1}{2}}{u}^T\left(t\right)Ru\left(t\right)$$

(21)

where R and Q are both the symmetric positive definite matrices, $\tilde{A}=\left[\begin{array}{cc}I& 0\\ A& I\end{array}\right]$, $\tilde{z}=\tilde{A}{[r^T,{\dot{r}}^T]}^T$.

To minimize (20) and achieve optimal control, it is sufficient to set a function $V(\tilde{z}\left(t\right),t)$ that is fulfilled by the Hamilton–Jacobi–Bellman equation

$${\displaystyle \frac{\partial V\left(\tilde{z}\left(t\right),t\right)}{\partial t}}+\underset{u}{\min }H\left[\tilde{z},u,{\displaystyle \frac{\partial V\left(\tilde{z}\left(t\right),t\right)}{\partial t}}\right]=0$$

(22)

$$H\left(\tilde{z},u,{\displaystyle \frac{\partial V\left(\tilde{z}\left(t\right),t\right)}{\partial t}}\right)={\displaystyle \frac{\partial V\left(\tilde{z}\left(t\right),t\right)}{\partial \tilde{z}}}\dot{\tilde{z}}\left(t\right)+L\left(\tilde{z},u\right)$$

(23)

Furthermore, the $V\left(\tilde{z}\left(t\right),t\right)$ obeys the following differential equation

$$-{\displaystyle \frac{\partial V\left(\tilde{z}\left(t\right),t\right)}{\partial t}}={\displaystyle \frac{\partial V\left(\tilde{z}\left(t\right),t\right)}{\partial \tilde{z}}}\dot{\tilde{z}}\left(t\right)+L\left(\tilde{z},u_o\right)$$

(24)

Therefore, the minimum term $\underset{u}{\min }H$ is attained when $u\left(t\right)=u_o\left(t\right)=-R^{-1}{B}^{T}P\tilde{z}\left(t\right)$ such that

$$H^{*}=\underset{u}{\min }H=\min \left({\displaystyle \frac{\partial V\left(\tilde{z}\left(t\right),t\right)}{\partial \tilde{z}}}\dot{\tilde{z}}\left(t\right)+L\left(\tilde{z},u\right)\right)$$

(25)

On the other hand, in accordance with the universal approximation theorem, a neural network is used to achieve the approximation of continuous nonlinear functions in the dynamic system. The relationship between the inputs and outputs can be clearly represented by the following matrix equation

$$j\left(q\right)=W^T\phi \left(q\right)+\epsilon$$

(26)

$$\widehat{j}\left(q\right)={\widehat{W}}^T\phi \left(q\right)$$

(27)

where, $q\in \mathsf{\Omega }\in R$ is the input vector, $W\in R^l$ is the optimal weight vector, with $l>1$, $\widehat{W}$ is the neural network estimated weights, $\widehat{j}$ is an estimated value, $\epsilon$ is an inherent approximation error, satisfying $\left|\epsilon \right|\le {\epsilon }_m$, ${\epsilon }_m>0$ is the upper bound of the neural network approximation error, $\phi \left(q\right)={\left[{\phi }_1\left(q\right),{\phi }_2\left(q\right),\cdots ,{\phi }_n\left(q\right)\right]}^T$ is the smoothing vector. The Gaussian function is usually chosen as the radial basis function, expressed as

$${\phi }_k\left(q\right)={\displaystyle \frac{1}{\sqrt{2\pi {\overline{\varphi }}_k}}}\exp \left(-{\displaystyle \frac{{\left(q-v_{lk}\right)}^T\left(q-v_{lk}\right)}{2{\overline{\varphi }}_k^2}}\right)$$

(28)

where $k=1,\cdots ,n$, $v_{lk}={\left[v_{l1},v_{l2},\cdots ,v_{ln}\right]}^T$ is the center of the attraction domain, ${\overline{\varphi }}_k$ is the standard deviation of the Gaussian function.

3. Design of the Adaptive Controller

Without a loss of generality, one can consider the mobile robot system described in (14) and (16) to design the controller. In the proposed controller, the nonlinear dynamics of the system are approximated through an adaptive neural network. The local unknown upper limits associated with each subsystem disturbance are estimated using the method of incorporating the robust term in the controller. And an assumption is given as follows.

Assumption: Suppose that ${\lambda }_{min}\left\{\delta \right\}$ is the smallest characteristic value of positive definite matrix $\delta ={[\delta }_{ii}]$, and the uncertain disturbance term $\mathsf{\Psi }\left(t\right)$ is constrained by the ${\lambda }_{min}\left\{\delta \right\}$, i.e., $\parallel \mathsf{\Psi }\parallel \le {\lambda }_{min}\left\{\delta \right\}$. Note that it is not necessary to know the upper bound. For the unknown nonlinear function, the matrix is defined as follows

$$\mathsf{\Lambda }=\left[\begin{array}{cccc}0& & & \\ & W_1& & \\ & & W_2& \\ & & & W_3\end{array}\right],{‖\mathsf{\Lambda }‖}_F\le {\mathsf{\Lambda }}_M$$

(29)

$$\widehat{\Lambda }=\left[\begin{array}{cccc}0& & & \\ & {\widehat{W}}_1& & \\ & & {\widehat{W}}_2& \\ & & & {\widehat{W}}_3\end{array}\right],\tilde{\mathsf{\Lambda }}=\mathsf{\Lambda }-\widehat{\mathsf{\Lambda }}$$

(30)

where $W_i (i=1, 2, 3)$ are the ideal interconnection weights, and ${\widehat{W}}_i (i=1, 2, 3)$ are the estimated weights of the neural network for the nonlinear function.

Theorem 1. 

For the nonholonomic mobile robot system (14) and (16) under unmodeled dynamics and parameter uncertainties, one can design the control law as follows

$$\dot{\widehat{\mathsf{\Lambda }}}=Q\mathsf{\Psi }{\xi }^T-sQ‖\xi ‖\widehat{\mathsf{\Lambda }}$$

(31)

$$u_r={\widehat{\delta }}^{T}sign\left({\tilde{z}}^T\left(t\right)P^{T}B\right)$$

(32)

$$\dot{\widehat{\delta }}={\eta }_{\delta }\left\{-\alpha \parallel {\tilde{z}}^T\left(t\right)P^{T}B\parallel \widehat{\delta }+\mathrm{diag}\left[\left|B^{T}P\tilde{z}\left(t\right)\right|\right]\right\}$$

(33)

where $\Psi ={\left[0,{\varphi }_1,{\varphi }_2,{\varphi }_3\right]}^T$,$s$is a positive constant (s > 0),$\widehat{\delta }\left(t\right)$represents the adaptive estimated value, P =$\left[\begin{array}{cc}K& 0\\ 0& \overline{M}\left(q\right)\end{array}\right]$, and K denotes a positive definite symmetric matrix. The bounded stability of the closed-loop system can be ensured when all states of the controlled robot system are limited, and the error eventually reaches a uniform boundedness.

Proof. 

Define a function of the Lyapunov candidate term as follows

$$\begin{array}{cc}V\hfill & ={\displaystyle \frac{1}{2}}{\tilde{z}}^{T}P\tilde{z}+{\displaystyle \frac{1}{2}}{\xi }^T\xi +{\displaystyle \frac{1}{2}}tr\left({\tilde{\mathsf{\Lambda }}}^T{Q}^{-1}\tilde{\mathsf{\Lambda }}\right)\hfill \\ & +{\displaystyle \frac{1}{2}}\eta {\tilde{f}}^2+{\displaystyle \frac{1}{2{\eta }_{\delta }}}tr\left({\tilde{\delta }}^T\tilde{\delta }\right)\hfill \end{array}$$

(34)

where $\tilde{\delta }=\delta -\hat{\delta }$ represents the estimated error, $tr(\cdot )$ represents a trace operator, $\tilde{f}(0)\ge f>0$, $\xi ={\left[r_1,r_2,r_3,r_4\right]}^T$ is the error matrix, $V_1={\displaystyle \frac{1}{2}}{\xi }^T\xi$, and $Q=\left[\begin{array}{cccc}0& & & \\ & {\mathsf{\Gamma }}_1& & \\ & & {\mathsf{\Gamma }}_2& \\ & & & {\mathsf{\Gamma }}_3\end{array}\right]$.

Derive the Lyapunov function (34), one can obtain

$$\begin{array}{cc}\dot{V}=\hfill & {\tilde{z}}^T\left(PA-PB{R}^{-1}{B}^{T}P+{\displaystyle \frac{1}{2}}\dot{P}\right)\tilde{z}\hfill \\ & +{\xi }^T\xi +\eta \tilde{f}\dot{\tilde{f}}+tr\left({\tilde{\mathsf{\Lambda }}}^T{Q}^{-1}\dot{\mathsf{\Lambda }}\right)\hfill \\ & -{\displaystyle \frac{1}{{\eta }_{\delta }}}tr\left\{{\tilde{\delta }}^T\dot{\widehat{\delta }}\right\}\hfill \end{array}$$

(35)

where ${\tilde{W}}_i^T=W_i^T-{\widehat{W}}_i^T,i=\mathrm{1,2},3,\dot{\tilde{\delta }}=-\dot{\widehat{\delta }}$.

Thus, the following equation is derived

$$\begin{array}{ccc}\dot{V}\hfill & =\hfill & -{\displaystyle \frac{1}{2}}{\tilde{z}}^T\left(t\right)\left({\displaystyle \frac{1}{2}}PB{R}^{-1}{B}^{T}P+{\displaystyle \frac{1}{2}}{\widehat{A}}^{-T}Q{\widehat{A}}^{-1}\right){\tilde{z}}^T\left(t\right)\hfill \\ & & -{\xi }^T{K}_e\xi +{\xi }^T\epsilon +{\xi }^T\tilde{\mathsf{\Lambda }}\mathsf{\Psi }\hfill \\ & & +tr\left({\tilde{\mathsf{\Lambda }}}^T{Q}^{-1}\dot{\mathsf{\Lambda }}\right)+\tilde{f}{r}_4{v}_l+\eta \tilde{f}\dot{\tilde{f}}\hfill \\ & & +\alpha ‖r^T‖tr\left\{{\tilde{\delta }}^T\widehat{\delta }\right\}-tr\left\{\tilde{\delta }diag\left[\left|{\tilde{z}}^T\left(t\right)PB\right|\right]\right\}\hfill \\ & =\hfill & -{\displaystyle \frac{1}{2}}{\tilde{z}}^T\left(t\right)\left({\displaystyle \frac{1}{2}}PB{R}^{-1}{B}^{T}P+{\displaystyle \frac{1}{2}}{\widehat{A}}^{-T}Q{\widehat{A}}^{-1}\right){\tilde{z}}^T\left(t\right)\hfill \\ & & -{\xi }^T{K}_e\xi +{\xi }^T\epsilon +tr\left({\tilde{\mathsf{\Lambda }}}^T{Q}^{-1}\dot{\mathsf{\Lambda }}+\tilde{\mathsf{\Lambda }}\mathsf{\Psi }{\xi }^T\right)\hfill \\ & & +\tilde{f}{r}_4{v}_l+\eta \tilde{f}\dot{\tilde{f}}+f^T\left[-{\delta }^{T}sign\left(B^{T}P\tilde{z}\left(t\right)\right)+\mathsf{\phi }\right]\hfill \\ & & +\alpha ‖{\left[f{B}^{T}P\tilde{z}\right(t\left)\right]}^T‖tr\left\{{\tilde{\delta }}^T\widehat{\delta }\right\}\hfill \end{array}$$

(36)

Using the Cauchy–Schwarz inequality, one can obtain

$$\begin{array}{cc}& f^T\left[-{\delta }^{T}sign\left(B^{T}P\tilde{z}\left(t\right)\right)+\mathsf{\phi }\right]\hfill \\ \le \hfill & ‖f^T‖\left(-{\lambda }_{min}\left\{\delta \right\}+‖B^{T}P\tilde{z}\left(t\right)‖\right)\hfill \\ \le \hfill & 0\hfill \end{array}$$

(37)

by virtue of the previous assumptions. Moreover, according to the Frobenius norm ${‖·‖}_F$, one can obtain

$$\begin{array}{cc}tr\left\{{\tilde{\delta }}^T\widehat{\delta }\right\}\hfill & =tr\left\{{\tilde{\delta }}^T\left(\delta -\tilde{\delta }\right)\right\}=tr\left\{{\tilde{\delta }}^T\delta \right\}-{‖{\tilde{\delta }}^T‖}_F^2\hfill \\ & \le {‖{\tilde{\delta }}^T‖}_F\left({‖\delta ‖}_F-{‖{\tilde{\delta }}^T‖}_F\right)\hfill \\ & \le {‖{\tilde{\delta }}^T‖}_F\left({\delta }_b-{‖{\tilde{\delta }}^T‖}_F\right)\hfill \end{array}$$

(38)

where ${‖\delta ‖}_F\le {\delta }_b<\mathrm{\infty }$.

Substituting (37) and (38) into (36) yields

$$\begin{array}{c}\dot{V}\le -‖f‖\left[{\lambda }_{min}\left\{F\right\}‖f‖+\alpha {‖{\tilde{\delta }}^T‖}_F\left({‖{\tilde{\delta }}^T‖}_F-{\delta }_b\right)\right]\\ -{\xi }^T{K}_e\xi +n‖\xi ‖tr\left[{\tilde{\mathsf{\Lambda }}}^T\left(\mathsf{\Lambda }-\tilde{\mathsf{\Lambda }}\right)\right]+M\end{array}$$

(39)

where $M=\tilde{f}{r}_4{v}_l-\eta \tilde{f}\dot{\tilde{f}}$, ${‖\tilde{z}‖}^2={‖r‖}^2+{‖rf‖}^2$. To ensure that $M\le 0$, the following definition is made

$$\dot{\tilde{f}}=\left\{\begin{array}{c}{\eta }^{-1}{r}_4{v}_l,r_4{v}_l>0\\ {\eta }^{-1}{r}_4{v}_l,r_4{v}_l\le 0,\tilde{f}>f\\ {\eta }^{-1},r_4{v}_l\le 0,\tilde{f}\le f\end{array}\right.$$

(40)

Substituting (40) into the M function, one can obtain (i) If $r_4{v}_l>0$, then $M=0$. (ii) If $r_4{v}_l\le 0$ and $\tilde{f}>f$, then $M=0$. (iii) Since $r_4{v}_l\le 0$, $\tilde{f}\le f$ and $\tilde{f}=f-\tilde{f}>0$, then $M\le 0$.

According to the Schwarz inequality, one can obtain

$$tr\left[{\tilde{\mathsf{\Lambda }}}^T\left(\mathsf{\Lambda }-\tilde{\mathsf{\Lambda }}\right)\right]\le {‖\tilde{\mathsf{\Lambda }}‖}_F{‖\mathsf{\Lambda }‖}_F-{‖\mathsf{\Lambda }‖}_F^2$$

(41)

Since $K_{emin}{‖\xi ‖}^2\le {\xi }^T{K}_e\xi$, the function $\dot{L}$ will be negative if the following (42)–(45) are satisfied

$$\begin{array}{c}{K}_{emin}‖\xi ‖-{\epsilon }_N+n\left({‖\tilde{\mathsf{\Lambda }}‖}_F^2-{‖\tilde{\mathsf{\Lambda }}‖}_F{\mathsf{\Lambda }}_M\right)\hfill \\ =K_{emin}‖\xi ‖-{\epsilon }_N+n{\left({‖\tilde{\mathsf{\Lambda }}‖}_F-{\mathsf{\Lambda }}_M\right)}^2-{\displaystyle \frac{n}{4}}{\mathsf{\Lambda }}_M^2\ge 0\hfill \end{array}$$

(42)

$${‖\tilde{\mathsf{\Lambda }}‖}_F\ge {\displaystyle \frac{1}{2}}{\mathsf{\Lambda }}_M+\sqrt{{\displaystyle \frac{{\mathsf{\Lambda }}_M^2}{4}}+{\displaystyle \frac{{\epsilon }_N}{n}}}$$

(43)

$$‖\xi ‖\ge {\displaystyle \frac{{\epsilon }_N+\frac{n}{4}{\mathsf{\Lambda }}_M^2}{K_{emin}}}$$

(44)

$$‖f‖\ge {\displaystyle \frac{{\delta }_b^2}{4}}{\displaystyle \frac{\alpha }{{\lambda }_{min}\left\{F\right\}}} or {‖{\tilde{\delta }}^T‖}_F\ge {\delta }_b$$

(45)

In accordance with the Lyapunov stability theorem [35], this proves that $‖\xi ‖$, $‖f‖$, and ${\tilde{\delta }}^T$ are uniformly ultimate boundedness and $\dot{V}\le 0$. In the case where α is relatively small or zero, $\dot{V}\le -{\lambda }_{min}\left\{F\right\}{‖f‖}^2-{\xi }^T{K}_e\xi ++M+n‖\xi ‖tr\left[{\tilde{\mathsf{\Lambda }}}^T\left(\mathsf{\Lambda }-\tilde{\mathsf{\Lambda }}\right)\right]$, which indicates that the tracking error tends towards zero using the adjustment $\xi$ and $f$. Conversely, when α is not zero, the trajectory error is observed to converge to an arbitrarily small neighborhood around the zero defined by (42)–(45), which is a function of the unknown upper bound for all subsystems. This convergence can be achieved through adjustments to the $\alpha , f$, and $\xi$. □

It is worth noting that, in contrast to other robust control methods already in use for robots, the robust term, which can be capable of being generalized to a wide range of robotic systems that are sensitive to local external perturbations, and effectively compensate for unknown upper limits corresponding to each subsystem, is proposed in the paper.

4. Design of the Adaptive Optimization Algorithm

The GWO, which has the advantages of a simple structure, good balance, and so on, is a new heuristic algorithm [37], and its good stability and rapid convergence is proved by the implementation of the algorithm on several various applications [38,39,40]. The GWO is derived from the cooperative behavior exhibited by gray wolves during their predation. The hunting activity involves the wolves forming distinct sub-groups known as grades [23]. The items are labeled ${\alpha }_w$, ${\beta }_w$, ${\delta }_w$, and ${\omega }_w$ in descending order; every grade of the gray wolves, ${\alpha }_w$, ${\beta }_w$, ${\delta }_w$, and ${\omega }_w$, plays a crucial part in the optimization process. The wolf of king ${\alpha }_w$ is the optimal solution and is the result of the optimization process. Similarly, the second, third, and fourth perfect solutions are denoted by ${\beta }_w$, ${\delta }_w$, and ${\omega }_w$. In this paper, the GWO is used along with a fuzzy control system, which provides online comprehensive exploration capacity for the parameter adjustment of the controller.

In this case, assuming the trajectory that the mobile robot must track is $q_{d,a}={\left[x_a{y}_a{\theta }_a\right]}^T$, the error posture of the mobile robot kinematics is defined as follows

$$\left[\begin{array}{c}{e}_x\\ e_y\\ e_{\theta }\end{array}\right]=\left[\begin{array}{ccc}-\cos \theta & \sin \theta & 0\\ -\sin \theta & -\cos \theta & 0\\ 0& 0& 1\end{array}\right]\left[q_{d,a}-q\right]$$

(46)

The position, velocity, and angular velocity of the mobile robot are represented by $(\left(x,y,\theta \right),v,\omega )$; then, the equation of motion can be obtained as follows [24]

$$\left[\begin{array}{c}{\dot{x}}_a\\ {\dot{y}}_a\\ {\dot{\theta }}_a\end{array}\right]=\left[\begin{array}{cc}\cos \theta & 0\\ \sin \theta & 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}\upsilon \\ \omega \end{array}\right]$$

(47)

To ensure that the kinematic error posture of (46) is consistently asymptotically stable, let Equation (47) be expressed as follows

$$\left[\begin{array}{c}{\dot{x}}_a\\ {\dot{y}}_a\\ {\dot{\theta }}_a\end{array}\right]=\left[\begin{array}{cc}\cos \theta & 0\\ \sin \theta & 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{\rho }_x{e}_x+V_a\cos \left(e_{\theta }\right)\\ {\dot{\theta }}_a+{\rho }_y{e}_y{V}_a+{\rho }_{\theta }{V}_a\sin \left(e_{\theta }\right)\end{array}\right]$$

(48)

where $V_a=\sqrt{{\dot{x}}_a^2+{\dot{y}}_a^2}$, ${\rho }_x$, ${\rho }_y$, and ${\rho }_{\theta }$ are all positive constants.

A.

Improvement of the GWO

Now, let the generalized error be represented as a function objective $E={[E_1\left(t\right),E_2\left(t\right)]}^T$, where $E_1\left(t\right)=r\left(t\right)$ is related to $A$ and $f\left(t\right)$. The error in the operation space is given as $E_2\left(t\right)={[e_x\left(t\right),e_y\left(t\right),e_{\theta }\left(t\right)]}^T$, which is related to ${\rho }_x$, ${\rho }_y$, and ${\rho }_{\theta }$. The optimal solution will be searched using the proposed improved GWO. The GWO is improved as follows

(1)

Constructing a nonlinear convergence factor $a_0$.

For the conventional GWO, the convergence factor $a$ gradually decreases from 2 to 0 in a linear manner throughout the iterations [41], and is usually set to take the value in the range of [−2,2] [23]. However, the convergence procedure of the algorithm is not inherently linear, making it challenging to adapt to the actual search scenario and leading to a reduction in the search efficiency of the algorithm. A new nonlinear convergence factor to improve the convergence accuracy is proposed as follows

$$a_0=3\left(1-{\displaystyle \frac{e^{\frac{t}{T}}-1}{e-1}}\right)$$

(49)

where t denotes the current iteration number and T represents the maximum iteration number of the population. The convergence factor curves before and after the improvement are shown in Figure 4. In the initial phase of the algorithm, $a_0$ decreases at a slower rate, allowing for a prolonged global search process. And in the later stages, $a_0$ decreases with a faster speed to enhance the quality of local optimal solution, promotes extensive local searches, and ultimately improves the overall search accuracy of the algorithm. Thus, this new nonlinear convergence method can balance the local and global search ability of the algorithm well.

(2)

Combined with the lion swarm optimization algorithm.

Due to the leadership role of the wolf ${\alpha }_w$ in the traditional GWO, the individuals in the whole wolf pack will eventually converge to the optimal solution ${\alpha }_w$ wolves, which makes the algorithm have a strong global search capability. However, as a result of reduced diversity within the population, it is relatively simple to become trapped in local optimality. Thus, one can incorporate the lion swarm algorithm into the GWO, thereby enhancing the global and local search capabilities, while reducing the possibility of trapping in local optimization.

The adult lion’s proportionality factor ${\beta }_l$ is a random number between [0,1] that affects the optimization of the algorithm, and in order to prevent the algorithm from converging slowly, ${\beta }_l$ is generally less than 0.5 [42].

The activity range of the lioness is affected by the disturbance factor ${\alpha }_l$, which decreases from large to small. This can accelerate the convergence speed of the algorithm and improve the efficiency of finding the optimal solution. ${\alpha }_l$ is expressed as follows

$${\alpha }_l=step\times \mathrm{e}\mathrm{x}\mathrm{p}{\left(-{\displaystyle \frac{30t}{T}}\right)}^{10}$$

(50)

where $step=0.1(\overline{high}-\overline{low})$ denotes the maximum moving step length within the lion’s range of activity, and $\overline{high}$ and $\overline{low}$ denote the average of the maximum and minimum values of each dimension within the activity range of adult lions, respectively.

The disturbance factor ${\alpha }_c$ can narrow or expand the search range of the lion cubs. After catching food, ${\alpha }_c$ tends to a linear downward trend and converges gradually, which improves the global search capability and stability of the algorithm. ${\alpha }_c$ is calculated as follows

$${\alpha }_c=step\times \left({\displaystyle \frac{T-t}{T}}\right)$$

(51)

Incorporating the above perturbation factors into the GWO, the positions of ${\alpha }_w$, ${\beta }_w$, and ${\delta }_w$ will have a certain degree of randomness, instead of converging purely to ${\alpha }_w$ only, which not only increases the diversity of the population so that the GWO is less likely to trap into a local optimum, but also enlarges the search space and improves the optimization effect. The hybrid lion swarm algorithm after the ${\alpha }_w$, ${\beta }_w$, and ${\delta }_w$ are updated as follows

$$X_1={\alpha }_l\left(X_{\alpha }-{\mathsf{\Lambda }}_1\times D_{\alpha }\right)$$

(52)

$$X_2={\beta }_l\left(X_{\beta }-{\mathsf{\Lambda }}_2\times D_{\beta }\right)$$

(53)

$$X_3={\alpha }_c\left(X_{\delta }-{\mathsf{\Lambda }}_3\times D_{\delta }\right)$$

(54)

where ${\mathsf{\Lambda }}_i$ are the coefficient vectors of ${\mathsf{\Lambda }}_i=2a_0\times r_i-a_0$,

$r_i$ are the random numbers of [0,1], and $D_{\alpha }$, $D_{\beta }$, and $D_{\delta }$ are the distances between ${\alpha }_w$ wolves, ${\beta }_w$ wolves, ${\delta }_w$ wolves, and ${\delta }_w$ wolves, respectively. $X_{\alpha }$, $X_{\beta }$, and $X_{\delta }$ denote the positions of ${\alpha }_w$, ${\beta }_w$, and ${\delta }_w$, respectively. Let the STEP of ${\alpha }_l$ and ${\alpha }_c$ in (52) and (54) take 1.

(3)

Incorporation of fuzzy control system.

A Mamdani fuzzy control system with fuzzy If–Then rules is employed to guide the gray wolves’ exploration of new zones. The fuzzy system could be regarded as an overseer that expands the field of search and alters the zones studied, which provides the gray wolves with a more intelligent and comprehensive exploration ability to find the best solution. In the process of online optimization, the new area that is updated with regard to every controller parameter $K_M(k,u_0)$ (M = 1, 2) is described through the following equation

$$K_{A_i}\left(k,u_0\right)=K_i\left(k,u_0-u_1\right)+h_{k}o\left(k,u_0\right)j\left(k,u_0\right)$$

(55)

where $K_{A_i}\left(k,u_0\right)$ denotes the controller parameter of ith gray wolf, $u_1$ denotes the search period, $h_k$ indicates the number of repetitions of the fuzzy process for each parameter, $o\left(k,u_0\right)$ denotes the output of the fuzzy system, and $j\left(k,u_0\right)$ indicates the proportional function of the error.

The fuzzy rule arranges the gray wolf individual into a new desired space to explore. Accordingly, the fuzzy system utilizes the integral of error ${\int }_T\parallel E_i(k,t)\parallel$, the paradigm $‖\tau (t)‖$ is input, and the explore space o (k, t) is output.

Thus, for the input $\left(x\right(k,t)\ge 0)$, set the linguistic values as (Z, P), and for the output $o(k,t)$, set (N, Z, P). The fuzzy rules are displayed in

Table 1. Fuzzy inference system (FIS) rule.

$‖\mathsf{\tau }\mathbf{(}\mathit{t}\mathbf{)}‖$	$‖{\mathit{E}}_{\mathit{i}}\mathbf{(}\mathit{k}\mathbf{,}\mathit{t})‖$
	Z	P
Z	Z	P
P	Z	N

and the Max–Min inference is according to Figure 5. The triangular–trapezoidal fuzzifier is used, the center of the region is the defuzzifier, and we define the output as follows

$$o\left(k,t\right)={\displaystyle \frac{{\mu }_k^P\left(x_k\right)-{\mu }_k^N\left(x_k\right)}{{\mu }_k^P\left(x_k\right)+{\mu }_k^Z\left(x_k\right)+{\mu }_k^N\left(x_k\right)}}$$

(56)

Next, the new search space is distributed to the remaining gray wolves as

$$\begin{array}{c}{K}_{A_i}\left(k,t\right)={\displaystyle \frac{N-i+1}{N}}\left(K_{A_1}\left(k,t\right)-K_{A_N}\left(k,t-T\right)\right)\\ +K_{A_N}\left(k,t-T\right)\end{array}$$

(57)

where i = 2, 3, …, N.

(4)

Add inertia weight $v$

In [13,36], a novel inertia weight $\omega$ is proposed, which plays a crucial role in the PSO as it reflects the capacity of particles to maintain their preceding time of motion. The inertia weight can improve the convergence and optimization of the PSO. Currently, the most widely used approach for dynamic inertia weights is the linear decreasing weight strategy [43]. The mathematical expression of this strategy is as follows

$$\omega \left(t\right)={\omega }_{max}-{\displaystyle \frac{{\omega }_{max}-{\omega }_{min}}{T}}\times t$$

(58)

where ${\omega }_{max}$ denotes the weight maximum and ${\omega }_{min}$ denotes the weight minimum.

Inspired by the above idea, let us add a dynamic inertia weight $v$ to the GWO, where the value of v is larger when the number of iterations is relatively small, making it easier for the gray wolf to perform a global search. In the later stage of the algorithm search, the value of v decreases as the value of t increases, which is more conducive for a gray wolf in a local search.

$$v\left(t\right)=v_n-{\displaystyle \frac{v_1-v_n}{T}}\times t$$

(59)

where $v_1$ represents the initial value of the inertia weight and $v_n$ represents the final value of the inertia weight. The position update formula becomes

$$X\left(t+1\right)=v\left(t\right)\times \left(X_1+X_2+X_3\right)$$

(60)

B.

Algorithmic flow for improving the GWO

The flowchart of the proposed improved GWO is shown in Figure 6. The algorithm will be used to optimize the variable parameters of the controller by the population of the gray wolves. Accordingly, the proposed improved GWO can be described as follows.

(1)

Initialization

Assume that two populations of gray wolves (M = 1, 2) are established and related with the controller parameters. The gray wolves in the populations are described as $A_1,A_2,\cdots ,A_N$, where N represents the total population within the gray wolves. To commence the training process, the controller parameters are initialized to zero, meaning that the gray wolf population begins from the origin. Then, set the population size N, initial number of iterations t = 1, maximum number of iterations T, and the initial convergence factor $a_0$.

(2)

Update Search Zone

The fuzzy system can be used to search new spaces that the gray wolf will explore. According to (56) and (57) to update the search zone, and incorporating expert knowledge in its predatory behavior, the algorithm can obtain a larger search space, a more intelligent search method, and a more accurate result for finding the optimum.

(3)

Updating The Location of Individuals

Calculate the gray wolf population fitness values and determine individual positions $X_{\alpha }$, $X_{\beta }$, and $X_{\delta }$ of ${\alpha }_w$ wolves, ${\beta }_w$ wolves, and ${\delta }_w$ wolves. According to (49), update the convergence factor ${\alpha }_0$. The individual gray wolf positions $X_1$,$X_2$, and $X_3$, after incorporation into the loin swarm optimization, are updated according to (52)–(54).

(4)

Stopping Step

Determine whether the amount of iterations t reaches the maximum value T. If $t<T$, then make t = t + 1 and repeat step 2 and step 3; otherwise, the algorithm ends to output the optimal result.

It may be noted that for the optimization problems considered in this paper, the three highest classes ${\alpha }_w$, ${\beta }_w$, and ${\delta }_w$ are sufficient to provide acceptable optimal solutions. Therefore, the rank ${\omega }_w$ is not used in the optimization algorithm.

5. Simulation and Analysis of Results

To examine the efficiency of the presented controller and the improved GWO, one can conduct simulations with the MATLAB/Simulink. The simulations carried out for the control scheme depicted in Figure 3 and the kinematic equations of the nonholonomic mobile robot system can be described in (1)–(3). This investigation aims to assess the performance of the proposed controller and its self-tuning optimization capabilities. Therefore, make the following assumptions: all of the tires have the same physical characteristics, the turning angle of the left and right tires is consistent, and the body is rigid. In this case, the model parameters of the system are as follows: r = 0.15 m, $b_0=0.45$ m, $I_x=0.00375$ $\mathrm{kg}·{\mathrm{m}}^2$, $I_z=0.03375$ $\mathrm{kg}·{\mathrm{m}}^2$, $m_a=6$ kg, $m=9$ kg.

To illustrate the controller’s reliability across various trajectories, we considered two distinct types: both linear trajectory (behavior 1) and periodic sinusoidal trajectory (behavior 2).

Behavior 1: Consider the desired trajectory, denoted as $q_d={\left[0.2t 0.2t {\displaystyle \frac{\pi }{4}}\right]}^T$, i.e., the trajectory is defined by a straight line at an angle of ${45}^{\circ }$. Furthermore, the selected different initial conditions are as follows: ${q\left(0\right)}_1={\left[0.5 0.2-{\displaystyle \frac{\pi }{4}}\right]}^T$, ${q\left(0\right)}_2={\left[-0.7-0.2 {\displaystyle \frac{\pi }{2}}\right]}^T$, ${q\left(0\right)}_3={\left[1 0.5-{\displaystyle \frac{\pi }{6}}\right]}^T$ and ${q\left(0\right)}_4={\left[-0.5 1 {\displaystyle \frac{\pi }{3}}\right]}^T$.

Behavior 2: Consider the desired trajectory on the plane {X, Y} to be given by $x_d=-2\cos ({\displaystyle \frac{\pi }{9.9}}t)$ and $y_d=-\sin \left({\displaystyle \frac{\pi }{4.74}}t\right)$, i.e., a curve resembling the lemniscates of Bernoulli can be obtained. We choose the initial state as $q\left(0\right)={\left[-1.6 0.36-{\displaystyle \frac{\pi }{4}}\right]}^T$.

The simulation results of the desired trajectories and the actual trajectories of the proposed straight lines and periodic curves are shown in Figure 7 and Figure 8. From the simulation results above, it can be seen that even though the initial starting point is not on the desired trajectory, the actual trajectory can still be basically coincident with the set desired trajectory, and for the circumstances under consideration, sufficient tracking and convergence performance can be attained in as little as t = 3 s, showcasing the rapid responsiveness of the proposed control strategy. Moreover, for different desired trajectories, the parameter values of the control law must be changed to have the same accuracy, which highlights the adaptability and robustness of the proposed controller scheme. This adaptability ensures that the controller remains effective under varying operational conditions, reinforcing its practical applicability in diverse trajectory tracking scenarios.

Comparative Study: When comparing the different self-adjusting algorithms, two concepts of efficiency are commonly considered: convergence rate and accuracy. Speed is typically assessed by the amount of iterations required to achieve the expected solution, while precision is the difference between the optimal result obtained through the algorithm and the expected result [21]. In the existing literature, the optimization algorithms generally work on the same principle within a confined search space and begin with a known initial state. Next, look for an optimal solution, which is accepted if the optimal solution found improves the behavior of the system.

To highlight the advantages of the proposed control method and the application of the improved GWO, a comparison was made with the adaptive fractional-order parallel fuzzy proportional fair integral differential controller [44] proposed by Kartik Singhal et al. through simulation. Some of the parameter values for the improved GWO are chosen as N = 20, t = 1, T = 30, $a_0=2$, $v_1=2$, and $v_n=0$. Setting the straight-line trajectory is the same as in behavior 1 and selecting the initial position is $q\left(0\right)=[0 1 {\displaystyle \frac{\pi }{4}}]$. The simulation results are shown in Figure 9, which shows that the designed controller in this paper has a faster convergence and better tracking effects. Moreover, the tracking errors of the simulation result plots are all bounded and acceptable, indicating that the designed optimized controller can make the actual trajectory converge to the desired trajectory quickly.

In addition, the traditional GWO [35] is contrasted with the improved GWO proposed in this paper, while a circular trajectory is devised to satisfy the trajectory diversity. The circular trajectory is defined by the cartesian space; that is, $x_d=0.69\cos \left({\displaystyle \frac{2\pi }{25}}t\right)$, $y_d=-0.69\sin \left({\displaystyle \frac{2\pi }{25}}t\right)$, and the initial position of the mobile robot at moment t = 0 is $q\left(0\right)=[2 0.5-{\displaystyle \frac{\pi }{2}}]$. Both algorithms use the same robot system, desired trajectory, and parameters. Figure 10 shows the tracking trajectories and errors of the system, respectively.

On the other hand, in order to more comprehensively highlight the effectiveness of the proposed algorithm, the proposed algorithm is compared with the PSO proposed in [45]. The parameters such as the number of population iterations of the two algorithms are the same as the above parameters. The parameters and external disturbances of the applied model are also the same. The simulation results are shown in Figure 11.

From the results in Figure 10, it can be observed that the proposed optimization algorithm has achieved a satisfactory tracking performance while ensuring minimal error in trajectory tracking during motion control. The traditional GWO exhibits greater oscillation and fluctuation, whereas the controller optimized using the improved algorithm demonstrates better convergence and pronounced stability. Figure 11a provides a comparative analysis of the trajectory tracking capabilities between the PSO algorithm and the novel algorithm proposed in this study. Both algorithms demonstrate a robust trajectory tracking performance, effectively managing the trajectories under examination. However, a closer inspection reveals a significant distinction in performance outcomes. The proposed algorithm exhibits superior efficacy in handling both linear trajectories and trajectories involving sharp turns. This enhanced performance is evidenced by the smoother tracking transitions observed in the results, indicating that the proposed algorithm achieves a higher degree of tracking fidelity and stability compared to the PSO algorithm. Further, Figure 11b illustrates a noteworthy advantage of the proposed algorithm over the PSO method in terms of operational efficiency. The proposed approach not only delivers faster response times but also significantly reduces the overall operational time. This is a crucial improvement, as it underscores the proposed algorithm’s capability to achieve timely and efficient trajectory tracking, thereby enhancing the practical applicability of the method in real-time scenarios. The enhanced response speed and reduced computational burden of the proposed algorithm represent a substantial advancement in optimizing trajectory tracking performance. This indicates and validates the effectiveness of the algorithm presented in this paper.

6. Conclusions

In this paper, considering non-ideal states such as sideways tilting, a neuro adaptive robust controller based on the improved GWO is proposed for the real-time control of mobile robots with unmodeled dynamics and unknown external perturbations. The neural network and the robust term introduced to estimate the unknown disturbance of each subsystem not only achieves the effect that the algorithm can adapt to any target trajectory without knowing the system model in advance, but also improves the controller performance. Then, the traditional GWO is improved and combined with a fuzzy system to integrate a new real-time adaptive algorithm. A significant advantage of this algorithm over some of the real-time control methods proposed by other researchers is it exhibits the capability to self-adjust the variables in real time to accommodate every reference trajectory. The simulation results show that the designed controller has good robustness and proves the effectiveness of the proposed algorithm.