# An Adaptive Control Based on Improved Gray Wolf Algorithm for Mobile Robots
## 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, \(I_{x}=0.00375\) kg\(\cdot\)m\({}^{2}\), \(I_{z}=0.03375\) kg\(\cdot\)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 \(q_{d}=\left[0.2t\ 0.2t\ \frac{\pi}{4}\right]^{T}\), i.e., the trajectory is defined by a straight line at an angle of 45\({}^{\circ}\). Furthermore, the selected dif  
Figure 6: Flow chart of improved gray wolf algorithm. ferent initial conditions are as follows: \(q(0)_{1}=\left[0.5\;0.2-\frac{\pi}{4}\right]^{T}\), \(q(0)_{2}=\left[-0.7-0.2\;\frac{\pi}{2}\right]^{T}\), \(q(0)_{3}=\left[1\;0.5-\frac{\pi}{6}\right]^{T}\) and \(q(0)_{4}=\left[-0.5\;1\;\frac{\pi}{3}\right]^{T}\). Behavior 2: Consider the desired trajectory on the plane \(\{X,\,Y\}\) to be given by \(x_{d}=-2\text{cos}\big{(}\frac{\pi t}{9.9}\,t\big{)}\) and \(y_{d}=-\text{sin}\left(\frac{\pi}{74}t\right)\), i.e., a curve resembling the leminiscates of Bernoulli can be obtained. We choose the initial state as \(q(0)=\left[-1.6\;0.36-\frac{\pi}{4}\right]^{T}\). The simulation results of the desired trajectories and the actual trajectories of the proposed straight lines and periodic curves are shown in Figures 7 and 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 \(t=3\;\text{s}\), 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 [PERSON] 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 \(v_{n}=0\).