# An Adaptive Control Based on Improved Gray Wolf Algorithm for Mobile Robots
## 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. Figure 1: _Cont._  
For the mobile robot model shown above, consider a kinematic system for a mobile robot as follows [32]  
\[mv_{I}\big{(}\dot{\gamma}+\omega\big{)}-m_{a}l_{x}\ddot{\psi}+\tau_{o}(t)=F_{y}G( \theta(t)) \tag{1}\]  
\[I_{z}\ddot{\theta}=a\big{(}F_{y1}+F_{y2}\big{)}\cos\theta-b\big{(}F_{y3}+F_{y4} \big{)} \tag{2}\]  
\[I_{x}\ddot{\psi}+D_{s}\dot{\psi}+K_{s}\psi=m_{a}v_{I}\big{(}\dot{\gamma}+\omega \big{)}l_{x}+m_{a}gl_{x}\psi \tag{3}\]  
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, \(v_{I}\) 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, \(l_{x}\) 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. \(\tau_{o}(t)\) represents the external and unknown disturbances, \(F_{y}=\big{[}F_{yi}\big{]}\in R^{1\times 4}\), where \(F_{yi}\) is the cornering force of the i-th tire, \(G(\theta(t))=[\cos\theta(t),\cos\theta(t),1,1]^{T}\) 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 [PERSON]. [PERSON] tire model [33] to establish the dynamic model of the tire. The tire model under nonlinear operating conditions can be formulated as (4). \[F_{yi}=D_{i}\text{sin}\Big{\{}\text{Ctan}^{-1}\Big{[}B_{i}\alpha_{i}-E_{i} \Big{(}B_{i}\alpha_{i}-\text{tan}^{-1}(B_{i}\alpha_{i})\Big{)}\Big{]}\Big{\}} \tag{4}\]  
where \(B_{i}=b_{3}\text{sin}\big{(}b_{4}\text{tan}^{-1}(b_{5}F_{zi})\big{)}/(CD_{i})\) 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, \(b_{1}\sim b_{8}\) are all constants, \(F_{zi}\) is the vertical force applied to the ith tire. Figure 1: (**a**) Motion coordinate system of a mobile robot. (**b**) Diagram of vehicle model in X–Y plane. (**c**) Diagram of vehicle model in Y–Z plane. 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  
\[\alpha_{f}=\left(\gamma+\frac{l_{r}\omega}{v_{l}}\right)-\theta \tag{5}\]  
\[\alpha_{r}=\gamma-\frac{l_{r}\omega}{v_{l}} \tag{6}\]  
The vertical forces on each tire for the system during normal driving without roll are as follows  
\[F_{z1}=F_{z2}=\frac{mgl_{r}}{2L} \tag{7}\]  
\[F_{z3}=F_{z4}=\frac{mgl_{f}}{2L} \tag{8}\]  
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  
\[