## 2 Preliminaries  
### Dynamics Model  
The skid-steering vehicle investigated in this study is shown in Figure 1, which is equipped with four independently driven wheels, forming a differential steering system. \(OXY\) is the reference coordinate system, \(oxy\) is the fixed coordinate system attached to the vehicle, and \(o\) represents the origin of the fixed coordinate system. \(B\) is the width of the vehicle. The distances from the center of gravity (CG) to the front and rear wheels are \(L_{f}\) and \(L_{r}\), respectively. As shown in Figure 1, the motion of the skid-steering vehicle is caused by the friction forces, which are generated by the wheel-ground interactions. The dynamics model is formulated as follows:\[m\big{(}v^{\prime}_{xa}-v_{ya}\omega_{qa}\big{)} =\sum_{i=1}^{i=4}f_{x_{i}}+d_{x}, \tag{1}\] \[m\big{(}v^{\prime}_{ya}+v_{xa}\omega_{qa}\big{)} =\sum_{i=1}^{i=4}f_{y_{i}}+d_{y},\] \[J\omega^{\prime}_{qa} =\sum_{i=1}^{i=4}\tau_{t_{i}}+d_{\phi},\]  
where \(f_{x_{i}}\) is the longitudinal force, and \(f_{y_{i}}\) is the lateral force on the \(i\)th wheel, \(m\) is the mass of the vehicle, \(J\) is the inertia of the vehicle around CG, \(v_{xa}\) is the longitudinal speed, \(v_{ya}\) is the lateral speed, \(\omega_{qa}\) is the yaw rate, \(\tau_{t_{i}}\) is the torque caused by the traction force \(\mathbf{f}_{i}=f_{x_{i}}\widehat{x}+f_{y_{i}}\widehat{y}\) (where \(\widehat{x}\) is the unit vector in the longitudinal direction, and \(\widehat{y}\) is the unit vector in the lateral direction), and \(d_{x}\), \(d_{y}\), and \(d_{\phi}\) are disturbances in the different directions, respectively. From Equation (1), it can be seen that the vehicle motion is determined by \(f_{x_{i}}\) and \(f_{y_{i}}\) resulting from the wheel-ground interaction. A large number of semi-empirical complex friction models have been used to describe the wheel-ground interaction, such as the LuGre and Dugoff models. Most such models are very complex and are not necessary for the design of RL-based control strategies. In this work, we choose an appropriate friction model, described in [39], where \(f_{x_{i}}\) and \(f_{y_{i}}\) are defined as follows:  
\[f_{x_{i}} =\mu_{f}N_{i}S_{f}(s_{i})\frac{s_{x_{i}}}{s_{i}}, \tag{2}\] \[f_{y_{i}} =\mu_{f}N_{i}S_{f}(s_{i})\frac{s_{y_{i}}}{s_{i}},\] \[s_{i} =\sqrt{s_{x_{i}}^{2}+s_{y_{i}}^{2}},\]  
where \(N_{i}\) is the force of the driving wheel in the vertical direction, \(\mu_{f}\) is the friction coefficient, \(S_{f}(s_{i})=\frac{2}{n}\times\arctan(90s_{i})\) is the dynamic feature of the friction force, \(s_{x_{i}}\) is the longitudinal slip, and \(s_{y_{i}}\) is the lateral slip. The slips in different directions are defined as follows:  
\[s_{x_{lf}} =r\omega_{\tau_{lf}}-v_{x}+\frac{B}{2}\omega_{qa}, s_{y_{lf}} =v_{y}+L_{f}\omega_{qa}, \tag{3}\] \[s_{x_{lf}} =r\omega_{\tau_{lf}}-v_{x}+\frac{B}{2}\omega_{qa}, s_{y_{r}} =v_{y}-L_{r}\omega_{qa},\] \[s_{x_{rf}} =r\omega_{\tau_{rf}}-v_{x}-\frac{B}{2}\omega_{qa}, s_{y_{rf}} =v_{y}+L_{f}\omega_{qa},\] \[s_{x_{rr}} =r\omega_{\tau_{rr}}-v_{x}-\frac{B}{2}\omega_{qa}, s_{y_{rr}} =v_{y}-L_{r}\omega_{qa},\]  
where \(\omega_{\tau_{i}}\) is the wheel rotation speed. As shown in Figure 2, the dynamics model for each driving wheel can be described as:  
\[m\big{(}v^{\prime}_{ya}+v_{xa}\omega_{qa}\big{)} =\sum_{i=1}^{i=4}f_{y_{i}}+d_{y},\]0  
where \(j\) is the wheel inertia, \(T_{i}\) is the driving torque, \(T_{z_{i}}\) is the rolling resistance torque, \(r\) is the wheel radius, and \(d_{\tau_{i}}\) is the disturbance from the external environment. Based on the definitions of the wheel dynamics model and wheel slip, the dynamics model of the wheel slip in the longitudinal direction can be described as:  
\[m\big{(}v^{\prime}_{ya}+v_{xa}\omega_{qa}\big{)} =\sum_{i=1}^{i=4}f_{y_{i}}+d_{y},\]1  
The lateral slip speed \(v_{qa}\) is very small, relative to the longitudinal speed \(v_{xa}\). From an engineering perspective, we simplify the problem by disregarding the lateral motion of the vehicle.