## 2 Kinematic and Dynamic Models
### Kinematic Model  
A kinematic model is firstly established for a skid-steer WMR to acquire its trajectory and orientation variation under tracking control. Figure 1 shows a description of the kinematic model with the error definition for path tracking. The position and orientation of the wheeled robot can be explained with the world coordinate system _XOY_. \(C\) is the center of gravity (CG) of the robot, which is taken as the reference point of path tracking in this study. The wheeled robot consists of two driving rear wheels and two driven front wheels, and steering is achieved by the speed difference between the two rear wheels. A generalized coordinate vector \(\mathbf{q}=\left[x,y,\theta,\varphi_{L},\varphi_{R}\right]^{T}\) of the wheeled robot is defined, where \(x\) and \(y\) denote the coordinates along the X and Y orientations of CG of the robot \(\theta\) is the orientation angle of the robot with respect to the \(x\)-axis; and \(\varphi_{L}\) and \(\varphi_{R}\) are the rotation angles of the left and right driving wheels, respectively. The kinematic equation of a skid-steer wheeled robot is described as  
\[\dot{\mathbf{q}}=\mathbf{S}(\mathbf{q})\dot{\mathbf{v}} \tag{1}\]  
where \(\mathbf{v}=\left[\varphi_{L},\varphi_{R}\right]^{T}\) and \(\mathbf{S}(\mathbf{q})\) are formulated as  
\[\mathbf{S}(\mathbf{q})=\left[ \begin{array}{ccc}\frac{r\cos\theta}{2}&\frac{r\sin \theta}{2}&-\frac{r}{B}&1&0\\ \frac{r\cos\theta}{2}&\frac{r\sin\theta}{2}&\frac{r}{B}&0&1\end{array} \right]^{T} \tag{2}\]  
where \(r\) is the radius of a driving wheel, and \(B\) is the distance between the centerlines of left and right wheels. The non-holonomic constraint that a wheeled robot is subjected to is given by  
\[\mathbf{A}(\mathbf{q})\dot{\mathbf{q}}=0 \tag{3}\]  
where \(\mathbf{A}(\mathbf{q})\) is  
\[\mathbf{A}(\mathbf{q})=\left[ \begin{array}{ccc}\sin\theta&-\cos\theta&0&0&0\\ -\cos\theta&-\sin\theta&\frac{B}{2}&r&0\\ -\cos\theta&-\sin\theta&\frac{B}{2}&0&r\end{array} \right] \tag{4}\]  
The non-holonomic constraint is under the assumption that the system is \"pure rolling without slipping\" [25]. Wheeled robots working in a field environment often slip or skid, and thus, the kinematic equation is rewritten by taking the slip ratio into account. The kinematic model of a skid-steer wheeled robot under slipping conditions could be obtained as  
\[\dot{\mathbf{q}}=\left[ \begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta}\\ \dot{\varphi}_{L}\\ \dot{\varphi}_{R}\end{array} \right]=\left[ \begin{array}{ccc}\frac{r\cos \theta}{2}(1-f_{L})&\frac{r\cos\theta}{2}(1-f_{R})\\ \frac{r\sin\theta}{2}(1-f_{L})&\frac{r\sin\theta}{2}(1-f_{R})\\ -\frac{r}{B}(1-f_{R})&\frac{r}{B}(1-f_{R})\\ 1&0\\ 0&1\end{array} \right]\dot{\mathbf{v}} \tag{5}\]  
where \(f_{L}\) and \(f_{R}\) are the slip ratios of left and right wheels, respectively, which are defined as  
\[f_{s}=1-\frac{v_{s}}{\dot{\varphi}_{s}r}\;\;,\;s=L\;or\;R \tag{6}\]  
Figure 1: Description of the kinematic model with error definition for path tracking. where \(v_{L}\) and \(v_{R}\) are the linear velocities of the left and right wheels, respectively. Path tracking errors of the robot are composed of position errors in both the lateral and longitudinal directions, as well as orientations with respect to the reference path, which are defined as  
\[e_{p}=\lambda\ \sqrt{\left(x-x_{r}\right)^{2}+\left(y-y_{r}\right)^{2}} \tag{7}\]  
\[e_{\theta}=\theta-\theta_{r} \tag{8}\]  
where \(e_{p}\) and \(e_{\theta}\) are the position and orientation errors, respectively, of the robot with respect to the reference path; \((x_{r},y_{r})\) represents the reference position; \(\theta_{r}\) is the reference orientation. \(\lambda\) is equal to 1 when the robot is at the left side of the reference path, while is \(-1\) when it is at the right side.