## 2 Problem Formulation
### Kinematic Model
#### 2.2.1 Discrete kinematics Model of Mobile Platform
Therefore, the discrete kinematics model of the mobile platform at \((k+1)\) time can be described as follows:  
\[\left[ \begin{array}{c}p_{xc}(k+1)\\ p_{yc}(k+1)\\ q_{zc}(k+1)\end{array} \right]=\left[ \begin{array}{c}p_{xc}(k)\\ p_{yc}(k)\\ q_{zc}(k)\end{array} \right]+d_{k}\cdot\frac{R_{c}}{2}\left[ \begin{array}{cc} \cos(q_{zc}(k))&\cos(q_{zc}(k))\\ \sin(q_{zc}(k))&\sin(q_{zc}(k))\\ 1/B_{c}&-1/B_{c}\end{array} \right]\left[ \begin{array}{c}\dot{q}_{cl}(k)\\ \dot{q}_{cr}(k)\end{array} \right] \tag{5}\]  
where \(d_{k}\) is the sampling time during the movement of the mobile platform. (2) Practical kinematics discrete model  
For a practical mobile platform, the midpoint of the two driving wheels cannot wholly coincide with the actual gravity center of the mobile platform. As shown in Figure 1, an offset may be created between the midpoint of the driving axis and the gravity center of the mobile platform, \(D_{cz}\) is the offset at the vertical height (the \(z\) axis of the driving-fixed frame), \(D_{cx}\) is the offset at the horizontal direction (the \(x\) axis of the driving-fixed frame), and \(D_{cy}\) is the offset at the \(y\) axis of the driving-fixed frame. Assuming that the mobile platform is moving in the horizontal plane, the impact of \(D_{cx}\) needs to be considered, but that of \(D_{cz}\) should be ignored in the kinematic modeling of the mobile platform. Therefore, if the gravity center coincides with the midpoint of the driving wheels, Formula (6) can be used to describe the position and posture of the gravity center of the mobile platform at \((k+1)\) time, which is expressed as follows:\[\left[ \begin{array}{c}p_{xd}(k+1)\\ p_{yd}(k+1)\\ q_{zd}(k+1)\end{array} \right]=\left[ \begin{array}{c}p_{xd}(k)\\ p_{yd}(k)\\ q_{zd}(k)\end{array} \right]+d_{k}\frac{R_{c}}{2}\left[ \begin{array}{cc}\cos(q_ {zd}(k))&\cos(q_{zd}(k))\\ \sin(q_{zd}(k))&\sin(q_{zd}(k))\\ 1/B_{c}&-1/B_{c}\end{array} \right]\left[ \begin{array}{c}q_{cl}(k)\\ q_{\sigma}(k)\end{array} \right] \tag{6}\]  
where \(p_{xd}\) and \(p_{yd}\) indicate the midpoint location of the driving wheel axis in the \(xy\) plane, and \(q_{zd}\) indicates the angle rotation around the \(z\) axis of the driving-fixed frame. Suppose the gravity center does not coincide with the midpoint of the two driving wheels. In this case, to the actual position of the gravity center, the corresponding offset needs to be added to the kinematics model, which is expressed as follows:  
\[\left[ \begin{array}{c}p_{xc}(k)\\ p_{yc}(k)\\ q_{zc}(k)\end{array} \right]=\left[ \begin{array}{c}p_{xd}(k)\\ p_{yd}(k)\\ q_{zd}(k)\end{array} \right]+\left[ \begin{array}{cc}\cos(q_{zd}(k))&-\sin(q_ {zd}(k))&0\\ \sin(q_{zd}(k))&\cos(q_{zd}(k))&0\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}D_{cx}\\ D_{cy}\\ 0\end{array} \right] \tag{7}\]  
where \(p_{xc}\) and \(p_{yc}\) are the position of the gravity center of the mobile robot in the \(xy\) plane, and \(q_{zc}\) is the rotation angle of the mobile robot. **Remark 1**. : _When selecting the position and posture of the gravity center as the output variable, the offset between the midpoint of driving wheels and the gravity center of the mobile platform needs to be considered. To reduce the complexity of the trajectory tracking problem, the position and posture of the midpoint of driving wheels in the world frame are regarded as the desired output variables._