## 2 Problem Formulation
### Kinematic Model  
#### 2.2.1 Discrete kinematics Model of Mobile Platform  
(1) Ideal kinematics discrete model  
For an ideal mobile platform, the midpoint of the driving wheels coincides with the gravity center of the mobile platform on the \(xy\) plane. Assuming the robot is moving on the \(xy\) plane, the linear velocity and angular velocity can be expressed as follows:  
\[\left\{ \begin{array}{c}\dot{P}_{xc}(t)=V_{c}(t)\cos(q_{zc}(t))\\ \dot{P}_{yc}(t)=V_{c}(t)\sin(q_{zc}(t))\\ \dot{\theta}_{cz}(t)=W_{c}(t)\end{array} \Leftrightarrow\left[ \begin{array}{ ccc}\dot{P}_{xc}(t)\\ \dot{\theta}_{yc}(t)\\ \dot{\theta}_{zc}(t)\end{array} \right]=\left[ \begin{array}{cc}\cos(\theta_{ zx}(t))&0\\ \sin(\theta_{zc}(t))&0\\ 0&1\end{array} \right]\right[ \begin{array}{c}V_{c}(t)\\ W_{c}(t)\end{array} \right] \tag{1}\]  
where \(\dot{P}_{xc}\), \(\dot{P}_{yc}\) is the linear velocity of the mobile platform along the \(x\) axis or the \(y\) axis of the world frame, \(\dot{\theta}_{zc}\) is the angular velocity of rotation around the \(z\) axis of the world frame, \(V_{c}\) is the linear velocity of the mobile platform, and \(W_{c}\) is the rotation angle velocity of the mobile platform. Figure 1: The basic parameters of the mobile platform. (**a**) The top view and corresponding parameters. (**b**) The side view and corresponding parameters. Considering control by two driving wheels, the linear velocity and angular velocity of the system can be directly described by the rotation speed of two driving wheels, which is explicitly expressed as follows:  
\[\left\{ \begin{array}{l}V_{c}(t)=\frac{R_{c}(W_{L}(t)+W_{R}(t))}{2}\\ W_{c}(t)=\frac{R_{c}(W_{L}(t)-W_{R}(t))}{2B_{c}}\end{array} \right.\Leftrightarrow \left[ \begin{array}{c}V_{c}(t)\\ W_{c}(t)\end{array} \right]=\frac{R_{c}}{2}\left[ \begin{array}{cc}1&1\\ 1/B_{c}&-1/B_{c}\end{array} \right]\left[ \begin{array}{c}W_{L}(t)\\ W_{R}(t)\end{array} \right] \tag{2}\]  
where \(W_{L}\) and \(W_{R}\) are the actual rotational speed of the left and right driving wheels, respectively; \(R_{c}\) is the radius of two driving wheels; and \(B_{c}\) is the distance between two driving wheels of the mobile platform. To control the mobile platform, we performed the kinematic modeling by combining Formulas (1) and (2) for directly driving the wheels, which can be expressed as follows:  
\[\left[ \begin{array}{c}\dot{P}_{xc}(t)\\ \dot{p}_{yc}(t)\\ \dot{\theta}_{zc}(t)\end{array} \right]=\frac{R_{c}}{2}\left[ \begin{array}{ cc}\cos(\theta_{zc}(t))&\cos(\theta_{zc}(t))\\ \sin(\theta_{zc}(t))&\sin(\theta_{zc}(t))\\ 1/B_{c}&-1/B_{c}\end{array} \right]\left[ \begin{array}{c}W_{L}(t)\\ W_{R}(t)\end{array} \right] \tag{3}\]  
In the actual movement, we need to convert continuous motion into discrete motion for controlling the mobile platform. In this case, the rotation speeds of the left wheel and the right wheel are regarded as the control variables; the linear velocity along the \(x\) axis and \(y\) axis of the world frame is denoted as \(\dot{p}_{xc}(k)\) and \(\dot{p}_{yc}(k)\), respectively; and the angular velocity around the \(z\) axis of the world frame is \(q_{zc}(k)\). Thus, the linear velocity and the angular velocity of the mobile platform can be expressed as follows:  
\[\left[ \begin{array}{c}\dot{p}_{xc}(k)\\ \dot{p}_{yc}(k)\\ \dot{q}_{zc}(k)\end{array} \right]=\frac{R_{c}}{2}\left[ \begin{array}{cc} \cos(q_{zc}(k))&\cos(q_{zc}(k))\\ \sin(q_{zc}(k))&\sin(q_{zc}(k))\\ 1/B_{c}&-1/B_{c}\end{array} \right]\left[ \begin{array}{c}\dot{q}_{cl}(k)\\ \dot{q}_{cr}(k)\end{array} \right] \tag{4}\]  
where \(q_{zc}\) is the rotation angle of the mobile platform at this time; \(\dot{q}_{cl}\) and \(\dot{q}_{cr}\) are the rotation velocity of the left and right wheel at this time, respectively. When the rotation velocity of the driving wheels is regarded as the mobile platform's control variable, \(p_{xc}\) and \(p_{yc}\) are the position of the mobile platform along the \(x\) axis and the \(y\) axis of the world frame, respectively; and \(q_{zc}\) is the rotation angle of the \(z\) axis.