## 2 Modeling and Parking Path Planning  
### Establishment of the Kinematic Model  
Vehicle parking is a low-speed (usually below 5 km/h) movement, and when a vehicle's wheels roll at a low speed, it does not undergo lateral sliding. Thus, the lateral force can be neglected, and there is no wheel side slip angle. Therefore, in this application, the limitations we considered were the response speed and the control accuracy of the associated actuator. The vehicle kinematic model is usually simplified, that is, the vehicle parking kinematics model is established based on the vehicle kinematics model for the parking movement. This paper establishes the following simplified model for low-speed parking. In this chapter, the kinematics model of a vehicle is established, and the path planning method for an automatic parking system is studied based on this model. As shown in Figure 2, \((x_{r},y_{r})\) present the midpoint coordinates of the rear axle of the vehicle, while \((x_{f},y_{f})\) refer to the midpoint coordinates of the vehicle's front axle, \(W\) represents the width of the vehicle, \(H\) refers to the width of the road, \(L_{1}\) and \(L_{2}\) denote the width and length of the target parking space, respectively, \(h\) represents the distance between the midpoint of the rear axle and a lateral obstacle, \(S\) refers to the distance between the midpoint of the rear axle and the end of an obstacle in front of the target parking space, \(\theta\) represents the driving direction angle of the vehicle, and \(\varphi\) is the Ackerman angle, besides, the clockwise direction is positive. Assuming that the lateral velocity of the vehicle rear wheel (be perpendicular to the wheel direction) is zero, the vehicle movement equation in the vertical direction can be obtained as follows:  
\[\dot{x}_{r}\cdot\sin\theta-\dot{y}_{r}\cdot\cos\theta=0 \tag{1}\]  
According to the Ackerman steering geometrical principle, the Ackerman angle \(\varphi\) in the process of car steering is approximately equal to the steering angle of the midpoint of the vehicle front axle, and the center angle of the front axle of the vehicle is approximately in linear proportion to the steering wheel angle \(\gamma\). \[\gamma=K\cdot\varphi\]  
where \(K\) refers to the ratio constant. Figure 2: Simplified Vehicle Model for Parking. The midpoint of the rear axle is taken as the origin; then, the coordinate system is established, as shown in Figure 2. The coordinate of the front axle midpoint can be expressed as follows:  
\[ \begin{array}{l}\dot{x}_{f}=v_{f}\cdot\cos(\theta+\varphi)\\ \dot{y}_{f}=v_{f}\cdot\sin(\theta+\varphi)\end{array}  \tag{2}\]  
Integrating the above equation, we can get  
\[ \begin{array}{l}x_{f}=\int v_{f}\cdot\cos(\theta+\varphi)\cdot dt\\ y_{f}=\int v_{f}\cdot\sin(\theta+\varphi)\cdot dt\end{array}  \tag{3}\]  
Based on the positional relationship between the midpoints of the front and the rear axles of the vehicle, the follows equation can be obtained. \[ \begin{array}{l}x_{r}=x_{f}+L\cdot\cos\theta\\ y_{r}=y_{f}+L\cdot\sin\theta\end{array}  \tag{4}\]  
Thus, the speed relationship between the midpoint of the front and rear axis can be obtained as follows:  
\[ \begin{array}{l}\dot{x}_{r}=\dot{x}_{f}-L\dot{\theta}\cdot\sin\theta\\ \dot{y}_{r}=\dot{y}_{f}+L\dot{\theta}\cdot\cos\theta\end{array}  \tag{5}\]  
Then, the vehicle movement equation can be written as:  
\[\dot{x}_{f}\cdot\sin\theta-\dot{y}_{f}\cdot\cos\theta-L\cdot\dot{\theta}=0 \tag{6}\]  
Substituting Equation (2) into Equation (6), we can get:  
\[\dot{\theta}=-^{\prime}v_{f}\cdot\sin\varphi^{\prime}/L \tag{7}\]  
Substituting Equations (2) and (7) into Equation (5), the follow equation is obtained:  
\[ \begin{array}{l}\dot{x}_{r}=v_{f}\cdot\cos\theta\cos\varphi\\ \dot{y}_{r}=v_{f}\cdot\sin\theta\cos\varphi\end{array} \]  
Then, the coordinates of the rear axle midpoint and the vehicle driving direction angle at time \(t\) can be expressed as follows:  
\[\left\{