## 4 Parking Space Size Constraint Model
### The Constraint
#### 4.2.2 Constraints on the Collision between the G Point and Road Boundary  
In the process of entering the parking space, the vehicle is constrained by the dimensions of the three sides of the parking space. The collision of the dimensions of the right parking space is constrained by the outer contour point B' of the right rear wheel and the vertex J of the parking space. However, when the turning radius of the vehicle is too large, a possible collision will occur with any point of the right parking space boundary OL. At the same time, when the car body goes back to the parking space, the rear contour line of the vehicle may collide with LK, and this distance can be controlled manually or by parking by wheels. Therefore, only the boundary collisions of the above two sizes are discussed and analyzed. Constraint collision between the outer contour point B' on the right side of the rear wheel and the vertex J of the parking space. The width of the parking space is set as C. According to the analysis of the Ackerman steering model, when the vehicle enters the parking space along the extreme right angle during a one-stage parking operation, only the contour point B' of the rear wheel will collide with the vertex J(c,0) of the parking space, as shown in Figure 4.  
As can be seen from Figure 4, point B' in the parking trajectory is a section of arc length with the vehicle's instantaneous steering center O\({}_{1}\)(u\({}_{0}\),v\({}_{0}\) + F - R) as the center of the circle. Therefore, it only needs to ensure that the vertex \(\rfloor\) of the parking spot is always in the trajectory circle where O\({}_{1}\)B is located, so:  
\[\left|\text{O}_{1}\text{J}\right|-\left|\text{O}_{1}\text{B}\right|\leq 0 \tag{10}\]  
\[\sqrt{\left(\text{u}_{0}-\text{c}\right)^{2}+\left(\text{v}_{0}+\text{f}- \text{R}\right)^{2}}-\text{R}+\text{f}+\text{e}\leq 0 \tag{11}\]  
Similarly, the left-hand side of the inequality is f(x). It should be noted that the minimum turning radius of the rear wheel of the vehicle is generally larger than the width of the lane, so the y-coordinate of the center of the instantaneous turning circle is generally negative. Therefore, when the vehicle completes the 90\({}^{\circ}\) rotation operation in advance in the lane of traffic, the constraint conditions above cannot be met. Constraint collision between the left rear vertex D of the vehicle and the size line on the left side of the parking space. As shown in Figure 5, the collision between the outer contour line of the vehicle and the left dimension line should meet two constraint conditions: (1) the constraint condition of point O of parking space exposure is satisfied; (2) a safe distance \(\delta_{\text{min}}\) is kept from the size line on the left side after alignment. The road boundary  
The driver enters the garage through the rearview mirror to judge whether the parking operation can be successfully completed by the two vertices O and J of the parking space exposure. Therefore, in order to avoid the spatial constraint of the parking space exposure O point, the distance between the instantaneous steering center O\({}_{1}\) and O is always greater than the distance between O\({}_{1}\) and D, namely  
\[\left|\text{O}_{1}\text{D}\right|\leq\left|\text{O}_{1}\text{O}\right| \tag{12}\]  
Figure 4: Schematic diagram of collision with the J constraint on the right vertex of the parking space. Figure 5: Collision between the rear vertex and the dimension line on the left side of the parking space. coordinates of the left rear vertex D of the vehicle: \[\left\{ \begin{array}{c}x_{\rm D}=\mathrm{u}_{0}-{\rm R}\sin\theta-\sqrt{\mathrm{ b}^{2}+\mathrm{e}^{2}}\cos(\theta-\arctan\mathrm{e}_{\rm B}^{\rm e})\\ y_{\rm D}=(\mathrm{v}_{0}+\mathrm{f})-{\rm R}(1-\cos\theta)-\sqrt{\mathrm{b}^{2} +\mathrm{e}^{2}}\sin(\theta-\arctan\mathrm{e}_{\rm B}^{\rm e})\end{array}