## 2 Optimization Model for Reference Trajectory in ARP
### Optimization Model for Generating the Reference Trajectory in ARP  
The parking process cycle is denoted as \(\Delta t\), and the detailed optimization model for generating the reference trajectory in ARP is presented below:  
\[\min L(T_{R})=\int_{0}^{T_{\max}}v(t)dt\approx\sum_{it=1}^{nt}\Delta L_{it} \tag{3}\]  
\[S.T.\left\{ \begin{array}{l}\Delta L_{it}\leq\Delta L_{\max}\\ \Delta L_{it}\leq\Delta L_{\max}\\ \forall P_{is}\in\Omega_{C,is}\\ \forall P_{g}(ig)\
otin\Omega_{c,it}\\ \forall P_{g}(ig)\in\Omega_{g}\\ |\Delta y_{ns}|=|y(P_{ns})-y_{H}|\leq\Delta y_{H_{\max}}\end{array} \right. \tag{4}\]  
where, \(\Omega_{g}\) is the coverage region of both boundary lines of the garage, and \(P_{g}\) is the set of the boundary line collision-avoidance detection location points. \(P_{g}(ig)\) is the \(ig\)-th boundary line collision-avoidance detection location point, \(ig\in\{1,2,\cdots,ng\}\), \(ng\) is the number of detection location points, obviously \(\forall P_{g}(ig)\in\Omega_{g}\). \(\Delta L_{it}\), \(\Delta L_{jit}\), and \(\Omega_{c,it}\) are the parking interval distance, attitude angle difference, and region by vehicle body coverage in the \(it\)-th cycle. \(\Delta L_{\max}\) represents the threshold for parking interval distance, and \(\Delta\angle_{\max}\) is the threshold for the difference in attitude angle. \(T_{\max}\) represents the duration of parking, while \(nt\) denotes the total number of cycles, \(nt\)=\(\left\lceil\frac{T_{\max}}{\Delta t}\right\rceil\), \(it\in\{1,2,\cdots,nt\}\), and \(\left\lceil\right\rceil\) is the carry up integer operator. During parking maneuvers, all collision detection points must stay outside the vehicle's coverage area to avoid garage boundary collisions. That is, \(\forall P_{g}(i)\
otin\Omega_{c,it}\). Point \(P_{is}\) represents the central location of the \(i\)-th fixed-point region, \(is\in\{2,\cdots,ns\}\), while region \(\Omega_{\mathbb{C},is}\) denotes its corresponding adjustable region, and so \(\forall P_{is}\in\Omega_{\mathbb{C},is}\). The variable \(\Delta y_{ns}\) signifies the error in parking position, \(\Delta y_{H_{\max}}\) indicates the threshold for the absolute value of this error, and \(\Delta y_{H_{\max}}\) corresponds to the ordinate value of the anticipated parking position. Trajectory \(T_{R}\) serves as the reference path for parking, with its length \(L(T_{R})\) being key indicators that substantially influence the quality of parking optimization [21]. Consider an interval \(\Delta=[a,b]\) with \(n+1\) interpolation points \((x_{0},x_{1},\cdots,x_{n})\) defined within this range, and the cubic spline interpolation function \(s(x)\) is characterized by the following two properties. The cubic spline function has two key advantages. First, the function is described by a cubic polynomial in each subinterval, implying that adjustments to a particular subinterval do not affect other subintervals. This locality facilitates the achievement of rapid outcomes when optimizing or adjusting the coordinates of specific region central points, without the need for recalculating the entire trajectory. Then, the cubic spline function is twice continuously differentiable over its entire domain, ensuring the smoothness of curvature variation along the trajectory. This high-order continuity enables the generation of exceptionally smooth curves. Based on these properties, applying cubic spline theory to handle the set of fixed-region centroids enables the quick generation of smooth parking reference trajectories, superior in both aspects to the geometric programming method [22]. Assuming that the center points' coordinates of all fixed-point regions within the standardized parking plane system have been acquired, smooth parking reference trajectories can be rapidly generated by cubic spline interpolation fitting. This exhibits a notable advantage over geometric methods. The algorithm in this study realizes the steering control of the vehicle based on the vehicle kinematics model. The so-called vehicle kinematics model refers to the following:  
\[\left\{