## 3 Proposed Method
### Trajectory Prediction with Clothoid  
Using regular GPS to provide a rough pose estimation for the intelligent vehicles is common in outdoor environments. However, regular GPS signals are obstructed in underground parking scenes, rendering them ineffective for rough localization. Therefore, in this paper, a trajectory prediction was employed to provide a rough pose estimation for the vehicles in underground parking scenes, allowing for a quick determination of whether the vehicle was within the map coverage area and facilitating the selection of an appropriate localization method. This paper categorized the vehicle trajectories into two types, namely straight lines and curves, based on the variation in the heading angle of the vehicle. The corresponding trajectory prediction methods were then proposed accordingly. Figure 4(a) illustrates the trajectory of a vehicle moving in a straight line with a constant heading angle. \(\left\{L_{e}\,|\,e\in\left[q,\,m-1\right]\right\}\) represents the historical localization nodes on this trajectory, with known global poses. Among them, \(L_{q}\) is the initial localization node of this trajectory and \(q\) represents the index of this node. \(L_{m}\) is the current localization node and \(m\) represents the index of this node. The global pose of \(L_{m}\) was predicted as follows:  
Figure 4: The illustration of coordinate transformation from the pixel coordinate system to the vehicle coordinate system. \[\left\{\begin{array}{l} \begin{array}{l}\mathbf{R}_{(m,g)}^{\text{ G}}=\mathbf{R}_{m-1}^{\text{ G}}=\cdots=\mathbf{R}_{q+1}^{\text{ G}}=\mathbf{R}_{q}^{\text{ G}}\\ \mathbf{t}_{(m,g)}^{\text{ G}}=\mathbf{t}_{m-1}^{\text{ G}}+\Delta\mathbf{t}_{(m,g)}^{ \text{ G}}\end{array} \right.,\text{ and}\bigg{\{} \begin{array}{l}q\leq m-2\\ m\geq 3\end{array} \right., \tag{3}\]  
where \(\mathbf{P}_{(m,g)}^{\text{ G}}=(\mathbf{R}_{(m,g)}^{\text{ G}},\mathbf{t}_{(m,g)}^{\text{ G}})\) represents the predicted global pose of the current localization node \(L_{m}\), and \(\Delta\mathbf{t}_{(m,g)}^{\text{ G}}\) denotes the estimated translation vector from node \(L_{m-1}\) to node \(L_{m}\). Due to the short and nearly equal time intervals between the adjacent localization nodes, and the assumption of the vehicle moving at a constant speed within a short period, \(\Delta\mathbf{t}_{(m,g)}^{\text{ G}}\) was calculated based on the mean of the historical translation vectors:  
\[\Delta\mathbf{t}_{m,\text{ g}}^{\text{ G}}\approx\frac{\sum_{ \begin{subarray}{c}\tau=q+1 \end{subarray} }^{m-1}\Delta\mathbf{t}_{\epsilon}^{\text{ G}}}{m-q-1}, \tag{4}\]  
where \(\Delta\mathbf{t}_{\epsilon}^{\text{ G}}\) represents the translation vector between the adjacent historical nodes. The curved trajectory of the vehicle was approximated as a Clothoid. As shown in Figure 5b, this represented a continuous curvature trajectory originating from zero curvature. The Cartesian position of the current localization node \(L_{m}\) in the coordinate system \(\{L_{q}\}\), which was situated at a distance of \(s_{m}\) from the origin along the Clothoid, was calculated using the following formulas [51]:  
\[\left\{ \begin{array}{l}x(s_{m})=sign(\alpha)\cdot\sqrt{\frac{\pi}{|\alpha|} }\cdot C\bigg{(}\frac{\alpha\,s_{m}}{\sqrt{\pi\,|\alpha|}}\bigg{)}\\ y(s_{m})=\sqrt{\frac{\pi}{|\alpha|}}\cdot S\bigg{(}\frac{\alpha\,s_{m}}{\sqrt {\pi\,|\alpha|}}\bigg{)}\end{array} \right., \tag{5}\]  
where \(C(s)\) and \(S(s)\) are the Fresnel cosine and sine function [52], respectively, and the function \(sign(\alpha)\) returns the sign of the sharpness \(\alpha\). The sharpness \(\alpha\) was defined as \(dc/ds\), which indicated the change in the curvature \(c\) concerning the distance \(s\) traveled on the curve. The curvature \(c\) was defined as \(d\theta/ds\), representing the change in the heading angle of the vehicle concerning the distance \(s\) traveled on the curve.