## 3 Proposed Method
### Trajectory Prediction with Clothoid
Since the vehicle was assumed to move at a constant speed over a short period, and a vehicle following the Clothoid at a constant speed experienced a constant rate of angular acceleration, the sharpness \(\alpha\) was a constant value that was calculated using the following formula:  
\[\alpha=\frac{\sum_{ \begin{subarray}{c}\tau=q+1\end{subarray} }^{m-1}\frac{c_{ \epsilon}-c_{q}}{s_{\epsilon}}}{m-q-1}=\frac{\sum_{ \begin{subarray}{c}\tau=q+1 \end{subarray} }^{m-1}\frac{\frac{\theta_{\epsilon}-\theta_{q}}{s_{\epsilon}}-c _{q}}{m-q-1}}{m-q-1}=\frac{\sum_{ \begin{subarray}{c}\tau=q+1\end{subarray} }^{ m-1}\frac{\theta_{\epsilon}}{s_{\epsilon}^{2}}}{m-q-1},\text{ and}\left\{ \begin{array}{l}\theta_{q}=0\\ c_{q}=0\\ q\leq m-2\\ m\geq 3\end{array} \right., \tag{6}\]  
Figure 5: The vehicle trajectory can be classified into two types: (**a**) represents the trajectory of straight-line driving, and (**b**) represents the trajectory of curved driving approximated as a Clothoid. where \(\theta_{e}\) represents the heading angle of the localization node in the coordinate system \(\{L_{q}\}\). Since \(L_{q}\) is the origin, both \(\theta_{q}\) and \(c_{q}\) are zero. Meanwhile, the time intervals between the adjacent localization nodes were very short, such as a localization frequency of 10 Hz, and the time interval between the adjacent localization nodes was only 0.1 s. As a result, the curve interval \(\Delta s_{e}\) between the adjacent localization nodes could be approximated using a straight line \(\Delta l_{e}\). Therefore, the calculation formula for \(s_{m}\) was as follows:  
\[\left\{ \begin{array}{l}s_{m}\approx\sum_{e=\ q+1}^{m}\Delta\ l_{e}\\ \Delta\ l_{m}\approx\frac{\sum_{e=\ q+1}^{m}\Delta\ l_{e}}{m-q-1}\ \,\ \text{and}\left\{\begin{array}{l}q\leq m-2\\ m\geq 3\end{array}\right.\.\end{array} \right. \tag{7}\]  
And then, to calculate the predicted heading angle of the localization node \(L_{m}\) in the coordinate system \(\{L_{q}\}\) based on the predicted position \((x(s_{m}),y(s_{m}))\):  
\[\theta_{\left(m,\ g\right)}=arctan(\frac{y(s_{m})}{x(s_{m})}). \tag{8}\]  
Finally, the predicted global pose of the localization node \(L_{m}\) was calculated based on the global pose of \(L_{q}\):  
\[\mathbf{P}_{\left(m,\ g\right)}^{\text{G}}=\mathbf{P}_{q}^{\text{G}}\cdot \mathbf{P}_{\left(m,\ g\right)}^{\left(\text{V},\ q\right)}, \tag{9}\]  
where \(\mathbf{P}_{q}^{\text{G}}\) represents the global pose of \(L_{q}\), and \(\mathbf{P}_{\left(m,\ g\right)}^{\left(\text{V},q\right)}\) derived from \((x(s_{m}),y(s_{m}))\), and \(\theta_{\left(m,\ g\right)}\) represents the predicted pose in the coordinate system \(\{L_{q}\}\). A check was performed by utilizing the global predicted pose of the current localization node \(L_{m}\) to determine whether it fell within the map coverage area. As illustrated in Figure 6, \(\delta\) represents the estimated error of \(L_{m}\)'s global predicted pose, while \(\psi\) denotes the configured search radius. Hence, the sum of \(\delta\) and \(\psi\), \((\delta+\psi)\), represented the actual search radius. If there existed map nodes falling within the actual search range of \(L_{m}\), it was considered to be within the map coverage area, and a localization candidate set was constructed based on these map nodes. Otherwise, \(L_{m}\) was regarded as being outside the map coverage area. As depicted in Figure 6, \(M_{\text{candi}}=\{M_{(k,i-1)},M_{(k,i)},M_{(k,i+1)},M_{(k,i+2)}\}\) formed the localization candidate set for \(L_{m}\).