## 2 Trajectory Prediction Methods at Intersection Right Turning
### Method to Obtain Triclothoidal Curve
\[A_{3}=\frac{\Delta\kappa}{3\kappa_{0}(1+\Delta\kappa/6\kappa_{0})}\bigg{(}- \frac{5}{2}A_{1}-\frac{3}{2}A_{2}+\Delta\psi\bigg{)}+\frac{1}{1+\Delta\kappa/ 6\kappa_{0}}(-A_{1}-A_{2}), \tag{10}\]  
\[L=\frac{1}{\kappa_{0}}\bigg{(}-\frac{5}{2}A_{1}-\frac{3}{2}A_{2}+\Delta\psi \bigg{)}-\frac{1\Delta\kappa}{6\kappa_{0}^{2}(1+\Delta\kappa/6\kappa_{0})} \bigg{(}-\frac{5}{2}A_{1}-\frac{3}{2}A_{2}+\Delta\psi\bigg{)} \tag{11}\]  
where \(\Delta\psi=\psi_{f}-\psi_{0},\Delta\kappa=\kappa_{f}-\kappa_{0}\). In the case where \(\Delta\kappa=0\) and \(\kappa_{0}\
eq 0\), Equations (10) and (11) are reduced to:  
\[\kappa(L)=\kappa_{0}+\left(\kappa_{1}{}^{\prime}+\kappa_{2}{}^{\prime}+ \kappa_{3}\right)\frac{L}{3}, \tag{5}\]0  
\[\kappa(L)=\kappa_{0}+\left(\kappa_{1}{}^{\prime}+\kappa_{2}{}^{\prime}+ \kappa_{3}\right)\frac{L}{3}, \tag{5}\]1  
In the case where \(\Delta\kappa\
eq 0\) and \(\kappa_{0}=0\), we can get  
\[\kappa(L)=\kappa_{0}+\left(\kappa_{1}{}^{\prime}+\kappa_{2}{}^{\prime}+ \kappa_{3}\right)\frac{L}{3}, \tag{5}\]2  
\[\kappa(L)=\kappa_{0}+\left(\kappa_{1}{}^{\prime}+\kappa_{2}{}^{\prime}+ \kappa_{3}\right)\frac{L}{3}, \tag{5}\]3  
In addition to these, in the case of \(\Delta\kappa=0,\kappa_{0}=0\), \(L\) can be eliminated from Equations (4) and (5) as:  
\[\kappa(L)=\kappa_{0}+\left(\kappa_{1}{}^{\prime}+\kappa_{2}{}^{\prime}+ \kappa_{3}\right)\frac{L}{3}, \tag{5}\]4  
\[\kappa(L)=\kappa_{0}+\left(\kappa_{1}{}^{\prime}+\kappa_{2}{}^{\prime}+ \kappa_{3}\right)\frac{L}{3}, \tag{5}\]5Applying variable transformation \(\tau=(3/L)\)s in Equations (2) and (3), we can obtain the following equations:  
\[ \begin{split}\frac{3}{L}&=\int_{0}^{1}\cos\biggl{(} \frac{1}{2}A_{1}\tau^{2}+\frac{1}{3}\kappa_{0}L\tau+\psi_{0}\biggr{)}d\tau+\int_ {0}^{1}\cos\biggl{\{}\frac{1}{2}A_{2}\tau^{2}+\biggl{(}A_{1}+\frac{1}{3}\kappa _{0}L\biggr{)}\tau\\ &+\frac{1}{2}A_{1}+\frac{1}{3}\kappa_{0}L+\psi_{0}\biggr{\}}d \tau+\int_{0}^{1}\cos\biggl{\{}\frac{1}{2}A_{3}\tau^{2}+\biggl{(}A_{1}+A_{2}+ \frac{1}{3}\kappa_{0}L\biggr{)}\tau\\ &+\frac{1}{2}A_{2}+\frac{3}{2}A_{1}+\frac{2}{3}\kappa_{0}L+\psi_ {0}\biggr{\}}d\tau,\end{split}  \tag{18}\]  
\[ \begin{split} 0&=\int_{0}^{1}\sin\biggl{(} \frac{1}{2}A_{1}\tau^{2}+\frac{1}{3}\kappa_{0}L\tau+\psi_{0}\biggr{)}d\tau+\int _{0}^{1}\sin\biggl{\{}\frac{1}{2}A_{2}\tau^{2}+\biggl{(}A_{1}+\frac{1}{3} \kappa_{0}L\biggr{)}\tau\\ &+\frac{1}{2}A_{1}+\frac{1}{3}\kappa_{0}L+\psi_{0}\biggr{\}}d \tau+\int_{0}^{1}\sin\biggl{\{}\frac{1}{2}A_{3}\tau^{2}+\biggl{(}A_{1}+A_{2}+ \frac{1}{3}\kappa_{0}L\biggr{)}\tau\\ &+\frac{1}{2}A_{2}+\frac{3}{2}A_{1}+\frac{2}{3}\kappa_{0}L+\psi_ {0}\biggr{\}}d\tau.\end{split}  \tag{19}\]  
By substituting Equations (10)-(15) into the above Equations (18) and (19), the remaining unknown variables reduce to two, namely \(A_{1}\) and \(A_{2}\). Moreover, in the case of \(\Delta\kappa=0,\kappa_{0}=0\), by substituting Equations (16) and (17) into Equation (19), \(A_{1}\) becomes the only unknown variable. These unknown variables can be obtained by solving the above-defined equations or simultaneous equations with one or two variables. The remaining unknown variables of the Triclothoidal curve are determined by Equations (10) and (11). Although iterative calculations are required to solve the simultaneous equations, we can provide reasonable initial assumptions. If we assume that \(L\approx 1\) and \(\sin(\psi(s))\approx\psi(s)\) and substitute it into Equation (19), the initial guesses for iterative calculations will be as follows:  
\[ \begin{split} A_{1 guess}&=\frac{1}{108\kappa_{0}( 1+\Delta\kappa/6\kappa_{0})}\\ &\quad\quad\Bigl{(}-108\kappa_{0}\Delta\psi-54\kappa_{0}^{2}+3 \kappa_{0}\Delta\kappa-18\Delta\kappa\Delta\psi+2\Delta\kappa^{2}-324\kappa_{0 }\psi_{0}-54\Delta\kappa\psi_{0}\Bigr{)},\end{split}  \tag{20}\]  
\[A_{2 guess}=-2A_{1 guess}+\Delta\psi-\kappa_{0}-\frac{1}{6}\Delta\kappa, \tag{21}\]  
where subscript _guess_ represents the initial guesses for iterative calculations.