## 2 Trajectory Prediction Methods at Intersection Right Turning
### Method to Obtain Triclothoidal Curve  
The proposed Triclothoidal curve is as shown in Figure 3. It must be noted that the states of a point on a Triclothoidal curve whose distance is \(s\) from the initial point is represented by \((x(s),y(s),\psi(s),\kappa(s))\). The curvature rates of the clothoidal curves that are components of the Triclothoidal curve are \(\kappa_{1}{}^{\prime},\kappa_{2}{}^{\prime},\kappa_{3}{}^{\prime}\). Subscripts \(ij\) on variables \(\kappa_{ij},\psi_{ij},\psi_{ij}\) and \(\kappa_{ij}\) show the states at the connecting point of the \(i\)th and \(j\)th clothoidal curves. It is assumed that each clothoidal curve has the same length and that the length of the Triclothoidal curve is \(L\). Integrating Equation (1) from the initial point, the states at the terminal point can be obtained by the following equations:  
\[ \begin{split} x(L)&=\int_{0}^{\frac{t}{3}}\cos \left(\frac{1}{2}\kappa_{1}{}^{\prime}s^{2}+\kappa_{0}s+\psi_{0}\right)ds+ \int_{0}^{\frac{t}{3}}\cos\left(\frac{1}{2}\kappa_{2}{}^{\prime}s^{2}+\kappa _{12}s+\psi_{12}\right)ds\\ &+\int_{0}^{\frac{t}{3}}\cos\left(\frac{1}{2}\kappa_{3}{}^{ \prime}s^{2}+\kappa_{23}s+\psi_{23}\right)ds,\end{split}  \tag{2}\]  
\[ \begin{split} y(L)&=\int_{0}^{\frac{t}{3}}\sin \left(\frac{1}{2}\kappa_{1}{}^{\prime}s^{2}+\kappa_{0}s+\psi_{0}\right)ds+ \int_{0}^{\frac{t}{3}}\sin\left(\frac{1}{2}\kappa_{2}{}^{\prime}s^{2}+\kappa _{12}s+\psi_{12}\right)ds\\ &+\int_{0}^{\frac{t}{3}}\sin\left(\frac{1}{2}\kappa_{3}{}^{ \prime}s^{2}+\kappa_{23}s+\psi_{23}\right)ds,\end{split}  \tag{3}\]  
\[\psi(L)=\psi_{0}+\kappa_{0}L+\left(\frac{5}{2}\kappa_{1}{}^{\prime}+\frac{3} {2}\kappa_{2}{}^{\prime}+\frac{1}{2}\kappa_{3}{}^{\prime}\right)\left(\frac{ L}{3}\right)^{2}, \tag{4}\]  
\[\kappa(L)=\kappa_{0}+\left(\kappa_{1}{}^{\prime}+\kappa_{2}{}^{\prime}+ \kappa_{3}\right)\frac{L}{3}, \tag{5}\]  
Figure 3: Schematic explanation of Triclothoidal curve. Figure 2: Definitions of the coordinate and states. where the states at the connecting points are  
\[\psi_{12}=\frac{1}{2}{\kappa_{1}}^{\prime}\bigg{(}\frac{L}{3}\bigg{)}^{2}+\kappa_ {0}\frac{L}{3}+\psi_{0}, \tag{6}\]  
\[\kappa_{12}={\kappa_{1}}^{\prime}\frac{L}{3}+\kappa_{0}, \tag{7}\]  
\[\psi_{23}=\frac{1}{2}{\kappa_{2}}^{\prime}\bigg{(}\frac{L}{3}\bigg{)}^{2}+ \frac{3}{2}{\kappa_{1}}^{\prime}\bigg{(}\frac{L}{3}\bigg{)}^{2}+\frac{2}{3}L \kappa_{0}+\psi_{0}, \tag{8}\]  
\[\kappa_{23}={\kappa_{2}}^{\prime}\frac{L}{3}+{\kappa_{1}}^{\prime}\frac{L}{3} +\kappa_{0}. \tag{9}\]  
We must determine the four variables \({\kappa_{1}}^{\prime},{\kappa_{2}}^{\prime},{\kappa_{3}}^{\prime}\), and \(L\) so that the values of Equations (2)-(5) match the terminal values provided by the boundary conditions of the curve generation problem or the intersection layout. Note that, considering the definition of coordinates, \(x\), \(y\) positions at the initial and terminal points are always \((0,0)\) and \((0,1)\), respectively; hence, the boundary conditions that we can set arbitrarily are \(\psi_{0}\), \(\psi_{f}\), \(\kappa_{0}\), and \(\kappa_{f}\). In other words, the Triclothoidal curve is uniquely defined if we determine the tangent angles and curvatures of the curve both at the initial and terminal points. The relationships between these arbitrarily definable boundary conditions and intersection layouts are discussed in the next section. The following are problem reductions where four unknown variables are determined by solving nonlinear equations with, at most, two variables. Introducing the new variables \(A_{i}={\kappa_{i}}^{\prime}(L/3)^{2},i=1,2,3\) and substituting them into Equations (4) and (5), \(A_{3}\) and \(L\) can be derived with the functions of \(A_{1}\) and \(A_{2}\) as shown in Equations (10) and (11).