## 3 Definitions of Parameters and Coordinates for Trajectory Generation at Intersection  
One of the most imperative features of the Triclothoidal curve is that the curve is uniquely determined by the boundary conditions, namely, the position, angle, and curvature at both the initial and terminal points of the curve. Besides, these conditions are the only inputs to generate the curve and no other inputs are required. Therefore, the trajectory prediction with the Triclothoidal curve requires only the ego vehicle's current position and a simple map. Thus, it has a substantial advantage for implementing the ADAS without being equipped with additional sensors or much intersection information. Hence, this section focuses on the Triclothoidal curve and explains the definitions of parameters and coordinates to apply the Triclothoidal curve generation to trajectory prediction at intersection turning. To apply the Triclothoidal curve to the trajectory prediction, at first, the initial and terminal points for intersection turning are defined. As shown in Figure 4, the initial point is defined as the center of the rear wheel axle of the ego vehicle, and the terminal point is defined as a point on the central part of the escaping road of the intersection. When the escaping road has multiple lanes, the terminal point is defined as the central part of those lanes. Here, introducing a vehicle coordinate system \((X,Y)\), an origin of the coordinates is defined as a center of the rear wheel axle of the ego vehicle. In this coordinate system, the following boundary conditions are defined to generate the Triclothoidal curve. The positions at the initial point for the Triclothoidal curve generation are defined as \((0,0)\), and the initial direction of the curve is defined as the ego vehicle's longitudinal direction. Therefore, the initial angle of the curve on the coordinate system is always 0. The initial curvature can be geometrically derived as \(\tan\delta/l_{wb}\) under the ego vehicle's steering angle \(\delta\) and wheelbase \(l_{wb}\). In order to define the conditions at the terminal point of the curve, and to adapt to the various sizes of the intersection, the terminal point is defined as an adjustable point on the lane center of the escaping road (point A in Figure 4). The positions at the terminal point \(\left(X_{f},Y_{f}\right)\) can be adjusted by a distance parameter \(d_{pre}\). This parameter \(d_{pre}\) is defined as the distance from point B to the terminal position. The point B is a crossing point of the ego vehicle's longitudinal centerline and the lane center of the escaping road. The distance parameter \(d_{pre}\) is an important parameter for controlling the accuracy of the predicted trajectory. The method to decide the parameter \(d_{pre}\) is described in Section 5. The terminal direction of the curve is equal to the relative angle \(\theta\) from the ego vehicle's driving direction to the direction of the escaping road of the  
Figure 4: Parameter definitions for Triclothoidal curve generation. intersection. In this study, we regard the shape of the escaping road as a straight road and, therefore, the curvature at the terminal point is defined as 0. Then, the boundary conditions are transformed from the coordinate system \((X,Y)\) into the coordinate system \((x,y)\), which was explained in Section 2, as shown in Figure 4. The coordinates \((x,y)\) are rotated at a rotation angle \(\lambda\), and the distances in the coordinates are normalized by the direct distance between the initial and terminal points. By applying this coordinate transformation, the conditions at the initial and terminal points to generate the Triclothoidal curve \((x(s),y(s),\psi(s),\kappa(s))\) can be expressed as follows:  
\[(x(0),y(0),\psi(0),\kappa(0))=\left(0,0,-\lambda,\frac{\tan\delta}{l_{wb}} \sqrt{X_{f}^{2}+Y_{f}^{2}}\right), \tag{23}\]  
\[(x(L),y(L),\psi(L),\kappa(L))=(1,0,\theta-\lambda,0), \tag{24}\]  
where the angle \(\lambda\) is \(\arctan\left(Y_{f}\Big{/}X_{f}\right)\).