## 2 Trajectory Prediction Methods at Intersection Right Turning  
This section explains methods for generating a geometric path for trajectory prediction. More specifically, this section discusses a circular arc, a Biare curve, a clothoidal curve, and our proposed Triclothoidal curve. The following text provides a brief description of these curves wherein details of the proposed method are included. A comparison of the prediction accuracy of these curves is provided in a subsequent section. One of the simplest geometric shapes we can assume as a right-turn path is a circular arc. We can generate a circular arc by determining only two variables; namely, the direction of the initial point (current location of the ego vehicle) and the turning radius. However, the path of the circular arc can specify either the position or direction at the terminal point, and therefore, we can use only one of these two terminal conditions. On the contrary, in the intersection turning scenarios, specifying both location and direction at the terminal  
Figure 1: Structure of this paper. point has the advantage of predicting the trajectory accurately. Besides, the intersection geometries enable us to define the position and direction at the terminal point. Here, the following methods can specify both position and direction at the terminal point. Combining two arcs (Biarc) is one way to obtain enough degrees of freedom to satisfy both the position and the direction at the endpoint. [PERSON] et al. showed that such a Biarc can be generated in a straightforward manner (i.e., there is no need to conduct an iterative calculation to obtain the curve) by determining the initial and terminal positions and directions [28]. The clothoidal curve is another candidate of the path that can specify both the position and the direction at the endpoint. Furthermore, the linearly changing curvature, which is a feature of the clothoidal, has an advantage in terms of driving comfort [29]; hence, it is expected to be closer to the actual route. [PERSON] and [PERSON] proposed a method to obtain the clothoidal curve easily [30]; therefore we can adopt it in the ADAS. By using [PERSON]'s method, a curve can be obtained by solving a nonlinear equation of one variable. This is a sufficiently small calculation cost even if real-time calculations are assumed. To generate a more probable predicted route, it is preferable to consider the amount of steering and not only the current vehicle direction. This means that the curvature at the initial position should be specifiable. However, none of the above three methods can specify the curvature at the initial point due to a lack of degrees of freedom in the curve. Therefore, we propose to obtain the required degrees of freedom by combining three clothoidal curves (Triclothoidal curve). We also propose an easier method for determining such a curve. Our method can generate the predicted route by solving simultaneous equations with, at most, two variables. The cost of this calculation is reasonable when compared to the case of the clothoidal curve. Other methods to generate paths for the ground vehicles, trajectory generation methods with Bezier curve, spline curve, and B-spline curve, have been proposed [31; 32; 33]. However, although these curves can define the positions of the initial and terminal points, they require the definition of three or more control points. Thus, these methods must determine another control point in addition to the initial and terminal points. [PERSON] et al. proposed a trajectory planning method that determines the control point of the Bezier curve by defining the design parameter of the intersection. However, this study provided neither theoretical definitions of the design parameter nor geometrical relationships between the intersection characteristics [34]. Besides, [PERSON]'s proposal did not consider the steering angle, and the curvature at the initial point was always zero.