## 3 Proposed Method
### Trajectory Prediction  
The future maneuver of adjacent vehicle is predicted according to the estimated driving intention of the vehicle. When keeping is estimated as the current intention, the lateral position is determined at the center of the current lane. On the other hand, the lateral position is set to the center of the adjacent lane when adjustment is estimated. If the estimated intention is changing or arrival, the lateral position of the target vehicle is predicted through the sinusoidal model [28]. Although numbers of lane change models have been proposed, the common models are isokinetic migration model, arc model, trapezoidal acceleration model, and sinusoidal model [29]. The isokinetic migration model is simple and easy to calculate; however, the generated maneuver would be unrealistic. The arc and trapezoidal acceleration models have the poor flexibility since they require to determine planning parameters. On the other hand, the sinusoidal model determines the lateral acceleration according to the duration of lane-changing, and the duration can be calculated without any particular parameters. The proposed method derives the duration using the lateral speed at the moment, when the lane-changing is detected, and the lane width. This model generates a trajectory, such as a sine curve, as shown in Figure 5. The acceleration in the lateral direction can be derived as follows,  
\[a_{i,lat}^{t_{c}}=\frac{2\pi H}{t_{lat}^{2}}\sin\frac{2\pi}{t_{lat}}t_{c}, \tag{7}\]  
where \(a_{i,lat}^{t}\) indicates the lateral acceleration of the vehicle \(i\), \(t_{c}\) is the time from the beginning of lane departure, \(H\) is the lane width, and \(t_{lat}\) is the lane-changing duration. \(t_{lat}\) is calculated using the lateral speed at the moment which the intention is estimated as changing. Therefore, the calculation of lateral acceleration does not require any particular parameters.  
The longitudinal position of adjacent vehicle is predicted based on the GM model, and the acceleration value using the estimated values of operation characteristic variables is calculated. Then, the position and speed of the vehicle \(i\) are updated as follows.  
\[\vartheta_{i,lon}^{t+1}=v_{i,lon}^{t}+\hat{a}_{i,lon}^{t}\Delta t, \tag{8}\]  
\[\hat{x}_{i,lon}^{t+1}=x_{i,lon}^{t}+\hat{\vartheta}_{i,lon}^{t}\Delta t, \tag{9}\]  
where \(\hat{x}_{i,lon}^{t}\) denotes the longitudinal acceleration derived by Equation (5). Based on the estimated driving intention, the proposed method identifies the preceding vehicle for the target vehicle to follow. When the estimated intention is keeping, the driver follows the preceding vehicle of the current lane. Then, if the driver has the intention of changing, arrival, or adjustment, he/she may aim to follow the preceding vehicle of the adjacent lane. The proposed approach reflects such tendency of drivers,  
Figure 5: Sinusoidal model for lane-changing trajectory: the proposed method uses the sinusoidal model to predict the lateral movement, when the target vehicle performs a lane change. In the figure, \(H\) is the final lateral displacement and \(t_{lat}\) is the lane-changing duration.  
and the preceding vehicle is selected as shown in Figure 6. It is assumed that the driver keeps the current acceleration until the trajectory prediction is over when there is no preceding vehicle.  
The trajectory for the time horizon of two seconds is predicted since the reaction times of drivers is commonly reported within two seconds [8]. Consequently, the sequence of future positions of the adjacent vehicles for two seconds in advance is generated.