# Goal Estimation of Mandatory Lane Changes Based on Interaction between Drivers
## 3 Proposed Method
### Feature Extraction for the Interaction
The repulsive potential energy of vehicle \(i\) at point \(j\) can be derived from  
\[U_{ij}=\alpha f\big{(}a_{i},\theta(\Delta x_{ij},\Delta y_{ij})\big{)}h\big{(} r(\Delta x_{ij},\Delta y_{ij})\big{)}v_{i}, \tag{3}\]  
\[f\big{(}a_{i},\theta(\Delta x_{ij},\Delta y_{ij})\big{)}=\frac{1}{2\pi I_{0}( \beta a_{i})}\exp\big{[}\beta a_{i}\cos\theta(\Delta x_{ij},\Delta y_{ij}) \big{]}, \tag{4}\]  
\[h\big{(}r(\Delta x_{ij},\Delta y_{ij})\big{)}=\frac{1}{2\pi\sigma^{2}}\exp \big{[}-\frac{r(\Delta x_{ij},\Delta y_{ij})}{2\sigma}\big{]}, \tag{5}\]  
\[r(\Delta x_{ij},\Delta y_{ij})=\sqrt{\Delta x_{ij}^{2}+\Delta y_{ij}^{2}}, \tag{6}\]  
\[\theta(\Delta x_{ij},\Delta y_{ij})=\arctan\bigg{(}\frac{\Delta x_{ij}}{\Delta y _{ij}}\bigg{)}, \tag{7}\]  
where \(f\big{(}a_{i},\theta(\Delta x_{ij},\Delta y_{ij})\big{)}\) denotes the von [PERSON] distribution, \(h\big{(}r(\Delta x_{ij},\Delta y_{ij})\big{)}\) is the repulsive potential model, \(\alpha\) is a coefficient, \(a_{i}\) represents the acceleration of vehicle \(i\), \(v_{i}\) is the speed of vehicle \(i\), \(\Delta x_{ij}\) and \(\Delta y_{ij}\) denotes the distance \(j\) from the vehicle \(i\), \(I_{0}\) represents the Vessel function, and \(\sigma\) is the standard deviation of the distance between the point \(j\) and the vehicle \(i\). The von Mises distribution adjusts the level of drift according to the value of \(a_{i}\). When \(a_{i}\) is zero, the potential field has uniform distribution. Figure 5 depicts the aspect of generated potential fields by the proposed method. Figure 5 shows the potential field when the vehicle \(i\) is rapidly accelerated, in contrast, Figure 5 represents the distribution when the vehicle \(i\) is rapidly decelerated. The red color indicates a high energy level whereas the blue color indicates a low level. Figure 5 shows the potential field when the vehicle \(i\) is slightly accelerated. Furthermore, Figure 5 represents the distribution when the vehicle \(i\) is slightly decelerated. It can be confirmed that the potential field is drifted according to the accelerating direction. In addition, the energy level is determined by the value of acceleration (deceleration). Figure 4: Aspects of dynamic characteristic potential field method: the potential field changes the distribution according to the moving direction of obstacle. (**a**) if the obstacle is not moving, the potential field is generated as the uniform distribution. Conversely, (**b**) if the obstacle is moving, the potential field is drifted to the moving direction. The advantage of using the dynamic potential field method is a description ability about discontinuous conditions. Most previous methods use the relative distance and speed as the features, however, it can cause the unstable estimation since the information is discontinuously changed if the corresponding vehicle changes a lane or overtakes other vehicles. On the other hand, the potential field is continuously changed since that is distributed on the whole lanes. Therefore, discontinuous changes do not occur by using the dynamic potential field. The proposed method calculates the difference of potential energies between the target and adjacent vehicles as the feature to describe the interaction between drivers. It is assumed that vehicles run along the lane. Consequently, the potential energy field generated from each vehicle has a square distribution with the same lane width, as shown in Figure 5. Especially, the proposed method focuses on the region where two potential fields are overlapped as shown in \[U_{r}=\frac{1}{2\pi\sigma^{2}}\exp\left[-\frac{r}{2\sigma}\right], \tag{2}\]0 Since the region can be considered as the interaction between two drivers, the proposed method calculates the difference of potential energies within the region and uses the value as the feature.