## 2 Intent-Estimation- and Motion-Model-Based Collision Avoidance Method
### Trajectory Prediction with Intent and Motion Model  
The maneuver intent can be estimated with the GMM-based method, but it is difficult to predict the real trajectory as there are too many realizations for a maneuver. With the initial state, the intended state and the maneuvering information, an ideal trajectory can be obtained. The prediction can be divided into two steps. First, a set of trajectories is generated considering the randomness of driving habits. Then, an appropriate trajectory is selected by considering comfort and duration. The Frenet frame Frenet (1966) along the lane center line is used to predict the trajectories and then converted to the initial Cartesian coordinates. The lateral component is represented with \(d(t)\), and the longitudinal component is \(s(t)\). The initial state and final state of the maneuver can be represented with \(ST_{0}\) and \(ST_{1}\). \[ST_{0}=(s_{0},\dot{s}_{0},\ddot{s}_{0},d_{0},\dot{d}_{0},\ddot{d}_{0}) \tag{13}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Road Type** & **Maneuver Type** & **Normal Moving** & **Left Lane Change** & **Right Lane Change** \\ \hline Straight Road & 50 & 50 & 50 \\ Winding Road & 50 & 50 & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Data used in the training process. Figure 5: Relationship between the length of feature vector M and the accuracy of intent estimation. The accuracy grows with the length while the growth rate is decreasing. \[ST_{1}=(s_{1},\dot{s}_{1},\ddot{s}_{1},d_{1},\dot{d}_{1},\ddot{d}_{1}) \tag{14}\]  
Going a step further, the initial state can be represented with Equation (15):  
\[\left[ \begin{array}{c}d_{0}\\ \dot{d}_{0}\\ \ddot{d}_{0}\\ s_{0}\\ \ddot{s}_{0}\\ \ddot{s}_{0}\\ \end{array} \right]=\left[ \begin{array}{c}d_{0}^{*}\\ v_{0}\sin(\theta_{0}-\theta_{\frac{\gamma}{T}})\\ \sqrt{(a^{2}+\gamma_{0}v_{0}^{2})}\sin(\theta_{0}-\theta_{\frac{\gamma}{T}})\\ 0\\ v_{0}\cos(\theta_{0}-\theta_{\frac{\gamma}{T}})\\ \sqrt{(a_{0}^{2}+\gamma_{0}v_{0}^{2})}\cos(\theta_{0}-\theta_{\frac{\gamma}{T} })\\ \end{array} \right] \tag{15}\]  
where \(d_{0}^{*}\) is the distance between the initial position of the vehicle and the lane center line. The orientation of the lane center line's tangent vector \(\overrightarrow{T}\) is represented by \(\theta_{\frac{\gamma}{T}}\). The normal acceleration of the vehicle is represented as \(\gamma_{0}v_{0}^{2}\). For the final state of the trajectory, the vehicle is supposed to be moving along the lane center line, and the longitudinal acceleration is a constant value during the maneuvering time. The state can be modeled with the following vector:  
\[\left[ \begin{array}{c}d_{1}\\ \dot{d}_{1}\\ \ddot{d}_{1}\\ \ddot{s}_{1}\\ \end{array} \right]=\left[ \begin{array}{c}d_{1}^{*}\\ 0\\ 0\\ 0\\ a_{0}\\ \end{array} \right] \tag{16}\]  
If the vehicle is performing a lane change, the value of \(d_{1}^{*}\) is the lane width; otherwise, it is set to zero. The duration of a maneuver is limited, and it is set as \(t^{K}\). A set of trajectories can be obtained by sampling the duration: \(t_{1}=\left\{t^{i}\right\}_{i=1\cdots K}\). The lateral movement of the vehicle can be modeled with a quintic curve [17], as represented in Equation (17):  
\[d(t)=c_{5}t^{5}+c_{4}t^{4}+c_{3}t^{3}+c_{2}t^{2}+c_{1}t+c_{0} \tag{17}\]  
Combing Equation (16) and (17), the following equation can be obtained:  
\[\left[ \begin{array}{ccccc}t_{0}^{5}&t_{0}^{4}&t_{0}^{3}&t_{0}^{2}&t_{0}&1\\ t_{1}^{5}&t_{1}^{4}&t_{1}^{3}&t_{1}^{2}&t_{1}&1\\ 5t_{0}^{4}&4t_{0}^{3}&3t_{0}^{2}&2t_{0}&1&0\\ 5t_{1}^{4}&4t_{1}^{3}&3t_{1}^{2}&2t_{1}&1&0\\ 20t_{0}^{3}&12t_{0}^{2}&6t_{0}&2&0&0\\ 20t_{1}^{3}&12t_{1}^{2}&6t_{1}&2&0&0\\ \end{array} \right]*\left[ \begin{array}{c}c_{5}\\ c_{4}\\ c_{3}\\ c_{2}\\ c_{1}\\ c_{0}\\ \end{array} \right]=\left[ \begin{array}{c}d_{0}\\ d_{1}\\ \bullet\\ d_{0}\\ \bullet\\ d_{1}\\ \bullet\\ d_{0}\\ \bullet\\ d_{1}\\ \end{array}