## 2 Intent-Estimation- and Motion-Model-Based Collision Avoidance Method
### Trajectory Prediction with Intent and Motion Model
\right] \tag{18}\]  
For the longitudinal movement, a quartic line is used, which can be represented as follows:  
\[s(t)=c_{4}t^{4}+c_{3}t^{3}+c_{2}t^{2}+c_{1}t+c_{0} \tag{19}\]  
The following equation can then be established:  
\[\left[ \begin{array}{ccccc}t_{0}^{4}&t_{0}^{3}&t_{0}^{2}&t_{0}&1\\ 4t_{0}^{3}&3t_{0}^{2}&2t_{0}&1&0\\ 4t_{1}^{3}&3t_{1}^{2}&2t_{1}&1&0\\ 12t_{0}^{2}&6t_{0}&2&0&0\\ 12t_{1}^{2}&6t_{1}&2&0&0\\ \end{array} \right]*\left[ \begin{array}{c}c_{4}\\ c_{3}\\ c_{2}\\ c_{1}\\ c_{0}\\ \end{array} \right]=\left[ \begin{array}{c}d_{0}\\ d_{1}\\ \bullet\\ d_{0}\\ \bullet\\ d_{1}\\ \bullet\\ d_{0}\\ \bullet\\ d_{1}\\ \end{array} \right] \tag{20}\]where \(\dot{s}=v_{0}+a*t\) and a set of trajectories can be obtained by solving the previous two equations with different ending times \(t_{1}\). To choose the most likely trajectory, two conditions should be considered: the normal acceleration and the duration of the maneuver. The cost function can be summarized into Equation (21):  
\[C_{t}^{K}=\max(a_{n}(t))+\varepsilon*t_{1}^{i},(i=1,2,3,\cdots K) \tag{21}\]  
where \(\varepsilon\) is a coefficient that is used to measure the effects of comfort and time, which is selected by analyzing collected data. In this paper, the value of \(\varepsilon\) is set as 0.5 to achieve the best result. The selected maneuver trajectory \(T_{M}\) is the trajectory with the smallest cost. All of the generated trajectories are shown in Figure 6. The selected trajectory is marked with the blue dotted line, while the other trajectories are in red. \[T_{M}=\text{argmin}(C_{t}^{K})_{i=1\cdots K} \tag{22}\]  
The long-term trajectory prediction is a desirable result that ignores the current moving state, which is indispensable for trajectory prediction. To increase the accuracy, the long-term predictions and the short-term predictions should be combined. The short-term is achieved by taking advantage of the current moving state of the vehicle and the constant yaw rate and acceleration motion model [18]. The velocity can be obtained as follows:  
\[\left[ \begin{array}{c}v_{x}\\ v_{y}\end{array} \right]=\left[\begin{array}{c}(a_{0}*t+v_{0})*\cos(w_{0}*t+ \theta_{0})\\ (a_{0}*t+v_{0})*\sin(w_{0}*t+\theta_{0})\end{  
whereas the influence of the current moving state decreases, and the trajectory can be obtained with Equation (25):  
\[T_{Accu}=F(t)T_{M}+(I-F(t))T_{S} \tag{25}\]  
Using this function, the problem becomes a curve-fitting problem, which is related by the parameter \(t\), a proportion of the maneuvering time ranging from zero to one. With our collected data, the influence curve is shown as a red line in Figure 7. \(F(t)\) decreases much more slowly in the first and third segments than in the second segment, which is in agreement with the practical situation. The trajectory prediction error of a typical lane change, which fluctuates with t, is shown in Figure 8. The error in \(T_{S}\) increases from 0 to more than 5 m, which is too large for collision avoidance, and the error in \(T_{M}\) first increases to more than 1 m, but then decreases. On the other hand, the maximum error in \(T_{Accu}\) is approximately 0.8 m, which is acceptable during the collision detection process. The average errors of \(T_{S}\), \(T_{M}\) and \(T_{Accu}\) are 1.98 m, 0.72 m and 0.41 m, which indicate that the accuracy of the fused trajectory is higher. Figure 8: Trajectory error comparison. The error of \(T_{S}\) increases throughout the maneuver; while the errors of \(T_{Accu}\) and \(T_{M}\) increase firstly and then decrease. \(T_{Accu}\) is more in line with the real trajectory. Figure 7: Influence curve. The influence of the ideal trajectory decreases slightly in the first stage and third stage while it falls fast in the second stage.