# A Novel Maneuver-Based Driving Envelope Generation Approach for Driving Safety Assessment
## 2 The Motion Constraints
### The Evasive Motion Model  
Motion planning in a dynamic environment involves many aspects, such as collision avoidance, the drivability of the planned trajectory, and the motion of the other traffic participants in the future, etc. In [32], the minimum distance required for avoiding the front dynamic obstacle was analyzed. However, the paper only considered avoiding obstacles with a constant steering angle. The constraints on the terminal state of the host vehicle after avoiding obstacles have not been considered. It is not guaranteed to adjust the state of the vehicle in the lane boundary after a collision-avoidance maneuver. In a real traffic scenario, after an evasive maneuver or lane-change maneuver, the vehicle should adjust the heading angle to avoid collision with the road boundary. In this paper, we utilize an evasive motion model in [9]. The motion of the host vehicle is decomposed into lateral movement and longitudinal movement in the reference frame. The lateral motion model in the reference frame is described as follows. \[a_{lat}(t)= \begin{cases}\frac{2\pi H}{t_{lat}^{2}}\sin(\frac{2\pi}{t_{lat}}t ),t\leq t_{lat}\\ 0,\text{otherwise}\end{cases}  \tag{2a}\] \[v_{lat}(t)= \begin{cases}-\frac{H}{t_{lat}}\cos(\frac{2\pi}{t_{lat}}t)+H/t_{ lat},t\leq t_{lat}\\ 0,\text{otherwise}\end{cases} \] (2b) \[y(t)= -\frac{H}{2\pi}\sin(\frac{2\pi}{t_{lat}}t)+\frac{H}{t_{lat}}t+y_{0}\] (2c) \[H= y_{target}-y_{0} \tag{2d}\]where \(a_{lat}\) is the lateral acceleration in the reference frame, \(t_{lat}\) is the required total time to finish the lateral motion, \(H\) is the total lateral moving distance of the evasive maneuver, \(y_{0}\) is the starting lateral position in the reference frame, \(y_{target}\) is the terminal lateral position in the reference frame. As mentioned in [33], the comfort zone of lateral acceleration should be within \(a\in[-2,2]\). The longitudinal motion model in the reference frame can be defined as follows. \[x_{lon}(t)= v_{lon}(0)t+\frac{1}{2}at^{2} \tag{3a}\] \[v_{lon}(t)= v_{lon}(0)+at \tag{3b}\]  
The lateral acceleration, curvature, and lateral velocity of the motion model are differentiable. Assuming that \(H\) is 3.5 m, the values of lateral acceleration \(a_{lat}\), lateral velocity \(v_{lat}\), lateral position \(y_{lat}\), curvature \(k\), and derivative of curvature \(k\) can be obtained, as shown in Figure 1.  
Given the maximum lateral accelerations \(a_{\max,lat}\) and the total lateral movement distance \(H\), the required time \(t_{lat}\) to finish the evasive maneuver can be obtained. \[t_{lat}=\sqrt{2\pi H/a_{\max,lat}} \tag{4}\]  
According to natural driving data, the longitudinal and lateral accelerations of the human driver are typically smaller than physically feasible accelerations. For most drivers, the maximum longitudinal and lateral acceleration is around 3-4 m/s\({}^{2}\) at low speed, and can be reduced to 1-2 m/s\({}^{2}\) at high speed [34]. The heading angle in the reference frame can be obtained using longitudinal velocity and lateral velocity. \[\varphi(t)=arctan(v_{lat}(t)/v_{lon}(t)) \tag{5}\]  
Figure 1: The lateral motion of the evasive motion model. The dashed blue line and red solid line respectively represent the values by setting \(t_{lat}\) as 5 s and 3 s.  
The position of the center of the rear axis can be calculated by the evasive motion model. Thus, the position of each vertex as shown in Figure 2 can be obtained. \[(x_{fl},y_{fl})= (x-w_{h}sin\varphi+l_{a}cos\varphi,y_{lat}+w_{h}cos\varphi+l_{a} sin\varphi); \tag{6a}\] \[(x_{fr},y_{fr})= (x+w_{h}sin\varphi+l_{a}cos\varphi,y_{lat}-w_{h}cos\varphi+l_{a} sin\varphi);\] (6b) \[H= y_{target}-y_{0} \tag{2d}\]0 (6c) \[H= y_{target}-y_{0} \tag{2d}\]1  
where, \(w_{h}\) is the half width of the host vehicle, \(l_{a}\) and \(l_{b}\) are the distances from the rear axis to the front most and rear most of the vehicle.