# A Novel Maneuver-Based Driving Envelope Generation Approach for Driving Safety Assessment
## 2 The Motion Constraints
### The Constraint of Collision Avoidance with Obstacle
#### 2.3.3 The Case of Lane-Change Maneuver  
In [9], the longitudinal safe distance to start a lane-change maneuver is derived. However, the longitudinal safe distance during a lane change maneuver is not derived. Inspired by [9], the safe distance during the lane-change process and the lane-change abortion process have been discussed in our previous work [36]. In this paper, the prediction uncertainty is considered to obtain the safe distance for lane change. A lane-change scenario is shown in Figure 5, the front vehicle \(L_{0}\) and rear vehicle \(F_{0}\) are in the current lane, the front vehicle \(L_{d}\) and the rear vehicle \(F_{d}\) are in the left target lane. Figure 4: The safe distance of evasive maneuver for static obstacle. (**a**) The extreme trajectory of evasive maneuver using evasive motion model. (**b**) The critical position of evasive maneuver. If considering the uncertainty of prediction, the occupancy area of traffic participants at each prediction step can be obtained. The constant acceleration lane keeping (CALK) model [37] can be used to represent the motion of lane-keep considering uncertainty. \[\left[ \begin{array}{c}s\\ n\\ v_{s}\\ a_{s}\end{array} \right]_{\hat{x}}=\left[ \begin{array}{cccc}1&0&\Delta t&\frac{ \Delta t^{2}}{2}\\ 0&1&0&0\\ 0&0&1&\Delta t\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{c}s\\ n\\ v_{s}\\ a_{s}\end{array} \right]_{k-1}+\left[ \begin{array}{c}w_{as}\Delta t^{2}/2\\ w_{as}\Delta t^{2}/2\\ w_{as}\Delta t\\ w_{as}\end{array} \right] \tag{10}\]  
where \(w_{as}\sim N(0,\sigma_{as}^{2})\), \(w_{an}\sim N(0,\sigma_{an}^{2})\). Since the vehicle is in lane keeping mode, the lateral position is bounded in the lane. The critical situation for calculating the longitudinal safe distance considering uncertainty is defined as Figure 6. The vertex of the host vehicle just touches the vertex of the occupancy set of the traffic participant. The time it takes for the host vehicle to move from the current lateral position to the lateral position defined in Figure 6 can be obtained according to Section 2.2. First of all, we calculate the time to arrive at the lateral position defined in Figure 6. Then, the longitudinal safe distance for each vehicle is obtained based on the time to arrive at the critical time. The minimum longitudinal safe distance \(D_{Fd,s^{\prime}}\), \(D_{Ld,s^{\prime}}\), \(D_{F0,s}\), \(D_{D0,s}\) for vehicles \(F_{d}\), \(L_{d}\), \(F_{0}\), \(L_{0}\) during the process of the lane-change maneuver can be derived as Equation (11). \[D_{Fd,s}= \underset{t}{max}\bigg{(}\int_{0}^{t}\int_{0}^{\lambda}(a_{Fd}( \tau)-a_{M}(\tau))d\tau d\lambda+(v_{Fd}(0)-v_{M}(0))t\bigg{)},\forall t\in[t_{ c,Fd},T] \tag{11a}\] \[D_{Ld,s}= \underset{t}{max}\bigg{(}\int_{0}^{t}\int_{0}^{\lambda}(a_{M}( \tau)-a_{Ld}(\tau))d\tau d\lambda+(v_{M}(0)-v_{Ld}(0))t\bigg{)},\;\forall t\in[ t_{c,Ld},T]\] (11b) \[D_{F0,s}= \underset{t}{max}\bigg{\{}\bigg{(}\int_{0}^{t}\int_{0}^{\lambda}(a_{ F0}(\tau)-a_{M}(\tau))d\tau d\lambda+(v_{F_{0}}(0)-v_{M}(0))t\bigg{)},0\bigg{\}}, \forall t\in[0,t_{c,F0}]\] (11c) \[D_{L0,s}= \underset{t}{max}\bigg{\{}\bigg{(}\int_{0}^{t}\int_{0}^{\lambda}(a_{ M}(\tau)-a_{Ld}(\tau))d\tau d\lambda+(v_{M}(0)-v_{Ld}(0))t\bigg{)},0\bigg{\}}\;\forall t\in[ t_{c,L0}] \tag{11d}\]  
Figure 5: A lane-change scenario in double-lane highway. Figure 6: The critical situations for lane-change considering uncertainty of prediction. (**a**) The critical situation for \(F_{d}\) at \(t_{c,Fd}\). (**b**) The critical situation for \(L_{d}\) at \(t_{c,Ld}\). (**c**) The critical situation for \(F_{0}\) at \(t_{c,F0}\). (**d**) The critical situation for \(L_{0}\) at \(t_{c,L0}\). where \(t_{c,F,d}\), \(t_{c,Ld}\), \(t_{c,F0}\), \(t_{c,L0}\) are the time it takes for the host vehicle to move from current state to the defined critical lateral position in Figure 6. \(a_{M}\) is the acceleration of the host vehicle, \(a_{Fd}\), \(a_{Ld}\), \(a_{F0}\), \(a_{L0}\) are the accelerations of the vehicles \(F_{d}\), \(L_{d}\), \(F_{0}\), \(L_{0}\).