# A Novel Maneuver-Based Driving Envelope Generation Approach for Driving Safety Assessment
## 2 The Motion Constraints
### The Constraint of Collision Avoidance with Obstacle
#### 2.3.4 The Case of Lane-Change Abortion  
During the lane change process, if the other vehicles take non-cooperative driving behavior, the condition for lane-change may no longer be satisfied due to the insufficient gap. The driving assistant system should evaluate whether the host vehicle can safely cancel the lane-change maneuver. For a lane-change abortion maneuver, there are two cases as shown in Figure 8. In the first case, the host vehicle cancels lane-change maneuver in a lane-keep mode if it will not enter the left target lane, as shown in Figure 8a. Only the front vehicle \(L_{0}\) and the rear vehicle \(F_{0}\) should be considered in the first case. In the second case, the host vehicle cancels lane-change maneuver in a lane-change mode if it will enter the target lane, as shown in Figure 8b. The surrounding vehicles in both lanes should be considered in the second case. During time \([t_{0},t_{1}]\), the vehicle adjusts the lateral velocity until the lateral velocity equals to zero at \(t_{1}\). The two cases can be classified according to the lateral position \(y(t_{1})\) at time \(t_{1}\). Assuming that the host vehicle moves at a constant lateral acceleration \(a_{lat,adj}\) during time \([t_{0},t_{1}]\). The time \(t_{1}\) can be obtained \(t_{1}=v_{lat}/a_{lat,adj}\). We can have the lateral position \(y(t_{1})\) at time \(t_{1}\). \[y(t_{1})=y_{0}+v_{lat}t_{1}-\frac{1}{2}a_{lat}t_{1}^{2} \tag{12}\]  
Figure 7: The minimum and maximum longitudinal position for lane-change scenario. In the second case, at time \(t_{1}\), the left lane-change abortion scenario in Figure 8b is the same as a right lane-change scenario. Considering the uncertainty of prediction, the defined critical lateral position for a right lane-change maneuver is shown in Figure 9.  
The minimum longitudinal safe distances for lane-change abortion are as follows. \[D_{Ld,s}= \max_{t}\biggl{(}\int_{0}^{t}\int_{0}^{\lambda}\left(a_{m}(\tau)-a _{Ld}(\tau)\right)d\tau d\lambda+(v_{m}-v_{Ld})t\biggr{)},\forall t\in[0,t_{1} +t_{c,Ld}] \tag{13a}\] \[D_{Fd,s}= \max_{t}\biggl{(}\int_{0}^{t}\int_{0}^{\lambda}\left(a_{Fd}(\tau)-a _{m}(\tau)\right)d\tau d\lambda+(v_{Fd}-v_{m})t\biggr{)},\forall t\in[0,t_{1} +t_{c,Fd}]\] (13b) \[D_{L0,s}= \max_{t}\biggl{(}\int_{0}^{t}\int_{0}^{\lambda}\left(a_{m}(\tau)-a _{L0}(\tau)\right)d\tau d\lambda+(v_{m}-v_{L0})t\biggr{)},\forall t\in[t_{1} +t_{c,L0},T]\] (13c) \[D_{P0,s}= \max_{t}\biggl{(}\int_{0}^{t}\int_{0}^{\lambda}\left(a_{P0,s}( \tau)-a_{m}(\tau)\right)d\tau d\lambda+(v_{F0}-v_{m})t\biggr{)},\forall t\in[ t_{1}+t_{c,P0},T] \tag{13d}\]  
where, \(t_{1}\) is the time it takes to adjust the lateral velocity to zero, \(t_{c,x}\) is the time it takes to move laterally from the lateral position \(y(t_{1})\) to the lateral position defined in Figure 9. \(a_{Ld}\), \(a_{Fd}\), \(a_{10}\), \(a_{F0}\) are the accelerations of vehicle \(L_{d}\), \(F_{d}\), \(L_{0}\), \(F_{0}\). \(v_{Ld}\), \(v_{Fd}\), \(v_{10}\), \(v_{F0}\) are the speeds  
Figure 8: The stages for lane-change abortion. (**a**) The first case, the host vehicle cancels the lane-change maneuver in a lane-keep mode. (**b**) The second case, the host vehicle cancels the lane-change maneuver in a lane-change mode. \[y(t_{1})=y_{0}+v_{lat}t_{1}-\frac{1}{2}a_{lat}t_{1}^{2} \tag{12}\]0 The defined critical lateral position for a right lane-change scenario. (**a**) The lateral position for vehicle \(L_{d}\) at \(t_{c,Ld}\). (**b**) The lateral position for vehicle \(F_{d}\) at \(t_{c,Fd}\). (**c**) The lateral position for vehicle \(L_{0}\) at \(t_{c,L0}\). (**d**) The lateral position for vehicle \(F_{0}\) at \(t_{c,F0}\). of vehicle \(L_{d}\), \(F_{d}\), \(L_{0}\), \(F_{0}\). \(v_{m}\) and \(a_{m}\) are the acceleration and speed of the host vehicle. \(T\) is the prediction horizon.