**4.3 Summary**

before, until the car reaches the center of adjacent lane at *k* ¼ *T*3. Between *k* ¼ *T*3 and *k* ¼ *T*5, the angular acceleration remains at 0 for a straight line motion in opposite lane for overtaking. A negative angular acceleration having the same magnitude as before is applied between *k* ¼ *T*5 and *k* ¼ *T*6 followed by positive angular acceleration between *k* ¼ *T*6 and *k* ¼ *T*7. Car 1 continues its motion in a

*Models and Technologies for Smart, Sustainable and Safe Transportation Systems*

There were quite a few assumptions made to obtain the simplified model discussed above and some of those assumptions could be removed in order to obtain a rather complicated yet general framework. For instance, it might not be possible for the AV to store all the history of past states and actions so an attempt at modelling with limited information could be made. With this disclaimer, a brief overview of some of the possible solution methods for this problem is presented

In this approach, taken from [42], it is assumed that the AV will travel at a constant speed throughout the overtaking maneuver. The estimates of positions and velocities of HDVs are obtained using Kalman filtering and a constant N is introduced to characterize the behavior of the AV with higher values corresponding to higher level of aggressiveness. Two feasible sets for the linear and angular velocities respectively of Car 1 are obtained by ensuring that the estimated position of AV after performing the maneuver will stay outside the estimated minimum safety region around the HDVs. Using the feasibility sets, probability of collision of Car 1 with the HDVs is obtained and the decision to overtake is based on those probabilities. If the decision to overtake is made, the linear and the angular velocities are chosen to minimize the probability of collision and the maneuver is

*4.2.2 Reachability analysis-based with martingale-based HDV modelling*

In this approach, taken from [43], the focus is on obtaining safety guarantees while overtaking. In this approach, the restriction on the constant speed of HDVs is also lifted, which was alluded to previously in Section 4.2. There are two different reachability analysis-based algorithms presented: one is a robust time-optimal

straight line after *k* ¼ *T*7 with zero angular acceleration.

**4.2 Solution methods**

*Sample trajectory for car overtaking problem.*

*4.2.1 Minimizing probability of collision*

performed as detailed in Section 4.1.3.

below.

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**Figure 4.**

The key takeaway from this section is that there is a need to place focus on explicitly considering the role of human drivers on the road while developing algorithms for autonomous vehicles. It was shown by the study of the simple car overtaking example that there are scenarios where the need to model human drivers on the road increases manifold. The algorithm presented in Section 4.2.2 explicitly models the behavior of human-drivers with martingales and provides overtaking algorithms with safety guarantees. The solution complexity of this approach is rather high due to the reachability analysis-based solutions so further research could be directed at improving the solution complexity or coming up with approaches that yield lower complexity while maintaining the safety guarantees. Furthermore, there is a prospect of exciting research in the direction of modeling human behavior with other approaches which may lead to various other interesting algorithms. The aim of this section was to provide motivation to the reader to explore research avenues that incorporate explicit modelling of human driving patterns yielding algorithms that will expedite the introduction of Autonomous Vehicles onto our roads.
