How should an autonomous vehicle overtake a slower moving vehicle: design and analysis of an optimal trajectory

Over the past few years, there has been much research on various aspects of control of autonomous vehicles. However, it seems that the problem of overtaking a slower moving vehicle has been somewhat neglected. This note deals with the three-phase overtaking maneuver and with designing a smooth and ergonomic optimal lane-change trajectory to be used under normal conditions. It is shown that the absolute shape, size, and time of the first-phase trajectory do not depend on the velocity of the leading, slower moving vehicle. Only the absolute point for initiating the diversion is affected. The relatively simple mathematical model for each lane-change trajectory is based on minimizing the total kinetic energy during the maneuver, superimposed on a "minimum-jerk trajectory." For high enough initial velocities, explicit formulas are obtained for the optimal distance and the optimal time of the maneuver. It is also shown that the total time is bounded from above and below, regardless of the velocity. By using the results of the suggested model, an autonomous vehicle, equipped with appropriate sensors, can estimate the best time and place to begin and end the overtaking and its total time and distance. This may help to make a decision whether to overtake or not.