Making safe and successful lane changes (LCs) is one of the many vitally important functions of autonomous vehicles (AVs) that are needed to ensure safe driving on expressways. Recently, the simplicity and real-time performance of the potential field (PF) method have been leveraged to design decision and planning modules for AVs. However, the LC trajectory planned by the PF method is usually lengthy and takes the ego vehicle laterally parallel and close to the obstacle vehicle, which creates a dangerous situation if the obstacle vehicle suddenly steers. To mitigate this risk, we propose a time-to-collision-aware LC (TTCA-LC) strategy based on the PF and cubic polynomial in which the TTC constraint is imposed in the optimized curve fitting. The proposed approach is evaluated using MATLAB/Simulink under high-speed conditions in a comparative driving scenario. The simulation results indicate that the TTCA-LC method performs better than the conventional PF-based LC (CPF-LC) method in generating shorter, safer, and smoother trajectories. The length of the LC trajectory is shortened by over 27.1\%, and the curvature is reduced by approximately 56.1\% compared with the CPF-LC method.

It has recently been shown that the problem of testing global convexity of polynomials of degree four is {strongly} NP-hard, answering an open question of N.Z. Shor. This result is minimal in the degree of the polynomial when global convexity is of concern. In a number of applications however, one is interested in testing convexity only over a compact region, most commonly a box (i.e., hyper-rectangle). In this paper, we show that this problem is also strongly NP-hard, in fact for polynomials of degree as low as three. This result is minimal in the degree of the polynomial and in some sense justifies why convexity detection in nonlinear optimization solvers is limited to quadratic functions or functions with special structure. As a byproduct, our proof shows that the problem of testing whether all matrices in an interval family are positive semidefinite is strongly NP-hard. This problem, which was previously shown to be (weakly) NP-hard by Nemirovski, is of independent interest in the theory of robust control.