Autonomous cars and fixed-wing aerial vehicles have the so-called non-holonomic kinematics which non-linearly maps control input to states. As a result, trajectory optimization with such a motion model becomes highly non-linear and non-convex. In this paper, we improve the computational tractability of non-holonomic trajectory optimization by reformulating it in terms of a set of bi-convex cost and constraint functions along with a non-linear penalty. The bi-convex part acts as a relaxation for the non-holonomic trajectory optimization while the residual of the penalty dictates how well its output obeys the non-holonomic behavior. We adopt an alternating minimization approach for solving the reformulated problem and show that it naturally leads to the replacement of the challenging non-linear penalty with a globally valid convex surrogate. Along with the common cost functions modeling goal-reaching, trajectory smoothness, etc., the proposed optimizer can also accommodate a class of non-linear costs for modeling goal-sets, while retaining the bi-convex structure. We benchmark the proposed optimizer against off-the-shelf solvers implementing sequential quadratic programming and interior-point methods and show that it produces solutions with similar or better cost as the former while significantly outperforming the latter. Furthermore, as compared to both off-the-shelf solvers, the proposed optimizer achieves more than 20x reduction in computation time.

Planning frameworks for autonomous vehicles must be robust and computationally efficient for real time realization. At the same time, they should accommodate the unpredictable behavior of the other participants and produce safe trajectories. In this paper, we present a computationally efficient hierarchical planning framework for autonomous vehicles that can generate safe trajectories in complex driving scenarios, which are commonly encountered in urban traffic settings. The first level of the proposed framework constructs a Model Predictive Control(MPC)routine using an efficient difference of convex programmingapproach, that generates smooth and collision-free trajectories. The constraints on curvature and road boundaries are seamlessly integrated into this optimization routine. The second layer is mainly responsible to handle the unpredictable behaviors that are typically exhibited by the other participants of traffic. It is built along the lines of time scaled collision cone(TSCC)which optimize for the velocities along the trajectory to handle such disturbances. We additionally show that our framework maintains optimal balance between temporal and path deviations while executing safe trajectories. To demonstrate the efficacy of the presented framework we validated it in extensive simulations in different driving scenarios like over taking, lane merging and jaywalking among many dynamic and static obstacles.