Bi-Convex Approximation of Non-Holonomic Trajectory Optimization

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.