Environment Warped Gait Trajectory Optimization for Complex Terrains

Contact-aware gait trajectory optimization is a challenging non-convex programming problem, especially for complex terrain shapes, where prominent numerical algorithms can fail to find a solution or fall into local minima. To alleviate this issue, we propose an environment warping technique that changes the coordinates for decision variables. Given a terrain of some general shape, our method first generates a locally injective, as-conformal-as-possible function that maps the ambient space around the terrain to a warped space. We then formulate the trajectory optimization in the warped space by remapping all the decision variables. Our method frees the numerical optimizer from tuning the trajectories to fit changing terrain shapes, leading to better numerical conditioning and fewer local minima. Numerical results show that our method outperforms direct trajectory optimization in terms of both success rates and quality of solutions.