Bringing the gap between what legged robots can do requires to carefully plan motion and foothold planning over challenging terrains. Motion and foothold planning requires to consider the robot’s dynamics and the terrain model in a suitable numerical optimization formulation. However, mathematically speaking, this problem is hard to solve due to contact discontinuities, nonlinear, high-dimensional and under-actuated dynamics.

To this end, we have proposed different numerical optimization formulations: nonlinear [1], stochastic [2,3] and mixed-integer convex [4] optimizations. In [1], we proposed a hierarchical numerical optimization which helps to plan jumping maneuvers and its contact event. We have studied non-convex terrain model which is used to plan motions and footholds through a stochastic optimization [2,3] . We use a cart-table with flywheel model and the robot stability is guaranteed by keeping the centroidal moment pivot inside the support polygon [3]. Finally, we have proposed to plan different quadrupedal gaits, and their foothold locations, by formulating both through integer variables [4].