Continuous-Time Motion Planning and Steering with Global Convergence

Motion planning is central to robotics, and it is increasingly used to correct the motions proposed by learned planners and policies. Both require constraints to hold at every instant of the motion, not only at sampled times. To our knowledge, no algorithm is known to provably reach a globally optimal motion once a smoothness cost is combined with such constraints. We cast the problem as a single nonconvex program over polynomial curves. It is elementary to state, yet nonconvex. Robots avoid one another and static or moving obstacles, and a learned proposal is steered by keeping the motion close to it. Each constraint becomes one polynomial that must stay nonnegative over the whole motion. We propose GlobMP (Global Motion Planning), a gradient projection method that alternates cost reduction with feasibility restoration. Under mild assumptions, we prove that, from a fine grid start, it converges globally. We demonstrate its effectiveness and superior performance over strong baselines in real and synthetic robotics experiments. These include synthetic teams of up to $200$ robots, multi-robot planning on the Robotarium testbed and an extended benchmark, maze navigation, and steering a tactile-reactive policy on a bimanual humanoid.