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To move through the world, mobile robots typically use a receding-horizonstrategy, wherein they execute an old plan while computing a new plan toincorporate new sensor information. A plan should be dynamically feasible,meaning it obeys constraints like the robot's dynamics and obstacle avoidance;it should have liveness, meaning the robot does not stop to plan so frequentlythat it cannot accomplish tasks; and it should be optimal, meaning that therobot tries to satisfy a user-specified cost function such as reaching a goallocation as quickly as possible. Reachability-based Trajectory Design (RTD) isa planning method that can generate provably dynamically-feasible plans.However, RTD solves a nonlinear polynmial optimization program at each planningiteration, preventing optimality guarantees; furthermore, RTD can struggle withliveness because the robot must brake to a stop when the solver finds localminima or cannot find a feasible solution. This paper proposes RTD*, whichcertifiably finds the globally optimal plan (if such a plan exists) at eachplanning iteration. This method is enabled by a novel Parallelized ConstrainedBernstein Algorithm (PCBA), which is a branch-and-bound method for polynomialoptimization. The contributions of this paper are: the implementation of PCBA;proofs of bounds on the time and memory usage of PCBA; a comparison of PCBA tostate of the art solvers; and the demonstration of PCBA/RTD* on a mobile robot.RTD* outperforms RTD in terms of optimality and liveness for real-time planningin a variety of environments with randomly-placed obstacles.

Safe, Optimal, Real-time Trajectory Planning with a Parallel Constrained Bernstein Algorithm