Sampling-based Motion Planning for Optimal Probability of Collision under Environment Uncertainty

Motion planning is a fundamental capability in robotics applications. Real-world scenarios can introduce uncertainty to the motion planning problem. In this work we study environment uncertainty in general high-dimensional problems wherein the choice of appropriate metrics and formulations are shown to have significant effect on the probability of collision of the solution path. Several practically motivated cost functions have been proposed in literature to model and solve the problem but are shown in this work to suffer from higher probabilities of collision. The current work presents a theoretically sound formulation that was first mentioned in previous work on minimum constraint removal. In this work, approximating the optimal problem is shown to be better in achieving lower probability of collision. To demonstrate the formulation in a sampling-based setting, a mixed integer linear program seeded by greedy search over a roadmap with sampled environments is used to report paths with low probability of collision. Compared against minimizing the sum and minimizing max probability cost functions on a seven degree-of-freedom robotic arm in uncertain environments, we show clear benefits and promise towards motion planning for optimal probability of collision.