  Uncertain dynamic obstacles, such as pedestrians or vehicles, pose a major
challenge for optimal robot navigation with safety guarantees. Previous work on
motion planning has followed two main strategies to provide a safe bound on an
obstacle's space: a polyhedron, such as a cuboid, or a nonlinear differentiable
surface, such as an ellipsoid. The former approach relies on disjunctive
programming, which has a relatively high computational cost that grows
exponentially with the number of obstacles. The latter approach needs to be
linearized locally to find a tractable evaluation of the chance constraints,
which dramatically reduces the remaining free space and leads to
over-conservative trajectories or even unfeasibility. In this work, we present
a hybrid approach that eludes the pitfalls of both strategies while maintaining
the original safety guarantees. The key idea consists in obtaining a safe
differentiable approximation for the disjunctive chance constraints bounding
the obstacles. The resulting nonlinear optimization problem is free of chance
constraint linearization and disjunctive programming, and therefore, it can be
efficiently solved to meet fast real-time requirements with multiple obstacles.
We validate our approach through mathematical proof, simulation and real
experiments with an aerial robot using nonlinear model predictive control to
avoid pedestrians.
