In this paper we consider the problem of generating motion plans for a robot that are guaranteed to succeed despite uncertainty in the environment, parametric model uncertainty, and disturbances. Furthermore, we consider the case where these plans must be generated in real-time, because constraints such as obstacles in the environment may not be known until they are perceived (with a noisy sensor) at runtime. Previous work on feedback motion planning for nonlinear systems was limited to offline planning due to the computational cost of safety verification. Here we augment the traditional trajectory library approach by designing locally stabilizing controllers for each nominal trajectory in the library and providing guarantees on the resulting closed-loop systems. We leverage sums-of-squares (SOS) programming to design these locally stabilizing controllers by explicitly attempting to minimize the size of the worst case reachable set of the closed-loop system subjected to bounded disturbances and uncertainty. The reachable sets associated with each trajectory in the library can be thought of as "funnels" that the system is guaranteed to remain within. The resulting funnel library is then used to sequentially compose motion plans at runtime while ensuring the safety of the robot. A major advantage of the work presented here is that by explicitly taking into account the effect of uncertainty, the robot can evaluate motion plans based on how vulnerable they are to disturbances. We demonstrate our method using thorough simulation experiments of a ground vehicle model navigating through cluttered environments and also present extensive hardware experiments validating the approach on a small fixed-wing airplane avoiding obstacles at high speed.
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