In this talk, I’ll describe a new approach to planning that strongly leverages both continuous and discrete/combinatorial optimization. The framework is fairly general, but I will focus on a particular application of the framework to planning continuous curves around obstacles. Traditionally, these sort of motion planning problems have either been solved by trajectory optimization approaches, which suffer with local minima in the presence of obstacles, or by sampling-based motion planning algorithms, which can struggle with derivative constraints and sample-complexity in very high dimensions. In the proposed framework, called Graphs of Convex Sets (GCS), we can recast the trajectory optimization problem over a parametric class of continuous curves into a problem combining convex optimization formulations for graph search and for motion planning. The result is a non-convex optimization problem whose convex relaxation is very tight — to the point that we can very often solve very complex motion planning problems to global optimality using the convex relaxation plus a cheap rounding strategy. I will describe numerical experiments of GCS applied to a quadrotor flying through buildings and robotic arms moving through confined spaces. On a seven-degree-of-freedom manipulator, GCS can outperform widely-used sampling-based planners by finding higher-quality trajectories in less time, and in 14 dimensions (or more) it can solve problems to global optimality which are hard to approach with sampling-based techniques. Finally, I’ll discuss new extensions using GCS for planning on manifolds and task and motion planning.

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