Geometric trajectory filtering via numerical conformal mapping

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Shuo Han and Richard M Murray
2011 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Submitted

The paper studies the problem which we refer to as geometric trajectory filtering, where only trajectories that satisfy the local safety constraints are selected from a library of trajectories. The goal is to speed up primitive-based motion planning while still maintaining a relatively a large collection of motion primitives. One way to solve this problem is to obtain a proper (preferably smooth) function, referred to as the containment indicator function, that describes the shape of the free space. To construct the containment indicator function for an arbitrary shape, the paper uses conformal mapping to transform the original shape of interest to a simpler target shape (e.g. disk, rectangle), which can then be characterized by elementary functions. Computational methods for finding the desired conformal maps are studied. It is shown that they can be formulated as convex optimization problems, whose solution can be obtained efficiently.