We study path planning on grids with blocked and unblocked cells. Any-angle path-planning algorithms find short paths fast because they propagate information along grid edges without constraining the resulting paths to grid edges. Incremental path-planning algorithms solve a series of similar path-planning problems faster than repeated single-shot searches because they reuse information from the previous search to speed up the next one. In this paper, we combine these ideas by making the any-angle path-planning algorithm Basic Theta~* incremental. This is non-trivial because Basic Theta~* does not fit the standard assumption that the parent of a vertex in the search tree must also be its neighbor. We present Incremental Phi~* and show experimentally that it can speed up Basic Theta~* by about one order of magnitude for path planning with the freespace assumption.
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