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Given a continuous motion planning problem of finding a collision-free path from A to B, the A* search is known to be optimal on a finite sized grid-approximation, where each grid-cell for example has 4 or 8 neighbours.

Now, the A* path might be optimal, but it certainly doesn't have to be exactly the same as the shortest path in continuous space, since the A* path is confined to the grid-cells and integer grid coordinates. Now, I would expect that if you increase the resolution to infinity, the resulting A* path should be exact and equal to the continuous problem. But looking at the literature, I can only find "resolution-completeness" for the cell-based approximation. For example in "A Survey of Motion Planning Algorithms from the Perspective of Autonomous UAV Guidance" by C. Goerzen, Z. Kong, B. Mettler the Rectanguloid Approximate Cell Decompositon is non-optimal, but resolution-complete. I really don't understand how it's not resolution-optimal, meaning optimal if you increase the resolution to infinity.

My question is whether I am actually interpreting it wrongly or if such a grid-approximation is really never exactly optimal even for infinite resolutions.

user3386109
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bluetintsprite
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  • There is no such thing as infinite resolution. There is only the limit as the resolution approaches infinity. For a 45 degree line using a 4-neighbor path, the path chosen by A* is always exactly sqrt(2) times longer than the ideal path, regardless of how close the resolution is to infinity. That's because even as the resolution approaches infinity, you can let the zoom level approach infinity as well, and still see the stair steps in the A* path. – user3386109 Oct 10 '21 at 04:20
  • But with increasing resolution the A*-path does get closer (in terms of length) to the exact path, doesn't it? Could I then get as good of an approximation of the exact path length as I want, even if the A*-path will never be exact? – bluetintsprite Oct 10 '21 at 13:06

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