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The geometrical representation of path planning problems

**Leo Dorst, Indur Mandhyan, Karen I. Trovato**

*
invited paper in: Robotics and Autonomous Systems, Elsevier, vol.7, 1991
, pp.181-195.*
The path planning problem for arbitrary devices is first and foremost a
geometrical problem. For the field of control theory, advanced
mathematical techniques have been developed to describe and use geometry.
In this paper, we use the notions
of the flow of vector fields and geodesics in metric spaces to formalize
and unify path planning problems. A path planning algorithm based on flow
propagation is briefly discussed. Applications of the theory to motion planning
for a robot arm, a maneuvering car, and Rubik's Cube are given. These
very different problems (holonomic, non-holonomic and discrete,
respectively) are all
solved by the same unified procedure.

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