A New Framework for Optimal Path Planning of Rectangular Robots Using a Weighted $L_p$ Norm

This letter introduces a new framework for modeling the optimal path planning problem of rectangular robots. Typically constraints for the safe, obstacle-avoiding path involve a set of inequalities expressed using logical <sc> or</sc> operations, which makes the problem difficult to solve using existing optimization algorithms. Inspired by the geometry of the unit sphere of the weighted <inline-formula><tex-math notation="LaTeX">$L_p$</tex-math> </inline-formula> norm, the authors find exact and approximate constraints for safe configurations using only logical <sc>and</sc> operations. The proposed method does not require integer programming nor computation of a Minkowski sum in the configuration space. In particular, the authors analyze two different cases of obstacle geometry: circular obstacles and rectangular obstacles. Using the weighted <inline-formula><tex-math notation="LaTeX">$L_p$</tex-math> </inline-formula> norm requires six inequalities to represent the exact constraints for collision avoidance of circular obstacles using <sc>and</sc> operations, and eight inequalities for rectangular obstacles. Four shortest path planning examples are analyzed to validate the effectiveness of the proposed method.

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