Planning 픾^3 -continuous paths for state-constrained mobile robots with bounded curvature of motion

The bounds on the mobile robot curvature of motion and path curvature continuity constraints usually result either from mechanical construction limitations or practical motion smoothness requirements. Most path planning primitives compatible with those constraints force planning algorithms to utilize costly numerical methods for computation of maximal path curvature or positional path constraints verification. In this paper a novel path primitive is proposed, which can be concatenated with the line and circle segments to form a path with bounded curvature such that its perfect realization by a unicycle robot guarantees continuous time-derivative of its curvature of motion. Satisfaction of prescribed curvature bounds and positional path constraints resulting from obstacles in the environment is formally guaranteed using explicit analytic formulas presented in the paper. It is shown that the proposed approach yields an arbitrarily precise \( {{\mathbb{G}}^3} \)-continuous approximation of the Reeds-Shepp paths. Presented analysis is further utilized to formulate the global path planning problem in a continuous domain as a tractable optimization problem. Computational effectiveness of the proposed method has been additionally verified by quantitative comparison of constraint satisfaction checking speed with the \( {\eta^3} \)-splines.