From G2 to G3 Continuity: Continuous Curvature Rate Steering Functions for Sampling-Based Nonholonomic Motion Planning

Motion planning for car-like robots is one of the major challenges in automated driving. It requires to solve a two-point boundary value problem (BVP) in real time while taking into account the nonholonomic constraints of the vehicle and the obstacles in the non-convex environment. This paper introduces Hybrid Curvature Rate (HCR) and Continuous Curvature Rate (CCR) Steer: Two novel steering functions for car-like robots that compute a curvature rate continuous solution of the two-point BVP. Hard constraints on the maximum curvature, maximum curvature rate, and maximum curvature acceleration are satisfied resulting in directly driveable $\mathrm{G^{3}}$ continuous paths. The presented steering functions are bench-marked in terms of computation time and path length against its $\mathrm{G^{1}}$ and $\mathrm{G^{2}}$ continuous counterparts, namely Dubins, Reeds-Shepp, Hybrid Curvature, and Continuous Curvature Steer. It is shown that curvature rate continuity can be achieved with only small computational overhead. The generic motion planner Bidirectional $\mathrm{RRT^{\ast}}$ is finally used to present the effectiveness of HCR and CCR Steer in three challenging automated driving scenarios.