Generation of Paths With Minimum Curvature Derivative With $ { \mmb{\eta } }^{3}$ -Splines

This paper deals with the generation of smooth paths planned by means of $ { \mmb{\eta } }^{3}$ -splines, a recently devised planning primitive used for the automated steering of wheeled mobile robots. The shape of $ { \mmb{\eta } }^{3}$-splines can be easily modified by acting on a set of free parameters. This capability can be used, for example, to satisfy an assigned optimality criterion. In this paper, it will be used to minimize the curvature variability in order to reduce the lateral solicitations affecting an autonomous robot. Evidently, curvature derivative could be minimized by means of an optimization algorithm. However, this approach cannot be suitably used in an online application which continuously requires the curve updating. For this reason, a heuristic method, based on closed form expressions, has been devised and proposed in the paper in order to efficiently generate almost optimal curves on the sole basis of the interpolating conditions. As a further characteristic, the proposed heuristic expressions permit obtaining, when appropriate interpolating conditions are given, $ { \mmb{\eta } }^{3}$-splines which at best emulate circular arcs and clothoids.