In a motion generation context, path geometric characteristics assume a relevant role. A novel path planning primitive was proposed in [1] able to generate continuous curvature paths and named η2-splines. Such path primitive is a parameterized quintic spline that allows interpolating any arbitrary sequence of points on the plane. The new primitive has been explicitly thought for the inversion based control of nonholonomic car-like vehicles [2], [3] and omnidirectional robots [4]. Recently, the attention has been focused on planning primitives whose curvature is also continuously differentiable. Paths which possess this characteristic are named G3-paths. G3-continuity is required especially for unicycle-like robots: in [5] it has been shown that, in order to generate continuously differentiable control signals, it is necessary to plan G3-paths. This requirement is not strictly necessary in the case of other autonomous vehicles, however the use of paths whose curvature is continuously differentiable leads to the generation of smooth command signals, which is, undoubtedly, a positive characteristic. For this reason in [6], a new planning primitive, named η3-splines, has been proposed for the generation of G3-paths. η3-splines are planned by means of closed form expressions and always fulfill any arbitrarily assigned set of interpolating conditions. An important advantage of η3-splines is represented by the possibility of refining their shape, still satisfying the assigned interpolating conditions, by acting on a set of six free parameters. Such a possibility can be used, for example, for the generation of curves which satisfy appropriate optimality criteria. A solution for the optimal planning of η3-splines is proposed in [7].
