η3D -splines for the generation of 3D Cartesian paths with third order geometric continuity

The paper proposes a new planning primitive, named η 3D -splines, based on 7th order polynomials, which is suited to generate three-dimensional paths characterized by third-order geometric continuity. The third order continuity represents an important property, since it allows continuous-jerk reference signals for the joints actuators of robotic systems. Differently from other approaches in the literature, η 3D -splines are efficiently evaluated by means of closed form expressions as function of the assigned interpolation conditions. This allows an intuitive real-time generation of composite paths: from the knowledge of the geometric characteristics of the curve which is currently executed, and by choosing a novel end-point together with the desired interpolating conditions, a new path can be efficiently generated by simultaneously maintaining the overall third order geometric continuity. Additionally, the η 3D -splines can be shaped by acting on a set of six free parameters, so as to emulate other planning primitives, like, for example, linear segments, circular arcs, clothoids, helical curves, and conic spirals. Furthermore, by means of the same parameters, all possible 7th order polynomials, which fulfill the given interpolating conditions, can be generated. The accompanying video shows an anthropomorphic manipulator executing a composite trajectory generated by means of the η 3D -splines.