The paper proposes a new planning primitive, named ${\eta }^{3D}$-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, ${\eta }^{3D}$-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 ${\eta }^{3D}$-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 ${\eta }^{3D}$-splines.

该论文提出了一种新的规划原语，名为 ${\eta }^{3D}$-splines，基于七阶多项式，适用于生成以三阶几何连续性为特征的三维路径。三阶连续性代表了一个重要的特性，因为它允许机器人系统的关节致动器的连续加加速度参考信号。与文献中的其他方法不同，${\eta }^{3D}$-样条通过封闭形式表达式作为指定插值条件的函数进行有效评估。这允许直观地实时生成复合路径：根据当前执行的曲线的几何特征的知识，并通过选择新的端点和所需的插值条件，可以有效地生成新路径同时保持整体的三阶几何连续性。此外，该${\eta }^{3D}$-splines 可以通过作用于一组六个自由参数来塑造，以模拟其他规划基元，例如，线性段、圆弧、回旋曲线、螺旋曲线和圆锥螺旋。此外，通过相同的参数，可以生成满足给定内插条件的所有可能的 7 阶多项式。随附的视频显示了一个拟人机械手执行复合轨迹，该轨迹由${\eta }^{3D}$-样条。