The paper deals with motion planning for a spin-rolling sphere when the sphere follows a straight path on a plane. Since the motion of the sphere is constrained by the straight line, the control of the sphere's spin motion is essential to converge to a desired configuration of the sphere. In this paper, we show a new geometric-based planning approach that is based on a full-state description of this nonlinear system. First, the problem statement of the motion planning is posed. Next, we develop a geometric controller implemented as a virtual surface by using the Darboux frame kinematics. This virtual surface generates arc-length-based inputs for controlling the trajectories of the sphere. Then, an iterative algorithm is designed to tune these inputs for the desired configurations. The feasibility of the proposed approach is verified by simulations.

翻译：该文件涉及球体在平面上遵循直线路径时旋转球的动作规划。 由于球体运动受直线限制, 控制球体的旋转运动对于接近所需的球体配置至关重要。 在本文中, 我们展示了一种新的基于几何规划方法, 其基础是对这一非线性系统的全面描述。 首先, 提出了运动规划的问题说明 。 其次, 我们开发了一个几何控制器, 通过使用 Darboux 框架动力学作为虚拟表面。 这个虚拟表面生成了以弧长为基础的输入来控制球体的轨迹。 然后, 一个迭代算法设计了这些输入来调节理想的配置。 通过模拟来验证拟议方法的可行性 。