This paper presents a new kinematic model based on the Darboux frame for motion control and planning. In this work, we show that an underactuated model of a spin-rolling sphere on a plane with five states and three inputs can be transformed into a fully-actuated one by a given Darboux frame transformation. This nonlinear state transformation establishes a geometric model that is different from conventional state-space ones. First, a kinematic model of the Darboux frame at the contact point of the rolling sphere is established. Next, we propose a virtual surface that is trapped between the sphere and the contact plane. This virtual surface is used for generating arc-length-based inputs for controlling the contact trajectories on the sphere and the plane. Finally, we discuss the controllability of this new model. In the future, we will design a geometric path planning method for the proposed kinematic model.

本文提出了一种基于 Darboux 框架的新运动学模型，用于运动控制和规划。在这项工作中，我们展示了在具有五个状态和三个输入的平面上的自旋滚球的欠驱动模型可以通过给定的 Darboux 框架转换转换为完全驱动的模型。这种非线性状态转换建立了一个不同于传统状态空间模型的几何模型。首先，建立滚动球接触点处的达布坐标系的运动学模型。接下来，我们提出了一个被困在球体和接触平面之间的虚拟表面。该虚拟表面用于生成基于弧长的输入，以控制球体和平面上的接触轨迹。最后，我们讨论了这个新模型的可控性。将来，