Darboux-Frame-Based Parametrization for a Spin-Rolling Sphere on a Plane: A Nonlinear Transformation of Underactuated System to Fully-Actuated Model

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 system as a spin-rolling sphere on a plane with three inputs and five states can be transformed into a fully-actuated model by the given Darboux-frame transformation. This nonlinear state transformation is a geometric model that is different from conventional state-space models. First, a kinematic model of the Darboux frame at the contact point of a rotating object i.e., the sphere, is established. Next, we propose a virtual surface that is trapped between sphere and plane surfaces. This virtual surface generates arc-length-based inputs for controlling the trajectories on the sphere and plane. Finally, we discuss the controllability of this new system after our introduced transformation. In the future, we will design a proper geometric path planning method for the obtained kinematic model.