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

TL;DR

This paper introduces a Darboux-frame-based nonlinear transformation that converts an underactuated spin-rolling sphere model into a fully-actuated one, enabling advanced control and planning strategies.

Contribution

It proposes a novel Darboux frame transformation that creates a geometric, fully-actuated model from an underactuated spin-rolling sphere system.

Findings

Transformation successfully converts underactuated to fully-actuated model.

The geometric model differs from traditional state-space models.

Controllability of the new model is established.

Abstract

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.