A coordinate-free approach to instantaneous kinematics of two rigid objects with rolling contact and its implications for trajectory planning

This paper adopts a coordinate-free approach to investigate the kinematics of rigid bodies with rolling contact. A new equation of angular velocity of the moving body is derived in terms of the magnitude of rolling velocity and two sets of geometric invariants belonging to the respective contact curves. This new formulation can be differentiated up to any order. Furthermore, qualitative information about trajectory planning can be deduced from this equation if the characteristics of rolling objects and the motion are taken into consideration

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 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.Comment: 17 pages, 7 figures, Accepted at Mechanism and Machine Theory Elsevie

A geometric motion planning for a spin-rolling sphere on a plane

The paper deals with motion planning for a spin-rolling sphere when the sphere follows an optimal straight path on a plane. Since the straight line constrains the sphere’s motion, controlling the sphere’s spin motion is essential to converge to a desired full 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. Finally, the feasibility of the proposed approach is verified by simulations