Curvature theory for point-path and plane-envelope in spherical kinematics by new adjoint approach

Abstract

Planar kinematics has been studied systematically based on centrodes, however axodes are underutilized to set up the curvature theories in spherical and spatial kinematics. Through a spherical adjoint approach, an axode-based theoretical system of spherical kinematics is established. The spherical motion is re-described by the adjoint approach and vector equation of spherical instant center is concisely derived. The moving and fixed axodes for spherical motion are mapped onto a unit sphere to obtain spherical centrodes, whose kinematic invariants totally reflect the intrinsic property of spherical motion. Based on the spherical centrodes, the curvature theories for a point and a plane of a rigid body in spherical motion are revealed by spherical fixed point and plane conditions. The Euler-Savary analogue for point-path is presented. Tracing points with higher order curvature features are located in the moving body by means of algebraic equations. For plane-envelope, the construction parameters are obtained. The osculating conditions for plane-envelope and circular cylindrical surface or circular conical surface are given. A spherical four-bar linkage is taken as an example to demonstrate the spherical adjoint approach and the curvature theories. The research proposes systematic spherical curvature theories with the axode as logical starting-point, and sets up a bridge from the centrode-based planar kinematics to the axode-based spatial kinematics.