Modeling Higher Pairs in Spherical Mechanisms

Abstract

The kinematics of spherical mechanisms (SMs) shares many features with that of planar mechanisms (PMs). Specifically, both types of mechanisms can be studied by projecting their geometry onto a reference surface: a plane (the motion plane) for PMs and a sphere (the reference sphere) for SMs. Despite this correspondence, spherical geometry is inherently more complex than planar geometry. Consequently, in many cases, the use of general spatial notations for modeling SMs is preferred over leveraging their specific geometric properties. The direct modeling of higher pairs is one such case. Unlike lower pairs, modeling higher pairs requires adding auxiliary equations to the mechanism's standard loop equations to build the complete system of constraint equations. Such equations can be written in a standard way for PMs, but they seem difficult to write without using spatial notations for SMs. This paper presents a general notation that makes the direct modeling of higher pairs possible for SMs without using spatial kinematics tools, and the kinematic analyses of SMs easy to implement by directly using their projections onto the reference sphere.