The Kinematic Image of RR, PR, and RP Dyads

TL;DR

This paper characterizes the geometric constraints of dyads with revolute and prismatic joints using projective dual quaternion models, introducing a new invariant to distinguish different joint configurations.

Contribution

It provides necessary and sufficient conditions for admissible transformations and introduces the fiber projectivity invariant for classifying dyad constraint varieties.

Findings

Constraint variety of revolute dyad is a ruled quadric with a null quadrilateral.

Replacing a revolute with a prismatic joint alters the null quadrilateral into complex and real lines.

The fiber projectivity invariant effectively characterizes different dyad configurations.

Abstract

We provide necessary and sufficient conditions for admissible transformations in the projectivised dual quaternion model of rigid body displacements and we characterise constraint varieties of dyads with revolute and prismatic joints in this model. Projective transformations induced by coordinate changes in moving and/or fixed frame fix the quadrics of a pencil and preserve the two families of rulings of an exceptional three-dimensional quadric. The constraint variety of a dyad with two revolute joints is a regular ruled quadric in a three-space that contains a "null quadrilateral". If a revolute joint is replaced by a prismatic joint, this quadrilateral collapses into a pair of conjugate complex null lines and a real line but these properties are not sufficient to characterise such dyads. We provide a complete characterisation by introducing a new invariant, the "fiber projectivity", and we present examples that demonstrate its potential to explain hitherto not sufficiently well understood phenomena.