Tilt Rotations and the Tilt Phase Space

TL;DR

This paper formalizes the concept of tilt rotations in 3D space, introduces the tilt phase space for mathematical analysis, and explores their properties and applications in balancing and contact analysis.

Contribution

It introduces the rigorous concept of tilt rotations, develops their parameterizations, and demonstrates their utility in representing and analyzing 3D rotations in a vector space.

Findings

Tilt rotations can be represented in the tilt phase space as a vector space.

The tilt phase space allows for meaningful addition and interpolation of rotations.

Properties like rotational velocities and composition are effectively analyzed using this framework.

Abstract

In this paper, the intuitive idea of tilt is formalised into the rigorous concept of tilt rotations. This is motivated by the high relevance that pure tilt rotations have in the analysis of balancing bodies in 3D, and their applicability to the analysis of certain types of contacts. The notion of a 'tilt rotation' is first precisely defined, before multiple parameterisations thereof are presented for mathematical analysis. It is demonstrated how such rotations can be represented in the so-called tilt phase space, which as a vector space allows for a meaningful definition of commutative addition. The properties of both tilt rotations and the tilt phase space are also extensively explored, including in the areas of spherical linear interpolation, rotational velocities, rotation composition and rotation decomposition.