Tilt Rotations and the Tilt Phase Space

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.