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.

Bipedal Walking with Corrective Actions in the Tilt Phase Space

“…Note that although it is not generally acceptable to just add 3D rotations, the special properties of the tilt phase space allow us to do just that in a meaningful, unambiguous, self-consistent and mathematically supported way. In fact, the tilt phase space turns tilt rotations into a well-defined vector space over R, explaining why the scaling and addition of tilt phases that is used in this paper is actually mathematically valid and geometrically meaningful [10].…”

“…One significant difference between [2] and this paper is the full 3D treatment given to the corrective actions, made possible in part by the use of the tilt phase space [10] instead of fused angles [11]. While fused angles work very well for separate treatments of the sagittal and lateral planes, the tilt phase space has advantages for concurrent treatments, in particular in relation to magnitude axisymmetry [10]. This is important to ensure that feedback magnitudes are the same scale no matter what continuous direction the robot is tilted in.…”

“…More details on all of the used rotation representations can be found in the respective papers [10] [11] and code releases. 1…”

“…Once the separate P and D components have been calculated, they are combined using tilt vector addition [10] and elliptically soft-coerced to obtain the final activations P a and P s (see Section IV-B). Note that although it is not generally acceptable to just add 3D rotations, the special properties of the tilt phase space allow us to do just that in a meaningful, unambiguous, self-consistent and mathematically supported way.…”