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The modified Rodrigues parameters (MRP) are a less explored attituderepresentation characterized by two numerically different triplets that areinfinite for different principal rotations. By judiciously switching betweenthese triplets, the MRP yield a minimal non-singular attitude description withadvantageous inherent properties that an increasing number of control-relatedapplications explore. The space of MRP results from the Alexandroffcompactification of $\mathbb{R}^3$, $\mathbb{\bar{R}}^3$, which, similar to thespace of unit quaternions, is a double cover of the rotation group$\mathrm{SO(3)}$. In this direction, this technical report highlights someinstrumental properties of the corresponding covering map and proposes a hybriddynamic path-lifting algorithm to uniquely and consistently extract the MRPvector from a given rotation matrix. The resulting output evolves inside athree-dimensional unit sphere enclosed by a hysteresis region. With this hybridalgorithm in place, a theoretical basis for equivalence of stability betweenresults in the base space $\mathrm{SO}(3)$ and the covering space$\mathbb{\bar{R}}^3$ is developed. To this end, two closed-loop hybrid systemsare formulated: one encompassing the lifted attitude dynamics and the otherencapsulating the attitude dynamics described in the base space and the hybridpath-lifting algorithm. As the main contribution, this report demonstrates thatan asymptotic stability result obtained with a feedback controller designed inthe covering space for the former system translates into an equivalentasymptotic stability result in the base space when the same feedback controlleris applied to the latter hybrid system.

Path-lifting Mechanism for MRP and Equivalence of Stability between results in $\bar{\mathbb{R}}^3$ and $\mathrm{SO}(3)$