# A geodesic feedback law to decouple the full and reduced attitude

## TL;DR

This paper introduces a geodesic feedback law that decouples the stabilization of full and reduced attitudes, enabling smooth maneuvers on SO(n) and the n-sphere with adjustable convergence trade-offs.

## Contribution

A novel feedback control method that simultaneously stabilizes full and reduced attitudes along geodesic paths, with explicit transient dynamics and stability region analysis.

## Key findings

- Exact solutions for attitude dynamics on SO(3)
- Characterization of the system's asymptotic behavior
- Trade-off parameters for convergence speed control

## Abstract

This paper presents a novel approach to the problem of almost global attitude stabilization. The reduced attitude is steered along a geodesic path on the n-sphere. Meanwhile, the full attitude is stabilized on SO(n). This action, essentially two maneuvers in sequel, is fused into one smooth motion. Our algorithm is useful in applications where stabilization of the reduced attitude takes precedence over stabilization of the full attitude. A two parameter feedback gain affords further trade-offs between the full and reduced attitude convergence speed. The closed loop kinematics on SO(3) are solved for the states as functions of time and the initial conditions, providing precise knowledge of the transient dynamics. The exact solutions also help us to characterize the asymptotic behavior of the system such as establishing the region of attraction by straightforward evaluation of limits. The geometric flavor of these ideas is illustrated by a numerical example.