# Attitude Observer on SO(3) with Time-Varying Reference Directions

## TL;DR

This paper develops a Lyapunov stability analysis for an attitude observer on SO(3) that remains exponentially stable even with time-varying reference directions, supported by numerical and experimental validation.

## Contribution

It introduces an alternative Lyapunov analysis demonstrating exponential stability of the attitude observer with dynamic reference directions, extending previous fixed-reference results.

## Key findings

- The observer maintains exponential stability with changing reference directions.
- Large observer gain relative to reference change rate ensures stability.
- Experimental results validate the theoretical analysis.

## Abstract

This paper introduces an advanced Lyapunov stability analysis for an attitude observer that has been developed on the special orthogonal group. In particular, when the attitude observer is constructed based on multiple direction measurements toward known reference points, a local exponential stability has been established by linearization, under the assumption that those reference points are fixed in the inertial frame. Several modifications have been proposed to deal with reference directions changing over time. Here, we present an alternative Lyapunov analysis to show that the attitude observer still exhibits exponential stability for time-varying reference directions, under the assumption that the observer gain is sufficiently large relative to the rate of change of the reference directions. These are illustrated by a numerical example, followed by experimental results with visual marker detection in an indoor space.