# Global Attitude Estimation using Uncertainty Ellipsoids

**Authors:** Taeyoung Lee, Amit K. Sanyal, Melvin Leok, N. Harris McClamroch  
**arXiv:** [math/0512163](https://arxiv.org/abs/math/0512163)  
**Date:** 2005-12-07  
**Summary page:** https://www.summarizepaper.com/en/paper/math--0512163v1/

## Key points

- The paper explores the problem of attitude estimation for mechanical systems.
- Attitude estimation is a crucial step in controlling the orientation or attitude of such systems.
- Current algorithms use coordinate representations, which have associated problems.
- This paper proposes a deterministic attitude estimation algorithm that globally minimizes the attitude estimation error.
- The algorithm assumes that the initial attitude, initial angular velocity and measurement noise lie within given ellipsoidal bounds.
- An uncertainty ellipsoid that bounds the attitude and angular velocity of the rigid body is obtained from these assumptions.
- A Lie group variational integrator and its linearization are used to propagate point estimates and uncertainty ellipsoids over time.
- This approach ensures that the resulting attitude estimation is optimal in terms of minimizing both the attitude estimation error and the size of the uncertainty ellipsoid.

## Summary

The paper "Global Attitude Estimation using Uncertainty Ellipsoids" by Taeyoung Lee, Amit K. Sanyal, Melvin Leok, and N. Harris McClamroch explores the problem of attitude estimation for mechanical systems. Attitude estimation is a crucial step in controlling the orientation or attitude of such systems. The current algorithms for attitude estimation use coordinate representations for the group of rigid body orientations. However, all coordinate representations have associated problems; minimal coordinate representations exhibit kinematic singularities for large rotations while non-minimal coordinates like quaternions require satisfaction of extra constraints. This paper proposes a deterministic attitude estimation algorithm that globally minimizes the attitude estimation error. The algorithm assumes that the initial attitude, initial angular velocity and measurement noise lie within given ellipsoidal bounds. An uncertainty ellipsoid that bounds the attitude and angular velocity of the rigid body is obtained from these assumptions. The center of this uncertainty ellipsoid provides point estimates while its size measures the accuracy of these estimates. To propagate these point estimates and uncertainty ellipsoids over time, a Lie group variational integrator and its linearization are used. This approach ensures that the resulting attitude estimation is optimal in terms of minimizing both the attitude estimation error and the size of the uncertainty ellipsoid. Overall, this paper presents an effective solution to the problem of global attitude estimation using uncertainty ellipsoids for mechanical systems with bounded measurement errors.