Equivariant Filter for Relative Attitude and Target's Angular Velocity Estimation

TL;DR

The paper tackles the problem of jointly estimating the relative attitude between a chaser and a target and the target's angular velocity using two fixed target-frame vectors. It formulates an Equivariant Filter on the Lie group $SE(3)$ by extending the system with virtual inputs, defining equivariant state/actions and a lift, and deriving a Riccati-based correction scheme with provable convergence. Observability is established via augmented measurements, and the filter is shown to converge faster and be more robust than a traditional EKF in simulations. Experimental validation uses fiducial markers with both a conventional camera and an event camera, demonstrating accurate estimates at high and low measurement rates and highlighting the method's practicality for real-world rendezvous scenarios.

Abstract

Accurate estimation of the relative attitude and angular velocity between two rigid bodies is fundamental in aerospace applications such as spacecraft rendezvous and docking. In these scenarios, a chaser vehicle must determine the orientation and angular velocity of a target object using onboard sensors. This work addresses the challenge of designing an Equivariant Filter (EqF) that can reliably estimate both the relative attitude and the target angular velocity using noisy observations of two known, non-collinear vectors fixed in the target frame. To derive the EqF, a symmetry for the system is proposed and an equivariant lift onto the symmetry group is calculated. Observability and convergence properties are analyzed. Simulations demonstrate the filter's performance, with Monte Carlo runs yielding statistically significant results. The impact of low-rate measurements is also examined and a strategy to mitigate this effect is proposed. Experimental results, using fiducial markers and both conventional and event cameras for measurement acquisition, further validate the approach, confirming its effectiveness in a realistic setting.

Figures (16)

• Figure 1: Chaser and target scenario with relevant reference frames.
• Figure 2: Chaser and target attitudes, expressed in Euler angles.
• Figure 3: True relative attitude and target's angular velocity ${(\mathbf{R},\boldsymbol{\omega})}$ (dashed) and estimated ${(\hat{\mathbf{R}},\hat{\boldsymbol{\omega}})}$ (solid).
• Figure 4: Filter correction terms.
• Figure 5: Norm of the error on the group (with the identity subtracted).

Theorems & Definitions (1)

• proof