Equivariant Filter Cascade for Relative Attitude, Target's Angular Velocity, and Gyroscope Bias Estimation

TL;DR

This paper tackles rendezvous with an uncooperative target by introducing a cascaded, two-stage Equivariant Filter (EqF) approach on SE(3) to jointly estimate the chaser's attitude and gyroscope bias, and the relative attitude plus the target's angular velocity. The first stage uses star-tracker data to correct gyroscope bias, providing unbiased angular velocity for the second stage, which relies on relative-vector observations from the target. The two filters are analyzed for stability and shown to be locally uniformly asymptotically stable, with simulations demonstrating robust performance at low sensor update rates and highlighting the critical role of bias estimation. The work also discusses the impact of measurement rate and sensor choices (e.g., potential use of event cameras) on estimation accuracy, offering a practical pathway for on-orbit servicing and debris removal missions.

Abstract

Rendezvous and docking between a chaser spacecraft and an uncooperative target, such as an inoperative satellite, require synchronization between the chaser spacecraft and the target. In these scenarios, the chaser must estimate the relative attitude and angular velocity of the target using onboard sensors, in the presence of gyroscope bias. In this work, we propose a cascade of Equivariant Filters (EqF) to address this problem. The first stage of the cascade estimates the chaser's attitude and the bias, using measurements from a star tracker, while the second stage of the cascade estimates the relative attitude and the target's angular velocity, using observations of two known, non-collinear vectors fixed in the target frame. The stability of the EqF cascade is theoretically analyzed and simulation results demonstrate the filter cascade's performance.

Figures (5)

• Figure 1: Equivariant Filter Cascade
• Figure 2: First Stage EqF: True (dashed lines) and estimated (solid lines) chaser's attitude and gyroscope bias.
• Figure 3: Second Stage EqF: true (dashed lines) and estimated (solid lines) relative attitude and target's angular velocity.
• Figure 4: Target's angular velocity, $\boldsymbol{\omega}$, with biased filter input.
• Figure 5: Norm of the errors on the group ${\mathrm{G}}$, in log-scale, for different measurement rates. Solid lines represent the scenario with star tracker measurements at 1Hz and vector measurements at 10Hz, while dashed lines represent the scenario with measurements at 100Hz.