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

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.

💡 Research Summary

The paper addresses the critical problem of estimating the relative attitude and angular velocity of an uncooperative target spacecraft during rendezvous and docking operations, while simultaneously compensating for gyroscope bias on the chaser spacecraft. Traditional approaches based on extended Kalman filters (EKF) or dual‑quaternion filters often struggle with non‑linearity, bias‑induced unobservability, and the need to enforce geometric constraints in an ad‑hoc manner. To overcome these limitations, the authors propose a cascade of two Equivariant Filters (EqFs), each exploiting the symmetry properties of the underlying dynamics on the Lie group SE(3).

The first filter estimates the chaser’s attitude (R_C) and the constant gyroscope bias (b) using low‑rate star‑tracker measurements. The state space is the product manifold (SO(3)\times\mathbb{R}^3). By defining right‑action maps for state, input, and output, the system is shown to be equivariant with respect to SE(3). A group‑level lift (\Lambda^{(1)}) yields linear error dynamics on the Lie algebra, and a Riccati equation provides a time‑varying covariance matrix (\Sigma^{(1)}). The filter correction term (\Delta^{(1)}) is derived from the linearized output residual and the gain matrices, ensuring that the error converges exponentially. The bias estimate (\hat b) is then subtracted from the raw gyroscope reading to produce an unbiased angular‑velocity input for the second stage.

The second filter estimates the relative attitude (R) (chaser relative to target) and the target’s angular velocity (\omega) using two non‑collinear vectors measured in the chaser frame (e.g., visual features on the target). The same manifold structure and symmetry group are used, with a lift (\Lambda^{(2)}) that incorporates the unbiased angular‑velocity input and two virtual inputs set to zero. The error system is again linearized, and a Riccati‑based correction (\Delta^{(2)}) is computed. Because the input to this filter has already been bias‑corrected, the relative‑attitude and angular‑velocity estimates become observable and converge rapidly.

Stability analysis is performed for the full cascade. By constructing Lyapunov functions for each stage and showing that the linearized error matrices (\mathring A^{(i)}_t) are Hurwitz, the authors prove exponential convergence of both filters. Moreover, they demonstrate that the convergence of the first filter guarantees that the bias error driving the second filter diminishes, creating a feedback loop that reinforces overall stability.

Simulation results use realistic sensor models: a star tracker with low update rate and a camera providing noisy vector observations. Initial attitude errors of several tens of degrees and bias errors of up to 0.5 rad/s are introduced. The cascade converges within a few seconds, achieving attitude errors below 0.2 deg and angular‑velocity errors below 0.01 rad/s, outperforming a comparable EKF implementation that diverges when bias is present. The authors also show robustness to measurement noise and to variations in the assumed constant target angular velocity.

Key contributions of the work include: (1) a novel two‑stage EqF architecture that separates bias/attitude estimation from relative‑pose/velocity estimation; (2) a simplified symmetry group choice (SE(3) rather than semi‑direct products) that reduces derivation complexity; (3) rigorous theoretical guarantees of global exponential stability for the cascade; and (4) demonstration of superior performance over traditional EKF methods in low‑rate, biased‑sensor scenarios. The proposed cascade is computationally lightweight, making it suitable for small‑satellite platforms with limited processing capability, and it provides a solid foundation for future real‑time on‑orbit servicing, debris removal, and autonomous docking missions.