The Advantages of Left-Invariant Extended Kalman Filtering for Spacecraft Attitude Estimation

Abstract

Accurate attitude estimation is a critical capability for small satellite missions, where size, weight, and power constraints limit sensor quality and computational resources. The Multiplicative Extended Kalman Filter (MEKF) is the industry-standard algorithm for attitude estimation but lacks theoretical guarantees of convergence, making it vulnerable to divergence under large initialization errors and infrequent measurements. The Invariant Extended Kalman Filter (IEKF) is a variant of the Extended Kalman Filter for systems whose state space is a Lie group that has local stability guarantees unavailable to the MEKF. This thesis demonstrates that the spacecraft attitude estimation problem satisfies the conditions required to apply the IEKF framework and derives the resulting Left-Invariant Extended Kalman Filter (LIEKF) using a rate-gyroscope and star tracker operating at different sampling rates. The MEKF and LIEKF are evaluated in simulation across stable and unstable dynamic conditions under varying initial attitude errors, gyroscope bias drift rates, and star tracker measurement rates. Under nominal conditions both filters perform comparably, but under adverse conditions the MEKF is susceptible to divergence while the LIEKF maintains a reliable state estimate. Monte Carlo analysis demonstrates that the LIEKF consistently converges more reliably than the MEKF as measurement availability decreases. As the LIEKF imposes no additional computational cost over the MEKF, it is a compelling drop-in replacement for small satellite attitude estimation applications.