On Quaternion Modeling and Kalman Filtering from Vector Observations

Abstract

This paper presents novel results on quaternion state-space modeling and estimation from rate gyro measurements and vector observations. The spectral properties of the Lyapunov differential equation for the quaternion second-order moment are investigated and expressions for its solution are developed. The properties of a four-dimensional rank-degenerate measurement model of the quaternion are analyzed. Four different approaches are suggested in order to reduce the dimension of the measurement equation, hinging on information optimization or on the spectral properties of the measurement matrix. While the measurement dimension’s reduction saves computations, it does not impair the exact statistical treatment of state-dependent process and measurement noises. As a result, the proposed filters are optimal and statistically consistent without tuning. The performance of the maximum information rate quaternion filter is verified via extensive Monte Carlo simulations. The test cases of spinning and rapidly tumbling rotations of a small satellite on a low Earth orbit are simulated. The filters process measurements of the sun vector, of the Earth magnetic field, and of rate gyros. The novel filters converge where a standard extended Kalman filter fails to do so under similar conditions.