Stochastic H_∞ Filtering of the Attitude Quaternion

Several stochastic H∞ filters for estimating the attitude of a rigid body from line-of-sight measurements and rate gyro readings are developed. The measurements are corrupted by white noise with unknown variances. Our approach consists of estimating the quaternion while attenuating the transmission gain from the unknown variances and initial errors to the current estimation error. The time-varying H∞ gain is computed by solving algebraic and differential linear matrix inequalities for a given transmission threshold, which is iteratively lowered until feasibility fails. Thanks to the bilinear structure of the quaternion state-space model, the algorithm parameters are independent of the state. The case of a gyro drift is addressed, too. Extensive Monte-Carlo simulations show that the proposed stochastic H∞ quaternion filters perform well for a wide range of noise variances. The actual attenuation, which improves with the noise variance and is worst in the noise-free case, is better than the guaranteed attenuation by one order of magnitude. The proposed stochastic H∞ filter produces smaller biases than nonlinear Kalman or unscented filters and similar standard deviations at large noise levels. An essential advantage of this H∞ filter is that the gains are independent of the quaternion, which makes it insensitive to modeling errors. This desired feature is illustrated by comparing its performances against those of unmatched nonlinear optimal filters. When provided with too high or too low noise variances, the multiplicative Kalman filter and the unscented quaternion filter are outperformed by the H∞ filter, which essentially delivers identical error magnitudes.