Analysis of the Stochastic Stability and Asymptotically Stationary Statistics for a Class of Nonlinear Attitude Estimation Algorithms

Abstract

Attitude estimation algorithms are critical components of satellite control systems, aircraft autopilots, and other applications. Attitude estimation systems perform their task by fusing attitude and gyroscope measurements; however, such measurements are typically corrupted by random noise and gyroscopes may have significant bias. Variations of the extended Kalman filter are commonly used, but this technique relies on instantaneous linearization of the underlying nonlinear dynamics and global stability cannot be guaranteed. Nonlinear attitude observers with guaranteed global stability have been derived and experimentally demonstrated, but only for the deterministic setting where no stochastic effects are present.

The first part of this thesis extends a deterministic nonlinear attitude estimator by introducing additional dynamics that allow learning variations of gyro bias as a function of operating temperature, a common source of bias variation in rate gyro readings. The remainder of the thesis formally addresses the problem of stochastic stability and asymptotic performance for this family of estimators when the measurements contain random noise. Analysis tools from stochastic differential equation theory and stochastic Lyapunov analysis are used together to demonstrate convergence of the filter states to a stationary distribution, and to bound the associated steady-state statistics as a function of filter gains and sensor parameters.

In many cases these bounds are conservative, but solutions have been found for the associated stationary Fokker-Planck PDEs for two cases. When only the gyro measurement contains noise, the attitude estimation errors are shown to converge to a bipolar Bingham distribution. When the gyro measurement is further assumed to have constant bias, the estimation errors are shown to converge to a joint bipolar Bingham and multivariate Gaussian distribution. Knowledge of the stationary distributions allow for exact computation of steady-state statistics. Further, the analysis suggests a method for modeling a continuous quaternion noise process with specified statistics on SO(3); this model is used for analyzing estimator performance when both the gyro and the attitude measurements contain noise. Bounds and exact predictions for the different noise models are validated using a high fidelity numerical integration method for nonlinear stochastic differential equations.