Novel Quaternion Stochastic Modeling and Filtering

A novel continuous-time stochastic differential equation (SDE) for spacecraft attitude quaternion kinematics with state-multiplicative noise and a novel continuous-time exact optimal quaternion filter are developed in the framework of Itô (mean-square) calculus. The quaternion Itô SDE contains dissipative terms that ensures the mean-square stability of the process. Closed-form deterministic propagation equations are developed for the second-order moment of the quaternion. The quaternion filter produces the linear minimum-variance unbiased estimate of the quaternion from continuous observations with additive white noise. The filter gain computations include coupled Riccati equations of the estimation error matrix and of the quaternion second-order moment. These computations are not estimate-dependent and can therefore be performed off-line. The special case of gyro error white noise with independent identically distributed components is considered. The case of correlated components can be addressed straightforwardly. Additional biases and varying drifts can easily be handled via state augmentation. The gain computation would however be estimate-dependent and would need to be performed on-line.