Four-sigma point marginal geometric simplex estimator for gyro-less attitude dynamics

Gyro-less attitude and angular rate estimations are of great importance in small, low-cost spacecraft, where high performance gyroscopes are not available due to multiple limitations. Recent development of accurate, high-bandwidth attitude sensors such as high data-rate star trackers, makes this approach implementable. The gyro-less estimator propagates the estimated states by nonlinear attitude dynamics model, and obtains measurements from nonlinear observer. The state estimation of such nonlinear systems requires approximation of the system equations, as done by the classical extended Kalman filter (EKF), or approximation of the probability densities, as done by Unscented Kalman filter (UKF) including linear regression Kalman filters (LRKFs). In particular, LRKFs utilize a set of deterministically chosen sample points, namely the sigma points, to approximate the state and measurement probability density function. However, the sigma points required in LRKFs grow linearly with the dimension of the state space, which becomes an obstacle for their onboard implementation for overload computational complexity. In this paper, we propose a partial-state sigma-point filter that retains the benefits of LRKFs' ability to deal with nonlinearity and is free from Jacobian matrices derivation. In the proposed filter, the state is decomposed into two parts: the directly observed part of the attitude, and the indirectly observed parts of the angular rate and acceleration. The state and covariance for attitude are approximated based on the geometric simplex transform, which utilize symmetric 4-sigma points to capture the first two moments of a vector in three-dimensional Euclidean space. The remaining states and process noises are treated based on marginalized conditional sigma-point transformation. Therefore, the proposed algorithm requires only four-sigma points to realize a ten-state estimation (attitude, angular rate and angular acceleration). In addition, similar precision to classical UKF is guaranteed and a significantly lower computational complexity is achieved. The performance of the new algorithm is demonstrated by simulations.