Polynomial Updates for the Unscented Kalman Filter

Most nonlinear filters used in spacecraft navigation are based on a linear approximation of the optimal minimum mean-square-error estimator. The unscented Kalman filter (UKF) handles nonlinear dynamics through a sigma-point transform, but the resulting state estimate remains a linear function of the measurement. This paper proposes a polynomial approximation of the Bayesian update, leading to an improved version of the classical UKF that retains the structure of the standard UKF but enriches the measurement update with higher-order (polynomial) terms. To compute the moments required by this polynomial estimator, we employ a conjugate unscented transformation (CUT), which accurately captures higher-order central moments of the state and measurement. Numerical examples, including Clohessy–Wiltshire and circular restricted three-body dynamics with non-Gaussian measurement noise, illustrate that the resulting polynomial-CUT filters improve both state estimation accuracy and covariance consistency when compared with their linear counterparts.