Reformulation of the second-order filter for nonlinear systems with application in gyro calibration

In this paper, a novel method is developed to estimate the first and second central statistical moments of a nonlinear function using a Taylor series expansion. The method is derived analytically, and a matrix representation is introduced to simplify the computation of the second-order terms in the Taylor series. The mean of the nonlinear function is approximated exactly to the second order, while the covariance follows this approximation under the assumption that third order statistical moments are neglected and fourth-order statistical moments are approximated using Isserlis’ theorem. The filter operates within a linear framework, similar to many other filters, and features a predictive and corrective structure. It is classified as a nonlinear and non-Gaussian filter with a symmetric Probability Density Function (PDF). The performance of the Second-Order Filter (SOF) is compared to that of the classical Unscented Kalman Filter (UKF). Monte Carlo simulations demonstrate that the SOF effectively reduces the three-sigma bounds, with estimations rarely falling outside these bounds. In contrast, the UKF shows poorer performance than the SOF, and its estimations occasionally fall outside the three-sigma bounds. However, the computational burden of the SOF is greater than that of the UKF. Finally, the application of the filter is demonstrated in the laboratory calibration of gyro sensors, involving 12 unknown parameters as additional states, along with four quaternions and vector measurements.