Nonlinear estimation with sparse temporal measurements

Abstract

Nonlinear estimators based on the Kalman filter, the extended Kalman filter (EKF) and unscented Kalman filter (UKF) are commonly used in practical application. The Kalman filter is an optimal estimator for linear systems; the EKF and UKF are sub-optimal approximations of the Kalman filter. The EKF uses a first-order Taylor series approximation to linearize nonlinear models; the UKF uses an approximation of the states' joint probability distribution. Long measurement intervals exacerbate approximation error in each approach, particularly in covariance estimation. EKF and UKF performance under varied measurement frequency is studied through two problems, a single dimension falling body and simple pendulum. The EKF is shown more sensitive to measurement frequency than the UKF in the falling body problem. However, both estimators display insensitivity to measurement frequency in the simple pendulum problem. The literature's lack of consensus as to whether the EKF or UKF is the superior nonlinear estimator may be explained through covariance approximation error. Tools are presented to analyze EKF and UKF measurement frequency sensitivity. Covariance is propagated forward using the approximations of the EKF and UKF. Each propagated covariance is compared for similarity with a Monte Carlo propagation. The similarity of the covariance matrices is shown to predict filter performance. Portions of the state trajectory susceptible to EKF divergence are found using the Frobenius norm of the Jacobian matrix, limiting the need to consider covariance propagation along the entire state trajectory. Long measurement intervals also reveal a commonly overlooked challenge in UKF application: sigma point selection methods may produce sigma point vec-tors that violate physical state constraints. Although the UKF can function under this condition over short measurement intervals, unexpected failure may occur without consideration of physical constraints. A novel constrained UKF, using the scaled unscented transform, is proposed to address this issue.