Log-Euclidean metric based covariance propagation on SPD manifold for continuous-discrete extended Kalman filtering

Abstract

For nonlinear systems with continuous dynamic and discrete measurements, a Log-Euclidean metric (LEM) based novel scheme is proposed to refine the covariance integration steps of continuous-discrete Extended Kalman filter (CDEKF). In CDEKF, the covariance differential equation is usually integrated with regular Euclidean matrix operations, which actually ignores the Riemannian structure of underlying space and poses a limit on the further improvement of estimation accuracy. To overcome this drawback, this work proposes to define the covariance variable on the manifold of symmetric positive definite (SPD) matrices and propagate it using the Log-Euclidean metric. To embed the LEM based novel propagation scheme, the manifold integration of the covariance for LEMCDEKF is proposed together with the details of efficient realization, which can integrate the covariance on SPD manifold and avoid the drawback of Euclidean scheme. Numerical simulations certify the new method's superior accuracy than conventional methods.

Keywords: Continuous–discrete systems; Covariance propagation; Extended Kalman filtering; Log-Euclidean metric; Symmetric positive definite.