Covariance Square Root Second-Order Mapping

TL;DR

This paper introduces the first square root method for second-order covariance mapping in recursive state estimation, improving accuracy and efficiency in nonlinear settings.

Contribution

It presents a novel square root computation technique for second-order covariance mappings, extending existing first-order methods to higher-order moments.

Findings

The new method is more accurate than traditional approaches.

It requires fewer floating point operations.

Numerical experiments demonstrate improved performance.

Abstract

In recursive state estimation, numerical error can play a major role in an algorithm's overall performance and reliability. Roundoff errors due to finite precision arithmetic can violate theoretical guarantees, leading to asymmetric and non-positive-semidefinite covariance matrices. In algorithms employing first-order covariance mappings, such as the extended Kalman filter, these issues have been mitigated by employing square root factorizations of the covariance matrix. However, existing techniques do not directly extend to higher-order moment mappings, which show great value in highly nonlinear settings. This paper presents the first known square root computation of the second-order covariance mapping. The square root computation is not only more accurate, as is shown in two distinct numerical experiments, but generally requires fewer floating point operations compared to the full covariance matrix computation.