The Square-Root Unscented and the Square-Root Cubature Kalman Filters on Manifolds

Estimating the state of a system by fusing sensor data is a major prerequisite in many applications. When the state is time-variant, derivatives of the Kalman filter are a popular choice for solving that task. Two variants are the square-root unscented Kalman filter (SRUKF) and the square-root cubature Kalman filter (SCKF). In contrast to the unscented Kalman filter (UKF) and the cubature Kalman filter (CKF), they do not operate on the covariance matrix but on its square root. In this work, we modify the SRUKF and the SCKF for use on manifolds. This is particularly relevant for many state estimation problems when, for example, an orientation is part of a state or a measurement. In contrast to other approaches, our solution is both generic and mathematically coherent. It has the same theoretical complexity as the UKF and CKF on manifolds, but we show that the practical implementation can be faster. Furthermore, it gains the improved numerical properties of the classical SRUKF and SCKF. We compare the SRUKF and the SCKF on manifolds to the UKF and the CKF on manifolds, using the example of odometry estimation for an autonomous car. It is demonstrated that all algorithms have the same localization performance, but our SRUKF and SCKF have lower computational demands.