A performance comparison between the extended Kalman filter and different approaches of sigma-point Kalman filters

Abstract

State estimation is an essential part of several engineering scenarios. Under the hypothesis of linear systems and additive Gaussian noise, the solution of the optimal estimation problem is the Kalman Filter (KF), formulated in the 1960s by Rudolph Emil Kalman. In the case of nonlinear systems, the sub-optimal algorithm called Extended Kalman Filter (EKF) linearizes the model using the first-order truncated Taylor series expansion around the known current state estimation. Depending on the system, these approximations often introduce large errors. Jacobian-free stochastic filtering methods have been used such as the sigma-point stochastic filters, which address these approximation problems with a sampling approach instead of the Jacobian formulation. In this thesis, the performance, benefits, and handicaps of the EKF and one approach of Sigma-Point Kalman Filter (SPKF) — a variant of the Unscented Kalman Filter (UKF) — are compared when applied to a vehicle localization system with sensors, i.e. GNSS and IMU. Each sensor has its own sampling time, resulting in delayed measurements that if not properly treated, may result in considerable estimation errors. For that reason, an approach of each stochastic filter in this work should consider these delays by reformulating the stochastic filter through the delay period. For industrial purposes, having a high-performance stochastic filter can improve the quality of autonomous driving software. The stochastic filters were implemented and compared in this project considering two scenarios: with simulated data and with a real data extracted from a test vehicle. For both, the comparison method used was the Mean Squared Error (MSE) and a computational cost analysis was done. In this case, tests of consistency were also performed, like the Durbin-Watson (DW) test and the Normalized Autocorrelation test. The filters have shown a similar performance for this application, but the Unscented Kalman Filter (UKF) variant algorithm have shown a slightly better performance, mainly for the states that are not measured by the sensors. For that reason, a hybrid algorithm with both filters - EKF and UKF - were proposed. This project is being developed at ZF Friedrichshafen AG in Friedrichshafen, Germany.