Comparative Analysis of Filtering Algorithms: EKF, UKF, PF, PFEKF, and PFUKF on a Common Problem

In this article, we explore how different filtering algorithms solve the same estimation problem and analyze their comparative differences. We examine five distinct filters: the Extended Kalman Filter (EKF) which linearizes nonlinear systems using Jacobian matrices, the Unscented Kalman Filter (UKF) that uses sigma points to capture nonlinear transformations more accurately, the Particle Filter (PF) employing sequential Monte Carlo methods with importance sampling, and two hybrid approaches - PF with EKF proposal (PFEKF) and PF with UKF proposal (PFUKF) that combine proposal distributions from traditional filters with particle filtering frameworks. First, we delve into the underlying mechanisms, advantages, and limitations of each filtering approach, including their mathematical formulations and computational requirements. We then demonstrate practical implementation of these filters on a common state estimation problem, comparing their performance metrics such as estimation accuracy, computational efficiency, and robustness to nonlinearities and non-Gaussian noise. The implementation typically involves defining system dynamics (state transition and measurement functions), initializing covariance matrices, and configuring resampling strategies for particle-based methods. Finally, we synthesize the findings to provide guidance on selecting appropriate filtering techniques based on problem characteristics, highlighting scenarios where each algorithm excels and discussing their respective limitations to help practitioners choose optimal solutions for their specific applications.