Estimating the State of a Dynamically Evolving System

Abstract

Estimating the state of dynamically evolving systems is a fundamental challenge across diverse fields such as robotics, navigation, economics, and environmental monitoring. This thesis explores and compares three prominent state estimation methods: the Kalman Filter (KF), the Extended Kalman Filter (EKF), and the Unscented Kalman Filter (UKF), each tailored to handle specific complexities encountered in real-world applications.

The foundational Kalman Filter is rigorously examined first, deriving its algorithm through Bayesian inference and the fusion of multiple estimates. A comparative analysis of these approaches highlights the KF’s robustness in linear systems while acknowledging limitations in nonlinear environments.

The thesis then transitions to the Extended Kalman Filter, which extends the KF to nonlinear systems by linearizing state equations. Detailed mathematical derivation and comparative studies underscore the EKF’s enhanced capabilities in handling complex dynamics, yet reveal challenges in accuracy and computational cost.

Moving further, the Unscented Kalman Filter is introduced as a non-linear state estimation method utilizing the Unscented Transform. Detailed exploration and mathematical formulation demonstrate its effectiveness in addressing uncertainties, presenting a viable alternative to both KF and EKF in scenarios where linearization proves inadequate.

To validate these methodologies, simulations are conducted using real-world data from the KITTI dataset, comprising of GPS and IMU measurements. Ground truth trajectories and non-linear variables such as yaw rates and forward velocities are utilized, showcasing each filter’s ability to estimate and track dynamic system states accurately.

Results from simulations are analyzed using performance metrics including Normalized Estimation Error Squared (NEES) and Root Mean Squared Error (RMSE), providing quantitative insights into filter performance relative to ground truth. These evaluations emphasize the strengths and limitations of each method across various application domains, supporting informed decisions on filter selection based on specific system dynamics and measurement characteristics.

In conclusion, this thesis contributes to a comprehensive analysis and comparative study of state estimation methods essential for navigating the complexities of dynamically evolving systems. By bridging theoretical advancements with practical insights, it lays a foundation for future research and application in fields requiring precise state estimation amidst dynamic change.