Derivation Filtering and Smoothing Approaches

Last updated: 8. March 2024

This Derivation is divided in the following sections:

• Part 1: Kalman Filter I Discusses the basic principles of the Kalman Filter, its mathematical foundations, and its application in linear systems. References error correction and state prediction.
• Part 2: Kalman Filter II Expands on the initial Kalman Filter by introducing nonlinear systems handling via the Extended Kalman Filter (EKF). Details the linearization process required for EKF.
• Part 3: RTS Smoother Describes the Rauch-Tung-Striebel (RTS) smoother as an enhancement to the Kalman Filter for smoothing estimates. Outlines the backward pass algorithm used for state smoothing.
• Part 4: Extended Kalman Filter Focuses on the mathematical derivation specific to the Extended Kalman Filter, including the Jacobian matrices and error state correction. Serves as a basis for the next part.
• Part 5: 6D EKF-RTS (Orientation Only) Applies the Extended Kalman Filter methodology to a six-axis sensor fusion to estimate an orientation, integrating additional dynamics and control inputs.
• Part 6: 6D EKF-RTS Extends the previous derivation to include the estimation of sensitivity and position states. Details the state transition matrix and the error state correction.
• Part 7: Initial Parameter Estimation Introduces the considerations of an optimization algorithm for initial parameter estimation.
• Part 8: 6D EKF-RTS Explores different approaches to mitigate the issues of wrong estimations of biases and sensitivities in the 6D EKF-RTS.
• Part 9: 6D EKF-RTS Introduces a new architecture of an Extended Kalman Filter for the 6D sensor fusion, moving the measurements of the acceleration and angular velocity to the control input.