A Step by Step Mathematical Derivation and Tutorial on Kalman Filters

TL;DR

This paper provides a detailed, step-by-step mathematical derivation of the Kalman filter using orthogonal projection and Bayesian methods, serving as a comprehensive tutorial for understanding its foundations.

Contribution

It offers a clear, detailed tutorial with proofs on deriving the Kalman filter through two different approaches, enhancing educational resources.

Findings

Two derivation methods explained in detail

Mathematical proofs provided for each approach

Enhanced understanding of Kalman filter foundations

Abstract

We present a step by step mathematical derivation of the Kalman filter using two different approaches. First, we consider the orthogonal projection method by means of vector-space optimization. Second, we derive the Kalman filter using Bayesian optimal filtering. We provide detailed proofs for both methods and each equation is expanded in detail.