Algorithmic and Computational Foundations of Robotics and Artificial Intelligence: The Power of the Covariance Matrix

Status: Alpha

Introduction

• Introduction and Motivation

Kalman-Filter for State-Space-Estimation

Principle and 1D-Case

• Bayes Filter
  • Foundations
    • Probabilities and Bayes Rule
    • Discrete Convolution
  • Bayes Filter
• Principle of the Kalman Filter
  • Foundations
    • Probability Densities and Normal Distribution
    • Random Walk and Central Limit Theorem
    • Expectation
    • Estimators
    • Continuous Convolution
  • Principle of the Kalman Filter (1D case)

The Multi-Dimensional Kalman Filter for 1D-Kinematics

• 1D-Kinematics of a Point Mass
• State Space Model: From 1D-Kinematics to Matrices
• Process Noise: Modeling Model Imperfection
  • Foundation: Multivariate Gaussian Distribution, Covariance Matrix and Correlation
  • Process Noise: Modeling Model Imperfection
• Observation Model
• Kalman Filter Summary

2D-Kinematics and the Extended Kalman Filter (EFK)

• 2D-Kinematics
• Rotations in 2D: Representing orientation and angular velocity
• The Extended Kalman Filter: Handling the non-linearity of 2D-rotations
• EKF for 2D Navigation
• Observability
  • Foundation:
    • Column Space, Null Space, Rank of a Matrix
  • Observability

Advanced: 3D-Kinematics and the Error State Kalman Filter

• Fundamentals: Quaternion kinematics for the error-state Kalman filter (external Link)
• Example:
  • Generation of syntetic data
  • Implementation of a ESKF for 3D-Kinematics

Principle Component Analysis (PCA)

• Principle Component Analysis
  • Foundations
    • Tensors and Transformation
    • Eigendecomposition, Eigenvalues and Eigenvectors
  • Principal Component Analysis
• Application: PCA for Image Compression

Evolution Strategies for Optimization and Control

• Introduction to Evolution strategy (ES)
• Covariance Matrix Adaptation Evolution Strategy (CMA-ES)
  • Foundations
    • Vector-form of the empirical Covariance Matrix
    • Exponential Smoothing
    • Hessian Matrix
    • Conjugate Directions
  • Covariance Matrix Adaptation Evolution Strategy (CMA-ES)