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

Introduction¶

• Introduction and Motivation

Kalman-Filter for State-Space-Estimation¶

Focus on IMU Data Processing (IMU: Intertial Measurement Unit)

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
• 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)¶

• Foundation:
  • Column Space, Null Space, Rank of a Matrix
  • Determinant of a square matrix
• 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

Advanced: 3D-Kinematics and the Error State Kalman Filter (ESKF)¶

Exhaustive revision of concepts and formulas related to quaternions and rotations in 3D space, and their proper use in estimation engines such as the error-state Kalman filter:

• J. Sola, Quaternion kinematics for the error-state Kalman filter (external link)

Principle Component Analysis (PCA)¶

• Foundations
  • Tensors and Transformation
  • Vector-form of the empirical Covariance Matrix
  • Eigendecomposition, Eigenvalues and Eigenvectors
• Principal Component Analysis
• Application: PCA for Image Compression

Evolution Strategies for Optimization and Control¶

Status: Alpha

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