The Geometry of Motion

A Multi-Volume Series on the Mathematics of Movement

Mathematical and computational foundations for rigid-body dynamics, control, biomechanics, neural control, and simulation

Preface

The Geometry of Motion is an in-development multi-volume series on the mathematics of movement. It begins with rigid-body kinematics and dynamics, then extends through nonlinear control, biomechanics, neural control, and computational simulation.

The series is meant to connect formal derivation with interpretation. Each volume focuses on a coherent technical theme, but the notation and examples are intended to carry across volumes so that concepts can be revisited from several angles.

NoteStatus Note

This series is still under active development. Some volumes are more mature than others, and the PDF and HTML editions are not always identical in figure support or layout. Volumes 0–II are rendered as HTML chapters here. Volumes III–V currently exist only as a PDF edition — each is linked directly from its entry below so the preface does not point to dead ends.

Volumes & Table of Contents

Volume 0: Mathematical Foundations

• Volume 0 Overview
• Chapter 1: A Primer on Linear Algebra
• Chapter 2: The Transition to State Space
• Chapter 3: Configuration Space and Degrees of Freedom
• Chapter 4: Rotations, Quaternions, and SO(3)
• Chapter 5: Twists, Wrenches, and the Screw Axis
• Chapter 6: Exponential Coordinates and Matrix Logarithms
• Chapter 7: Recursive Algorithms
• Chapter 8: Spatial Vector Algebra and Inertia
• Chapter 9: The Product of Exponentials Formula
• Chapter 10: The Articulated Body Algorithm
• Chapter 11: Lagrangian Mechanics
• Chapter 12: Machine Learning and Neural Networks

Volume I: Tangent-Space Methods for Nonlinear Control and Biomechanics

• Volume I Overview
• Chapter 1: Foundations, Tangent Spaces, and Exactness
• Chapter 2: Variational Dynamics and the Moving Frame
• Chapter 3: Superposition from Linear to Affine
• Chapter 3b: Induced Acceleration in Biomechanics
• Chapter 4: Contraction Theory and Metric Certificates
• Chapter 5: Local Optimal Control
• Chapter 6: The Stability-Optimality Duality
• Chapter 7: Forward Dynamics, Counterfactuals, and Drift
• Chapter 8: Mechanical Applications
• Chapter 9: Parallel Mechanisms and Constrained Dynamics

Volume II: Control Is Motion

• Volume II Overview

Future Volumes (PDF Editions)

• Volume III: Biomechanics (PDF edition) (PDF only; not yet rendered as HTML chapters.)
• Volume IV: Neural Control (PDF edition) (PDF only; not yet rendered as HTML chapters.)
• Volume V: Simulation (PDF edition) (PDF only; not yet rendered as HTML chapters.)

How to Read the Series

Start with Volume 0 if you want the mathematical background.

Start with Volume I if your main interest is tangent-space methods, nonlinear control, and the site’s control-theoretic claims.

Use Volume II and later volumes when you want the application-facing material on movement, biomechanics, neural control, and simulation.

Note on Editions

Some figures, especially diagram-heavy PDF material, render better in the PDF edition than in the HTML edition. Where the HTML edition uses simplified substitutes, the underlying text remains the primary source of the argument.