Volume I: Tangent-Space Methods for Nonlinear Control
Volume I is the active textbook foundation for AffineDrift control theory. Its canonical LaTeX source lives under articles/The_Geometry_of_Motion/Volume_I/, with the rendered tangent-space reading path carried by the compact seven-part series.
Publication state: available as a research-development manuscript; not validated as a complete course, reference standard, or empirical authority.
Open Tangent-Space Series Open Compact Part 1 View Reviewed LaTeX Source (.tex)
Scientific Status
This volume is a living technical manuscript for research and engineering workflows. It is not presented as peer-reviewed consensus material. Mathematical statements require their stated hypotheses; conclusions about golfers additionally require independent measurements and model validation.
Mathematical Reference Appendices
The notation reference and appendices distinguish coordinate geometry, the sensitivity of a specified motion, and the additional conditions needed for stability or optimization. Parallel transport compares vectors; a state-transition map propagates perturbations. A positive-definite metric, a matrix exponential, or a successful numerical solve does not by itself establish stable face delivery.
The revised reference supplies controlled Jacobians, branch-aware rotation maps, power-preserving twist/wrench transformations, metric-weighted sensitivities, and the hypotheses behind matrix and solver formulas. Both printed Python examples are executable manufactured checks. This reference review does not certify every chapter or establish an empirical golf result.
Read the Reviewed Manuscript PDF
Use the PDF bookmarks to find the notation reference and Appendices A and B. Manuscript and PDF links pin the technically reviewed edition dated 5 October 2026. Notebook links retain their separately identified historical snapshots.
Source Traceability
- Primary manuscript source:
articles/The_Geometry_of_Motion/Volume_I/ - Chapter source files:
articles/The_Geometry_of_Motion/Volume_I/chapters/ch0*_*.tex - Website reading path:
articles/tangent-hyperplanes-series/part-*.qmd
Notebook Workflow
The bridge manifest labels the listed Jupyter notebooks scaffolded. That state means a file and tutorial title exist; it does not establish numerical correctness, successful execution, dependency availability, or reproducibility. The revision-pinned notebook and Colab links preserve historical scaffolds, separately from the current manuscript sources. Colab execution still depends on an external runtime.
Chapter 1: Foundations: Tangent Spaces and Exactness
The core thesis: nonlinear systems preserve exact linear superposition in instantaneous tangent mappings at fixed state and mode.
Chapter 2: Variational Dynamics and the Moving Frame
State-transition operators, perturbation flow, and sensitivity transport along nonlinear trajectories.
Chapter 3: Superposition: From Linear to Affine
Exact instantaneous input superposition at a fixed state and mode in control-affine systems, plus the bounded zero-torque counterfactual intervention contract.
Read Superposition in Control-Affine Systems Source .tex Open Notebook in Colab View Notebook on GitHub
Additional Canonical Source: Chapter 3B, Induced Acceleration Analysis
Separates model-conditioned acceleration contributions while retaining coordinate, constraint, and interaction terms.
Chapter 4: Contraction Theory and Metric Certificates
Riemannian contraction metrics, differential stability, and metric certificates on moving hyperplanes.
Chapter 5: Local Optimal Control: LQR, DDP, and Riccati
Finite-horizon local quadratic control and the Riccati recursion used in trajectory optimization.
Chapter 6: The Stability–Optimality Duality
Explores a proposed relationship between contraction stability certificates and value-function curvature in tangent coordinates; equivalence requires additional hypotheses and is not asserted globally.
Chapter 7: Forward Dynamics, Counterfactuals, and Drift
Forward dynamics models, hybrid transitions in tangent space, and zero-torque counterfactual drift analysis.
Read Chapter 7 Source .tex Open Notebook in Colab View Notebook on GitHub
Chapter 8: Applications: Biomechanics, Robotics, and Beyond
Practical implementation checklist, residual-aware feedback, and validation workflows across high-dimensional systems.
Additional Canonical Source: Chapter 9, Parallel Mechanisms
Develops the manuscript’s parallel-mechanism extension; the chapter is source material, not a validated hardware-design procedure.
Follow-Up and Challenge Paths
Citation
@online{untitled,
author = {},
title = {Volume {I:} {Tangent-Space} {Methods} for {Nonlinear}
{Control}},
url = {https://affinedrift.com/books/tangent-space-methods.html},
langid = {en}
}