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 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; readers should treat claims as working hypotheses unless externally verified.
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 GitHub and Colab links below preserve the reviewed source snapshot; 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.
Read Chapter 1 Source .tex Open Notebook in Colab View Notebook on GitHub
Chapter 2: Variational Dynamics and the Moving Frame
State-transition operators, perturbation flow, and sensitivity transport along nonlinear trajectories.
Read Chapter 2 Source .tex Open Notebook in Colab View Notebook on GitHub
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.
Read Chapter 4 Source .tex Open Notebook in Colab View Notebook on GitHub
Chapter 5: Local Optimal Control: LQR, DDP, and Riccati
Finite-horizon local quadratic control and the Riccati recursion used in trajectory optimization.
Read Chapter 5 Source .tex Open Notebook in Colab View Notebook on GitHub
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.
Read Chapter 6 Source .tex Open Notebook in Colab View Notebook on GitHub
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.
Read Chapter 8 Source .tex Open Notebook in Colab View Notebook on GitHub
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.