Tangent-Space Series
Tangent-Space Series
Exploratory series on tangent-space and tangent-hyperplane formulations for nonlinear dynamics and control
This section gathers the AffineDrift tangent-space series on nonlinear dynamics, contraction-style reasoning, and related control questions. It is exploratory work. Some legacy page titles and file names still use the older “Tangent Hyperplane” label; this landing page uses “Tangent-Space Series” as the broader editorial label for the same material. The central local claim is that, at a fixed state and contact mode, the mapping from applied generalized forces to instantaneous generalized acceleration is linear within the tangent description being used. Finite-time conclusions require additional assumptions, residual bounds, and empirical validation where the argument is applied to measured movement.
State: Experimental. This series is original AffineDrift synthesis and has not been peer reviewed. Linearity applies to the first differential or declared instantaneous force-to-acceleration map under fixed state, constraints, and contact mode; it does not make the nonlinear finite-time flow linear. Claims about force attribution, accumulation over time, and control design remain in-development arguments.
How to Read This Material
- Start here: the seven-part compact series below is the current editorial reading path. It supersedes the older standalone manuscript as the main reader journey; that designation is an editorial choice, not a comparative validation result.
- Use the Critiques and Responses section alongside these pages; it contains the main limitations and objections that currently matter most.
- For deeper derivations (Hamiltonian/DDP/iLQR/MPC, case studies, appendices), see the full reference manuscript and the named advanced articles further down.
Canonical Series
The seven-part compact series is the single canonical reading path. Each part builds on the previous one.
- Part 1: Geometry: tangent spaces as exact infinitesimal descriptions, the exactness claim, and second-order residuals.
- Part 2: Dynamics: variational dynamics along a reference trajectory.
- Part 3: Control: LQR, iLQR, and DDP as local methods on the moving tangent space.
- Part 4: Residuals and Curvature: residual scaling and the limits of linearized predictions.
- Part 5: Contraction: contraction certificates as tangent-space stability statements.
- Part 6: Hybrid Systems: saltation maps, mode changes, and the limits of smooth tangent reasoning.
- Part 7: Residual-Aware Control: residuals as trust-region, controller-switching, and validation signals.
Full Reference Manuscript
- Full Reference Manuscript: Exact Instantaneous Superposition in Nonlinear Dynamics: the single-file reference manuscript. It restates the framework and adds the deep derivations (integral superposition, Hamiltonian/DDP/iLQR/MPC, case studies, appendices) that the compact series summarizes.
Advanced / Reference Articles
These named articles develop specific advanced topics in depth. They are source-backed companions to the canonical series, not a parallel numbered reading path.
- Contraction Theory Meets Tangent Spaces: source article behind compact Part 5, with longer derivations and implementation sketches.
- Hybrid Tangent Spaces: source article behind compact Part 6, with saltation and contact-implicit detail.
- Residual-Aware Control: source article behind compact Part 7, with algorithm sketches and applications.
Companion Pages
Scope Limits
- The strongest statements here are local statements about instantaneous mappings and linearized objects.
- Claims about finite-time superposition, accumulated error, or controller performance depend on how residual terms are bounded and how mode changes are handled.
- The hybrid and contraction-oriented extensions remain under active development.