Volume II: Control Is Motion
Volume II extends the tangent-space framework into trajectory-native control design. This page publishes the current chapter map and source links for the active manuscript in articles/The_Geometry_of_Motion/Volume_II/.
Publication state: available as a provisional research-development manuscript; not validated as a complete course, controller design standard, or empirical golf authority.
The ‘Control Is Motion’ Paradigm
Control Is Motion (CIM) is a paradigm for nonlinear control design built on the following principle:
Control inputs are best understood not as setpoint commands but as geometric deformations of a reference trajectory in state space. The objective of a controller is to shape the geometry of motion — its orbit, its convergence tube, and its timing — rather than to steer toward a static target.
Mathematically: For a smooth control-affine system \(\dot{x} = f(x) + G(x)u\) with a nominal trajectory \(x^*(t)\), the CIM paradigm defines local control effectiveness through a chosen transverse projection:
\[ \delta \dot{x}_\perp = A_\perp(t) \delta x_\perp + B_\perp(t) \delta u \]
where \(\delta x_\perp\) is the component of perturbation transverse to \(x^*(t)\), and \(A_\perp(t), B_\perp(t)\) are projected linearization matrices evaluated along the reference trajectory. This is a local, frame-dependent analysis tool, not a global linear model. Control authority is measured by the ability to modify transverse error while respecting the trajectory’s phase, not by proximity to a static target.
Key distinctions from classical control: - The “reference” is a trajectory (orbit), not a point. - Stability is orbital (convergence to \(x^*(t)\) for all \(t\)), not asymptotic to a fixed equilibrium. - Control design must respect the manifold structure of state space; coordinate-free formulations are preferred. - Drift \(f(x)\) encodes the passive geometry of motion; control \(G(x)u\) deforms it.
Required guardrail: A phase coordinate is useful only when its domain, monotonicity condition, reset behavior, and failure cases are stated. A shoulder angle, foot-contact event, or other physical marker is not automatically a global clock replacement; it must remain well defined and locally ordered over the portion of state space being controlled.
Relationship to established theory: CIM is not a new mathematical framework. It is a design philosophy that organizes established results on maneuver regulation and transverse dynamics (Hauser & Hindman, 1995; Shiriaev, Freidovich & Gusev, 2010), virtual constraints and hybrid zero dynamics (Westervelt et al., 2007), and orbital stabilization under one conceptual lens. Volume II develops this philosophy systematically using the tangent-space machinery of Volume I.
Scientific Status
This volume is an active manuscript under ongoing development. Content is intended for transparent technical iteration and should be interpreted as provisional unless corroborated by primary literature and experiments.
Publication Guardrail
The golf examples in this volume are modeling programs, not settled empirical claims. Strong statements about passive wrist release, funnel narrowing, face orientation, or repeatability require a declared mechanical model, phase coordinate, contact and grip assumptions, actuator limits, disturbance model, and either a reproducible notebook or an explicit citation to external validation data. A computed funnel certifies invariance only for the model and controller used to produce it. Claims that a release or late-downswing contraction is “self-correcting” should also state whether the argument comes from open-loop model contraction, closed-loop feedback, hybrid zero dynamics, or a numerical funnel on a declared disturbance set. Chapter 5 and Chapter 7, in particular, should distinguish open-loop underactuated behavior, closed-loop correction, and verified invariance certificates from coaching language that treats the release as automatic or recovery as inevitable.
Source Traceability
- Primary manuscript source:
articles/The_Geometry_of_Motion/Volume_II/main.tex - Chapter source files:
articles/The_Geometry_of_Motion/Volume_II/chapters/*.tex - Notebook bridge manifest:
notebooks/geometry_of_motion/manifest.json
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: Throwing Away the Target
Reframes tracking as trajectory shaping rather than endpoint chasing.
Chapter 2: Curves in State Space
Treats motion as geometry on state manifolds and their tangent bundles.
Chapter 3: Configuration Manifolds
Builds the manifold representation needed for coordinate-robust trajectory control.
Chapter 4: Orbital Stability and Transversality
Defines cycle stability and phase-consistent perturbation treatment.
Chapter 5: Underactuation and Passive Dynamics
Explains how passive structure and limited actuation cooperate in simplified skill models.
Chapter 6: Trajectory Optimization
Develops iterative optimization on curved state spaces with dynamics constraints.
Chapter 7: Funnel Synthesis
Constructs model- and controller-dependent trajectory tubes that certify convergence under stated disturbances.
Chapter 8: Phase-Variable Control
Uses intrinsic phase coordinates to coordinate movement timing and corrections.
Chapter 9: Stochastic Trajectories and Motor Variability
Adds uncertainty-aware trajectory reasoning for biological and robotic systems.
Chapter 10: Learning to Move
Connects adaptation and learned policies to geometric control primitives.
Chapter 11: Case Study - Complete Golf Swing
Integrates the volume’s methods in a full motion-analysis workflow.
“Complete Golf Swing” is the canonical manuscript chapter title, not a claim that the workflow, model, or evidence is complete.