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 an organizing viewpoint for nonlinear control design built on the following principle:
Control inputs are understood not exclusively as setpoint regulation 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 transverse containment tube, and its phase progression — rather than solely steering toward a static equilibrium.
Task-dependent problem formulations: Trajectory-centric control is not a universal replacement for classical setpoint regulation or time-indexed tracking. It represents a task-dependent formulation within established control theory, especially valuable for underactuated, ballistic, or rhythmic maneuvers where maintaining the geometric path and energy structure takes precedence over enforcing a rigid clock schedule. When precise time-synchronization or fixed-point holding is demanded by the task, classical timed tracking and equilibrium regulation remain the rigorous and appropriate choices.
Mathematically: For a smooth control-affine system \(\dot{x} = f(x) + G(x)u\) with a nominal trajectory \((x^*(t), u^*(t))\), perturbations \(\delta x(t) = x(t) - x^*(t)\) and \(\delta u(t) = u(t) - u^*(t)\) linearize as \(\delta \dot{x} = A(t) \delta x + B(t) \delta u\). When local transverse coordinates \(z \in \mathbb{R}^{n-1}\) are introduced via a time-dependent moving projection \(z = P(t)\delta x\), the chain rule introduces moving-frame terms:
\[ \dot{z} = \left(\dot{P}(t) + P(t)A(t)\right) \delta x + P(t)B(t) \delta u \]
An arbitrary matrix projection \(P(t)\) is not automatically an autonomous, closed reduced system. To obtain a decoupled transverse subsystem:
\[ \dot{z} = A_\perp(t) z + B_\perp(t) \delta u \]
one requires an explicit transverse chart (such as a moving family of transverse Poincaré sections \(\Pi(\theta)\) parametrized by a scalar phase \(\theta(x)\) with monotonic forward velocity \(\dot{\theta} > 0\)) that systematically eliminates the along-trajectory tangential component \(\delta x_\parallel\). Control authority is measured by the ability to regulate transverse deviation \(z\) while coordinating the trajectory’s phase, rather than by proximity to a static setpoint.
Distinctions from classical setpoint control: - Reference object: The nominal target is a geometric curve or orbit \(\mathcal{O} = \{x^*(t)\}\), not an isolated static point. - Orbital stability versus timed tracking: Stability of an orbit requires that the distance to the orbit for trajectories starting close to \(\mathcal{O}\) remains small for all forward time (\(\operatorname{dist}(x(t), \mathcal{O}) < \epsilon\)), and asymptotic orbital stability requires that the distance to the orbit tends to zero (\(\lim_{t \to \infty} \operatorname{dist}(x(t), \mathcal{O}) = 0\)). This is distinct from timed tracking convergence (\(\|x(t) - x^*(t)\| \to 0\)): an orbitally stable cycle can attract nearby trajectories while a non-zero asymptotic phase offset \(\phi\) persists (\(\|x(t) - x^*(t + \phi)\| \to 0\) with \(\phi \neq 0\)), leaving a persistent time-synchronized tracking error. - Finite-horizon athletic maneuvers: A finite golf stroke (downswing duration \(\sim 250\text{--}350\,\mathrm{ms}\)) is not an infinite-time periodic orbit. For such motions, the appropriate certificates are finite-horizon reachability containment, transverse tube invariance, or terminal hitting-set tolerances over a bounded phase interval \([0, \theta_{\mathrm{impact}}]\), rather than asymptotic limit-cycle convergence. - Geometry and manifold constraints: Control design respects the manifold geometry of the configuration space; coordinate-invariant formulations avoid chart singularities. - Zero-input drift versus physical passivity: The zero-input vector field \(f(x)\) represents the unforced mathematical drift of the state equations, but it is not inherently physically passive. Establishing physical passivity requires declaring a system port \((u, y)\), a non-negative storage function \(V(x) \ge 0\) (such as mechanical kinetic plus potential energy), and verifying the dissipation inequality \(\dot{V} \le u^\top y\). Algebraic input coordinate shifts (such as non-zero nominal bias forces) move terms between drift and input, which can destroy passivity with respect to the shifted port.
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 distinct mathematical axiom system. It is an organizing perspective that builds on established control literature: - Maneuver regulation and path following for unconstrained nonlinear systems (Hauser & Hindman, 1995). - Transverse linearization, moving coordinate frames, and periodic orbit stabilization (Shiriaev, Freidovich & Gusev, 2010; Manchester, 2010, 2011). - Virtual constraints and hybrid zero dynamics for underactuated locomotion (Westervelt et al., 2007). - Transverse LQR, trajectory funnels, and phase-independent regulation in underactuated mechanics (Tedrake, MIT Underactuated Robotics).
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. Manuscript links pin the technically reviewed edition dated 5 October 2026. The revision-pinned notebook and Colab links preserve historical scaffolds; 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 finite-horizon invariance, reachability containment, and disturbance attenuation under stated assumptions.
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.
Follow-Up and Challenge Paths
Citation
@online{untitled,
author = {},
title = {Volume {II:} {Control} {Is} {Motion}},
url = {https://affinedrift.com/books/control-is-motion.html},
langid = {en}
}