Tangent Hyperplane Framework: Reading Path

tangent-space
A guided route through local sensitivity, finite prediction error, optimal control, stability, and impact dynamics.

Overview

This collection investigates how local differential models help explain and control nonlinear motion. A derivative tells us how a small change in state or input affects a chosen model along a reference motion. To turn that sensitivity into a useful prediction, we must also know how discrepancies grow, what feedback changes, and what happens at contact events.

These questions are intertwined in a golf swing. A change in joint torque can alter the club’s approach, the time of impact, and the contact state. A controller may reduce some deviations while leaving others important to strike location or clubface orientation. An accurate local derivative is one part of that argument; it does not establish finite-motion accuracy, a globally optimal swing, or the golfer’s neural strategy.

The pages combine established mathematics, worked examples, and proposals for investigation. Read each result with its assumptions and evidence. The contribution sought here is a useful synthesis for modeling and explanation; novelty, experimental performance, and coaching benefit require separate demonstrations.

Prerequisites: Linear algebra, multivariable calculus, and ordinary differential equations support the technical route. State-space control is useful for the optimization articles; Lyapunov theory helps with contraction. The accessible companions offer intuition, but the technical pages supply the conditions that analogies omit.

The Tangent-Space Series hub provides the canonical seven-part reading sequence. This guide supports deeper study of the full reference manuscript and advanced articles alongside that sequence.

Learning Tracks

Track Reading and Activity What to Check Before Moving On
Conceptual Understanding Read an accessible companion, then its main article’s introduction and limitations. Explain the difference between a derivative and a finite prediction without claiming that nonlinear motion becomes linear.
Applied Practitioner Study the main derivation, reproduce a small example, then implement a model of your own system. Declare state, inputs, units, constraints, numerical tolerances, and an independent prediction check.
Technical Review Read the main article with its limitations and relevant published critiques; derive equations and construct counterexamples. Identify exactly which hypotheses support each bound, stability claim, and optimization result.

Reading time depends strongly on background and whether you reproduce the calculations. These are study routes, not promises of expertise after a fixed number of hours. Some articles contain executable examples and exercises; others contain pseudocode or proposed applications. Check the local description before treating an example as a complete implementation.

Part I: Foundations

1. Unified Thesis

For the full derivations, consult the Full Reference Manuscript after the relevant compact series parts. Its central sequence is differential structure, propagation along a trajectory, and use of local models in control synthesis.

Focus on three distinctions:

  • A tangent space is a vector space attached to a point. The nonlinear vector field assigns a velocity at each point; the existence of tangent spaces does not make that field linear in a neighborhood.
  • The variational equation is linear along a fixed nominal trajectory. Its solutions superimpose; finite changes to nonlinear trajectories generally have an additional remainder.
  • A local quadratic optimization problem can be solved accurately while remaining an approximation to a constrained, nonlinear finite-horizon problem.

The introductory summary is an optional orientation. Its analogies should be checked against the technical derivations. Use the published critiques to identify competing explanations and open questions. Internal critique manuscripts are excluded from the rendered reading path; their existence is not independent peer review or proof that the issues they discuss are resolved. Read the assumptions and limitations in each current technical article.

Following the Golf Swing Questions

For biomechanics, choose a physical question before choosing an algorithm. Examples include how a torque perturbation changes club delivery, how feedback could reduce deviations, and how impact timing changes the measured outcome. Specify the intervention, observation time, task output, and quantities held fixed.

Use the Unified Thesis for the local model, Residual-Aware Control for finite prediction error, and Hybrid Tangent Spaces when contact changes the dynamics. Read Contraction Theory when the question concerns convergence under a specified input policy.

The earlier golf-application manuscript is in the repository’s draft archive. It is not a separate current prerequisite. A coaching decomposition can suggest hypotheses, but decomposing a motion into coordinates or basis functions does not show that independently changing those parts will produce additive effects in a golfer.

Part II: Advanced Theory

2. Residual-Aware Control

Residual-Aware Control develops the relationship between a local Taylor remainder, its propagation, and the eventual task error. Read it after the variational and state-transition sections of the Unified Thesis.

It distinguishes the mathematical remainder of a specified model from discrepancies caused by measurement noise, uncertain parameters, discretization, or omitted physics. Its scalar calculations illustrate bounds and interval choices. Proposed applications to golf, aerial vehicles, or walking still need system-specific evaluation; they are not evidence of a demonstrated performance improvement.

A foot strike is an event in a rigid-impact model, not simply a large smooth Hessian. Gimbal lock is a degeneracy of an attitude parameterization, not an intrinsic curvature singularity of physical orientation space. Those situations call for an appropriate event model or coordinate representation before interpreting a large residual.

Companion: Accessible Explanation. Use the main article’s bound derivations and observation assumptions to evaluate its proposals.

3. Contraction Theory and Tangent Spaces

Contraction Theory Meets Tangent Spaces asks whether neighboring solutions approach one another under the same specified dynamics and input policy. It develops the metric conditions and conditional links to linear-quadratic value functions.

This is different from asking whether successive DDP iterations reduce a cost. Physical time and optimizer iteration are different independent variables. A contraction condition can constrain a controller-design problem, but adding a penalty to an objective does not prove the resulting controller satisfies that condition throughout a region.

Study the actual closed-loop Jacobian, the metric’s time dependence, and the region in which the inequalities hold. Then ask how the resulting state-distance statement relates to the task output. Stability alone does not establish desirable club delivery, low effort, contact feasibility, or a correct model of human control.

Companion: Accessible Explanation. Check the main article’s counterexamples and regional conditions before drawing stability conclusions.

4. Hybrid Tangent Spaces

Hybrid Tangent Spaces combines smooth evolution within a mode with guards and resets at events. It derives how an event-time shift contributes to sensitivity and explains when the saltation matrix is applicable.

A guard specifies an event condition; a reset specifies the change in state. Neither is generally a rotation. Dissipative impact can reduce velocity magnitude, and nearby motions can strike at different times. The resulting common-time sensitivity includes both reset derivatives and the time spent under different mode dynamics.

Read this article for rigid foot impacts, club–ball contact idealizations, or switching models. Smooth compliant contact, ideal rigid impact, and nonsmooth friction are different modeling choices. Grazing, simultaneous events, sticking, or an accumulation of infinitely many events require additional analysis; a formula for one isolated transverse event does not cover every contact regime.

Companion: Accessible Explanation. Read the main article’s sensitivity and Zeno-limit discussion with its worked event calculations.

Key Equations Reference

The notation below is local to this guide. Follow each article’s definitions when switching between continuous and discrete time, physical and scaled coordinates, or different definitions of residual. Assume the relevant solutions exist on the stated interval.

First Variations and Their Propagation

For a differentiable family of solutions of \(\dot x=f(t,x,u)\), let \(\xi=\partial_\epsilon x_\epsilon|_0\) and \(\nu=\partial_\epsilon u_\epsilon|_0\). Along the nominal trajectory,

\[ \dot\xi=A(t)\xi+B(t)\nu, \qquad A=f_x,\quad B=f_u. \]

If \(\Phi(t,s)\) solves \(\partial_t\Phi=A(t)\Phi\) with \(\Phi(s,s)=I\), then

\[ \begin{aligned} \xi(t)={}&\Phi(t,t_0)\xi(t_0)\\ &+\int_{t_0}^{t}\Phi(t,s)B(s)\nu(s)\,ds. \end{aligned} \]

This is an exact equation for the derivative of the solution family under the required differentiability assumptions. It is not an exact finite-displacement rollout. For \(C^1\) dynamics, the instantaneous Taylor remainder is little-o of the joint state/input displacement. A uniform quadratic estimate needs stronger regularity, such as a bounded Hessian on the relevant segments.

Finite Prediction Error

In fixed, declared state/input scalings, write the continuous dynamics as \(\dot z=g(t,z,v)\). Let \((\bar z,\bar v)\) be a nominal state/input trajectory satisfying these dynamics. Set \(e=z-\bar z\), \(\Delta v=v-\bar v\), and let \(\eta\) solve the linear comparison model \(\dot\eta=A\eta+B\Delta v\) with \(\eta(t_0)=e(t_0)\), where now \(A=g_z\) and \(B=g_v\) along the nominal motion, and \(\Phi\) is the state-transition matrix of this scaled \(A\). Define

\[ \begin{aligned} N={}&g(t,\bar z+e,\bar v+\Delta v)\\ &-g(t,\bar z,\bar v)-Ae-B\Delta v,\\ r={}&e-\eta. \end{aligned} \]

Then \(\dot r=Ar+N\) and \(r(t_0)=0\). If the joint state/input Hessian has bilinear operator norm at most \(K\) on every segment used in this expansion,

\[ \begin{aligned} q(s)&=\|e(s)\|^2+\|\Delta v(s)\|^2,\\ \|r(t)\|&\leq\frac{K}{2}\int_{t_0}^{t}\|\Phi(t,s)\|q(s)\,ds. \end{aligned} \]

The norm is Euclidean in the declared scaled coordinates, with induced matrix and bilinear norms. The bound contains the actual finite deviation \(e\); replacing it by the linear prediction requires another bound. Dropping \(\|\Phi(t,s)\|\) requires a justified nonexpansion estimate in that norm. The residual derivation develops these conditions and the distinction from closed-loop propagation.

A scalar counterexample exposes the missing factor. For \(\dot x=x+x^2\), nominal \(x=0\), and \(x(0)=0.01\), the exact solution is \(x(t)=0.01e^t/[1-0.01(e^t-1)]\). At \(t=1\), its error relative to the linear prediction is about \(0.00047524\). The unweighted integral \(\int_0^1x(s)^2ds\) is only \(0.00032591\); it cannot bound that error. For this scalar example, weighting by \(e^{1-s}\) recovers the exact remainder. These are dimensionless manufactured values, not golf measurements.

Full DDP and the iLQR Approximation

For a discrete transition \(x_{k+1}=F_k(x_k,u_k)\), discrete stage cost \(\ell_k\), and next-step value \(V^+\), define \(Q=\ell_k+V^+\circ F_k\). Suppress the time index \(k\) in the derivatives below; \(F_i\) denotes output component \(i\), and the sum runs over those components. Where the derivatives exist, the chain rule gives

\[ \begin{aligned} Q_{uu}={}&\ell_{uu}+F_u^\top V^+_{xx}F_u\\ &+\sum_i V^+_{x_i}(F_i)_{uu}. \end{aligned} \]

Full DDP retains the last term; iLQR omits the value-gradient-weighted dynamics Hessians. For the scalar example \(F(u)=u^2\), \(V^+(x)=x\), and \(\ell=0\), the exact \(Q_{uu}\) is \(2\) at \(u=0\), while the iLQR curvature term is zero. This illustrates the derivative distinction, not a performance comparison. See the local Q-function derivation and Tassa, Mansard, and Todorov, Section II. Constraints, regularization, and acceptance of an update remain separate parts of the algorithm.

Contraction in a Specified Metric

Apply the differentiable feedback law \(u=\pi(t,x)\) to \(\dot x=f(t,x,u)\). Let \(A_{\mathrm{cl}}=f_x+f_u\pi_x\). The first variation obeys \(\dot\xi=A_{\mathrm{cl}}\xi\). A differential contraction certificate uses a smooth positive-definite metric \(M(t,x)\) and the matrix inequality

\[ \begin{gathered} A_{\mathrm{cl}}^\top M+MA_{\mathrm{cl}}+\dot M\\ \preceq-2\alpha M,\qquad\alpha>0. \end{gathered} \]

Here \(\dot M\) is the material derivative along the closed-loop flow. The squared differential length \(V=\xi^\top M\xi\) decays at rate at least \(2\alpha\), and its square root at rate at least \(\alpha\). Finite-distance conclusions require suitable connecting paths to remain in the certified region; uniform metric bounds support comparison with Euclidean distance. The identity-metric condition concerns the symmetric part of \(A_{\mathrm{cl}}\), not merely its eigenvalues. Read the contraction article with Lohmiller and Slotine, Sections 3.1–3.4.

Events and Common-Time Sensitivity

Let \(x^-\) and \(x^+\) denote states immediately before and after an event. For an isolated event \(h(t,x^-)=0\) with reset \(x^+=R(t,x^-)\), nearby trajectories generally reach the guard at different times. In compatible local coordinates, their first variations at a common clock time satisfy

\[ \begin{aligned} \xi^+&=S\xi^-,\\ S&=R_x+\frac{(f^+-R_xf^--R_t)h_x}{h_t+h_xf^-}. \end{aligned} \]

Here \(h_x\) is a row derivative; incoming and outgoing fields are evaluated on their respective sides of the event under the specified input policy. Smooth guards/resets, a nonzero denominator, and a locally consistent event sequence are essential. The reset derivative alone applies to an appropriate prescribed-time reset; it generally omits the timing effect for state-triggered events. See the saltation derivation and Kong and colleagues, Section III. Neither \(R_x\) nor \(S\) is generally an orthogonal rotation.

Glossary and Notation

Term Meaning in This Guide
Tangent Space \(T_x\mathcal M\) An \(n\)-dimensional vector space at a point of an \(n\)-dimensional smooth manifold; coordinate representations identify it with \(\mathbb R^n\).
Fréchet Derivative The linear map whose subtraction leaves a remainder little-o of the increment. The derivative is well defined; its finite affine prediction may have error.
State and Input \(x,u\) Local coordinates in \(\mathbb R^n\) and \(\mathbb R^m\); \(f(t,\cdot,\cdot)\) maps an appropriate domain in \(\mathbb R^n\times\mathbb R^m\) to \(\mathbb R^n\).
First Variation \(\xi,\nu\) Derivatives of a family of states and inputs, distinguished here from finite differences \(e,\Delta v\).
State Transition \(\Phi(t,s)\) Fundamental matrix of the specified time-varying homogeneous variational equation.
Residual A discrepancy requiring an explicit definition: instantaneous Taylor remainder, accumulated prediction error, or measured mismatch are different objects.
Costate \(\lambda\) An adjoint variable for a specified optimal-control problem, with terminal and possible event conditions. Its interpretation depends on the objective and constraints.
Hamiltonian \(H=L+\lambda^\top f\) A continuous-time optimal-control expression under this sign convention, with running cost \(L\); it is not automatically mechanical energy or the discrete DDP Q-function.
Contraction Decay of neighboring-solution separation in a declared metric under the same specified dynamics/policy and regional assumptions.
Hybrid System A model with continuous evolution and discrete modes or transitions; the model must specify solution and event conventions.

Norms need units or explicit scaling. Combining radians, translational positions, velocities, and muscle activations in an unscaled Euclidean norm changes the meaning of an error threshold. Big-O and little-o describe limiting behavior with a declared small parameter; their constants and validity interval cannot silently be carried to arbitrary horizons.

Common Pitfalls

An Exact Derivative Does Not Make a Finite Rollout Exact

Relinearization updates local information. It does not eliminate Taylor remainder, numerical integration error, parameter uncertainty, or sensor error. Special affine systems can have exact finite linear predictions; generic nonlinear systems require an error analysis.

Local Steps Do Not Guarantee Global Reachability

Input bounds, obstacles, contact feasibility, and underactuation restrict reachable motions. An optimizer can converge to a local solution or fail. Repeated small updates do not prove that every target is reachable or that a global optimum will be found.

A Residual Is Not Automatically Intrinsic Curvature

Even a scalar nonlinear field on a flat Euclidean line can have a nonzero Taylor remainder. Coordinate and vector-field nonlinearities must be distinguished from intrinsic manifold curvature. A measured mismatch can also arise from noise or model error. Reducing a task-relevant residual can be useful; interpreting it requires knowing which discrepancy was measured.

Optimization Progress and Physical Stability Need Separate Arguments

A lower objective value across iterations does not imply that neighboring physical trajectories converge in time. Conversely, a contracting closed loop need not solve the desired optimization problem. A Riccati construction can connect the two under stated conditions; shared Jacobians alone do not establish a general duality.

Smooth Analysis Must Be Matched to the Contact Model

Apply variational dynamics within smooth modes and the appropriate event sensitivity at transitions. Event detection and event-time refinement matter alongside integration accuracy. A contact-sequence change can invalidate a derivative computed for the previous sequence.

How the Pieces Fit in a Golf or Humanoid Model

Begin with a measurable output, such as clubface orientation at first contact or foot placement after a disturbance. Include enough state to predict its evolution: configuration alone generally omits velocity, while muscle activation, flexible equipment, or a feedback controller can introduce additional state and memory. Decide which of those effects the current model resolves.

Next separate propagation from intervention. Open-loop sensitivity asks what happens when a specified input is perturbed. Closed-loop sensitivity also differentiates the controller’s response to changed state; delayed or estimated feedback needs the corresponding model. Those two calculations can give different answers even for the same mechanical plant.

Then propagate deviations to the observation. If the observation occurs at contact, a shift in event time can matter as much as a change in the incoming state. For a smooth output \(y=\psi(t,x)\) evaluated at a variable event time, the first-order output change includes both the state effect and the time shift. Specify whether the state variation is measured at the event or at a common clock time before adding a timing term, so it is not counted twice.

Finally compare predicted task changes with uncertainty and observations. A small state-space norm does not automatically mean an acceptable strike; the output map can be sensitive in particular directions. A simulation can establish a consequence of its equations, but multiple neural strategies, parameter choices, or contact models may produce similar observed trajectories. Distinguishing them requires suitable measurements or interventions. This sequence links geometry, dynamics, control, and evidence without treating one calculation as proof of all four.

External Resources

Use these sources to check definitions and derivations, rather than treating the collection’s own critiques as external validation.

  • Khalil, Nonlinear Systems, Third Edition: The author’s contents locate sensitivity in Section 3.3, Lyapunov stability in Chapter 4, and feedback linearization in Chapter 13. Feedback linearization has its own hypotheses and is distinct from taking a first variation.
  • Lee, Introduction to Smooth Manifolds, Second Edition: The author’s front matter and contents place tangent vectors, differentials, and the tangent bundle in Chapter 3. Chapter 4 covers submersions, immersions, and embeddings.
  • Sastry, Nonlinear Systems: Analysis, Stability, and Control: Use the publisher’s contents to locate Mathematical Background, Lyapunov Stability Theory, Basics of Differential Geometry, and Linearization by State Feedback.
  • Lohmiller and Slotine: Their contraction analysis paper develops differential length and metric conditions. Read the regional assumptions with the inequalities.
  • Tassa, Mansard, and Todorov: Control-Limited Differential Dynamic Programming develops the local quadratic model and a particular treatment of box control limits.
  • Kong, Payne, Zhu, and Johnson: Saltation Matrices explains reset and timing contributions to hybrid sensitivity.

Study and Contribution

For an initial session, choose one question and one main article. Write down the proposed conclusion, its assumptions, and a small example that could falsify an overly broad version of the claim. For deeper study, reproduce the example, change an assumption, and explain the outcome before extending it to a golf or humanoid model.

The collection includes internal assessments and historical drafts. Their grades, defenses, and proposed future work are not independent evidence. A paper claiming a new result needs a specific statement and comparison with the relevant literature; a proposed application needs validation appropriate to its intended use. Availability of a derivation, code example, PDF, or reading path does not by itself establish that validation.

Report errors or missing assumptions through the AffineDrift repository. Useful contributions provide the exact claim, a derivation or counterexample, and the source or measurement needed to resolve it. Keep mathematical examples distinct from experimental results and identify unfinished implementations explicitly.

The useful habit is to connect a local statement to a finite prediction, then to a task outcome and the evidence that could test it.

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  author = {},
  title = {Tangent {Hyperplane} {Framework:} {Reading} {Path}},
  url = {https://affinedrift.com/articles/tangent-hyperplane-articles/TABLE_OF_CONTENTS.html},
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