# Critic's Corner: Testing the Tangent Hyperplane Argument

**Purpose:** State assumptions, test limits, and separate mathematical identities from engineering evidence.
**Audience:** Technical reviewers and readers of the tangent-space series.
**Reviewed:** October 1, 2026.

::: {.callout-note}
## Scope of This Review

This internal companion manuscript is excluded from the public site build. The [reference manuscript](Tangent_Hyperplanes_Unified_Thesis.qmd) develops the derivations; the [series overview](TABLE_OF_CONTENTS.qmd) identifies the public reading path. These examples check declared mathematical models. They are not measurements of golfers or evidence of a superior teaching method.
:::

## 1. What Does the Geometric View Contribute?

Derivatives, variational equations, local optimal-control methods, and differential stability analysis are established mathematics. The useful proposal here is pedagogical: follow a perturbation from its input, through the changing dynamics, to the output that matters. No priority claim or new convergence theorem follows from presenting these ideas together.

For a golf model, a torque-command change affects acceleration through the current mass matrix and constraints; the state perturbation is transported along the swing; the terminal task map converts it into a change in clubhead speed or face orientation. Muscle activation, tendon compliance, and contact may belong inside those dynamics rather than being independent corrections applied afterward. An instantaneous input gain alone cannot describe this chain.

This perspective does not reduce every method to one theorem. LQR optimizes a specified linear-quadratic problem. Lyapunov analysis can use nonlinear energy functions without linearizing. MPC repeatedly solves a finite-horizon problem and need not use tangent approximations. Shared use of local models in some implementations does not supply shared guarantees of feasibility, stability, or optimality.

## 2. In What Sense Is a Derivative Exact?

In a declared coordinate chart, differentiability at $x$ means there is a linear map $Df(x)$ satisfying

$$
f(x+h)=f(x)+Df(x)h+\rho(x,h),\qquad
\frac{\|\rho(x,h)\|}{\|h\|}\longrightarrow0.
$$

The derivative is a well-defined mathematical object; the finite first-order prediction is generally approximate. Differentiability gives a little-$o(\|h\|)$ remainder. A locally Lipschitz derivative, including a bounded continuous Hessian on the relevant segment, supplies an $O(\|h\|^2)$ bound. Merely assuming $C^1$ does not supply that stronger order: $f(h)=|h|^{3/2}$ at zero is a counterexample.

For a smooth map between manifolds, its tangent map sends one tangent space into another. Comparing finite states still requires a chart, retraction, or other declared construction. A Hessian of coordinate components is not, by itself, intrinsic Riemannian curvature. Nonlinear response also occurs on flat Euclidean state spaces.

## 3. What Does a Quantitative Error Bound Require?

### Separate the Three Errors

A first-order **prediction error** compares a nonlinear trajectory with its linearized prediction. A four-trajectory **interaction residual** compares combined and separate perturbations after subtracting the evolved nominal trajectory. A **physical-model error** compares modeled behavior with the actual system. None automatically bounds the other two.

Consider $\dot x=f(t,x,u)$ and a nominal solution $(\bar x,\bar u)$. Use fixed scales for state and input components so all norms below have a declared meaning. Assume $f$ is continuous in time and $C^2$ in $(x,u)$ on a tube containing both trajectories and their connecting segments. Put

$$
\Delta x=x-\bar x,\quad \Delta u=u-\bar u,\quad
A=f_x(t,\bar x,\bar u),\quad B=f_u(t,\bar x,\bar u).
$$

The finite linearized prediction $\xi$ solves $\dot\xi=A\xi+B\Delta u$ with $\xi(t_0)=\Delta x(t_0)$. Define $e=\Delta x-\xi$ and the vector-field remainder

$$
\rho=f(t,\bar x+\Delta x,\bar u+\Delta u)
-f(t,\bar x,\bar u)-A\Delta x-B\Delta u.
$$

Then $\dot e=Ae+\rho$. With $\Phi(t,s)$ the state-transition matrix of $A$, variation of constants gives the exact identity

$$
e(T)=\int_{t_0}^{T}\Phi(T,s)\rho(s)\,ds.
$$

If the joint Hessian in $(x,u)$ has bilinear operator norm at most $H(s)$ on every connecting segment,

$$
\|e(T)\|\leq\frac12\int_{t_0}^{T}
\|\Phi(T,s)\|H(s)\|(\Delta x(s),\Delta u(s))\|^2\,ds.
$$

This conditional bound needs the *actual evolving* deviations to remain in the tube. Inserting the initial perturbation everywhere does not establish that. If $\|\Phi(T,s)\|\leq K e^{\lambda(T-s)}$, $H\leq H_*$, and the joint deviation norm is at most $d$, the bound is $KH_*d^2(e^{\lambda L}-1)/(2\lambda)$, with $L=T-t_0$ and limit $KH_*d^2L/2$ at $\lambda=0$. A bound proportional to elapsed time alone needs controlled transport.

### A Counterexample to Omitting Transport

In nondimensional variables, take $\dot x=ax+u^2$, $x(0)=0$, and constant $u=\epsilon$ about the zero-input nominal. The first-order input response is zero, but

$$
e(T)=\epsilon^2\frac{e^{aT}-1}{a},\qquad
e(T)=\epsilon^2T\quad\text{if }a=0.
$$

For $a=2$, $T=1$, and $\epsilon=0.1$, the error is approximately $0.0319453$, exceeding the untransported estimate $\epsilon^2T=0.01$. Two constant perturbations $\epsilon v_1$ and $\epsilon v_2$ instead give the four-trajectory interaction $2\epsilon^2v_1v_2(e^{aT}-1)/a$. These are different residuals in the same flat state space.

### A Pendulum Bound With Consistent Units

For $\ddot\theta=-k\sin\theta$, where $k=g/\ell>0$ has units $\mathrm{s}^{-2}$ and angles are expressed in radians, the only nonzero second state derivative of the acceleration component is

$$
\frac{\partial^2(-k\sin\theta)}{\partial\theta^2}=k\sin\theta.
$$

The first derivative is $-k\cos\theta$; it is not a Hessian entry. An acceleration error integrated once is a velocity error, not an angle error.

Introduce dimensionless time $\tau=\sqrt{k}\,t$ and state $z=(\theta,w)$ with $w=\dot\theta/\sqrt{k}$. Then $z'=(w,-\sin\theta)$. At the downward equilibrium the linearized matrix is skew-symmetric, so its transition matrix has Euclidean norm one. For the unforced, undamped solution initialized at $(\theta_0,0)$ with $0<\theta_0<\pi$, conservation of $w^2/2+1-\cos\theta$ confines $|\theta|\leq\theta_0$ in the well around zero. Since $|\theta-\sin\theta|\leq|\theta|^3/6$,

$$
\|z(\tau)-z_{\mathrm{lin}}(\tau)\|_2\leq\frac{\theta_0^3\tau}{6},
\qquad z_{\mathrm{lin}}=(\theta_0\cos\tau,-\theta_0\sin\tau).
$$

For $k=10\,\mathrm{s}^{-2}$, $t=1\,\mathrm{s}$ and $\theta_0=0.1$, the bound is $0.00052705$ in this scaled norm. The angle error is therefore at most $0.00052705$ radians (about $0.0302$ degrees), and the angular-velocity error is at most $\sqrt{k}$ times that bound, approximately $0.00167\,\mathrm{rad/s}$. These component bounds are conservative and model-specific. Meeting a task tolerance is a separate decision; forcing, damping, an inverted equilibrium, or different scaling requires a new argument.

## 4. What Can DDP and iLQR Guarantee?

DDP retains second derivatives of the dynamics in its local quadratic optimization model; the usual iLQR approximation omits those terms. Here $Q_{uu}$ is the control Hessian of the local action-value model (stage cost plus predicted downstream value). Positive-definite $Q_{uu}$ makes the unconstrained quadratic control step uniquely solvable. It does not establish global convergence, constraint satisfaction, or closed-loop stability. See the iLQR/DDP discussion in [Tedrake's trajectory-optimization notes](https://underactuated.mit.edu/trajopt.html).

Quadratic local convergence belongs to a specified algorithm and regular solution. Its hypotheses include suitable smoothness and nondegeneracy, sufficiently close initialization, and appropriate step acceptance; $C^2$ alone does not universally give a quadratic Newton rate. Approximate Hessians, persistent damping, and inexact derivatives may change the rate. Do not transfer a full-DDP result to iLQR without checking its assumptions.

Regularization and line searches help construct and accept steps. Their parameters depend on scaling and conditioning; no universal damping interval exists. Active constraints need an algorithm that handles them and corresponding regularity conditions. They are not categorically incompatible with fast local convergence. Check objective change, stationarity, feasibility, and derivative/integration accuracy separately; a small objective change can also mean stagnation.

## 5. What Evidence Supports the Applications?

The illustrative calculations establish properties of their stated models. They do not establish that golfers use a proposed control strategy or that changing technique improves performance.

The earlier attribution of quadrotor experimental validation to Tassa, Erez, and Todorov (2012) was incorrect. Their [paper](https://roboti.us/lab/papers/TassaIROS12.pdf), especially its abstract and Sections II–IV, studies online trajectory optimization with simulated systems including a humanoid, acrobot, swimmer, and hopper. These are algorithmic demonstrations within their simulation setup, not hardware quadrotor experiments. As an editorial requirement, claims about industrial or human applications need an identified study with inspected methods and results.

For a golf claim, specify the observable, intervention, population, measurement uncertainty, calibration, comparison conditions, and failure criterion. A manufactured identity check verifies an implementation against mathematics; a simulation sensitivity study explores a model. Neither supplies a measured effect size or causal identification in humans. Claims that a geometric presentation improves learning likewise need educational evidence.

## 6. What Does a Linear Spacecraft Example Establish?

Within the linear Clohessy–Wiltshire model, the first-order model reproduces the model dynamics exactly; an LQR calculation can check signs, units, and implementation. It does not validate the circular-reference-orbit and small-relative-motion assumptions against nonlinear orbital dynamics, or test nonlinear residual handling.

An elliptical reference orbit can lead to a *linear time-varying* relative-motion model: time variation does not itself imply nonlinear state dependence. Comparing with a nonlinear orbital model requires specified perturbations, horizon, constraints, and controller. Neither orbital variation nor an Earth-oblateness ($J_2$) term proves DDP/iLQR outperforms a well-designed linear or gain-scheduled controller. Similarly, adding anatomical detail to a swing model does not by itself establish better prediction or control.

## 7. How Does This Relate to Prior Work?

| Method | Relevant Local Structure | Additional Requirement |
|---|---|---|
| EKF | Jacobians propagate local uncertainty approximations | Statistical assumptions and observability; nonlinear posteriors need not be Gaussian |
| Trajectory Linearization and Gain Scheduling | Local models vary along a trajectory or operating condition | Tracking and scheduling analysis; stable frozen models alone are insufficient |
| LQR and DDP/iLQR | Quadratic subproblems exploit dynamics and cost structure | Problem assumptions and the actual algorithm's convergence conditions |
| Nonlinear MPC | Repeated finite-horizon optimization | Feasibility, timing and stability analysis for the formulation |
| Contraction Analysis | Differential displacement evolves through the Jacobian | A controlled metric and contraction inequality on the relevant region |

For this contraction argument, fix the input policy and write the resulting smooth dynamics as $\dot x=f(x,t)$; a feedback policy is included in $f$. A differential displacement $\delta x$ then obeys $\dot{\delta x}=A(x,t)\delta x$, with $A=\partial f/\partial x$ across the region. A smooth metric $M(x,t)$ with $mI\preceq M\preceq\bar mI$ for fixed positive constants gives

$$
\frac{d}{dt}(\delta x^TM\delta x)
=\delta x^T(A^TM+MA+\dot M)\delta x.
$$

Here $\dot M=\partial_tM+\sum_i f_i\partial_{x_i}M$ is the derivative along the flow. The inequality $A^TM+MA+\dot M\preceq-2\lambda M$, $\lambda>0$, gives differential contraction; trajectory-distance conclusions also need connecting paths to remain in the contraction region. This is more precise than “Jacobian negativity.” See [Lohmiller and Slotine, Sections 3.3–3.4](https://web.mit.edu/nsl/www/preprints/contraction.pdf).

These are relationships, not claims that the fields lacked geometric interpretations. Nor should the nonlinear EKF's history be assigned wholesale to the linear Kalman–Bucy filter. A conceptual connection is useful only while its distinct assumptions remain visible.

## 8. What Happens at Contact and Other Nonsmooth Events?

Smooth-segment derivatives are insufficient when perturbations change event timing. For a time-independent differentiable guard $h(x)=0$, reset $x^+=R(x^-)$, and transverse isolated event with the same nearby event sequence, the first-order jump is described by the saltation matrix

$$
\Xi=DR+\frac{(f^+-DRf^-)\nabla h^T}{\nabla h^Tf^-}.
$$

The vector fields are evaluated on their respective sides. The denominator must be nonzero. The second term accounts for event-time shifts; differentiating the reset alone misses it. Time-dependent guards and resets require additional terms. Grazing, simultaneous events, changing event order, and accumulated events require separate analysis. See [Kong et al., Section III](https://arxiv.org/html/2306.06862v3#S3).

For example, let $\dot x=1$ before crossing zero, $\dot x=3$ afterward, with the identity reset. From $x(0)=-1$ at $T=2$, the endpoint is $3$. Increasing the initial state by $\delta$ advances the crossing by $\delta$ and changes the endpoint by $3\delta$. Both smooth flows have zero state Jacobian and unit transition factor, so endpoint sensitivity equals $\Xi=3$, although $DR=1$ and the event occupies no time. A golf impact cannot be dismissed on duration alone.

Regularization is a modeling choice, not a prerequisite for every gradient method. Stribeck friction may retain a discontinuous sign term; compliant contact may retain nonsmooth activation; replacing saturation by $\tanh$ changes the actuator law. State the actual equations, differentiability, admissible forces, and limiting behavior, then check sensitivity to regularization and integration resolution. Coulomb sticking may need a set-valued force law; ideal impacts may instead be represented by hybrid resets. These are related but distinct constructions.

## Review Standard

The defensible argument is conditional: local derivatives, transport, task maps, and event sensitivity help organize model-based questions about a swing. Demonstrate their usefulness for the particular model and task. Residuals remain errors when evaluating predictions, even when they also reveal nonlinear interactions.

For each application, record the reference trajectory, coordinate scales, perturbation, constraints and contact regime; distinguish symbolic results, numerical checks, simulation comparisons, and measurements. Report what would invalidate the conclusion. This internal review neither certifies the entire thesis nor establishes empirical validity or publication readiness.