Tangent Hyperplanes V: Contraction

Tangent Hyperplanes V: Contraction

tangent-space

Contraction certificates as tangent-space stability statements: metric-based convergence and differential guarantees for motion.

From One Reference Trajectory to a Region

A variational model describes sensitivities along a chosen trajectory. Contraction asks for a stronger property: do all admissible infinitesimal separations shrink throughout a stated region? Integrating that differential result along connecting curves can then bound finite trajectory separation. The region and its invariance are part of the proof.

This article develops the connection to Part 3’s control problem. The longer Contraction Theory Meets Tangent Spaces is companion material; a numerical example or proposed algorithm there does not substitute for a certificate for a particular golf model.

Differential Length and Its Derivative

Consider $\dot x=f(x,t)$ with Jacobian $A=D_xf$. Assume a continuously differentiable symmetric metric $M(x,t)$ with uniform bounds

\[ aI\preceq M(x,t)\preceq bI,\qquad 0<a\le b<\infty. \]

For the first variation $\dot\xi=A\xi$, define the squared differential length

\[ V(x,\xi,t)=\xi^TM(x,t)\xi. \]

Its length is $\sqrt V$. Along the flow,

\[ \dot V=\xi^T\bigl(\dot M+A^TM+MA\bigr)\xi, \]

where the total derivative is

\[ \dot M=\partial_tM+\sum_j f_j\partial_{x_j}M. \]

If, for a uniform rate $\lambda>0$,

\[ \dot M+A^TM+MA\preceq-2\lambda M, \]

then $V(t)\le e^{-2\lambda(t-t_0)}V(t_0)$ and differential lengths decrease at rate $\lambda$. Checking only $A^TM+MA$ omits the metric’s change along the trajectory. These are the differential ingredients of Lohmiller and Slotine’s contraction analysis.

Why Connecting Curves and Invariance Matter

For a differentiable curve $\gamma(s)$ joining two states, its metric length at time $t$ is

\[ L_M(\gamma,t)=\int_0^1 \sqrt{\gamma'(s)^TM(\gamma(s),t)\gamma'(s)}\,ds. \]

Assume solutions exist for the claimed horizon, and all points of the connecting curve evolve inside the region where the contraction inequality holds. Applying the differential bound to each curve tangent and integrating gives

\[ L_M(\gamma_t,t)\le e^{-\lambda(t-t_0)}L_M(\gamma_{t_0},t_0). \]

For a path-connected forward-invariant region, take the infimum over connecting curves in that region to obtain the same bound for its intrinsic metric distance. No argument can connect two components through an unverified part of state space. Uniform metric bounds permit comparison with Euclidean lengths; comparison with straight-line endpoint distance additionally needs a suitable connecting-path geometry, such as the straight segment lying in the region.

Sampling the inequality on a nominal trajectory is a diagnostic, not a proof throughout a surrounding tube. A regional certificate requires analytic bounds, verified numerical bounds, or another method that covers the stated domain. Time-varying tubes need an explicit argument that the evolving connecting curves remain admissible.

When a Riccati Matrix Supplies a Metric

Using the half-quadratic cost convention in Part 3, the LQR value Hessian is $S(t)$. With $K=R^{-1}B^TS$, the Riccati equation implies

\[ \begin{aligned} \dot S+(A-BK)^TS+S(A-BK)\\ {}=-Q-K^TRK. \end{aligned} \]

To conclude contraction in this metric, verify both $aI\preceq S\preceq bI$ and

\[ Q+K^TRK\succeq2\lambda S \]

over the claimed interval. Positive-semidefinite $S$ may assign zero length to nonzero variations and cannot serve as a full-state Riemannian metric. A zero terminal state cost often makes uniform positivity fail at the horizon endpoint.

Neither finite-horizon optimality nor a decreasing rollout objective supplies these hypotheses automatically. For example, $\dot x=x+u$ with only a positive input penalty has optimal $u=0$, $S=0$, and expanding state perturbations. More generally, for a nonlinear policy $u=\pi(x,t)$, use the full closed-loop Jacobian

\[ A_{\rm cl}=D_xf(x,\pi,t)+D_uf(x,\pi,t)D_x\pi(x,t). \]

A gain computed on one reference curve does not establish this regional inequality.

Task-Space Metrics Can Miss Joint Motion

Let a task map be $y=\psi(q)$ with Jacobian $J=D_q\psi$, and let its task metric be $W\succ0$. Its pullback quadratic form is

\[ M_q=J^TWJ. \]

This is positive definite only if $J$ has full column rank on the chosen tangent coordinates. A clubhead position task generally has fewer outputs than a whole-body joint model has coordinates. Joint variations in $\ker J$ then have zero task length, even though the body moved.

Such a form can certify a task-space or restricted-subspace statement when the relevant dynamics and assumptions support it. It cannot certify contraction of every joint direction. Adding a positive joint-space term defines a different metric whose contraction inequality must be checked again; numerical regularization is not a proof of unobserved stability.

A Contraction-Aware Design Proposal

A proposed optimization loop can compute local feedback, assess a candidate differential inequality, and penalize or reject steps that violate a specified stability condition. It must still handle feasibility, nonlinear rollout and the regional verification problem. This is a design pattern, not an implemented or experimentally validated algorithm established by this article.

For golf and humanoid models, state the contact mode, actuator and delay models, state estimates, parameter uncertainty, metric units, region and rate. A favorable margin belongs to that declared closed-loop model. It does not establish that an athlete’s nervous system optimizes the same metric, or that unmodeled muscle and soft-tissue behavior shares the certificate.

Part 6 considers the additional derivative map required at a state-triggered contact transition. Smooth contraction between events alone does not guarantee contraction across them.

Citation

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@online{untitled,
  author = {},
  title = {Tangent {Hyperplanes} {V:} {Contraction}},
  url = {https://affinedrift.com/articles/tangent-hyperplanes-series/part-5-contraction.html},
  langid = {en}
}

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--- title: "Tangent Hyperplanes V: Contraction" description: "Contraction certificates as tangent-space stability statements: metric-based convergence and differential guarantees for motion." aliases: - ../tangent-hyperplane-contraction/textbook-main.html - ../tangent-hyperplane-contraction/chapters/04-stability-optimality-duality.html - ../tangent-hyperplane-contraction/chapters/05-coordinate-free-riemannian.html - ../tangent-hyperplane-contraction/chapters/06-computation-and-certification.html categories: - tangent-space --- ## From One Reference Trajectory to a Region A variational model describes sensitivities along a chosen trajectory. Contraction asks for a stronger property: do all admissible infinitesimal separations shrink throughout a stated region? Integrating that differential result along connecting curves can then bound finite trajectory separation. The region and its invariance are part of the proof. This article develops the connection to [Part 3's control problem](./part-3-control.html). The longer [Contraction Theory Meets Tangent Spaces](../tangent-hyperplane-articles/Advanced/Contraction_Tangent_Unification.html) is companion material; a numerical example or proposed algorithm there does not substitute for a certificate for a particular golf model. ## Differential Length and Its Derivative Consider $\dot x=f(x,t)$ with Jacobian $A=D_xf$. Assume a continuously differentiable symmetric metric $M(x,t)$ with uniform bounds $$ aI\preceq M(x,t)\preceq bI,\qquad 0<a\le b<\infty. $$ For the first variation $\dot\xi=A\xi$, define the **squared** differential length $$ V(x,\xi,t)=\xi^TM(x,t)\xi. $$ Its length is $\sqrt V$. Along the flow, $$ \dot V=\xi^T\bigl(\dot M+A^TM+MA\bigr)\xi, $$ where the total derivative is $$ \dot M=\partial_tM+\sum_j f_j\partial_{x_j}M. $$ If, for a uniform rate $\lambda>0$, $$ \dot M+A^TM+MA\preceq-2\lambda M, $$ then $V(t)\le e^{-2\lambda(t-t_0)}V(t_0)$ and differential lengths decrease at rate $\lambda$. Checking only $A^TM+MA$ omits the metric's change along the trajectory. These are the differential ingredients of [Lohmiller and Slotine's contraction analysis](https://www.mit.edu/~nsl/preprints/contraction.pdf). ## Why Connecting Curves and Invariance Matter For a differentiable curve $\gamma(s)$ joining two states, its metric length at time $t$ is $$ L_M(\gamma,t)=\int_0^1 \sqrt{\gamma'(s)^TM(\gamma(s),t)\gamma'(s)}\,ds. $$ Assume solutions exist for the claimed horizon, and all points of the connecting curve evolve inside the region where the contraction inequality holds. Applying the differential bound to each curve tangent and integrating gives $$ L_M(\gamma_t,t)\le e^{-\lambda(t-t_0)}L_M(\gamma_{t_0},t_0). $$ For a path-connected forward-invariant region, take the infimum over connecting curves in that region to obtain the same bound for its intrinsic metric distance. No argument can connect two components through an unverified part of state space. Uniform metric bounds permit comparison with Euclidean lengths; comparison with straight-line endpoint distance additionally needs a suitable connecting-path geometry, such as the straight segment lying in the region. Sampling the inequality on a nominal trajectory is a diagnostic, not a proof throughout a surrounding tube. A regional certificate requires analytic bounds, verified numerical bounds, or another method that covers the stated domain. Time-varying tubes need an explicit argument that the evolving connecting curves remain admissible. ## When a Riccati Matrix Supplies a Metric Using the half-quadratic cost convention in Part 3, the LQR value Hessian is $S(t)$. With $K=R^{-1}B^TS$, the Riccati equation implies $$ \begin{aligned} \dot S+(A-BK)^TS+S(A-BK)\\ {}=-Q-K^TRK. \end{aligned} $$ To conclude contraction in this metric, verify both $aI\preceq S\preceq bI$ and $$ Q+K^TRK\succeq2\lambda S $$ over the claimed interval. Positive-semidefinite $S$ may assign zero length to nonzero variations and cannot serve as a full-state Riemannian metric. A zero terminal state cost often makes uniform positivity fail at the horizon endpoint. Neither finite-horizon optimality nor a decreasing rollout objective supplies these hypotheses automatically. For example, $\dot x=x+u$ with only a positive input penalty has optimal $u=0$, $S=0$, and expanding state perturbations. More generally, for a nonlinear policy $u=\pi(x,t)$, use the full closed-loop Jacobian $$ A_{\rm cl}=D_xf(x,\pi,t)+D_uf(x,\pi,t)D_x\pi(x,t). $$ A gain computed on one reference curve does not establish this regional inequality. ## Task-Space Metrics Can Miss Joint Motion Let a task map be $y=\psi(q)$ with Jacobian $J=D_q\psi$, and let its task metric be $W\succ0$. Its pullback quadratic form is $$ M_q=J^TWJ. $$ This is positive definite only if $J$ has full column rank on the chosen tangent coordinates. A clubhead position task generally has fewer outputs than a whole-body joint model has coordinates. Joint variations in $\ker J$ then have zero task length, even though the body moved. Such a form can certify a task-space or restricted-subspace statement when the relevant dynamics and assumptions support it. It cannot certify contraction of every joint direction. Adding a positive joint-space term defines a different metric whose contraction inequality must be checked again; numerical regularization is not a proof of unobserved stability. ## A Contraction-Aware Design Proposal A proposed optimization loop can compute local feedback, assess a candidate differential inequality, and penalize or reject steps that violate a specified stability condition. It must still handle feasibility, nonlinear rollout and the regional verification problem. This is a design pattern, not an implemented or experimentally validated algorithm established by this article. For golf and humanoid models, state the contact mode, actuator and delay models, state estimates, parameter uncertainty, metric units, region and rate. A favorable margin belongs to that declared closed-loop model. It does not establish that an athlete's nervous system optimizes the same metric, or that unmodeled muscle and soft-tissue behavior shares the certificate. [Part 6](./part-6-hybrid.html) considers the additional derivative map required at a state-triggered contact transition. Smooth contraction between events alone does not guarantee contraction across them. ## Related Articles ::: {.callout-note title="See Also"} - **[Tangent-Space Series](../../pages/tangent-hyperplanes.html)** — The series hub: full reading order, provenance, and companion material - **[Degrees of Freedom and the Curse of Dimensionality](../degrees-of-freedom-and-dimensionality.html)** — Contraction toward a low-dimensional attractor as a dimensionality-reduction mechanism - **[Tangent Hyperplanes IV: Residuals and Curvature](part-4-residuals-curvature.html)** — The previous part of the series - **[Tangent Hyperplanes VI: Hybrid Systems](part-6-hybrid.html)** — The next part of the series :::