Tangent Hyperplanes VI: Hybrid Systems

Tangent Hyperplanes VI: Hybrid Systems

tangent-space

How tangent-space reasoning changes across impacts, guards, and mode switches.

Smooth Motion and Discrete Events

A rigid impact model combines smooth motion between events with guards and reset maps at events. Other models, including compliant finite-duration contact, may represent the same physical interaction through continuous dynamics. Choose the representation before differentiating it; a physical collision does not force every useful model to have an instantaneous state jump.

This article connects smooth variational dynamics with event sensitivity. The longer Hybrid Tangent Spaces is companion material. The saltation construction and its limitations are established in Kong, Payne, Zhu and Johnson’s tutorial, especially Sections III-A and III-B.

Declare the Mode, Guard and Reset

In mode $i$, let $\dot x=f_i(x,t)$. A guard $g(x,t)=0$ triggers a transition to mode $j$ with reset $x^+=R(x^-,t)$. The vector fields here include the specified input policy; when that policy is state dependent, its derivatives belong in the closed-loop Jacobian.

Within a smooth mode, $\dot\xi=A_i(t)\xi$ propagates an initial-state variation. Independent input variations add the usual forcing term. At a state-triggered event, however, the perturbed trajectory generally reaches the guard at a different time. Applying only $D_xR$ misses that timing effect.

Assume a single isolated transition, differentiable fields, guard and reset near the event, and the same transition sequence for nearby trajectories. Write

\[ n=\nabla_xg(x^-,t_e),\qquad \gamma=\partial_tg+n^Tf^-\ne0. \]

Here $f^-=f_i(x^-,t_e)$ and $f^+=f_j(R(x^-,t_e),t_e)$. The condition $\gamma\ne0$ is transversality: the trajectory crosses the moving guard with nonzero normal speed.

Derive the Event-Time Correction

Let $\xi^-$ denote the variation at the nominal event time using the pre-event flow’s local continuation. Differentiating the guard equation yields the event-time variation

\[ \delta t_e=-\frac{n^T\xi^-}{\gamma}. \]

The perturbed state first flows for this extra time, undergoes the reset, and is then compared using the post-event continuation at the same nominal time. Therefore

\[ \xi^+=D_xR\xi^- +(D_xR f^-+\partial_tR-f^+)\delta t_e. \]

Substitution gives the saltation matrix:

\[ \begin{aligned} \xi^+&=\Xi\xi^-,\\ \Xi&=D_xR+ \frac{(f^+-D_xR f^--\partial_tR)n^T}{\partial_tg+n^Tf^-}. \end{aligned} \]

The autonomous guard/reset formula is the special case $\partial_tg=0$, $\partial_tR=0$. Explicit time dependence in the vector fields alone does not require these derivatives to be nonzero.

This is a same-time derivative map built from local flow continuations. It is not the difference between two physical trajectories evaluated at their separate impact instants. For an observation after both nearby events, compose the pre-event transition matrix, $\Xi$, and the post-event transition matrix. Direct parameter or input dependence in the guard or reset needs its own additional derivative terms.

Example: A Bouncing Point Mass

Take height $h$, vertical velocity $v$, gravity $g_0>0$, a stationary floor and restitution coefficient $e\in(0,1]$. Between bounces,

\[ f(h,v)=\begin{bmatrix}v\\-g_0\end{bmatrix}, \qquad R(h,v)=\begin{bmatrix}h\\-ev\end{bmatrix}. \]

The guard is $h=0$ with incoming $v^-<0$. Its normal is $(1,0)^T$, so

\[ \Xi=\begin{bmatrix} -e&0\\ -(1+e)g_0/v^-&-e \end{bmatrix}. \]

The off-diagonal entry records the effect of changed impact time on velocity at a common comparison time. $D_xR=\operatorname{diag}(1,-e)$ alone omits it. A perfectly inelastic event with $e=0$ requires an appropriate resting-contact mode rather than continued free fall through the floor. This is an idealized sensitivity example, not a club-ball collision model; restitution, contact geometry and relevant internal states must be supplied for that application.

Optimization and Stability Across Events

For a fixed regular mode sequence, compose smooth transition maps and saltation matrices in the forward sensitivity calculation, then use the corresponding derivatives in the backward optimization pass. If an update changes mode order, the previous derivative need not describe the new rollout. Recompute the event structure or use a formulation designed for such changes.

Contact-implicit formulations often use contact forces and complementarity conditions to allow contact timing to emerge. They do not necessarily introduce discrete mode variables. Relaxations and smoothing need convergence and physical-consistency checks; a successful numerical solve alone does not establish that the intended contact law was enforced.

Likewise, contraction between events is insufficient if an event expands perturbations too much. A compatible instantaneous metric comparison can use $\Xi^TM^+\Xi\preceq M^-$, together with the continuous-flow conditions and an appropriate event-sequence analysis. This is an additional condition, not a consequence of solving an optimal-control problem.

Grazing, Simultaneous Events and Zeno Behavior

As $\gamma$ approaches zero, timing sensitivity can become large; the denominator alone does not prove every matrix entry diverges because the numerator may also vanish. At exact grazing, the ordinary transverse-event derivation is unavailable. Nearby trajectories may touch, miss or enter different modes.

Simultaneous guards can create different event orders under arbitrarily small perturbations. A single saltation matrix may then need replacement by a piecewise or generalized derivative analysis. Infinitely many transitions in finite time (Zeno behavior) require a defined limiting or regularized model before an infinite composition is meaningful. State explicitly which cases the analysis excludes.

Golf Impact and the Measurement Question

A rigid impulse model treats club-ball contact as an idealized reset in velocity; a compliant model resolves deformation and force over a finite interval. Both require declared constitutive assumptions and validation appropriate to the outputs. Smooth pre-impact swing sensitivities do not by themselves predict ball launch: include the event timing and impact model, or integrate and differentiate the compliant contact model.

Also distinguish a clubhead state at a fixed time from its state when it reaches the ball. Changing torque or initial posture can change both the approach trajectory and contact time. Part 7 discusses how discrepancies can inform whether the current local prediction should be updated or rejected.

Citation

BibTeX citation: Download .bib

@online{untitled,
  author = {},
  title = {Tangent {Hyperplanes} {VI:} {Hybrid} {Systems}},
  url = {https://affinedrift.com/articles/tangent-hyperplanes-series/part-6-hybrid.html},
  langid = {en}
}

For attribution, please cite this work as:

“Tangent Hyperplanes VI: Hybrid Systems.” n.d. https://affinedrift.com/articles/tangent-hyperplanes-series/part-6-hybrid.html.

--- title: "Tangent Hyperplanes VI: Hybrid Systems" description: "How tangent-space reasoning changes across impacts, guards, and mode switches." aliases: - ../tangent-hyperplane-articles/Drafts_Original_Articles/Advanced_Integral_Expansions_Package/01_Hamiltonian_and_Adjoint.html - ../tangent-hyperplane-articles/Drafts_Original_Articles/Advanced_Integral_Expansions_Package/02_Numerical_Implementation_and_Attribution.html categories: - tangent-space --- ## Smooth Motion and Discrete Events A rigid impact model combines smooth motion between events with guards and reset maps at events. Other models, including compliant finite-duration contact, may represent the same physical interaction through continuous dynamics. Choose the representation before differentiating it; a physical collision does not force every useful model to have an instantaneous state jump. This article connects smooth variational dynamics with event sensitivity. The longer [Hybrid Tangent Spaces](../tangent-hyperplane-articles/Advanced/Hybrid_Tangent_Spaces.html) is companion material. The saltation construction and its limitations are established in [Kong, Payne, Zhu and Johnson's tutorial](https://arxiv.org/html/2306.06862v2), especially Sections III-A and III-B. ## Declare the Mode, Guard and Reset In mode $i$, let $\dot x=f_i(x,t)$. A guard $g(x,t)=0$ triggers a transition to mode $j$ with reset $x^+=R(x^-,t)$. The vector fields here include the specified input policy; when that policy is state dependent, its derivatives belong in the closed-loop Jacobian. Within a smooth mode, $\dot\xi=A_i(t)\xi$ propagates an initial-state variation. Independent input variations add the usual forcing term. At a state-triggered event, however, the perturbed trajectory generally reaches the guard at a different time. Applying only $D_xR$ misses that timing effect. Assume a single isolated transition, differentiable fields, guard and reset near the event, and the same transition sequence for nearby trajectories. Write $$ n=\nabla_xg(x^-,t_e),\qquad \gamma=\partial_tg+n^Tf^-\ne0. $$ Here $f^-=f_i(x^-,t_e)$ and $f^+=f_j(R(x^-,t_e),t_e)$. The condition $\gamma\ne0$ is transversality: the trajectory crosses the moving guard with nonzero normal speed. ## Derive the Event-Time Correction Let $\xi^-$ denote the variation at the nominal event time using the pre-event flow's local continuation. Differentiating the guard equation yields the event-time variation $$ \delta t_e=-\frac{n^T\xi^-}{\gamma}. $$ The perturbed state first flows for this extra time, undergoes the reset, and is then compared using the post-event continuation at the same nominal time. Therefore $$ \xi^+=D_xR\xi^- +(D_xR f^-+\partial_tR-f^+)\delta t_e. $$ Substitution gives the saltation matrix: $$ \begin{aligned} \xi^+&=\Xi\xi^-,\\ \Xi&=D_xR+ \frac{(f^+-D_xR f^--\partial_tR)n^T}{\partial_tg+n^Tf^-}. \end{aligned} $$ The autonomous guard/reset formula is the special case $\partial_tg=0$, $\partial_tR=0$. Explicit time dependence in the vector fields alone does not require these derivatives to be nonzero. This is a same-time derivative map built from local flow continuations. It is not the difference between two physical trajectories evaluated at their separate impact instants. For an observation after both nearby events, compose the pre-event transition matrix, $\Xi$, and the post-event transition matrix. Direct parameter or input dependence in the guard or reset needs its own additional derivative terms. ## Example: A Bouncing Point Mass Take height $h$, vertical velocity $v$, gravity $g_0>0$, a stationary floor and restitution coefficient $e\in(0,1]$. Between bounces, $$ f(h,v)=\begin{bmatrix}v\\-g_0\end{bmatrix}, \qquad R(h,v)=\begin{bmatrix}h\\-ev\end{bmatrix}. $$ The guard is $h=0$ with incoming $v^-<0$. Its normal is $(1,0)^T$, so $$ \Xi=\begin{bmatrix} -e&0\\ -(1+e)g_0/v^-&-e \end{bmatrix}. $$ The off-diagonal entry records the effect of changed impact time on velocity at a common comparison time. $D_xR=\operatorname{diag}(1,-e)$ alone omits it. A perfectly inelastic event with $e=0$ requires an appropriate resting-contact mode rather than continued free fall through the floor. This is an idealized sensitivity example, not a club-ball collision model; restitution, contact geometry and relevant internal states must be supplied for that application. ## Optimization and Stability Across Events For a fixed regular mode sequence, compose smooth transition maps and saltation matrices in the forward sensitivity calculation, then use the corresponding derivatives in the backward optimization pass. If an update changes mode order, the previous derivative need not describe the new rollout. Recompute the event structure or use a formulation designed for such changes. Contact-implicit formulations often use contact forces and complementarity conditions to allow contact timing to emerge. They do not necessarily introduce discrete mode variables. Relaxations and smoothing need convergence and physical-consistency checks; a successful numerical solve alone does not establish that the intended contact law was enforced. Likewise, contraction between events is insufficient if an event expands perturbations too much. A compatible instantaneous metric comparison can use $\Xi^TM^+\Xi\preceq M^-$, together with the continuous-flow conditions and an appropriate event-sequence analysis. This is an additional condition, not a consequence of solving an optimal-control problem. ## Grazing, Simultaneous Events and Zeno Behavior As $\gamma$ approaches zero, timing sensitivity can become large; the denominator alone does not prove every matrix entry diverges because the numerator may also vanish. At exact grazing, the ordinary transverse-event derivation is unavailable. Nearby trajectories may touch, miss or enter different modes. Simultaneous guards can create different event orders under arbitrarily small perturbations. A single saltation matrix may then need replacement by a piecewise or generalized derivative analysis. Infinitely many transitions in finite time (Zeno behavior) require a defined limiting or regularized model before an infinite composition is meaningful. State explicitly which cases the analysis excludes. ## Golf Impact and the Measurement Question A rigid impulse model treats club-ball contact as an idealized reset in velocity; a compliant model resolves deformation and force over a finite interval. Both require declared constitutive assumptions and validation appropriate to the outputs. Smooth pre-impact swing sensitivities do not by themselves predict ball launch: include the event timing and impact model, or integrate and differentiate the compliant contact model. Also distinguish a clubhead state at a fixed time from its state when it reaches the ball. Changing torque or initial posture can change both the approach trajectory and contact time. [Part 7](./part-7-residual-aware.html) discusses how discrepancies can inform whether the current local prediction should be updated or rejected. ## Related Articles ::: {.callout-note title="See Also"} - **[Tangent-Space Series](../../pages/tangent-hyperplanes.html)** — The series hub: full reading order, provenance, and companion material - **[Force and Mobility Ellipsoids](../force-mobility-matrices.html)** — State-dependent mobility across contact modes: the smooth-case picture that saltation maps interrupt - **[Tangent Hyperplanes V: Contraction](part-5-contraction.html)** — The previous part of the series - **[Tangent Hyperplanes VII: Residual-Aware Control](part-7-residual-aware.html)** — The next part of the series :::