Hybrid Impact-Contact and Event-Time Uncertainty

A source-bounded synthetic protocol comparing rigid, compliant, and hybrid club-ball contact models

This program asks how a declared contact law, mode transition, and uncertainty in the detected impact time change club–ball outcomes and any upstream attribution that uses them. It compares three reduced planar models against analytic and deterministic synthetic fixtures. The current result is a software and protocol feasibility study, not a calibrated club, ball, shaft, or golfer.

Scientific Authority Boundary

Every number below is synthetic-fixture evidence. The program provides no coaching, clinical, causal, population, or equipment-design authority. It does not identify injury risk, technique, intent, muscle contribution, or an optimal club. Qualification is outcome specific: no contact model is universally correct.

Primary-Source Register

Source	What It Supports Here	What It Does Not Authorize
Penner (2003) (Penner 2003)	Golf impact and flight-physics context	This reduced state, parameter set, or uncertainty interval
Cross (1999) (Cross 1999)	Impulse, restitution, friction, and implement–ball mechanics	Direct transfer of a bat/racket model to a golf fixture
Roberts, Jones, and Rothberg (2001) (Roberts et al. 2001)	A golf-specific contact-time measurement method and dependence on speed and ball construction	This event detector, solver law, or a link between measured time and perceived feel
Petersen and McPhee (2009) (Petersen and McPhee 2009)	Golf clubface/ball finite-element impact-model precedent	Promotion of this penalty law to a calibrated design model
McNally, McPhee, and Henrikson (2018) (McNally et al. 2018)	A declared comparison in which shaft coupling changes modeled launch conditions	A universal shaft correction or proof that either free-body or full-club treatment is always adequate
Kong et al. (2024) (Kong et al. 2024)	Guard/reset sensitivity and event-aware uncertainty transport in hybrid systems	This golf guard, reset map, or first-order envelope

The sources bound model choice and measurement questions; none validates the manufactured outputs. Roberts et al. found measured contact duration depended on ball construction and clubhead speed, and that measured duration did not track golfer perception in their study. That is a warning against replacing an instrumented event with a subjective label, not a population claim.

Preregistered Questions and Falsifiers

Protocol revision affinedrift.hybrid-impact-contact/v1 fixes three questions:

Question	Metric	Falsifier or Adverse Disposition
Does each model satisfy its own declared mechanics?	Paired-body momentum residual, restitution reset, force/impulse integral, and energy accounting	Negative if a balance or reset identity fails
Are reported outputs stable to numerical and event choices?	Fixed-step convergence plus a complete four-input sensitivity interval	Null if competing models or sensitivity choices do not support a distinct conclusion
Can the reduced single-contact result be promoted?	Calibrated measurements, outcome-specific validation, governed data, and held-out error	Unavailable until every physical and human gate exists

Negative, null, excluded, solver-failed, and unavailable cases stay in the ledger. A low residual or visually plausible launch cannot replace a failed contract.

Pre-Impact State and Frame Contract

The nominal contact frame is fixed before solving:

Field	Declaration
Origin	Nominal first contact point on the undeformed clubface
Normal	`+x clubface normal from club toward ball`
Tangent	`+y face tangent`
Spin	`+z right-hand rule`
Units	m, s, rad, N, kg
Club velocity	\((44, 0)\) m/s for the centered fixture; \((44, 2)\) m/s for the oblique fixture
Ball velocity and spin	\((0, 0)\) m/s and \(0\) rad/s
Reduced masses	Club \(0.200\) kg; ball \(0.04593\) kg
Ball geometry	Radius \(0.02135\) m; manufactured solid-sphere inertia

The masses and inertia are numerical fixtures, not equipment measurements. The state is a two-dimensional contact-coordinate reduction: off-center geometry, clubhead inertia tensor, face curvature, ball construction, shaft modes, and three-dimensional spin-axis dynamics remain outside it.

Let the closing speed be

\[ u_n = (v_c^- - v_b^-) \mathbin{\cdot} n > 0. \]

Zero contact, separating motion, a closing speed at or below the declared grazing threshold, and more than one candidate contact do not enter the nominal solver.

Rigid Impulse Model

The instantaneous model uses a Newton normal reset and a Coulomb-limited tangential impulse:

\[ J_n = \frac{(1+e)u_n}{1/m_c+1/m_b}, \qquad J_t = \operatorname{clip} \left( -\frac{s_t^-}{1/m_c+1/m_b+r_b^2/I_b}, -\mu J_n, \mu J_n \right). \]

Equal and opposite impulses update the club and ball. The ball angular update uses the declared \(+z\) sign. This model exposes a direct restitution parameter but has zero contact duration and cannot report a force waveform or deformation.

Compliant Contact Model

The finite-duration model integrates one Hunt–Crossley-like regularized normal law:

\[ F_n(\delta,\dot\delta) = k\,\delta^p \max\!\left(0,1+\chi\dot\delta\right), \qquad m_{\mathrm{eff}}\ddot\delta=-F_n. \]

The executable law is \(k\delta^p\max(0,1+\chi\dot\delta)\). The fixture uses \(p=1.5\), a fixed-step fourth-order Runge–Kutta integrator, and a declared separation event. Stiffness and damping are uncalibrated synthetic parameters. The law is useful for testing contact duration, peak force, impulse integration, and convergence; it is not a material model for a particular ball or face.

Hybrid Event Model

The hybrid model flows to a detected guard and then applies the rigid reset:

\[ x^- = \phi(\hat t_e+\Delta t_e, x_0), \qquad x^+ = R(x^-;e,\mu). \]

This makes event time part of the state-to-outcome map. The manufactured flow uses constant club acceleration only to expose sensitivity; it is not a swing estimate. Saltation analysis explains why a guard crossing carries timing information beyond the reset Jacobian (Kong et al. 2024). This implementation reports a complete finite parameter grid rather than claiming a first-order saltation approximation is adequate near every impact.

Solver and Event-Detection Policy

Boundary	Frozen Rule	Failure State
Contact guard	Signed gap crosses zero from positive to nonpositive while closing	No contact or separating
Sampling	200 kHz, bracketed linear timestamp interpolation	Missing bracket or nonfinite sample
Timing envelope	±50 µs event time; ±10 µs synchronization	Unreported timing uncertainty
Grazing	Closing speed must exceed 0.05 m/s	Grazing is rejected as ill-conditioned
Contact topology	Exactly one candidate contact	Multiple Contact is unavailable, not sequentialized silently
Compliant solve	Fixed step \(2\times10^{-6}\) s; separation before 3 ms and within 2,000 steps	Explicit solver failure; no partial output

Grazing makes event-time sensitivity singular or poorly conditioned. Multiple Contact can change transition order and impulse allocation. Neither case is rescued by choosing a preferred ordering after seeing the result.

Balance, Convergence, and Failure Contracts

For the centered manufactured state, both paired-body solvers have a reported linear-momentum residual below \(10^{-10}\) kg m/s. The rigid reset reproduces the declared restitution equation to numerical precision. Halving the compliant time step from 2 µs changes manufactured ball speed by approximately \(3.51\times10^{-6}\) m/s.

Model	Ball Normal Speed	Contact Time	Peak Normal Force	Energy Before / After	Interpretation
Rigid Impulse Model	63.693 m/s	0	unavailable	193.600 / 179.441 J	Analytic reset check
Compliant Contact Model	60.529 m/s	0.436 ms	13.196 kN	193.600 / 174.737 J	Uncalibrated force-law fixture
Hybrid Event Model, nominal time	63.693 m/s	0	unavailable	193.600 / 179.441 J	Nominal flow/reset agrees with rigid reset

Different synthetic energy loss or ball speed is not evidence that one model is closer to a real impact. Physical accuracy requires calibrated force, deflection, contact time, launch, and spin evidence over a declared envelope.

Event-Time and Parameter Uncertainty

The oblique fixture evaluates all \(3^4=81\) combinations of event time, restitution, friction, and face-normal angle. It reports the entire interval; there is no best-case selection.

Output	Nominal	Complete Manufactured Interval	Unit
Ball speed	63.695	62.199 to 65.176	m/s
Launch angle in the original frame	0.482	-0.347 to 1.304	degrees
Ball spin	-62.792	-86.895 to -38.670	rad/s

These bounds are conditional on only four finite ranges. They omit measurement bias, face curvature, impact location, shaft state, material-law error, three-dimensional inertia, ball construction, and correlated parameters. Therefore they are not confidence intervals or tolerance specifications.

Outcome-Specific Model Comparison

Intended Outcome	Minimum Needed Evidence	Model Limitation
Post-impact linear velocity	Calibrated pre/post states, frames, and timing	A rigid reset can fit velocity while hiding force history
Contact time and peak force	Calibrated high-bandwidth force/deformation measurement	An impulse model cannot provide these outputs
Spin and gear effect	Three-dimensional contact location, friction, inertia, and angular measurements	The planar fixture has one spin axis and no face curvature
Upstream attribution interval	Event-aligned pre-impact state covariance and a declared attribution model	Contact sensitivity does not identify a golfer’s intent or anatomical cause
Equipment response	Qualified ball, face, shaft, and boundary-condition models	Free-body and full-club assumptions are outcome and configuration dependent

Model selection is made per output and validation envelope. Agreement between two models on ball speed does not validate their force histories, spin, or upstream attribution.

Negative, Null, and Unavailable Results

Ledger State	Retained Result	Authority Limit
Supported	The rigid synthetic fixture satisfies its paired momentum and restitution equations	Software equation check only
Negative	The single-contact map rejects grazing input	Does not characterize real grazing contact
Null	The comparison identifies no universal model winner	Does not imply models are physically equivalent
Unavailable	Human, calibrated equipment, multiple-contact, and population conclusions	Cannot be replaced by a plausible simulation

Every numerical result retains the machine-readable origin synthetic-fixture; missing promoted evidence retains unavailable.

Upstream Attribution Boundary

An impact outcome is downstream of the pre-impact club/ball state, contact geometry, material law, and event time. If an UpstreamDrift analysis attributes delivery dynamics to a terminal impact metric, it must propagate the terminal metric’s interval and preserve the exact frame and event definition. A shift in ball speed after changing restitution or event timing is not evidence that an upstream joint torque, muscle, intention, or technique caused that shift.

Promotion and Further-Research Gates

The next evidence tier requires all of the following before any physical or human statement:

Calibrate clubface and ball geometry, mass properties, compliance, damping, friction, contact time, force bandwidth, and sensor synchronization.
Compare rigid, compliant, finite-element, and coupled shaft models on locked outcome-specific benchmarks, including off-center and three-dimensional hits.
Quantify solver convergence, parameter identifiability, correlated uncertainty, guard interpolation error, grazing, repeated, and simultaneous contacts.
Register equipment revisions, raw-data licensing, exclusions, null results, and held-out test criteria.
Obtain ethics, privacy, consent, safety, and independent approvals before any participant collection; software tests do not grant those authorities.

Until then, the human tier remains unavailable.

Reproducible Implementation

The contract, fixtures, solvers, uncertainty sweep, and tests live in src/affine_control/impact_contact_protocol.py, impact_contact_fixtures.py, impact_contact_models.py, impact_contact_uncertainty.py, and tests/test_hybrid_impact_contact_protocol.py. Exact SHA-256 evidence for this page and those files is recorded in the rendered-route claim audit. A checksum attests reviewed bytes; it does not establish scientific truth.

References

Cross, Rod. 1999. “Impact of a Ball with a Bat or Racket.” American Journal of Physics 67 (8): 692–702. https://doi.org/10.1119/1.19354.

Kong, Nathan J., J. Joe Payne, James Zhu, and Aaron M. Johnson. 2024. “Saltation Matrices: The Essential Tool for Linearizing Hybrid Dynamical Systems.” Proceedings of the IEEE 112 (6): 585–608. https://doi.org/10.1109/JPROC.2024.3440211.

McNally, William, John McPhee, and Erik Henrikson. 2018. “The Golf Shaft’s Influence on Clubhead-Ball Impact Dynamics.” Proceedings 2 (6): 245. https://doi.org/10.3390/proceedings2060245.

Penner, A. Raymond. 2003. “The Physics of Golf.” Reports on Progress in Physics, ahead of print. https://doi.org/10.1088/0034-4885/66/2/202.

Petersen, Willem, and John McPhee. 2009. “Shape Optimization of Golf Clubface Using Finite Element Impact Models.” Sports Engineering 12 (2): 77–85. https://doi.org/10.1007/s12283-009-0030-7.

Roberts, J. R., R. Jones, and S. J. Rothberg. 2001. “Measurement of Contact Time in Short Duration Sports Ball Impacts: An Experimental Method and Correlation with the Perceptions of Elite Golfers.” Sports Engineering 4 (4): 191–203. https://doi.org/10.1046/j.1460-2687.2001.00084.x.