Planar-to-Spatial Model Ladder
This page defines a comparison protocol for four nested model levels. It asks a limited question: which is the least complex declared level that satisfies a particular task metric? It does not rank entire models with one score.
The executable fixtures are analytic and synthetic; no participant data are used. They test the comparison protocol, not human performance. This work does not establish a global fidelity percentage, coaching advice, clinical guidance, causal attribution, or population-level conclusions.
Frozen Comparison Contract
All levels use a right-handed, pelvis-fixed frame and SI units. The shared state starts with club-path angle and rate. Each child appends coordinates while preserving the names, units, ordering, initialization, and event convention of its parent. The frozen event is the first descending crossing of club-path angle through zero radians.
| Level | Included Physics | Omitted Physics | Intended Use |
|---|---|---|---|
| Planar rigid | Planar rigid-link kinematics, inertial accounting, generalized loads | Out-of-plane motion, bilateral closure, shaft flexibility | Planar path and energy-accounting checks |
| Spatial open chain | Three-dimensional rigid-body kinematics and out-of-plane face orientation | Bilateral closure, hand-contact reaction split, shaft flexibility | Three-dimensional face and path checks |
| Spatial closed chain | Bilateral closure constraints and closed-chain reaction loads | Shaft flexibility and participant-specific tissue dynamics | Closure residual and bilateral load-share checks |
| Flexible shaft | One declared shaft bending coordinate and shaft strain energy | Higher shaft modes, aerodynamics, and participant-specific tissue dynamics | Shaft deflection and impact-state sensitivity checks |
The protocol revision is affinedrift.model-ladder/v1; the manufactured parameter revision is manufactured-analytic-parameters/v1. Results must label every quantity as measured, estimated, modeled, assumed, or unavailable. The source is MIT licensed. The fixtures contain no personal or capture data.
Exact Adjacent-Level Fixtures
The parity intervention sets every child-only coordinate to zero. The shared manufactured state is (club_path_angle, club_path_rate) = (0.25 rad, -3.0 rad/s). Exact prefix projections then produce these regression results:
| Projection | Maximum Shared-Coordinate Residual |
|---|---|
| Spatial open chain to planar rigid | 0 |
| Spatial closed chain to spatial open chain | 0 |
| Flexible shaft to spatial closed chain | 0 |
Zero projection residual validates the declared map only. It does not prove that the richer physics are negligible in a measured swing.
The manufactured flexible-shaft convergence fixture uses 1, 2, 4, and 8 modal terms. Its prescribed impact-speed errors are 0.08, 0.02, 0.005, and 0.00125 m/s. The exact factor-of-four decrease is an executable regression for the protocol; it is not a measured convergence rate for a physical shaft.
The Planar Rung, Animated
The viewer below plays one run of the lowest rung: the planar, rigid three-link teaching model in src/affine_control/golf_model.py. It starts at rest from a declared top-of-backswing pose. A constant torque of 100 N m acts at the hub and 30 N m at the middle joint, and the wrist joint gets none. The run stops when the club link first points straight down.
Each link is coloured by where its angular acceleration comes from at that instant: the drift (gravity and motion, with no torque) or the applied torque. The club link is driven only through the joints above it, so its colour shows how much of its swing the passive dynamics supply. The frames are computed in Python by scripts/generate_swing_viewer_data.py; the page only draws them.
- Drift: no torque
- Applied torque
The Spatial Rung, Arm Chain
The viewer below draws the cited three-dimensional arm chain JOHNSON_2015_ARM from src/affine_control/golf_model_3d.py: shoulder transverse flexion, forearm pronation and wrist deviation, with lengths and limits from Johnson (2015) (Johnson 2015). The motion is a prescribed sweep of the three joints through 80 percent of their nearer limit, not a swing. Drag the view slider to turn the scene and see the wrist leave the arm’s swing plane. Johnson’s 30 and 50 degree torso and swing-plane inclinations are viewer placement only and are not applied.
Read these caveats with the figure; they are part of the result.
| Joint | Axis | Limits (degrees) |
|---|
Task-Specific Metrics and Uncertainty
Each level records the same output categories, with metric-specific units and an uncertainty interval:
| Output Category | Example Declared Metric | Unit |
|---|---|---|
| Conserved quantities and residuals | Absolute energy residual | J |
| Kinematics | Club-path error | rad |
| Joint and generalized loads | Generalized-load error | N m |
| Club face and path | Face-orientation error | rad |
| Impact state | Impact-speed error | m/s |
| Runtime | Wall-clock time in a frozen benchmark environment | s |
| Conditioning | Scaled Jacobian condition number | 1 |
| Parameter sensitivity | Normalized local sensitivity | 1 |
The analytic fixture supplies deterministic modeled intervals for every category except runtime. Wall-clock runtime is unavailable until the engine, hardware, software revision, warm-up, repeat count, and timing method are frozen. Unavailable is a result, not a value to be imputed. A future engine comparison must also publish solver tolerances and convergence failures.
Minimum-Sufficient-Level Guidance
Selection starts at the simplest level and advances only when the entire declared uncertainty interval fails the task tolerance or a required output is unavailable.
| Task | First Level That Can Qualify in the Manufactured Fixture | When the Simpler Conclusion Fails |
|---|---|---|
| Planar club-path error | Planar rigid | The question requires out-of-plane orientation or closure loads |
| Three-dimensional face orientation | Spatial open chain | A planar model has no declared out-of-plane face coordinate |
| Bilateral load share | Spatial closed chain | An open chain cannot identify the declared closure reaction split |
| Shaft deflection at impact | Flexible shaft | A rigid shaft has no declared bending coordinate or strain energy |
| Participant transfer | None | No governed participant comparison is available |
This is task-specific sufficiency, not proof that a selected model is generally accurate. A more complex level may introduce conditioning, parameter, or solver uncertainty that outweighs its additional outputs.
Next Evidence Gate
The next stage is a governed cross-engine comparison against revision-pinned three-dimensional simulation or capture data. Before that stage can change any scientific conclusion, it must declare coordinate transformations, calibration, participant and trial inclusion, event alignment, task outcomes, uncertainty, solver settings, source revisions, privacy and licensing terms, and falsifiers. Until those records exist, the current findings remain manufactured protocol tests rather than empirical validation.
The executable contract lives in src/affine_control/model_ladder_protocol.py; exact fixtures live in src/affine_control/model_ladder_fixtures.py.
References
Citation
@online{untitled,
author = {},
title = {Planar-to-Spatial {Model} {Ladder}},
url = {https://affinedrift.com/models/model-ladder.html},
langid = {en}
}