Proximal-to-Distal Energy Transfer in the Golf Swing
For a book-like introduction to energy, forces, timing, torque, variability, and falsifiability, begin with How a Golf Swing Carries Energy. It uses the same open evidence while explaining each model boundary in plain language.
In a declared swing model, how does the timing of distal actuation change club delivery, and which parts of that result survive explicit counterfactual and robustness tests?
What the Simplified Model Shows About Early Wrist Drive
In the baseline two-link simulation, driving the wrist joint from the start produced 15% less clubhead speed than a passive wrist. The result illustrates how early uncocking can increase rotational inertia before proximal speed has developed. It is a model comparison, not an estimate of what any individual golfer will gain or lose.
Retain Early, Release Late
Within the finite tested command family, the highest modeled speeds occurred when the wrist remained folded early and received positive drive later. Relative to the passive baseline, the grid-selected late-drive programs were 12% to 14% faster. Most late club-energy gain in the model came through joint-force transfer rather than direct wrist work. This is not a continuous optimum or a coaching instruction.
The hand-path drift/control extension is published from the exact UpstreamDrift merge commit 69eb7e9d. The local evidence snapshot records the full 40-character commit, compact results for all three model tiers, regeneration commands, and SHA-256 digests for every figure reproduced here. The offline command python scripts/check_proximal_distal_evidence.py --require-pinned verifies the pin, artifact integrity, finite estimands, and force/couple/power/work closure. The open evidence supports model-mechanics statements; it does not identify muscle activation, metabolic effort, perceived effort, or a universal coaching strategy.
The canonical paper-wide audit is pinned to UpstreamDrift commit a1a61399. It adjudicates 1,063 of 1,063 narrative candidates against 295 bounded claim records, leaving zero unadjudicated candidates. The local audit snapshot records the exact registry and PDF hashes and the principal adverse findings.
Release review is now complete as well: all 40 of 40 release claims have been reviewed, up from ten open reviews at the previous pin. Review completion is a traceability property, not a scientific verdict. It means each release claim carries explicit atomic support, controls, falsifiers, boundaries, and a declared remaining gate — and all 40 still carry a scientifically open gate. No claim here is validated by having been reviewed.
The reviewed handwritten momentum-transfer agenda is also preserved as nine independently tracked points. Eight have bounded model answers, partial answers, or a supported rejection of a general rule. MTQ-06, timing precision, remains unresolved beyond an adverse registered planar comparison and at the human level. Casting has a bounded answer: no single event definition is scientifically sufficient, so the next comparison must preregister several definitions. Every point has a decisive next test, falsifier, data gate, model plan, and participant-held-out human stage.
| Agenda Point | Present Answer | Decisive Next Gate |
|---|---|---|
| Drift Contribution | Estimand-, frame-, and window-dependent | Common cross-tier ledger and held-out bilateral wrenches |
| Geometry Dependencies | Partial reduced spatial controls | Calibrated compliant contact and independent forward engines |
| Casting | Definition-dependent rather than one universal event | Preregistered event definitions under matched state, work, and load |
| Early Proximal Acceleration | Pointwise and nonmonotonic | Full-delivery-state-matched forward factorial intervention |
| Segment Release | Objective- and constraint-dependent | Spatial viable-region map with impact outcomes |
| Timing Precision | Unresolved beyond the adverse planar screen | Common-phase spatial and participant-held-out comparison |
| Self-Correction and Noise | No sustained recovery in 60 registered planar cases | Attraction regions with contact, saturation, observers, and holdout |
| Proximal-Velocity Maximization | Rejected as a general planar rule | Causal spatial dose response with full delivery-state matching |
| Slack | Partial answers only for typed synthetic constitutive classes | One class at a time with contact, shaft, tissue, and activation sensing |
Audit completion means that each statement has an explicit evidence boundary; it does not validate a universal human or coaching strategy. In particular, proximal or torso speed is not a standalone transfer rule, the current coupled shaft baseline fails its quantitative small-deflection screen, a universal passive-shaft speed benefit is rejected at the declared synthetic first-mode tier, a local task-Jacobian partition is not evidence of a neural synergy, and governed bilateral human grip-wrench validation remains unexecuted. The Biomechanics and Nonlinear Control collection review also remains blocked on manual NotebookLM reauthentication; collection output will not be treated as evidence without independent verification of the original sources.
1 Introduction
1.1 Motivation
Clubhead speed at impact is the dominant controllable determinant of drive distance, and it correlates strongly with playing standard (Fradkin et al. 2004; Hellström 2009; Hume et al. 2005). The golf downswing is among the most-studied examples of a sequenced multi-segment rotation in sport biomechanics (McPhee 2022; Dillman and Lange 1994). Across measurement systems and populations, the downswing of a skilled golfer shows a characteristic order: the pelvis reaches its peak angular speed first, then the thorax, then the lead arm, and finally the club, with each successive segment peaking faster than the one before (Cheetham et al. 2008; Tinmark et al. 2010). This kinematic sequence, an instance of the general proximal-to-distal sequencing observed in throwing and striking skills (Putnam 1993), is routinely interpreted as a visible signature of outward mechanical-energy transfer. That interpretation has also influenced applied swing descriptions (Smith et al. 2015).
That interpretation leaves a first-order question unanswered:
Within a declared model or measured swing, does earlier distal actuation improve delivery, or can delayed distal actuation permit a more effective late handoff?
For experimental contrast, the finite model inputs are labeled “early drive” and “retain early/release late.” They are not a survey of coaching doctrine or coaching cues. The comparison tests a behavior established in double-pendulum studies since the 1960s: higher modeled clubhead speeds can occur when distal drive is delayed (Cochran and Stobbs 1968; Jorgensen 1970; Milburn 1982; Pickering and Vickers 1999; Sprigings and Neal 2000; Sprigings and MacKenzie 2002). The contribution here is a reproducible force, power, work, and counterfactual audit rather than a claim of novelty for delayed release.
This study approaches the question with counterfactual dynamics. A pointwise ZTCF sample, or pointwise drift vector, is the acceleration the model evaluates at one achieved state with applied control removed. A stitched pointwise ZTCF trace collects those same-state samples without integrating them. A forward or branched ZTCF trajectory integrates the drift after a torque killswitch and is required to test persistence. The control contribution is the same-state difference between total and drift acceleration. A ZVCF is not the control contribution: it is a separate zero-velocity construction used to isolate configuration-dependent terms under a declared convention.
This article applies the canonical definitions in the Zero-Torque Counterfactual glossary. The pointwise ZTCF sample is \(f(\mathbf{x}) = M(\mathbf{q})^{-1}(-C(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}} - \mathbf{g}(\mathbf{q}))\), while the same-state control contribution is \(G(\mathbf{x})u\). A stitched trace evaluates \(f\) at successive achieved states; it must not be used to infer persistence. A forward or branched ZTCF integrates \(f\) after setting the declared applied torque channels to zero. ZVCF evaluates the declared model at zero velocity and must not be relabeled as pure control.
1.2 Research Questions
The document is organized around four questions:
- RQ1 — Mechanism. By what mechanical pathways does energy generated proximally reach the club, and what does each pathway imply about the timing of the handoff? (Section 2)
- RQ2 — Evidence. Where does the P→D transfer occur in measured swings, and how do skilled and less-skilled players differ — in sequence order, in magnitudes, or in transfer efficiency? (Section 3)
- RQ3 — Formalization. How can “transfer early” versus “retain-early/release-late” be stated as falsifiable claims about measurable quantities, and what do the ZTCF/ZVCF counterfactuals predict under each? (Section 5)
- RQ4 — Model test. What does a two-link model show when distal-handoff timing is manipulated directly and energy accounting and counterfactual decompositions are computed along each trajectory? (Section 9, Section 10, Section 12)
- RQ5 — Attribution. How much hand-path and joint force, impulse, power, and work is associated with pointwise drift versus same-state control, and how does that balance change with model topology, joint, and time window? (Section 10.5)
- RQ6 — Allocation and Preload. For an identical club-moment task, how do arm-dominant and wrist-dominant actuator allocations change internal loading, and under what declared transmission model does preload preserve torque continuity? (Section 6)
- RQ7 — Representation and Biology. Which conclusions survive frame and reference-point changes, how does one net joint moment map to many muscle activations, and which questions belong to MuJoCo, Pinocchio, Drake, OpenSim, or MyoSuite? (Section 7)
2 Mechanics of Sequenced Swings
2.1 The Summation-of-Speed Principle
Sequential segment motion in human movement dates to Bunn’s principle of summation of speed (Bunn 1972). When segments connected in series accelerate in sequence, the distal tip speed is not merely the sum of independent segment rotation rates; it is the compound result of base rotation carrying distal axes plus distal rotation relative to those axes (Putnam 1991, 1993).
In a planar 2-DOF chain with arm length \(L_1\) and club length \(L_2\), the clubhead velocity vector is: \[ \mathbf{v}_{\text{head}} = L_1 \dot{\theta}_1 \mathbf{u}_1^\perp + L_2 (\dot{\theta}_1 + \dot{\theta}_2) \mathbf{u}_2^\perp, \] where \(\mathbf{u}_1^\perp, \mathbf{u}_2^\perp\) are unit tangent vectors in the swing plane. The speed magnitude squared is: \[ v_{\text{head}}^2 = L_1^2 \dot{\theta}_1^2 + L_2^2 (\dot{\theta}_1 + \dot{\theta}_2)^2 + 2 L_1 L_2 \dot{\theta}_1 (\dot{\theta}_1 + \dot{\theta}_2) \cos\theta_2. \] The cross-term \(2 L_1 L_2 \dot{\theta}_1 (\dot{\theta}_1 + \dot{\theta}_2) \cos\theta_2\) reveals the geometric leverage of wrist cock \(\theta_2\): when \(\theta_2\) is near \(-\pi/2\) (wrist cocked), the cross-term is small; as \(\theta_2 \to 0\) (wrist uncocked), \(\cos\theta_2 \to 1\), maximizing distal speed for given segment rates.
2.2 Interaction Dynamics and Torque Handoffs
The equations of motion for the planar double pendulum express the causal coupling between joint torques and segment accelerations: \[ M(\mathbf{q}) \ddot{\mathbf{q}} + C(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}} + \mathbf{g}(\mathbf{q}) = \boldsymbol{\tau}. \] Explicitly expanding for the wrist acceleration \(\ddot{\theta}_2\): \[ \ddot{\theta}_2 = \frac{\tau_2 - C_{21}\dot{\theta}_1^2 - g_2(\mathbf{q})}{M_{22}} - \frac{M_{21}}{M_{22}}\ddot{\theta}_1. \] The term \(-\frac{M_{21}}{M_{22}}\ddot{\theta}_1\) is the inertial interaction acceleration. Because \(M_{21} = m_2 L_1 L_{c2} \cos\theta_2 + I_2 > 0\) for typical geometries, accelerating the arm (\(\ddot{\theta}_1 > 0\)) exerts an inertial torque on the wrist that acts to increase wrist cock (maintain lag). Conversely, decelerating the arm (\(\ddot{\theta}_1 < 0\)) uncocks the wrist and transfers energy to the club without requiring applied distal generalized torque in this model (Putnam 1993; Herring and Chapman 1992). This does not imply zero distal muscle force because co-contraction and passive structural loads are not identified by the generalized-torque counterfactual.
2.3 Parametric Energy Transfer
As White (White 2006) and Miura (Miura 2001) demonstrated, the dominant distal energy gain late in the downswing is parametric: the inward centripetal force exerted by the arm on the wrist joint performs mechanical work on the club segment as the hub path tightens and the club rotates toward alignment. Measured and optimized hub-path analyses provide golf-specific context for this geometric mechanism (Nesbit and McGinnis 2009; Nesbit and McGinnis 2014). The power delivered by this joint force is: \[ P_{\text{joint}} = \mathbf{F}_{\text{wrist}} \cdot \mathbf{v}_{\text{wrist}}. \] Proximal speed \(\dot{\theta}_1\) contributes to the centripetal acceleration field (\(L_1 \dot{\theta}_1^2\)), but high speed alone does not guarantee positive transfer. Force direction, hand velocity, club orientation, relative club rate, and swing phase determine the sign of interface power. The phase-resolved test below replaces the earlier unconditional statement with an explicit state-matched counterfactual.
3 Empirical Evidence in Golfers
Empirical kinetic and kinematic studies confirm the P→D energy flow structure. Nesbit and Serrano (Nesbit and Serrano 2005) showed that the torso and hips contribute 68.7–72.2% of total swing work, while shoulder and wrist joints contribute 24.3–28%. Power generation is concentrated early (torso), while clubhead speed conversion occurs late.
Recent findings by Rachnavy et al. (Rachnavy et al. 2026) (Frontiers in Sports and Active Living, 2026, DOI: 10.3389/fspor.2026.1790645) demonstrate that foot-ground interaction variables alone account for 35.5% of clubhead speed variance, rising to 75.4% when trunk sequencing and impulse-based energy transfer efficiency are included. Mediation analysis confirms that transfer efficiency, rather than sequence order alone, distinguishes high-performing swings.
Segmental kinetic energy sequencing studies by Kenny et al. (Kenny et al. 2008) and Anderson et al. (Anderson et al. 2006) further document outward energy migration, showing that energy arrives late in the club after proximal peaks have passed.
3.1 Force Along the Hand Path
MacKenzie, McCourt, and Champoux (MacKenzie et al. 2020) analyzed 76 amateur golfers using three-dimensional club inverse dynamics. They reported that average force along the hand path was strongly associated with clubhead speed (\(r=.96\) in their sample) and defined it through linear work divided by hand-path length. This is an important net mechanical result. It does not identify biological effort: the reconstructed grip force is an equivalent net force on the club, and a coupled model can attribute part of that load to configuration- and velocity-dependent interaction dynamics.
For hand velocity \(\mathbf{v}_H\) above a declared threshold \(\epsilon\), define
\[ \mathbf{e}_t = \frac{\mathbf{v}_H}{\lVert\mathbf{v}_H\rVert}, \qquad F_{\parallel} = \mathbf{F}\mathbin{\cdot}\mathbf{e}_t. \]
The MacKenzie-compatible path average is
\[ \overline{F}_{\parallel} = \frac{\int F_{\parallel}\lVert\mathbf{v}_H\rVert\,dt} {\int \lVert\mathbf{v}_H\rVert\,dt} = \frac{W_{\mathrm{linear}}}{L_H}. \]
It is distinct from time-average force and from impulse. The planned attribution therefore reports path length, signed path-average force, vector impulse, signed/positive/negative/absolute tangent impulse, force and couple power, and cumulative work as separate estimands.
Neither total force nor the modeled control residual identifies muscle activation, metabolic cost, or perceived effort. Muscle redundancy, co-contraction, passive tissue impedance, activation history, and internal two-hand loading can produce the same net wrench. Physiological interpretation requires independent measurements or an explicitly declared actuator-cost model.
4 Proximal Velocity and Drift-Mediated Transfer
The motivating observation is that releasing while the proximal system moves rapidly can appear to produce less grip braking and a cleaner outward fling. The reduced model can test a narrow form of that proposal. Its \(\dot{\theta}_1\) coordinate is proximal-link angular velocity. It is not an anatomical shoulder or thorax measurement. Drift means the achieved-state acceleration contribution when modeled controls are zeroed; it is not a claim that a human motion is passive.
Changing proximal rate is not a unique intervention, so the study registers two velocity contracts:
- Preserve Relative Club Rate: hold \(\dot q_2-\dot q_1\) fixed. The club is co-transported with the proximal link.
- Preserve Absolute Club Rate: hold laboratory-frame \(\dot q_2\) fixed. The relative rate changes as proximal velocity changes.
Five swing phases and nine proximal rates are evaluated under each contract, for 90 exact pointwise cases. Configuration and torque are matched within a phase. Reaction force is independently reconstructed for total, drift, and control contributions, with componentwise closure required before reporting power.
The result rejects a phase-independent benefit. With relative rate held, the drift-power slope is \(-8.25\) W/(rad/s) in mid-downswing, then \(+65.06\) in delivery and \(+283.74\) before impact. With absolute club rate held, the delivery slope reverses to \(-5.96\) W/(rad/s), while the pre-impact slope remains strongly positive at \(+184.58\) W/(rad/s). Thus high proximal speed helps most reliably in the late reference state, and the delivery conclusion depends on relative motion.
The operational braking measure is negative grip power \(P_G=\mathbf F_G\cdot\mathbf v_G\) and its time integral, not force magnitude. Mid-downswing reaches \(-148.8\) W under relative-rate matching and \(-296.5\) W under absolute-rate matching. Delivery has no negative sample under relative-rate matching but reaches \(-130.2\) W under absolute-rate matching. A large force perpendicular to grip velocity can transfer little power; a smaller opposing force can remove energy.
A trajectory-level grid then varies proximal-drive cut time, post-cut proximal torque, and wrist-release time. Only 26 of 60 programs reach club vertical inside the registered delivery window. Among those valid trials, greater proximal-link velocity at release correlates with lower impact speed (\(r=-0.692\)) and greater negative grip work (\(r=+0.943\)). A descriptive standardized regression that includes release time, proximal-drive cut time, and post-cut torque retains a negative release-velocity coefficient (\(\beta=-0.514\)). These are coupled model associations, not causal estimates, but they directly reject the idea that waiting for a higher proximal-link rate is sufficient.
The fastest valid program reaches 38.85 m/s with 6.11 J of negative grip work and a 315.39 N peak grip force. No sampled program is best on speed, braking work, and peak force simultaneously. Several nominal programs are identical because their proximal-drive cuts occur after impact; they remain visible for auditability.
The supported strategy is conditional: build proximal speed before favorable geometry, enter delivery with compatible relative club motion, reduce direct distal opposition only when drift power is positive, and optimize club speed together with negative grip work, peak grip force, face/path behavior, and timing robustness. Torso speed is a state and possible control target, not the objective itself. A rapidly rotating torso may coexist with positive transfer, but the fixed-hub model cannot establish its causal benefit.
A qualified two-hand extension must retain torso or hub angular motion, common- and differential-mode hand velocities, grip separation, bilateral forces, direct wrist moments, arm geometry, hand-path curvature, club relative rate, and shaft deformation. The current fixed-shoulder two-arm and translating base tiers do not answer the rotating-torso question. A forward rotating-base, two-hand model and synchronized human kinematics plus bilateral grip wrenches are required to test it.
This hypothesis is weakened if the late positive slope disappears under small parameter or configuration perturbations, if negative grip work or peak force dominates the speed gain, if matching distal laboratory-frame state removes the effect, or if a two-hand model attributes the gain to direct wrist work rather than interaction-force transport.
5 The Timing Question and Counterfactual Framework
5.1 Formalizing the Strategies
We contrast two explicit strategies for distal energy handoff: - S1 (Early Drive): Apply positive wrist torque \(\tau_2 > 0\) immediately after transition (\(t_{\text{on}} = 0\)). - S2 (Retain Early, Release Late): Apply zero or negative wrist torque \(\tau_2 \le 0\) early (retaining wrist cock against centrifugal opening), releasing positive torque \(\tau_2 > 0\) late in the downswing.
5.2 What Falsifies S2?
Strategy S2 would be falsified if early distal drive produced higher clubhead speed than late drive under matched proximal torque effort \(\tau_1(t)\).
5.3 Pointwise vs. Re-Simulation Counterfactuals
The same-state affine decomposition splits instantaneous acceleration into drift and control: \[
\ddot{\mathbf{q}}(t) = \underbrace{M(\mathbf{q})^{-1}(-C(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}} - \mathbf{g}(\mathbf{q}))}_{\text{Pointwise Drift}} + \underbrace{M(\mathbf{q})^{-1}\boldsymbol{\tau}}_{\text{Same-State Control}}.
\] The Python implementation in simulation_backends/ztcf_zvcf.py evaluates this split pointwise along sampled trajectories. It is an instantaneous tangent-field statement, not a future zero-torque trajectory. Separate matched-state forward killswitch ensembles quantify nonlinear trajectory divergence after the command is removed.
5.4 Completed Model Ladder and Reporting Contract
The completed extension uses three declared tiers: an exact planar double pendulum, a one-arm mobile-hub/three-link point-mass model, and a mechanically closed two-arm/floating-club model. The first two are forward trajectories. The two-arm case is a prescribed, constraint-consistent local kinematic sweep and must not be interpreted as realized swing work or a clubhead-speed prediction.
At every reported interface, total equals drift plus control for force, couple, path projection, power, impulse, and work to numerical tolerance. Joint tables retain signs and normalized-time quartiles rather than reducing vectors to one norm or relabeling quartiles as anatomical swing phases. Signed shares are not used near zero denominators; magnitude shares and cancellation indices are reported separately. In the two-hand tier, the common mode is \(\mathbf{F}_R+\mathbf{F}_L\) and the differential mode is \((\mathbf{F}_R-\mathbf{F}_L)/2\), preserving internal loading that a net club wrench cannot identify.
The exact three-dimensional sensor map sharpens that boundary. Two point forces provide six unknown components, but their net wrench has rank five: an equal-and-opposite axial force mode along the hand-separation line is invisible. One independently measured internal axial scalar closes that point-force rank gap. If each hand may also apply a full six-axis wrench, however, the mapping from 12 bilateral components to one six-component net club wrench has rank six and nullity six. The result is unchanged by a consistent proper rotation and persists across the registered 0.06–0.30 m grip-span sweep. This is structural, instantaneous linear identifiability; it does not identify muscle or scapular strategy, establish practical recovery under sensor noise, or validate a human allocation. The executable evidence therefore qualifies the measurement plan rather than replacing bilateral six-axis grip sensing.
A trajectory-level synthetic point-force sensor qualification then tests that plan under controlled failure modes. Recovering hand allocation from net wrench alone closes the net wrench numerically yet produces 11.86 N allocation RMSE and 29.05 N axial-mode RMSE. In the declared synthetic cases, calibrated cross-talk reduces allocation RMSE from 0.94 N to 0.15 N, while tracking an 8-mm contact migration removes the 2.02 N fixed-contact bias. The combined registered case has 1.02 N allocation RMSE and 3.87 N 95th-percentile error across 32 seeded trials and 301 samples. These results make net-wrench closure, cross-talk calibration, and contact-center tracking explicit acceptance tests. They are machine-readable synthetic evidence, not a device calibration or human validation result; full bilateral wrenches, distributed pressure, anatomy, intentionality, and coaching inference remain untested.
5.4.1 Subject-Scaled Contact Closure Is an Independent Gate
The next spatial audit asks a more elementary question than whether a contact Jacobian has full local rank: do the anatomical hand points occupy the declared club-contact points at the evaluated state? Six deterministic de Leva design profiles, three grip spans, and 61 states per case were evaluated as engineering test points rather than participants. Their prescribed hand points miss the declared contacts by 0.171–0.616 m, with a median miss of 0.405 m; no sample meets the registered 5 mm closure tolerance.
Every 6 by 20 bilateral contact Jacobian nevertheless has rank six, with condition number 5.35–6.40. Thus local rank does not prove contact closure. It answers whether six local closure directions are independently available, not whether the prescribed configuration already satisfies those closures. The pinned adverse evidence also retains the rank-five point-force map, rank-six axial augmentation, and linear grip-span/couple scaling. None of those algebraic controls repairs the open anatomical contacts.
The bounded closed-contact inverse kinematics follow-up holds the club coordinates at 13 prescribed phase samples and solves 14 reduced body and arm coordinates. It closes 234 of 234 profile–span–phase samples to at most \(1.16\times10^{-10}\) m. Every achieved contact Jacobian retains rank six, the minimum broad engineering joint-limit margin is 0.103 rad, the minimum coarse bounding-sphere clearance is 0.0309 m, and a deliberately unreachable 2.0 m grip-span control fails closure. The closed-contact evidence therefore resolves whether this declared reduced tree can satisfy the registered bilateral point-contact equations; it does not erase the adverse prescribed-state result.
This is a necessary-condition screen, not an anatomical or performance result. Its joint limits are engineering guards rather than clinical ranges; its collisions are coarse spheres rather than mesh anatomy; and it omits scapular motion, forearm rotation, multi-axis wrists, fingers, and distributed grip contact. It does not establish anatomical feasibility, force transfer, passivity, timing demand, useful slack, delivery benefit, or a human strategy. The next admissible rung is calibrated compliant forward contact from these closed states, repeated with subject-specific anatomy and independent-engine reversal, killswitch, power, and work–energy controls.
5.4.2 Paired Scapulothoracic Geometry Screen
The next intervention holds the trunk and all six club coordinates fixed and compares fixed shoulder centers with four bounded scapular coordinates per side: protraction, elevation, upward rotation, and winging. The mobile branch is nested exactly around the fixed branch. It is informed by an ellipsoid-constrained shoulder model, but it is a reduced kinematic surrogate, not a reproduction of subject anatomy.
Across 54 paired profile–span–phase states, the fixed branch reaches the 0.5 mm bilateral tolerance in 0 of 54 cases. The mobile branch reaches the residual in 31 of 54, but only 16 of 54 also pass the separate solver- termination gate. Twenty-eight cases activate at least one screening bound, and the maximum shoulder-center excursion is 0.101 m. A retained 2.0 m adverse grip span still fails, with 0.480 m bilateral error. The pinned MT-E09 evidence therefore supports a model-structure effect while retaining numerical and range boundaries.
Both contact Jacobians have rank six. With the added scapular coordinates, local coordinate nullity rises from two to ten, so bilateral contact position does not identify how motion is divided between scapular and glenohumeral coordinates. Scapular mobility changes modeled reachability, but this result does not establish anatomy, muscle action, contact force, power, work, passive transfer, club delivery, or a human strategy. Those claims require a validated articulated shoulder, subject-specific geometry where governed, calibrated distributed grip contact, paired forward dynamics, and held-out human data.
5.4.3 From Closed States to Bounded Articulated Forward Contact
The articulated tier is qualified one rung at a time, and each rung is a separate registered gate rather than a step toward a foregone conclusion.
First, the 20-coordinate articulated tree is checked as dynamics, not merely as geometry. Across the 234 closed common states, native MuJoCo and Pinocchio agree on the mass matrix, bias terms, and inverse dynamics to relative errors of \(8.6\times10^{-13}\), \(1.4\times10^{-12}\), and \(1.8\times10^{-12}\); the mass matrix is exactly symmetric and its smallest eigenvalue stays positive at \(1.39\times10^{-4}\). These are common-state operators, not a forward trajectory, and net joint dynamics do not identify muscle action.
Second, bilateral compliant point forces are projected into that tree at the same states with zero integration steps. Peak contact force reaches 2.71 N, action–reaction residual is exactly zero, virtual-power residual is \(3.9\times10^{-16}\) W, and contact dissipation power is negative everywhere (between \(-0.0906\) and \(-0.0892\) W), so the declared contact law removes energy rather than supplying it. A zero-preload control produces at most \(2.1\times10^{-7}\) N. Initial accelerations match across engines to \(4.9\times10^{-13}\) relative error.
Third, those states are integrated through a bounded 5 ms horizon — 756 trajectories over 18 states, seven variants, and three timesteps. Peak contact force reaches 17.76 N, maximum attachment separation stays at 1.23 mm against a 10 mm screening threshold, and the normalized work–energy residual refines monotonically from 0.00738 to 0.00170 to 0.000854 as the step halves. Engine parity holds to \(1.5\times10^{-14}\) on trajectories.
The horizon is the boundary. Five milliseconds is a numerical qualification interval, not late downswing and not impact; absence of a registered failure before 5 ms is right-censored and establishes nothing beyond it. The attachments are bilateral Kelvin–Voigt points, not unilateral biological contact.
Fourth, typed unilateral slack is exercised on the same tree: 1,944 trajectories over 18 conditions with 0.5 mm and 1.5 mm dead zones. Opening events occur in 108 cells and reattachment in 216, with at most two transitions per cell and peak contact force of 23.64 N. Residuals again refine monotonically (0.01923 to 0.01097 to 0.00480) and active-set parity failures are zero. Contact events can therefore be represented and compared under matched states — but the typed-slack evidence does not identify timing economy, self-correction, biological slack, or intent.
5.4.4 Distributed Grip and Passive Shaft Change the Model, Not the Verdict
Two further pathways replace idealized elements with distributed ones, and both were qualified against the same registered gates.
The distributed grip study replaces one point per hand with one, three, or five tension fibers while holding total grip stiffness at 1800 N/m and damping at 18 N\(\cdot\)s/m, so station count changes without silently stiffening the grip. Across 288 trajectories nested at 4, 10, 25, and 50 ms, peak station force reaches 4.03 N and the worst station carries 77.0% of the load. No opening transitions occur. The time residual refines 0.03119 to 0.01560, and the three-versus-five-station difference at the fine step is 0.00355, so the discretization is converging. This establishes discretization sensitivity; it does not identify measured grip pressure, finger anatomy, friction, or tissue.
Two consequences of that follow, and both cut against the tier’s own headline question. Because no station ever opened, the question this tier exists to answer — whether the single-point open/taut conclusions persist once force is spread across several stations — is untested here rather than answered. Opening and reattachment events occur only in the single-point typed-slack study above. And because friction is absent from the contact law entirely, a distributed grip in this model cannot slip; it can only stretch. Both are tracked upstream as open work in the distributed contact research registry, and neither should be read as evidence that distribution does not matter.
The passive shaft study adds two tip-normalized first bending modes plus tip twist. Bending inherits a 24-element finite-element authority at 5.2399 Hz with zero relative error against that reference; declared tapered hollow-section torsion is 70.1260 Hz. Across 384 trajectories, maximum tip deflection is 1.696 mm and maximum twist is 0.00186 rad — small, as the linear-domain screen requires. Two coarse-step torsion probes at 1.0 ms and 0.50 ms left the declared linear domain and are excluded and reported rather than quietly dropped; the atlas uses the 0.25/0.125 ms pair, whose residual refines 0.00780 to 0.00390.
The headline result is adverse to the intuitive reading. Of 384 coupled-versus-rigid cells, only 126 match within 5% on both peak load and dissipated work. Among those matched cells the final club-speed difference spans \(-0.0285\) to \(+0.0212\) m/s — negative and positive. A universal passive-shaft speed benefit is therefore rejected at this tier. The shaft atlas also retains its own limits: the structural values are synthetic references and not equipment calibration; a committed six-mode beam shows materially larger discrepancy under short high-frequency loading than under slow loading; ground reaction and free moment are absent from this tier entirely; and post-hoc matching is descriptive rather than a randomized identification of a causal shaft effect.
5.4.5 The Ground Pathway Fails Its Own Preregistered Screen
The final component of the ground pathway program adds a finite planar base: ground translation, a separately identifiable free pitch moment, and a coupled pathway, each against an exact fixed-base comparator. The atlas runs 384 primary trajectories plus 192 rigid-shaft and horizontal-restraint-removed controls across two engines and two timesteps. Peak ground force reaches 559.09 N and peak intrinsic free moment 14.58 N\(\cdot\)m. Every registered numerical gate passes: the worst normalized energy residual refines 0.01986 to 0.00995, engine parity holds to \(1.8\times10^{-10}\), and no primary or control cell fails a numerical or parity check.
The preregistered estimand then returns the sharpest negative result in the program. Requiring a 5% match on both peak grip load and total dissipated work, coupled versus fixed, admits 0 of 384 cells. The reason is structural rather than incidental: only the coupled pathway contains ground damping, so total dissipated work differs by a factor of 1.72–2.00 by construction. Peak load agrees closely — its error range is 0 to 0.49 — but the work criterion cannot be met. Loosening the tolerance does not rescue it at 10%, 25%, 50%, or 100%; only a 200% tolerance admits any cell at all, at which point every cell matches and the screen has stopped discriminating.
A labeled post-hoc screen that redefines dissipated work as grip plus shaft only, excluding ground damping, admits 60 cells at 5%. Among those, club-speed differences run from \(-0.00075\) to \(+0.01394\) m/s, with 20 positive and 40 negative. This is recorded as a sensitivity, not a replacement estimand; it was chosen after seeing the primary failure, and the upstream evidence labels it as such.
One reading must be resisted explicitly. At the 50 ms horizon the unmatched translation and coupled pathways are uniformly faster than the fixed base — all 48 cells positive, up to \(+0.177\) m/s. Those cells fail the registered load and work match. They are not evidence of a ground-pathway delivery benefit, and reporting them as such would invert the study’s actual finding.
The tier also demonstrates that initialization is not innocuous. At the 0.125 ms step, natural-zero, gravity-only, and conditional-equilibrium starts produce peak ground forces of 32.8, 565.5, and 510.3 N and 4 ms club speeds of 0.264, 1.908, and 0.946 m/s — an order of magnitude apart from the same model. The atlas uses natural-zero for exact killswitch comparability, and the conditional solve is retained as a separate branch because it balances only base generalized forces, not the whole mechanism.
The ground evidence models a linear planar translation and free pitch moment. It is not unilateral normal contact, Coulomb friction, foot segmentation, or measured pressure; it is not force-plate calibrated; and it ends at 50 ms, so it cannot describe the whole downswing or impact.
6 Arm–Wrist Allocation and Transmission Preload
6.1 Same Club Task, Different Internal Loads
A net club moment can be produced by direct wrist moments, by the moment of two separated hand forces about the club reference point, or by both. Club kinematics and net wrench do not identify that allocation. The executable open evidence study therefore evaluates every allocation at the same state and constrains every case to the same 8 N m control moment.
The finite-history implementation is pinned to UpstreamDrift merge e96a585a.
The two poles are mathematically defined actuator subspaces, not anatomical labels. The proximal pole permits four shoulder and elbow generalized torques and sets direct wrist moments to zero. The wrist pole permits two direct wrist moments and sets proximal generalized torques to zero. Within each subspace, the minimum-norm control is scaled to the common club task; convex mixtures span the interval between them.
Across 19 club angles and 21 allocation fractions, the maximum task error is \(5.33\times10^{-15}\) N m, and direct wrist moment plus the force-generated two-hand couple closes to the net moment within \(4.44\times10^{-15}\) N m. The internal demands are not equivalent: modeled RMS hand force spans 7.58–91.51 N and generalized-torque norm spans 5.74–25.45 N m. The allocation minimizing one metric varies with geometry and need not minimize another. Consequently, the model supplies a tradeoff surface; it does not establish a universal arm-led or wrist-led technique.
6.2 What “Slack” Means in This Test
“Slack” can mean a grip gap, series-elastic extension, low tangent stiffness, activation delay, or low activation. These are not interchangeable. The executed sensitivity study adopts one falsifiable definition only: a 0.012 rad rotational dead zone followed by 600 N m rad\(^{-1}\) series stiffness and an 18 ms first-order torque-development time. Those values are synthetic and are not identified tissue, grip, or wrist properties.
A newer one-class-at-a-time audit separately exercises contact disengagement, transmission dead zone, structural preload, biological series compliance, and control deadband under slow and multisine reversal excitation. The four mechanical surrogates close their work-energy ledgers to a maximum absolute residual of \(5.13\times10^{-10}\) J. The control deadband remains a delayed signal map, not mechanical energy storage. Although every registered local three-parameter sensitivity matrix is full rank, the contact and biological surrogates differ by only 1.96% normalized output RMSE under the multisine. One transmitted-output channel therefore does not identify the mechanism. The machine-readable audit and canonical figure retain that non-identification boundary.
Two programs share the same pre-transition net torque (6 N m) and post-transition net torque (10 N m). In the persistent-direction program, the arm channel stays positive while the wrist channel stays negative. In the role-reversal program, both channels reverse sign. From loaded equilibrium, the persistent-direction program has no zero-transmission interval and a 0.0717 N m s torque-error impulse. The role-reversal program crosses zero transmission for 11.5 ms in the arm channel and 22.0 ms in the wrist channel, with a 0.1954 N m s error impulse. A relaxed start increases the error to 0.2011 N m s for either program. The persistent-direction advantage appears in 11 of 12 dead-zone/time-constant sensitivity cases; the two programs are equal to numerical precision in all three zero-dead-zone cases.
A finite-history check begins both transmission channels at zero deflection, applies the preparation commands for 180 ms, and then changes only the desired commands at the transition. The internal deflections and transmitted torques cross that boundary continuously, without resetting state. After ten declared 18 ms time constants, its post-transition gap durations, delays, and error impulses reproduce the loaded-equilibrium comparison above. This operational preparation interval tests state carryover; it is not a model of an anatomical backswing or evidence of a specific preparatory muscle action.
The conditional implication is narrow: if the relevant transmission behaves like the declared gap-and-stiffness element, maintaining load direction can preserve immediate torque transmission. Biological series elasticity can also store energy, filter impact, and alter force-rise dynamics. Muscle preflexes and short-range stiffness provide plausible but testable alternatives to a literal gap element (Araz et al., 2023). Co-contraction can increase task-relevant stiffness but has energetic and control costs. Neither zero compliance nor maximum preactivation is presumed optimal.
6.3 Relation to Scapular and Wrist Activity
Scapular electromyography reports phase-dependent shoulder-girdle activity (Jobe et al., 1995), and wrist electromyography reports lead/trail asymmetries without establishing a downswing EMG–clubhead-speed relationship in the cited subelite sample (Robinson et al., 2023). Neither measurement alone identifies hand-force direction, bilateral force couple, joint torque, or mechanical work. A measured internal grip wrench is required to resolve closed-chain upper-limb kinetics (Choi and Park, 2020). The proximal control subspace must therefore not be relabeled as scapular retraction, and passive resistance must not be equated with muscular inactivity.
The decisive experiment combines bilateral six-axis grip wrenches, grip pressure, full-body and club kinematics, shaft strain, ground reaction, closed-chain inverse dynamics, forearm/shoulder/scapular EMG, and an estimate of transition-aligned tangent stiffness or force-development delay. The model is contradicted if matched club tasks lack the predicted allocation-dependent internal forces, inferred reversal produces no transmission gap under a documented gap or low stiffness, or any apparent advantage disappears on participant-level holdout.
The persistent-direction case performs better only inside the declared phenomenological transmission family. No human allocation, biological slack, scapular muscle strategy, injury effect, or clubhead-speed benefit has been measured. The complete equations, sensitivity grid, traces, and falsifiers are maintained in the open research epic.
7 Reference Frames, Biological Redundancy, and Engine Roles
7.1 One Physical Interaction, Several Coordinate Descriptions
A model result should not change because a reviewer expresses it in a laboratory frame, a club-attached frame, or about another point on the same rigid body. With force-first wrench \(\mathsf W_A=[\mathbf F^T\;\mathbf M_A^T]^T\) and linear-first twist \(\mathsf V_A=[\mathbf v_A^T\;\boldsymbol\omega^T]^T\), transport from point \(A\) to point \(B\) requires
\[ \mathbf M_B=\mathbf M_A-(\mathbf r_B-\mathbf r_A)\times\mathbf F, \qquad \mathbf v_B=\mathbf v_A+ \boldsymbol\omega\times(\mathbf r_B-\mathbf r_A). \]
Those paired changes preserve instantaneous power:
\[ P=\mathbf F\cdot\mathbf v+ \mathbf M\cdot\boldsymbol\omega. \]
The new executable audit rotates and transports arbitrary three-dimensional samples and checks Jacobian virtual work \(\boldsymbol\tau^T\dot{\mathbf q}=\mathsf W^T\mathbf J\dot{\mathbf q}\). Its largest power residual is \(1.14\times10^{-13}\) W, and the virtual-work residual is \(3.55\times10^{-15}\) W. These values establish numerical convention closure, not experimental accuracy.
This distinction is central to the AffineDrift viewpoint. A drift/control split depends on the selected state, inputs, and coordinates. The correctly transported physical power does not. The Reference-Point Problem, Rotation Representation Guide, Screw Theory Reference, and cross-formalism dynamics treatment develop the underlying representation issues in greater depth.
7.2 The Same Net Moment Does Not Identify One Muscle Strategy
For a moment-arm matrix \(\mathbf R_m(\mathbf q)\) and nonnegative muscle forces \(\mathbf f_m\), the net modeled joint moment is
\[ \boldsymbol\tau_m=\mathbf R_m(\mathbf q)\mathbf f_m. \]
When more muscle channels exist than independent joint moments, null-space changes in muscle force alter internal loading without altering \(\boldsymbol\tau_m\). The reduced Hill-type example holds a 10 N m task fixed across 41 agonist–antagonist allocations. Maximum moment error is \(1.78\times10^{-15}\) N m, while the declared stiffness proxy rises from 3.33 to 23.33 N m rad\(^{-1}\) and series-elastic energy rises from 0.208 to 5.208 J.
Therefore inverse dynamics cannot determine whether a golfer used one unique scapular, arm, or wrist strategy. Co-contraction may change stability, energy cost, or tissue loading, but selecting among those consequences requires an explicit objective and measurements. The inverse-dynamics limitation article, Constraint Forces, and Induced Acceleration Analysis provide complementary treatments.
7.3 Preparation History With Activation and Series-Force Dynamics
The earlier dead-zone study tested one operational definition of slack. A new reduced biological bridge instead carries neural excitation, activation, Hill-type force, and series-force state continuously through the transition. Persistent-direction and complete-role-reversal cases again share 6 N m before and 10 N m after the transition. No state is reset.
The persistent case produces a 0.08998 N m s post-transition error impulse; the role reversal produces 0.09084 N m s. This small conditional difference shows only that activation and series-force histories make the route into an identical net task observable. It does not prove scapular retraction, wrist passivity, or a preferred human technique. The Passive and Distributed Control chapter develops the related biological-control ideas.
7.4 Use Each Engine for a Declared Question
- MuJoCo tests contact-rich forward dynamics and achieved contact wrenches.
- Pinocchio supplies fast rigid-body Jacobians, inverse dynamics, and forward dynamics for cross-checks and sensitivity work.
- Drake is suited to constrained trajectory optimization and alternative contact formulations under a preregistered objective.
- OpenSim is the appropriate bridge to subject-scaled muscle paths, moment arms, inverse dynamics, and induced acceleration.
- MyoSuite supports activation-driven muscle/contact simulation and control.
The currently executed cross-engine evidence is the reduced MuJoCo/Pinocchio forward-contact tier. Drake, OpenSim, and MyoSuite are actionable repository capabilities and registered next validation tiers; they are not completed human experiments. Engine agreement tests assumptions shared by the implementations. It cannot replace bilateral grip wrenches, motion, shaft strain, EMG, or participant-held-out evaluation.
One additional representation-level test is now executed across all five adapters. A nonzero canonical pelvis pose and four nonzero joint rotations are encoded into each engine’s coordinate vector and decoded again. The largest translation residual is zero, the largest rotation residual is \(3.56\times10^{-15}\) degrees, and the largest joint residual is \(8.89\times10^{-16}\) degrees. This verifies pose-coordinate mapping only. It does not execute five dynamics engines or establish contact, wrench, trajectory, anatomical, or physiological parity.
The complete equations, evidence archives, backend boundaries, and falsification program are maintained in the advanced expansion research program (UpstreamDrift/issues/8505) and the canonical research monograph.
8 Transmission Pathways, Robust Speed, and Task Stability
The extended analysis separates five propositions that should not be merged: energy reaches the club; a particular pathway carries it; an intervention changes that pathway; the controller rejects declared perturbations; and a human can realize the controller repeatably and safely. A kinematic sequence does not identify a unique pathway, negative torque does not determine power sign, a pointwise drift sample is not a forward future, and nominal speed is not evidence of human self-stabilization.
The new executable study compares clock-triggered restrain-then-drive, arm-angle-triggered handoff, a higher-impedance state trigger, and early drive under common training and held-out perturbations. The perturbations vary initial arm and wrist states, command scale, activation delay, shaft stiffness, and grip separation. The objectives are lower-tail delivery speed, speed and face/path dispersion, peak hand force, and a squared-torque effort proxy—not a single nominal-speed score.
Relative to clock timing, the state trigger raises held-out 10th-percentile delivery speed from 4.94 to 5.28 m/s and reduces mean planar face/path proxy error from 9.44 to 2.29 degrees. It also raises 90th-percentile peak hand force from 166.9 to 208.1 N. Higher impedance changes the balance again. Every registered program remains Pareto-nondominated, so the evidence rejects a universal optimum.
The local three-outcome Jacobian of eight perturbation variables has rank three and nullity five. Most of the declared perturbation covariance lies in the local task-null subspace. This illustrates motor abundance: variability can be large in elemental coordinates while task outcomes remain comparatively stable. It is a local model property, not evidence of a neural synergy, and nonlinear curvature can make an infinitesimally null direction harmful at larger amplitude. Golf UCM evidence likewise suggests that skilled control is about structuring variability around club outcomes rather than minimizing all joint variability Morrison et al..
8.1 Common-Phase Timing Viability and Recovery
A follow-on falsification study removes a hidden comparison ambiguity by placing clock and state-triggered release on the same nominal phase coordinate. It executes five phase offsets under six declared load and perturbation cohorts, with paired reference and perturbed trajectories for every cell: 60 cases and 120 trajectories in total. A viable cell must jointly satisfy delivery-speed, face/path, peak-hand-force, squared-torque-effort, and numerical-closure guards.
Under the primary guards, the clock policy is viable at four of five phase points across the intersection of all loads, whereas the state trigger is robustly viable only at the latest sampled point. Strict and lenient guards retain the same ordering. Requiring sustained half-error recovery makes both regions empty: the study observes no sustained half-error recovery in any perturbed case. The evidence therefore does not support a larger state-triggered timing region or model self-correction advantage at this declared planar tier. It also does not show that people should use clock timing. Human timing demand, feedback strategy, and coaching remain outside this experiment.
The complete arrays, guards, cohort definitions, timestep sensitivity, and bounded interpretation are pinned in the timing-viability record and the canonical figure.
The practical research implication is therefore conditional: compare timing policies on a common phase coordinate while constraining hand force, effort, face, path, strike, and injury-relevant loads; neither trigger family presently earns a human-technique recommendation. Biological stability requires phase-registered perturbations, bilateral grip wrenches, force plates, full-body and club motion, shaft response, EMG, and participant-held-out impact/launch outcomes. Coactivation can increase impedance and reject disturbances, but it can also increase force and effort; it is not automatically beneficial de Rugy et al..
The complete gap register and path forward are available in the adversarial transmission review, with implementation tracked in the adversarial transmission research registry (UpstreamDrift/issues/8507).
9 Computational Methods
The numerical analyses use an open-source, backend-neutral double-pendulum model. The baseline is a planar, fixed-hub arm–club chain with a rigid shaft and deterministic fixed-step integration:
- E1 Timing Sweep: 92 torque programs sweeping wrist onset times \(t_{\text{on}} \in [0.0, 0.35]\) s, wrist drive levels \(\tau_2 \in \{0, 15\}\) N·m, and early restraint levels \(R \in \{0, 5, 10\}\) N·m across shoulder torques \(\tau_1 \in \{60, 100\}\) N·m.
- E1b Bounded Actuator Sweep: Linear torque-velocity motor bounds (\(\tau(\omega) = \tau_{\text{iso}}(1 - \omega/\omega_{\text{max}})\)).
- E1c Robustness Evaluation: Sensitivity analysis across 5 alternative impact criteria.
- E1d Parameter Sensitivity: One-at-a-time variation of arm length and mass, club length and head mass, swing-plane inclination, and joint damping across 13 declared cases; a program is reselected from the same finite grid within each case.
- E1e Command Rise-Time Sensitivity: Every program is repeated with 20, 35, and 50 ms first-order transitions after the initial preload. These are command filters, not muscle activation or continuous optimal control.
- E2 Drift/Control Split & E4 Robertson–Winter Accounting: Pointwise ZTCF decomposition and interface power balance \(\mathbf{F} \cdot \mathbf{v} + \tau \omega\), following the segment-energy accounting framework of Robertson and Winter (Robertson and Winter 1980), using analytic COM acceleration and an exact interpolated early/late integration boundary.
The registered arm-angle rule defines the primary delivery-zone estimand; it is not an anatomical validity threshold. Every row retains its unfiltered first club-vertical crossing and a reason-coded status.
10 Results
All reported values trace to versioned JSON/NPZ data and analysis code linked in Section 14.1.
10.1 E1 — Timing Sweep Results
| Program (\(\tau_s = 60\) N·m) | \(t_{\mathrm{on}}\) [s] | \(t_{\mathrm{imp}}\) [s] | Clubhead speed [m/s] | vs. Passive |
|---|---|---|---|---|
| Passive wrist | — | 0.370 | 34.2 | — |
| Early drive (S1 pole) | 0.000 | 0.327 | 28.9 | −15.3% |
| Grid-selected drive-only | 0.200 | 0.352 | 38.4 | +12.4% |
| Grid-selected restrain-then-drive (\(R=10\)) | 0.100 | 0.349 | 38.9 | +13.8% |
Key model-derived findings: 1. Early wrist drive is lower than passive (−15.3%). Driving from the top drops modeled speed from 34.2 m/s to 28.9 m/s. 2. Identical torque applied late adds +12.4%. Peak speed reaches 38.4 m/s at \(t_{\text{on}} = 0.200\) s. 3. Restrain-then-drive is highest among the tested programs (+13.8%). The grid-selected command (\(R = 10\) N·m) reaches 38.9 m/s (and 46.9 m/s at \(\tau_s = 100\) N·m).
10.2 E1b — Torque-Velocity Actuator Bounds
The S2 ordering survives actuator bounds intact: early drive (28.8 m/s) < passive (30.8 m/s) < selected late drive (34.6 m/s) < selected restrain-then-drive (34.8 m/s).
10.3 Kinematic Sequence & Segment Energies
The energy budget shows that the grid-selected programs put the least energy into the club early (5.0 J vs. 23.0 J early drive) and bank energy in the arm (55.1 J shoulder work early). The selected restraint command produces −1.9 J of modeled early wrist-actuator work because its imposed torque opposes opening velocity. That sign does not identify muscle action, neural intent, eccentric physiology, or a human technique.
10.4 E4 Interface Powers & E2 Counterfactual Split
Robertson–Winter accounting (Robertson and Winter 1980) closes pointwise with analytic COM acceleration and shows that late energy transfer is dominated by joint-force power (131.7 J) rather than wrist-actuator work (25.7 J). The E2 split shows an instantaneous S2 signature: control opposes drift early; drift dominates late. Forward killswitch claims come from separately integrated matched-state branches.
10.5 Hand-Path Attribution Across the Model Ladder
The MacKenzie-compatible primary estimand is the linear force-work delivered by the golfer to the club divided by achieved hand-path length, \(W_{\mathrm{linear}}/L_H\). It is evaluated at the wrist in the open chains and at the net mid-grip point in the closed loop. Table Table 2 reports the signed reference-case values. ZVCF is projected onto the achieved path only as a configuration-dependent diagnostic; because the ZVCF evaluation has zero velocity, it does not traverse that path itself.
| Model Tier | Trajectory Status | Path [m] | Total Work [J] | Drift Work [J] | Control Work [J] | Total \(\overline{F}_{\parallel}\) [N] | Drift [N] | Control [N] |
|---|---|---|---|---|---|---|---|---|
| Exact double pendulum | Forward simulation to first valid impact | 2.224 | 130.794 | 132.375 | −1.580 | 58.807 | 59.517 | −0.711 |
| One-arm, three-link point-mass | Forward simulation over the declared window | 2.734 | 27.802 | 28.781 | −0.979 | 10.170 | 10.528 | −0.358 |
| Two-arm floating-club closed loop | Prescribed local kinematic sweep | 0.0425 | −0.0038 | −0.0524 | +0.0485 | −0.0898 | −1.233 | +1.143 |
The first two cases demonstrate the central interpretive point. Drift supplies 101.2% and 103.5% of signed hand-path force-work, respectively, while the same-state control residual is slightly negative. Those percentages exceed 100% because control opposes drift; they are not mixture probabilities. On a magnitude basis, drift accounts for 98.8% of the double-pendulum split and 96.7% of the one-arm split. A large measured force along the hand path can therefore coexist with little or negative same-state control contribution in a declared mechanical model. This does not make the motion effortless: the counterfactual does not observe muscle force, co-contraction, passive tissue loading, stabilization, or metabolic cost.
10.5.1 Geometry and Force-Vector Mechanisms
The vector plate in Figure 25 makes the geometry explicit. Each arrow uses one declared force direction and one reference point; the tangent is the achieved hand-path direction. A component can be large while projecting weakly onto the tangent, and a control vector can point opposite the total or drift vector while still satisfying the same-state closure identity. The two-hand row retains both contact forces because equal-and-opposite differential loading can create a couple without changing the common resultant.
10.5.2 Impulse, Work, Power, Joint, and Time-Window Structure
Impulse and work answer different questions. Vector impulse retains the Cartesian direction of accumulated force, tangent impulse accumulates the signed path-directed force in time, and force-work weights the tangent projection by hand speed. The publication therefore exports signed, positive, negative, and absolute tangent impulse alongside cumulative work instead of substituting one quantity for another.
The joint/time atlas evaluates every modeled interface: shoulder and wrist in the double pendulum; shoulder, elbow, and wrist in the one-arm model; and both shoulders, elbows, and hand contacts in the two-arm closed loop. Its four windows are equal-duration normalized-time quartiles. They are bookkeeping windows, not inferred transition, delivery, or impact phases.
Instantaneous power shares use \(|P_d|/(|P_d|+|P_c|)\), mask near-zero denominators, and mark strong cancellation. This avoids presenting an unstable signed ratio as a physical fraction when drift and control nearly cancel.
10.5.3 Two-Hand Redundancy and the Late Negative-Couple Hypothesis
The two-hand reference case separates the common contact-force mode from the differential mode. The common mode governs net translation; the differential mode, acting through grip separation, contributes a couple. Consequently, multiple hand-force and wrist-torque allocations can reproduce the same net club wrench. A net inverse-dynamics solution cannot determine the unique biological allocation between the hands.
In the declared local sweep, drift and control almost cancel at the mid-grip: their force-work magnitudes split 51.9% versus 48.1%, with a cancellation index of 0.962. The late sign reversal appears in this local prescribed case, but its tiny 0.0425 m path and near-zero net work preclude a swing-performance claim. The result instead demonstrates why a two-hand model must retain contact-level forces and couples: a negative net torque can arise from a differential push–pull couple, wrist-joint torques, or combinations of both.
The bounded preview calculation tests one narrower signal-control hypothesis using the archived WSCG planar traces. At the pointwise ZTCF minimum (\(t=0.2148\) s), drift contributes −19.63 N·m of a −22.78 N·m target couple; the required control residual is only −3.15 N·m. For a first-order actuator with a 30 ms time constant, a 24 ms preview reduces late-window residual-couple tracking RMSE from 1.252 to 0.531 N·m (57.6%). Across 10–50 ms time constants, the best preview is 9–35 ms and the modeled RMSE reduction is 46.1–79.4%. This is a delayed-actuator signal result, not evidence that golfers use muscle preactivation and not a clubhead-speed optimization.
10.5.4 Closure and Sensitivity
All reference cases close total = drift + control for force, couple, power, and work below \(2.3\times10^{-13}\) in the exported records. The sensitivity panel shows how the declared estimands respond to local parameter perturbations; it does not represent population uncertainty or confidence intervals.
10.6 E1c — Impact-Criterion Robustness
Evaluating all 92 trajectories under 5 alternative criteria (fixed hand position, ball-position sweep \(v_x\), peak speed anywhere in first pass, matched arm angle \(\theta_1=0.5\) rad, and registered max-arm-angle sweep \(1.5–2.5\) rad) preserved the retain-early/release-late ordering. The 2.0 rad rule accepts 63 programs and places 29 first crossings outside the registered delivery zone. Across the threshold sweep, 58–69 programs are accepted and the selected winner at both shoulder torques is unchanged. This reduces sensitivity to the selected definition; it does not validate unmodeled impact physics.
10.7 E1d — Model-Parameter Sensitivity
The ordering early drive < passive < selected late drive < selected restrain-then-drive persisted in all 13 one-at-a-time parameter cases. Across those cases, the selected late-drive onset shifted from 0.175 to 0.225 s and the selected restrained onset from 0.100 to 0.150 s. Speed ranges were 26.4–31.3 m/s for early drive, 32.9–35.6 m/s for passive, 36.5–40.6 m/s for selected late drive, and 36.8–41.3 m/s for selected restrain-then-drive.
These are local model perturbations, not confidence intervals or a population distribution. The stable ordering supports the modeled mechanism; the shifting selection argues against a universal onset-time prescription.
10.8 E1e — Command Rise-Time Sensitivity
The registered ordering persists at both shoulder torques for 20, 35, and 50 ms post-preload command transitions. At 60 N·m, selected late drive changes from 38.40 to 38.07 m/s and selected restrain-then-drive from 38.85 to 38.66 m/s between 0 and 50 ms; the selected onset and restraint can change. This rejects a single-sample switching artifact within the declared filter family, not a muscle model or global optimum.
11 Ground-Reaction Drift Attribution
Ground-reaction force is an external contact wrench and a constrained-dynamics observable, not a directly commanded input. At one measured state, a frame-explicit constrained model can write its reaction as
\[ \lambda=\lambda_0+\lambda_v+\lambda_u+\lambda_e, \]
where the terms are the configuration-dependent, velocity-dependent, control-induced, and other-external-load contributions. This is the additive identity: configuration + velocity + control + other external load. A pointwise ground-reaction ZTCF retains configuration, velocity, and declared non-control external loads while setting controllable generalized torques to zero. A pointwise ground-reaction ZVCF retains configuration and applied loads while setting velocity to zero. ZTCF and ZVCF overlap in the configuration/external terms and must not be added as complementary causes.
The first executable benchmark uses the fixed shoulder of the same planar double pendulum as an ideal support. Across 161 achieved states, total, ZTCF, ZVCF, and zero-velocity/zero-control support reactions close their independent identities below \(2\times10^{-13}\) N. Using pointwise ZTCF alone to predict the modeled total support-reaction waveform gives componentwise \(R^2\) values of 0.871 and 0.814, but its RMSE is 64.6 N and 89.8 N. The waveform association therefore does not imply small amplitude error.
The vector impulses make the interpretation sharper. Total support impulse is \((22.92,35.20)\) N s, ZTCF impulse is \((17.91,60.12)\) N s, and the control contribution is \((5.01,-24.93)\) N s in the declared planar frame. The vertical ZTCF impulse exceeds the total because control opposes it. A “percent passive” ratio would exceed 100% and would not be a fraction of biological effort.
This is not a human force-plate validation. The fixed support has no feet, center of pressure, free moment, pelvis, or three-dimensional contact geometry. A combined resultant also cannot identify bilateral foot forces. A human test requires synchronized bilateral six-axis force plates, whole-body and club kinematics, inertial uncertainty, declared filtering and frames, and evaluation on each held-out participant. Componentwise RMSE and bias, impulse error, peak timing, loaded-sample COP error, free-moment error, residual pelvis wrench, and parameter sensitivity should all be reported. A center-of-mass acceleration baseline should be included so added model complexity must earn its claims.
The interpretation is consistent with evidence that golfers exhibit more than one viable COP strategy (Ball and Best 2007) and that selected ground-force and moment features are associated with clubhead speed without uniquely determining it (Han et al. 2019; Watson et al. 2026). It also corrects a common inference: measured GRF minus modeled ZTCF contains control-induced reaction, model error, measurement error, and omitted loads; it does not uniquely recover muscle torque.
12 Discussion
The numerical results support delayed distal release as a robust mechanism within the tested planar two-link model. Early wrist drive increases the system’s moment of inertia about the shoulder hub before proximal speed has developed, reducing the modeled shoulder work converted into arm speed.
Qualifications and scope limits: - The primary timing estimand is planar 2-DOF with a rigid shaft and fixed hub. The full resource separately executes moving-base/flexible and coupled-modal-club models, a distributed-shaft reference, a 20-DOF spatial common-state comparison, and native MuJoCo/Pinocchio reduced spatial forward contact. These extensions remain model-tier evidence, not subject-calibrated human validation. - Actuators use declared command-level surrogates. Physiological muscle activation, tissue history, and continuous constrained optimal control remain open. - The advanced biological bridge adds reduced Hill-type activation and series-force states, but it remains a synthetic mechanism study rather than a subject-scaled OpenSim or MyoSuite validation. - The local one-at-a-time screen does not estimate population effects. A separate 12-input Latin-hypercube/PRCC study exposes coupled effects, non-identifiability, and held-out strategy tradeoffs, but its engineering envelopes are not population distributions or Sobol variance decomposition. - No coaching recommendation follows from these finite model inputs.
13 Conclusions
- In the baseline model, early distal drive produces 15% less speed than the passive condition.
- Grid-selected late wrist torque produces 12% to 14% more speed than the passive baseline.
- The selected retention program performs −1.9 J of modeled early wrist-actuator work while preserving more arm energy; this is not a physiological inference.
- Late club-energy gain is dominated by joint-force transfer in the representative model trajectory.
- The qualitative ordering survives actuator bounds, alternative impact criteria, declared one-at-a-time parameter cases, and 20–50 ms command transitions, while selected timing shifts.
- In the fixed-support benchmark, pointwise drift explains much of the modeled reaction waveform but not its amplitude; opposing control makes a component impulse exceed the total, so drift attribution is not a biological-effort percentage.
These findings demonstrate a mechanism in a simplified model. They do not establish a universal golfer-level effect or coaching prescription.
14 Open Materials and Model Details
14.1 Code and Data Availability
The currently published timing-study code, tests, figures, parameter records, and machine-readable outputs remain available in the open-source research directory. The arm–wrist allocation implementation, complete sensitivity archive, and falsification plan are tracked in the UpstreamDrift research repository. The hand-path analysis is pinned to UpstreamDrift commit 69eb7e9d, and the ground-reaction extension is pinned to UpstreamDrift commit 06a0ca63, while this repository carries a compact, hash-verified evidence snapshot and the publication SVGs rather than a second simulation runtime. Run python scripts/check_proximal_distal_evidence.py --require-pinned to verify the local evidence boundary offline.
The repository name identifies provenance; the scientific claims do not depend on adopting the broader software project. The review adjudication records each criticism, its verification status, and the implemented response. Work is tracked in the upstream research registry.
14.2 Reproducibility
To re-run experiments and regenerate figures from source code:
python3 -m scripts.research.proximal_distal_energy.run_experiments
python3 -m scripts.research.proximal_distal_energy.make_figures
python3 -m scripts.research.proximal_distal_energy.e1b_bounded_torque
python3 -m scripts.research.proximal_distal_energy.e1c_impact_sensitivity
python3 -m scripts.research.proximal_distal_energy.e1d_parameter_sensitivity
python3 -m scripts.research.proximal_distal_energy.e1e_smooth_command_sensitivity
python3 -m scripts.research.proximal_distal_energy.run_hand_path_attribution_study
python3 -m scripts.research.proximal_distal_energy.run_torque_allocation_preload_study
python3 -m scripts.research.proximal_distal_energy.run_grf_drift_study
python3 -m scripts.research.proximal_distal_energy.run_advanced_biological_bridge
python3 -m scripts.research.proximal_distal_energy.run_transmission_robustness_study
python3 -m scripts.research.proximal_distal_energy.make_transmission_robustness_figures14.3 Model Parameters
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Upper segment length | \(L_1\) | 0.75 | m |
| Upper segment mass | \(m_1\) | 7.5 | kg |
| Upper COM distance | \(L_{c1}\) | 0.3375 | m |
| Lower segment length | \(L_2\) | 1.0 | m |
| Lower segment mass | \(m_2\) | 0.35 | kg |
| Lower COM distance | \(L_{c2}\) | 0.7557 | m |
| Projected gravity | \(g_{\text{proj}}\) | 8.033 | m/s\(^2\) |
Companion bibliography available in proximal-distal-energy-transfer-bibliography.md.