Source: full poster text including the results table. It is a one-page abstract, so detail is limited by the format.
Research theme
The 2022 companion paper showed that measuring strength at one joint angle badly underestimates true peak torque. The obvious remedy is to measure at several angles and fit the torque–angle curve. But nobody had established how many measurement angles you need, or where to put them. This poster answers that by simulation.
Method: torque was computed from a two-parameter quadratic — T(θ) = (1 − k₂(θ − θ_opt)²) · T₀ — using optimal angle, width and peak isometric torque taken from the literature, sampled at 10° intervals across the joint range (flexion 90°, extension 85°). Random noise of 0–10 Nm was subtracted at each sample point to simulate submaximal effort, generating 100 torque-angle curves per joint action. Then, for every possible combination of measurement sites from three angles up to the full set (10 available for flexion, 9 for extension), simulated annealing re-fitted θ_opt, k₂ and T₀ by minimising the unweighted unbiased RMS difference. The percentage error between the recovered parameters and the true (literature) values is the result.
What they measured
- T₀ — peak isometric torque (Nm): how strong the athlete is at their best angle
- θ_opt — optimal angle (°): where in the range that peak sits
- k₂ — width: how sharply strength falls away either side of the peak
Findings
Mean (± SD) percentage error between recovered and true parameters (Table 1):
3 sites 5 sites 7 sites Max (10/9) Knee flexion T₀ (peak torque) 6.1 ± 3 4.5 ± 1.1 3.9 ± 0.7 3.9 θ_opt (optimal angle) 7.9 ± 2 4.9 ± 1.4 3.7 ± 0.7 3.4 k₂ (width) 57 ± 31 23 ± 7.7 16 ± 3.5 14 Knee extension T₀ 3.0 ± 1.3 2.1 ± 0.4 2.1 ± 0.2 1.9 θ_opt 0.7 ± 0.9 0.1 ± 0.1 0.1 ± 0.0 0.1 k₂ 8.7 ± 7.9 2.6 ± 1.2 2.0 ± 0.4 2.4 More sites is better, but with sharply diminishing returns. Error “decreases exponentially such that measuring all available sites may not be necessary.” The jump from 3→5 sites buys the most; 7→max buys almost nothing (and for knee extension k₂, the max is marginally worse than 7 — 2.4 vs 2.0 — a sign that noise, not sampling, dominates at that point).
The width parameter k₂ is by far the hardest to recover, especially for the knee flexors. At three sites, k₂ error was 57 ± 31% for flexion versus 8.7 ± 7.9% for extension. Even at maximum sampling, flexion k₂ was still 14% out.
Why the flexors are harder: their torque–angle profile is predominantly ascending and flat, so the data contain little information about curvature. The knee extensors show a more acute ascending–descending curve, which pins the parameters down.
Where you put the sites matters as much as how many. Smaller errors came from combinations that included a site near the optimal angle — close to full extension (0°) for the knee flexors, near mid-range (60°) for the knee extensors.
But there is a trade-off, and it is the key practical finding. For the knee flexors, clustering sites near the optimum (e.g. [0°, 10°, 20°]) gave small errors in T₀ and θ_opt but larger errors in k₂ than combinations spanning more of the range (e.g. [0°, 50°, 80°]). You cannot optimise for peak strength and for curve shape with the same sampling strategy. For the knee extensors, both clustering near the optimum (e.g. [60°, 70°, 80°]) and purely ascending (e.g. [20°, 30°, 40°]) or purely descending (e.g. [80°, 90°, 100°]) sets gave larger errors across all parameters — i.e. you need to straddle the peak.
Conclusion: practitioners should choose their protocol based on the research question, the muscle group, and practical constraints — there is no single correct number of measurement angles.
Simulation/modelling result. No human participants.
What this means for video and motion analysis
No direct coaching cue. For anyone assessing a simulation-derived claim or designing a measurement pipeline:
- This is the answer to “so how should strength be measured?” — the constructive complement to the 2022 paper’s warning. Roughly five well-placed angles straddling the expected optimum recovers peak torque to ~2–5% and optimal angle to ~0.1–5%. That is the standard a torque-driven simulation’s strength inputs should meet.
- The parameter most likely to be wrong in a published model is curve width. Peak torque and optimal angle are recoverable; k₂ for the knee flexors is still 14% out even with maximum sampling. Since k₂ governs how fast an athlete’s strength falls off as the joint moves away from optimum — which is precisely what determines whether a fast, large-range movement is strength-limited — this is not a harmless residual. Any “optimal technique” that hinges on strength at extreme joint angles inherits this uncertainty.
- Sampling strategy is a design choice with a built-in trade-off. Cluster near the peak and you learn how strong; spread across the range and you learn what shape. Any protocol implicitly picks one. This generalises well beyond dynamometry: it is the same trade-off as sampling densely near an event of interest versus sampling broadly to characterise a whole trajectory.
- Diminishing returns are real and quantified. Going from 7 sites to all available sites gained essentially nothing, and in one case lost a little. More data is not monotonically better once noise dominates — a useful counterweight to “just collect more frames/more angles/more sensors”.
- Note that “submaximal effort” was modelled as up to 10 Nm of subtracted noise. Real athletes do not give maximum effort at every test angle, and that alone injects error into the fitted curve before any protocol choice is made.
Caveats and limits
- One-page poster abstract. No figures reproduced, no statistical inference, minimal methodological detail. This is the weakest-evidence item in the folder by format.
- No human participants. All curves are literature-derived and synthetically noised.
- Knee only — flexors and extensors. Not the hip, shoulder, trunk or elbow that dominate a bowling action.
- Isometric only. The torque–velocity relationship is untouched, and a bowling action is fast.
- Monoarticular quadratic model, so biarticular effects (which the group’s own earlier work shows can matter by 19% vs 3%) are excluded.
- Assumes the underlying relationship really is a quadratic. Errors are measured against a quadratic ground truth, so any mismatch between the quadratic and real muscle behaviour is invisible here.
- Noise was modelled as subtractive only (0–10 Nm), consistent with submaximal effort but not with symmetric measurement noise.
Relationship to other Felton work
- Direct sequel to Parkinson, Apps, Felton & Lewis (2022), which established the problem (single-angle measurement underestimates peak torque by up to 96%). Together they form a two-paper argument: don’t use one angle; use ~5 straddling the peak; expect the width parameter to remain the weak point.
- Supplies the practical protocol behind the strength-constraint requirements set out in McErlain-Naylor, King & Felton (2021), which asserts that torque-driven models’ credibility rests on subject-specific in-vivo dynamometry.
- Underpins the strength inputs to Felton’s cricket fast bowling simulation programme (Felton 2015 PhD; Felton, Yeadon & King 2020). No contradiction with that work — but it does put an uncertainty band on it that the cricket papers themselves do not report.
- Builds on King, Lewis & Yeadon (2012) and Lewis, Yeadon & King (2018), the biarticular torque representation papers cited throughout the 2021 review.