Source: full paper text including both results tables, not abstract only.
Research theme
A torque-driven simulation model needs to know how strong the athlete is — not as a single number, but as a torque–angle curve: how much force the muscles around a joint can produce at every joint position. In practice, strength is very often measured at just one joint angle (isometric peak torque at 90° of knee flexion is a common convention), and that single value is treated as “maximum strength”. But the shape and peak location of the torque–angle curve differ between individuals and between muscle groups. So a single measurement is almost certainly taken at a suboptimal angle, and will systematically underestimate true strength.
This paper quantifies that underestimation, by simulation rather than by testing people. A two-parameter quadratic torque–angle function was populated with parameters drawn from the biomechanics literature, then those parameters were perturbed across their realistic range, and the “measured” torque at commonly used test angles was compared with the model’s true peak.
What they measured
- The torque–angle relationship modelled as
T(θ) = (1 − k₂(θ − θ_opt)²) × 100, where θ_opt is the angle at which the muscle group is strongest and k₂ is the width/curvature of the curve (how quickly strength falls away either side of the peak). - Half range — a readable version of k₂: how many degrees from the optimal angle you must go before predicted torque reaches zero. Small half range = narrow, peaky curve.
- Torque error — the gap between the torque you would measure at a given test angle and the true peak torque, as a percentage.
Parameter sources (Table 1): Felton (2015) PhD, King, Lewis & Yeadon (2012), and King, Wilson & Yeadon (2006). Perturbation steps: 2° for θ_opt, 0.01 for k₂ — giving 672 strength-curve profiles for the knee flexors and 1232 for the knee extensors.
Test angles examined (the ones commonly used in the literature): 90°, 120°, 150° for knee flexion; 230°, 240°, 270° for knee extension. (Angles are defined so as to correspond to agonist muscle length — posterior angle for flexion, anterior for extension.)
Findings
Measuring at the wrong angle can miss almost all of the athlete’s strength. Worst-case underestimation reached 96% for knee flexion (measured at 90°) and 80% for knee extension (measured at 270°).
Full error table (Table 2), torque error vs true peak:
Joint action Test angle Mean ± SD Min Max Range Knee flexion 90° −36.1 ± 19.3% −6.1% −96.2% 90.1 Knee flexion 120° −12.5 ± 9.0% −1.0% −42.8% 41.8 Knee flexion 150° −1.8 ± 2.2% 0.0% −10.7% 10.7 Knee extension 230° −4.6 ± 4.7% 0.0% −20.0% 20.0 Knee extension 240° −1.3 ± 1.3% 0.0% −5.0% 5.0 Knee extension 270° −31.1 ± 16.1% −6.5% −79.9% 73.5 The common convention is close to the worst choice. Measuring knee flexion at 90° — a widely used protocol — carried a mean error of −36% and a worst case of −96%. The best test angles were 150° for knee flexion and 240° for knee extension.
Error grows both with distance from optimum and with curve narrowness. Narrower torque–angle curves (smaller half range) produce larger error at any given off-optimum angle. Half range varied 203°→92° for the knee flexors and 79°→45° for the knee extensors — so the knee extensors are far peakier, and small changes in curve width cost more error there.
Shape explains the asymmetry. The knee flexors have a predominantly ascending, flat profile with an apparent plateau at extended positions (optimum bounded 140–180°), so they are forgiving. The knee extensors are a sharper ascending–descending curve with the optimum near mid-range (bounded 230–250°), so they are unforgiving.
Zero error only when test angle = optimal angle, regardless of curve width — which is the whole point: you cannot know you’re at the optimum without characterising the curve.
Compounding measurement error. On top of all of the above, misalignment of the knee joint with the dynamometer axis introduces 0.3–17% error in isometric torque and 10–15° discrepancy between intended and true joint angle. So the modelled errors are a floor, not a total.
Biarticularity caveat. Because two-joint muscles make maximum torque a function of two joint angles, these errors will differ when the secondary (hip) angle changes — especially for the knee flexors, where biarticular contribution to net joint torque is larger.
Simulation/modelling result, not a measurement on human participants.
What this means for video and motion analysis
No direct coaching cue. This is a measurement-methodology paper. For anyone trusting a simulation-derived result or building a system:
- Why this matters to “optimal technique” predictions. A torque-driven simulation searches for the technique that produces the best outcome given the athlete’s strength limits. Those limits are the torque–angle curves. If a curve is fitted from a single measurement at a bad angle, the model thinks the athlete is up to a third weaker than they are — and it will therefore refuse to search techniques that would actually be within reach, and will converge on a different, more conservative “optimum”. The strength input silently changes the answer. This is the plain-terms reason the Parkinson papers exist in a cricket biomechanics catalogue.
- The error is directional, not random. A single-angle measurement can only underestimate (the errors are all negative). So the bias in downstream simulations is systematic: predicted optima will be biased toward what a weaker-than-real athlete could do.
- Read strength numbers with the protocol attached. “Peak isometric knee flexor torque” is nearly meaningless without the test angle. Two studies using 90° and 150° are reporting quantities that can differ by a factor of two on the same athlete.
- The general lesson for any measurement pipeline: when you sample a curved relationship at one point and call it the maximum, your error is set by the curvature you didn’t measure. The narrower the underlying response, the worse a single sample is. This is exactly the failure mode of a video pipeline that estimates a peak (peak speed, peak angle, peak load) from too few frames or too coarse a sampling — a peaky signal sampled sparsely is systematically underestimated.
- Sensor/axis alignment errors are not negligible. 10–15° of discrepancy between intended and actual joint angle is a large fraction of the distance that matters here. Any system that assumes it knows a joint angle to better than that should demonstrate it.
Caveats and limits
- No human participants. Entirely a modelling study. The curves are literature-derived and perturbed; the “errors” are between one model and another.
- Only three literature sources for the parameter bounds, one of which is Felton’s own PhD. The bounds (θ_opt 140–180° flexion, 230–250° extension) drive the headline results — different bounds would give different worst cases.
- Knee only. Not tested at the hip, shoulder, ankle or elbow — all of which feature in the cricket models.
- Monoarticular quadratic model. The authors flag that biarticularity would change the numbers and that hip angle should be controlled when measuring knee torque.
- Isometric only. Says nothing about the torque–velocity side of the strength profile, which matters just as much in a fast, dynamic action.
- 4-page conference paper, not peer-reviewed to journal standard.
Relationship to other Felton work
Uses Felton (2015), Factors limiting fast bowling performance in cricket (Loughborough PhD), as one of three parameter sources — i.e. the torque–angle parameters that drove the cricket bowling simulations are literally an input to this error analysis.
Directly followed by Parkinson, Felton, Lewis & Apps (2023), which asks the complementary question: how many measurement angles, and where, do you need to recover the whole curve accurately?
Supports the strength-constraint section of McErlain-Naylor, King & Felton (2021), which states that the major advantage of torque-driven models is that individual-specific strength parameters “can be readily obtained using an isovelocity dynamometer, providing assurance that torques exerted at each joint angle and velocity within any predicted optimal technique are realistic for the individual.”
TENSION: that assurance is exactly what this paper undercuts. The 2021 review presents in-vivo dynamometry as the thing that makes torque-driven models trustworthy; this 2022 paper shows that if the dynamometry protocol uses a single, conventional test angle, the resulting strength parameters can be wrong by 30–96%. The two are reconcilable — the review assumes a full multi-angle protocol, which Felton’s own cricket work did use — but the confident framing in the review should not be transferred to any study that measured strength at one angle.
Reinforces the biarticular findings the 2021 review cites (Lewis et al. 2012: 3% vs 19% error; King, Lewis & Yeadon 2012; Lewis, Yeadon & King 2018).