Critique: Identifiability and “Input” as a Residual

Critique and response context for Identifiability and “Input” as a Residual in AffineDrift’s control-affine golf-swing framework.

Critique: Identifiability and “Input” as a Residual

Summary of Concern

The framework defines Input Force via subtraction: \(\tau_{\text{input}} = \tau_{\text{total}} - \tau_{\text{drift}}\). Here, \(\tau_{\text{total}}\) is derived from Inverse Dynamics (ID) on measured motion, and \(\tau_{\text{drift}}\) is computed from the model.

This structure forces all unmodeled dynamics and measurement noise into the \(\tau_{\text{input}}\) term. specifically:

  1. Unmodeled Physics: Aerodynamic drag, tissue compliance, and marker wobble are not in \(f(x)\) (Drift). ID sees their effects in the motion. Thus, the residual \(\tau_{\text{input}}\) absorbs them.
    • Example: Drag opposes motion. ID calculates a higher “Net Torque” to explain the motion? No, ID calculates the torque required to produce the observed (damped) motion. Wait.
    • Correction: If drag slows the club, \(\ddot{q}\) is smaller. ID calculates \(\tau_{total} = M \ddot{q} + \dots\). So \(\tau_{total}\) is smaller. The model drift (without drag) predicts a certain \(\tau_{drift}\).
    • If \(\tau_{total}\) (measured) < \(\tau_{drift}\) (model), then \(\tau_{\text{input}}\) becomes negative (braking).
    • So the golfer is “credited” with braking the club, when actually the air did it.
  2. Noise: \(\ddot{q}\) is noisy. \(f(x)\) (Drift) depends on \(q, \dot{q}\) (less noisy). \(\tau_{\text{input}}\) depends on \(\ddot{q}\). So \(\tau_{\text{input}}\) inherits all the noise.

The claim that \(\tau_{\text{input}}\) isolates “Active Muscular Effort” is false. It isolates “Active Effort + Unmodeled Forces + Noise”.

Location

  • Page: articles/affine-nature-golf-swing.qmd
  • Section: sec-ztcf (Using ZTCF with inverse dynamics)
  • Claim: “Input forces arising solely from actively applied joint torques.”

Nature of the Issue

  • Empirical Insufficiency: Sensitivity to model mismatch.
  • Overgeneralization: Claiming “Exactness” of decomposition when it is structurally sensitive to unmodeled terms.

Why This Is a Problem

  • False Negatives/Positives: As shown above, aerodynamic drag could be misinterpreted as “Active Braking” by the golfer.
  • Noise Interpretation: High-frequency noise in \(\tau_{\text{input}}\) might be over-interpreted as “twitchy” control or high-frequency muscle inputs, when it is just differentiation noise.

Evidence / References

  • Southgate, D. F., et al. (2009). Sensitivity of inverse dynamics to errors in input data.
  • Hatze, H. (2002). The fundamental problem of myoskeletal inverse dynamics.

Severity

  • High: It compromises the physical interpretation of the results, specifically the “Soleness” of the attribution.

Suggested Remedies

  1. Relax “Solely” Claim: Change “arising solely from active torques” to “containing the active torque contribution (plus unmodeled residuals)”.
  2. Explicit Noise Model: Discuss how noise propagates.
  3. Aerodynamic Correction: Explicitly recommend including aerodynamics in the Drift model if outdoor swings are analyzed, to prevent “Active Braking” artifacts.
  4. Inline Edit: In Section sec-limitations (Data and parameter limitations), add: > “Because \(\tau_{\text{input}}\) is calculated as a residual (\(\tau_{\text{total}} - \tau_{\text{drift}}\)), it absorbs all unmodeled external forces (e.g., aerodynamics) and measurement errors. For example, unmodeled aerodynamic drag will reduce the observed acceleration, causing Inverse Dynamics to compute a lower total torque; the decomposition will attribute this deceleration to a negative ‘braking’ input by the golfer. Thus, \(\tau_{\text{input}}\) should be interpreted as the ‘Net Non-Conservative Forcing’ rather than pure muscular torque in the presence of modeling errors.”