This program asks whether phase-dependent effective mechanical impedance can be identified during swing-like motion and whether model bases or EMG envelopes add bounded explanatory information. It freezes device, event, signal, model, uncertainty, reliability, and adverse-outcome rules before a result is fit. The current evidence is analytic and synthetic; it is not a perturbation-device qualification or participant result.
The protocol provides no muscle-force identification, no unique reflex, passive, voluntary, or neural-strategy attribution, and no coaching or clinical authority. Mechanical impedance is an input–output property at a declared boundary. EMG overlap is a signal proxy. A fitted basis coefficient is model conditioned, not an observed biological source.
Primary-Source Register
| Westwick and Perreault (2012) (Westwick and Perreault 2012) |
Impedance-identification limits from bandwidth, noise, and acausal estimates |
This fixture, device, phase, or population |
| Lipps et al. (2020) (Lipps et al. 2020) |
Multidimensional shoulder-impedance measurement precedent |
Swing-specific transfer or endpoint–joint equivalence |
| van ’t Veld et al. (2021) (Veld et al. 2021) |
Parallel-cascade intrinsic/reflex modeling and EMG association precedent |
Unique physiological partition or neural source |
| Li et al. (2021) (Li et al. 2021) |
Dependence of co-contraction indices on delay, muscle pair, and formulation |
Treating a co-contraction index as measured stiffness |
| Carey et al. (2026) (Carey et al. 2026) |
Synthetic sensitivity of co-contraction indices to normalization and index choice |
Mechanical or participant validation |
| Hermens et al. (2000) (Hermens et al. 2000) |
Surface-EMG sensor and placement method recommendations |
Elimination of crosstalk or recovery of individual force |
These studies bound the protocol design; none is treated as validation of the manufactured fixtures below. Evidence from posture, ankle, shoulder, or gait does not establish that a swing-phase system is stationary, safe to perturb, or identifiable. Hermens et al. is classified as a journal method-recommendation article, not a formal measurement standard.
Preregistered Questions and Outcomes
Protocol revision affinedrift.active-impedance/v1 predeclares three questions:
| Is the phase-specific effective-impedance design identifiable? |
Full column rank, zero nullity, condition number below threshold, and held-out residual inside tolerance |
Negative after rank loss, poor conditioning, or failed prediction |
| Do intrinsic, reflex, passive, and voluntary bases remain separable under sensitivity choices? |
Complete interval over delay and alternative bases remains inside tolerance |
Null when the interval does not support a distinct partition |
| Does the protocol transport to people? |
Participant-held-out error and preregistered reliability |
Unavailable until every human and device gate exists |
Negative, null, excluded, failed, and unavailable results remain in the result ledger. A preferred fitted story cannot replace them.
Perturbation Device and Safety Envelope
The current device is a simulated torque-pulse device, revision synthetic-feasibility-only. The declaration includes control mode, trigger source and tolerance, maximum displacement, velocity, and torque, an independent hardware interlock, a held-enable/operator stop, dummy-load preflight, saturation and timestamp checks, and explicit stopping rules. Those fields are executable contracts; their numerical fixture values are not safe human limits.
Any physical program must first qualify device dynamics, command/measurement latency, load-cell and motion calibration, mechanical stops, fault response, fixture strength, contact retention, and event-trigger behavior over the full operating envelope. Discomfort, unexpected contact loss, saturation, clipping, limit crossing, or trigger disagreement stops the trial and blocks repetition until governed review. Passing software tests does not approve a device.
Phase and Response-Window Contract
The manufactured program declares transition and pre-impact phases with different event triggers and perturbation offsets. Each phase has three nonoverlapping operational windows:
| Baseline |
-80 to 0 ms |
Background EMG proxy and pre-perturbation torque |
Does not label passive or voluntary origin |
| Early Response |
0 to 80 ms |
Effective mechanical response, model-partitioned response, and EMG response proxy |
Latency plus a declared model is required; timing alone does not identify a reflex |
| Late Response |
80 to 250 ms |
Effective response, activation proxy, and model residual |
Reflexive, passive, voluntary, and task changes can overlap |
The bounds are frozen analysis labels, not universal neurophysiological latency claims. Alternative delays, window edges, trigger jitter, and phase definitions belong in the complete sensitivity interval.
Endpoint and Joint Impedance
Endpoint and joint impedance are distinct declared outputs. A local joint-space perturbation model can be written as
\[
\Delta \tau(t)
= I\,\Delta \ddot q(t)
+ B\,\Delta \dot q(t)
+ K\,\Delta q(t)
+ G\,r\!\left(t-d\right)
+ W\,v(t)
+ \varepsilon(t),
\]
where \(r(t-d)\) and \(v(t)\) are preregistered reflex-like and voluntary-like regression bases. Their coefficients are Model Partitioned quantities. The equation does not prove that either basis contains only the named biological contribution.
For a locally declared Jacobian \(J\), a stiffness mapping may include
\[
K_q = J^{T} K_x J + K_{\mathrm{geo}}.
\]
The geometric term, coordinate frames, reference point, constraints, and Jacobian rank must be declared. The mapping is not generally invertible, and a joint-space result cannot be relabeled as endpoint stiffness—or vice versa—by omitting those terms.
EMG Processing and Electrode Uncertainty
Every EMG channel freezes channel ID, muscle label, side, electrode location and orientation, interelectrode distance, sample rate, passband, normalization, electromechanical delay and uncertainty, crosstalk screen, and electrode-map revision. Raw amplitude is not comparable across channels, sessions, electrodes, or people without the declared normalization and uncertainty analysis.
Each governed pair record fixes distinct agonist and antagonist channel IDs on one side, their roles, normalization revision, CCI family, formula identifier, comparison scope, and a sensitivity plan. Unknown channels, cross-side pairs, or envelopes whose identifiers do not match the declared roles fail closed.
The manufactured index is in Carey’s Amplitude-Driven family. For two already processed, nonnegative envelopes \(a(t)\) and \(b(t)\), its frozen symmetric-envelope-overlap-v1 formula is
\[
C = \operatorname{mean}_t
\left[
\frac{2\min(a(t),b(t))}{a(t)+b(t)}
\right],
\]
with zero assigned when both samples are zero. It ranges from zero to one and is unchanged when the two channels are swapped. It changes when only one channel is rescaled. Therefore this EMG Proxy is not stiffness, force, metabolic cost, or a unique co-contraction mechanism. Carey et al. distinguish shape-, amplitude-, and temporal-driven CCI families and show that normalization and formula choice alter the estimate. Direct cross-index value comparison is prohibited; this protocol permits only relative trends within the same frozen formula and normalization revision. Index choice must match the preregistered hypothesis.
Li et al. report formulation-, electromechanical-delay-, and antagonist-pair- dependent CCI–stiffness associations in only two post-stroke gait participants. That bounded result motivates the sensitivity plan; it does not validate this formula, identify stiffness, or transport to swing-like motion. Muscle-pair selection, electrode placement, crosstalk, filtering, and delay remain explicit sensitivity inputs.
Excitation and Identifiability
The synthetic regression uses the frozen parameter order
\[
\theta = [I,\ B,\ K,\ G,\ W]^T
\]
and design columns for acceleration, velocity, displacement, delayed response, and voluntary basis. Before fitting, the protocol reports observations, parameters, rank, nullity, and two-norm condition number. One declared relative singular-value tolerance controls both the SVD rank/nullity gate and the least-squares solution. A result is qualified only when the matrix has full column rank and its condition number is below the preregistered threshold. Scaling, perturbation bandwidth and amplitude, noise, state nonstationarity, input–motion correlation, and closed-loop feedback can all alter practical identifiability.
A low residual does not rescue a rank-deficient design: infinitely many parameter allocations can predict the same response. Likewise, successful recovery from manufactured data proves the code path and declared design, not the adequacy of the biological model.
Synthetic Recovery and Confounding Fixtures
The full-rank fixture uses 96 deterministic samples and five independent basis columns. It exactly recovers two distinct phase vectors:
| Transition |
1.8 |
5.0 |
72.0 |
14.0 |
3.5 |
Full rank; exact manufactured recovery |
| Pre-Impact |
1.8 |
7.5 |
96.0 |
9.0 |
-2.0 |
Full rank; exact manufactured recovery |
Units inherit the declared joint model and each value is synthetic. A negative \(W\) is a regression coefficient, not evidence of braking intent or eccentric muscle action.
The negative control sets the voluntary basis equal to displacement. Rank drops by one, leaving a stiffness/voluntary null direction. Fitting fails closed even though the combined response remains exactly reproducible. This is the central counterexample to interpreting algebraic component labels as identified causes.
Evidence Tiers, Uncertainty, and Reliability
| Effective Mechanical |
Perturbation input and measured mechanical response identify a bounded input–output relation |
Unique tissue, reflex, voluntary, or neural source |
| Model Partitioned |
A declared full-rank model allocates response among named bases with complete sensitivity intervals |
Treating basis names as observed physiology |
| EMG Proxy |
Processed envelope overlap or activation association under a frozen pipeline |
Stiffness, individual force, effort, or strategy identity |
| Unavailable |
Required quantity or governance record is absent |
Substituting an estimate or preferred narrative |
Complete uncertainty spans perturbation amplitude and bandwidth, trigger timing, reflex delay, passive and voluntary bases, model order, filtering, electrode placement, normalization, muscle-pair choice, and residual structure. Reliability is phase stratified and must report repeatability plus held-out prediction, not only within-fit error. The manufactured result ledger deliberately retains one supported, one negative, one null, and one unavailable outcome.
Every ledger entry carries a machine-readable evidence origin, provenance record ID, revision, and synthetic-fixture marker. All numerical results on this page are labeled synthetic-fixture; the unavailable biological-actuator entry is labeled unavailable. A synthetic value cannot be relabeled as measured while retaining its fixture provenance, and no fixture value may be silently promoted to participant evidence.
Negative, Null, and Unavailable Results
Rank loss is negative evidence for the requested partition, even when total response prediction remains exact. A sensitivity interval that spans competing partitions is null for a distinct contribution. Missing bandwidth, phase, calibration, electrode, safety, governance, or held-out records makes the corresponding result unavailable. These dispositions falsify or bound a claim; they are not data-cleaning inconveniences.
References
Carey, Hannah D., Friedl De Groote, and Andrew Sawers. 2026.
“A Comparative Analysis of Co-Contraction Indices Using Synthetic EMG Data: Implications for Selection and Interpretation.” PLOS ONE 21 (5): e0343081.
https://doi.org/10.1371/journal.pone.0343081.
Hermens, Hermie J., Bart Freriks, Catherine Disselhorst-Klug, and Gunter Rau. 2000.
“Development of Recommendations for SEMG Sensors and Sensor Placement Procedures.” Journal of Electromyography and Kinesiology 10 (5): 361–74.
https://doi.org/10.1016/S1050-6411(00)00027-4.
Li, Geng, Mohammad S. Shourijeh, Di Ao, Carolynn Patten, and Benjamin J. Fregly. 2021.
“How Well Do Commonly Used Co-Contraction Indices Approximate Lower Limb Joint Stiffness Trends During Gait for Individuals Post-Stroke?” Frontiers in Bioengineering and Biotechnology 8: 588908.
https://doi.org/10.3389/fbioe.2020.588908.
Lipps, David B., Emma M. Baillargeon, Daniel Ludvig, and Eric J. Perreault. 2020.
“Quantifying the Multidimensional Impedance of the Shoulder During Volitional Contractions.” Annals of Biomedical Engineering 48 (9): 2354–69.
https://doi.org/10.1007/s10439-020-02509-w.
Veld, Ronald C. van ’t, Alfred C. Schouten, Herman van der Kooij, and Edwin H. F. van Asseldonk. 2021.
“Neurophysiological Validation of Simultaneous Intrinsic and Reflexive Joint Impedance Estimates.” Journal of NeuroEngineering and Rehabilitation 18 (1): 36.
https://doi.org/10.1186/s12984-021-00809-3.
Westwick, David T., and Eric J. Perreault. 2012.
“Estimates of Acausal Joint Impedance Models.” IEEE Transactions on Biomedical Engineering 59 (10): 2913–21.
https://doi.org/10.1109/TBME.2012.2213339.