11-EFT.WP.Core.DrawingKinetics v1.0 | Chapter 8 Metrology Workflow and Calibration

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Chapter 8 Metrology Workflow and Calibration

I. Scope and Objectives

• Establish a closed metrology loop from raw observations to publishable indicators for drawing/stretching systems, including time-base alignment, geometric calibration, sensor calibration, constitutive parameter identification, and conservation audits.
• Provide executable workflows and quality gates for Mx-11 (time-base and geometry calibration), Mx-12 (constitutive parameter identification), and Mx-13 (conservation check), ensuring cross-device and cross-node comparability on a unified ts (see Chapter 7 on spectrum and jitter, and Chapter 3 on conservation).

II. Terminology and Symbols

• Time-base mapping and jitter: tau_mono, ts = alpha + beta * tau_mono, beta > 0, jitter(ts).
• Geometry and kinematics: lambda(x,t), s(x,t) = ( d/dt ) ( ln( lambda ) ), v(x,t), A(x,t), path gamma(ell) and measure d ell.
• Conserved quantities: rho(x,t), rho_L(x,t) = rho * A, J(x,t) = rho_L * v.
• Tension and constitutive terms: T_fil(x,t), K_el, K_vis, optional dissipative/thermal parameter set params.
• Two arrival-time conventions and discrepancy:
  T_arr^(1) = ( 1 / c_ref ) * ( ∫ n_eff d ell );
  T_arr^(2) = ( ∫ ( n_eff / c_ref ) d ell );
  delta_form = T_arr^(2) - T_arr^(1).
• Quality gates and audits: check_dim(expr), eps_norm, eps_mass, eps_spec (spectrum–time variance discrepancy, see Chapter 7).

III. Postulates and Minimal Equations

• P11-7 (time-base consistency, continuing from Chapter 7)
  All observation streams must satisfy the linear mapping ts = alpha + beta * tau_mono prior to publication and be aligned on the unified ts.
• P11-1 (slender/volume approximation, continuing from Chapter 2)
  Within a prescribed interval one may adopt the volumetric approximation A * lambda ≈ const for geometric calibration and strain-rate inference.
• S12-12 (minimal model for time-base estimation)
  ts_i = alpha + beta * tau_i + e_i, with robust regression to estimate alpha, beta; jitter jitter_rms = sqrt( mean( e_i^2 ) ).
• S12-13 (winding geometry and speed conversion)
  v_meas(ts) = omega(ts) * R_eff(ts); R_eff(ts) = R_nom + Delta_R(ts). If a volumetric approximation is known, Delta_R can be fitted from layer thickness and turn count.
• S12-14 (tensiometer calibration)
  T_fil(ts) = k_T * ( adc(ts) - adc_0 ) + d_T(ts), where d_T is a slow drift term to be estimated and removed via high-pass filtering or a drift model.
• S12-15 (observabilizing the strain rate)
  s(ts) = ( d/dt ) ( ln( lambda(ts) ) ). When lambda is derived from the inlet/outlet speed ratio, compute using aligned v_in(ts), v_out(ts).

IV. Data and Manifest Gauges

1. Ingress data
   Raw sensor streams: adc_T(ts), encoder(ts), marker(ts), beacon(ts); device local time tau_mono and metadata (range, sensitivity, noise, etc.).
2. Publication manifest fields (CalReport minimal set)
   • timebase.alpha, timebase.beta, timebase.jitter_rms, timebase.fit_resid_p99.
   • geom.R_eff_model (parameterization and confidence intervals), geom.v_scale, geom.slip_factor.
   • sensorT.k_T, sensorT.adc_0, sensorT.drift_model, sensorT.resid_p95.
   • constitutive.params (e.g., K_el, K_vis, ...), constitutive.ci, constitutive.fit_score.
   • conservation.eps_mass, conservation.eps_norm.
   • arrival.T_arr_1, arrival.T_arr_2, arrival.delta_form, arrival.path = description of gamma(ell) plus c_ref, n_eff gauges.
3. Dimensions and units
   check_dim( T_fil ) == N; check_dim( v ) == m/s; check_dim( rho_L ) == kg/m; check_dim( s ) == 1/s; check_dim( T_arr ) == s.

V. Algorithms and Implementation Bindings

1. I10-4 calibrate_kinematics( trace:any, sensors:list, timebase:dict ) -> CalReport
   • Align tau_mono -> ts: estimate alpha, beta, output jitter_rms and residual distribution.
   • Geometric calibration: fit R_eff(ts) using encoder(ts) and the winding-layer model to obtain v_meas(ts) and its uncertainty.
   • Sensor calibration: with known weights or static-load curves estimate k_T, adc_0, fit d_T(ts) and de-drift.
   • Strain-rate reconstruction: compute lambda(ts) and s(ts), run check_dim and apply smoothing constraints.
   • Two arrival-time conventions: compute T_arr^(1), T_arr^(2) in parallel, report delta_form and the path/medium manifest.
   • Pre-conservation check: on short windows evaluate eps_mass, eps_norm; if thresholds are exceeded, roll back to geometry/time-base steps.
2. I10-2 estimate_tension( lambda:float, s:float, params:dict ) -> float
   Used in the inner loop of Mx-12 parameter iteration to test the explanatory power of the constitutive model for T_fil.
3. Auxiliary conventions
   • Idempotency: repeated calls with identical inputs must return the same CalReport; replays use fixed random seeds and locked window selection.
   • Errors: E_BC_INVALID, E_DIMENSION_MISMATCH, E_CONSERVATION_FAIL, E_UNSTABLE_STEP.

VI. Metrology Flows and Run Graph

1. Mx-11 timebase-geo-cal
   • Anchor-event acquisition: match beacon(ts) or pulse trains to a reference source.
   • Estimate alpha, beta: robust regression (e.g., Huber/Theil–Sen), compute jitter_rms and fit_resid_p99.
   • Geometric calibration: fit R_eff(ts) and slip_factor, output v_meas(ts) with confidence bands.
   • Quality gates: gate.timebase.jitter_max, gate.geo.scale_tol. If failed, roll back to sensor checks.
2. Mx-12 constitutive-fit
   • Set excitations: step/ramp/small-amplitude sweep scenarios (see Chapter 12) to increase parameter identifiability.
   • Objective: J(params) = sum_t ( T_fil_meas(ts) - T_fil_pred(ts; params) )^2 / sigma_T^2.
   • Constraints: params obey physical priors (non-negativity, stability region); record fit_score and parameter confidence intervals.
   • Consistency check: cross-verify with Chapter 7’s H_{T|v}(f); require C^2_{Tv}(f) above threshold in the principal energy band.
3. Mx-13 conservation-check
   • Compute rho_L = rho * A and J = rho_L * v.
   • Audit S12-1: discrete residuals of ( d/dt ) rho_L + ( d/dell ) J ≈ 0; output eps_mass.
   • Probabilistic/spectral normalization: output eps_norm and eps_spec (see Chapter 7) to pass publication gates.

VII. Verification and Test Matrix

1. Minimum required cases
   • Dual-anchor time base: synthetic sequences with known alpha_true, beta_true; require |beta - beta_true| / beta_true <= tol_beta.
   • Winding geometry: standard drum and known line diameter; validate v_meas bias and R_eff fit residuals.
   • Tension calibration: known weight steps; validate k_T linearity and adc_0 offset; after drift suppression resid_p95 passes its gate.
   • Constitutive parameters: Maxwell synthetic regression; parameter relative error and fit_score meet gates.
   • Conservation check: eps_mass <= gate.mass in steady windows; convergence of eps_mass versus T_obs under non-steady scenarios.
2. Boundary and extreme conditions
   Sparse/missing anchors, strong constitutive nonlinearity, slip jumps, quantization saturation, sensor saturation and hysteresis; provide fallbacks (densify anchors, shrink windows, regularize, excise saturated segments).

VIII. Cross-References and Dependencies

• Chapter 2: definitions of lambda, s, v, A and parameterization of the path gamma(ell).
• Chapter 3: S12-1 continuity and S12-2 momentum balance as the yardsticks for Mx-13.
• Chapter 4: constitutive families for T_fil provide parameterization and stability domains for Mx-12.
• Chapter 7: frequency-domain cross-verification H_{T|v}(f), C^2_{Tv}(f), and eps_spec.
• Core.Sea: time-base gauges and the two T_arr conventions. Core.Metrology: uncertainty budgets and traceability. Core.DataSpec: manifest schema. Core.Errors: alerts and threshold semantics.

IX. Risks, Limits, and Open Questions

• Strong non-stationarity and time-varying constitutive behavior can misfit fixed-parameter models; introduce time-varying or piecewise parameters.
• Complex winding geometry (non-uniform laydown, slip hysteresis) violates 1D R_eff models; incorporate image/pose-assisted metrology.
• Cross-domain multi-device synchronization may drift or jump; monitor alpha, beta online and trigger recalibration.
• When c_ref varies spatially, the two T_arr conventions diverge; confidence for delta_form should include path randomness and medium uncertainty.

X. Deliverables and Version Management

1. Artifacts
   • cal_timebase.json (alpha, beta, jitter and residual statistics), cal_geometry.json (R_eff_model, slip_factor), cal_tension.json (k_T, adc_0, drift_model).
   • constitutive_fit.json (params, ci, fit_score), conservation_report.json (eps_mass, eps_norm).
   • arrival_report.json (T_arr^(1), T_arr^(2), delta_form, gamma(ell), c_ref, n_eff).
   • Summary document CalReport.pdf and manifest entries aligned to schema.core.drawing/v1.
2. Version strategy
   Changes to time-base or geometry gauges are marked MOD; added uncertainty fields are marked ADD; any change impacting historical comparability must ship a migration script and compatibility flag compat.cal.v1.