2. Correlation metrics and acceptance criteria
Correlation engineering is only as credible as the metrics used to declare a match. A single scalar — "the curves look close" — is not an acceptance criterion. In airbag CAE, correlation is judged across a small family of physically meaningful channels, each answering a different question: Did the pulse reach the right magnitude? Did the occupant or structure receive the right momentum change? Did the event happen at the right time? Did the deformation travel the right distance? And, taken together, do the two signals agree in shape, not just at a few peaks?
This chapter defines the standard metric set, shows how each is computed from a measured-versus-simulated pair, and then explains how those metrics are aggregated into recognized rating schemes — normalized RMS error, CORA, and the ISO/SAE correlation rating approaches.
Sign convention and preprocessing
Before any metric is computed, fix the reference frame. For a frontal barrier pulse, acceleration is positive in the vehicle longitudinal direction (rearward occupant loading), and velocity is obtained by integrating acceleration from a known initial condition. Displacement is the double integral. If the measured channel is filtered (commonly SAE J211 CFC 60 for occupant accelerations, CFC 180 or 1000 for structural channels), the simulation must be filtered with the same class and cutoff. Mismatched filtering is the single most common cause of false correlation failures.
Time alignment is equally critical. Define t = 0 at a physical trigger (barrier contact, squib fire, or a defined acceleration threshold crossing) and apply the same definition to both signals. A 2 ms shift can move time-to-peak by more than the acceptance band on a short pulse.
The core metric set
Each metric below is computed on the aligned, equally filtered pair. Let m(t) be the measured signal and s(t) the simulated signal over the comparison window [t₁, t₂].
| Metric | Definition | What it diagnoses |
|---|---|---|
| Peak acceleration | max |a(t)| and its sign | Pulse severity; restraint loading level |
| Velocity change Δv | ∫a dt over window | Momentum transfer; energy input to occupant/structure |
| Displacement | ∬a dt² (or direct measurement) | Stroke, intrusion, travel distance |
| Time-to-peak | t at which peak occurs | Event phasing; timing of deployment vs. loading |
| Energy absorption | ∫F dx or ∫a·v dt | Work done by the structure/restraint |
| Normalized RMS error | ‖m−s‖ / ‖m‖ | Overall shape agreement |
Peak acceleration is reported as a signed value with its time stamp. A simulation that matches peak magnitude but places it 8 ms early is not correlated — it will misphase the restraint. Velocity change is the most robust single scalar because integration smooths high-frequency noise; a Δv error under 5% is a strong indicator that the overall impulse is right. Displacement exposes integration drift: if Δv matches but displacement diverges, the acceleration history has the wrong sign distribution over time. Time-to-peak is a phasing metric and should be reported in milliseconds, not as a percentage, because the acceptance band is absolute. Energy absorption is computed as the area under the force–displacement curve, or equivalently ∫a·v dt when mass is constant; it captures whether the structure dissipated the right amount of work.
Normalized RMS error
The normalized RMS error (NRMSE) gives a single number for shape agreement:
NRMSE = sqrt( Σ (m_i − s_i)² / Σ m_i² )where the sums run over all samples in the comparison window. A value of 0 means identical; 0.10 means the residual energy is 10% of the measured signal energy. NRMSE is sensitive to amplitude and phase but insensitive to sign convention errors only if both channels share the same convention. It is a useful screening metric but should never be the sole acceptance criterion, because a signal can have low NRMSE while missing a critical peak.
CORA and ISO/SAE rating methods
CORA (Correlation and Analysis) decomposes the comparison into independent sub-scores and combines them with weighting factors. The standard CORA approach evaluates:
- Corridor rating — does the simulated curve stay inside a tolerance band around the measured curve?
- Cross-correlation rating — how well do the shapes align after optimal time shift?
- Phase rating — agreement of peak timing and zero crossings.
- Magnitude rating — agreement of peak values and overall amplitude.
Each sub-rating is mapped to a 0–1 score, then combined into an overall CORA score. Typical acceptance thresholds in automotive practice are:
| Overall CORA score | Rating | Typical use |
|---|---|---|
| ≥ 0.90 | Excellent | Model validated for predictive use |
| 0.75 – 0.90 | Good | Acceptable for design iteration |
| 0.60 – 0.75 | Fair | Requires investigation; limited predictive use |
| < 0.60 | Poor | Not correlated; model must be revised |
The ISO/SAE correlation rating methods (ISO 18571 and the SAE J2954 family) formalize this further by defining objective rating metrics that are independent of the analyst. ISO 18571 specifies a set of metrics — including corridor, phase, magnitude, and slope — and a defined aggregation. The key advantage is reproducibility: two analysts running the same tool on the same data get the same score.
Worked example: frontal pulse correlation
Consider a 50 km/h frontal barrier test with a measured occupant chest acceleration pulse. The measured peak is 42 g at 68 ms; Δv over the 0–120 ms window is 14.2 m/s. The simulation predicts a peak of 39 g at 74 ms and Δv of 13.6 m/s. The metric table reads:
| Metric | Measured | Simulated | Error | Verdict |
|---|---|---|---|---|
| Peak acceleration | 42 g | 39 g | −7.1% | Pass (<10%) |
| Δv | 14.2 m/s | 13.6 m/s | −4.2% | Pass (<5%) |
| Time-to-peak | 68 ms | 74 ms | +6 ms | Marginal |
| Displacement | 0.31 m | 0.29 m | −6.5% | Pass |
| Energy absorption | 1.82 kJ | 1.71 kJ | −6.0% | Pass |
| NRMSE | — | — | 0.14 | Fair |
The +6 ms time-to-peak and NRMSE of 0.14 point to a phasing issue, likely in the restraint deployment timing or the structural crush sequence. The magnitude metrics pass, so the fix is not a stiffness change but a timing adjustment — for example, revisiting the airbag trigger time or the seat belt load limiter onset. This is exactly the diagnostic value of separating magnitude from phase metrics.
Choosing acceptance criteria for a program
Acceptance criteria are program-specific and must be written into the correlation plan before the first simulation runs. A defensible set for a frontal occupant restraint program might read:
- Δv error ≤ 5% over the full pulse window.
- Peak acceleration error ≤ 10% and time-to-peak within ±3 ms.
- Displacement error ≤ 8% at maximum stroke.
- Energy absorption error ≤ 8%.
- NRMSE ≤ 0.15 for primary channels; ≤ 0.20 for secondary channels.
- Overall CORA ≥ 0.80 for the occupant chest and pelvis channels; ≥ 0.75 for structural channels.
These thresholds are not universal. A side impact program with short, high-frequency pulses may tighten time-to-peak to ±2 ms and relax NRMSE to 0.20, because the signal is inherently noisier. The principle is to set criteria that reflect the physical sensitivity of the downstream decision — if a 3 ms timing error changes the injury outcome, then 3 ms is the criterion.
With the metric set and acceptance framework established, the next step is applying them consistently across a full correlation matrix — multiple channels, multiple tests, and multiple configurations — which is where the engineering judgment shifts from single-curve comparison to program-level validation.