Sensor Fusion Reliability Calculator — Fused Accuracy, Effective Latency & Fusion Gain Validator
Calculate fused accuracy using sensor accuracy, latency, sampling rate and correlation coefficient based on variance-weighted sensor fusion.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Sensor 1
Sensor 2
Fusion parameters
System dynamics
Performance requirements
Results
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Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
Fused accuracy formula:
σ12 = variance of Sensor 1
σ22 = variance of Sensor 2
cov12 = ρ · √(σ12 · σ22)
denominator = σ12 + σ22 − 2 · cov12
w1 = (σ22 − cov12) / denominator
w2 = (σ12 − cov12) / denominator
σfused,var2 = (σ12 · σ22 − cov122) / denominator
σfused = max(√σfused,var2, 0.5 · min(σ1, σ2))
Effective latency formula:
teff = w1 · t1 + w2 · t2
Fusion gain formula:
G = ((min(σ1, σ2) − σfused) / min(σ1, σ2)) · 100
where:
- σfused — fused accuracy (length)
- teff — effective latency (time)
- G — fusion gain (%)
- σ1 — Sensor 1 accuracy (length)
- σ2 — Sensor 2 accuracy (length)
- t1 — Sensor 1 latency (time)
- t2 — Sensor 2 latency (time)
- f1 — Sensor 1 sampling rate (frequency)
- f2 — Sensor 2 sampling rate (frequency)
- ρ — correlation coefficient (-)
- w1 — Sensor 1 fusion weight (-)
- w2 — Sensor 2 fusion weight (-)
When to use this calculator
When to use this calculator:
- Evaluate whether two sensors can meet a required fused accuracy after variance-weighted fusion.
- Check whether combined sensor latency is acceptable for slow, medium or fast system dynamics.
- Compare two sensors with different accuracy values, latency values and sampling rates before fusion.
- Detect cases where one sensor dominates the fused result with more than 90% weighting.
- Identify sampling-rate inconsistency when the faster sensor rate is more than 2 times the slower sensor rate.
How to interpret the result
Fused accuracy is defined as the final accuracy error after combining two sensors using variance, covariance and fusion weights.
Effective latency is defined as the weighted latency of both sensors. It depends on Sensor 1 latency, Sensor 2 latency and the calculated fusion weights.
Fusion gain is defined as the percentage improvement of fused accuracy compared with the more accurate individual sensor.
Fused accuracy depends on both sensor accuracy values and the correlation coefficient. Increasing sensor accuracy error increases fused accuracy error. Increasing correlation decreases the useful gain from fusion.
- Safe — used when fused accuracy is not greater than required accuracy, effective latency is within the selected dynamics limit, no noise amplification is detected, the sampling rate ratio is not greater than 2, and no sensor has more than 90% dominance.
- Warning — used when fused accuracy is not greater than required accuracy, but at least one additional condition fails: latency outside the dynamics limit, noise amplification, sampling rate ratio greater than 2, or sensor dominance above 90%.
- Unsafe — used when fused accuracy is greater than required accuracy.
- Invalid — used when input validation fails, correlation is outside (-1, 1), latency is too high relative to sampling time, the fusion denominator is not greater than 0, fused variance is not positive, effective latency is invalid, or fusion gain is not finite.
The result is used to evaluate whether two sensors improve accuracy without violating latency, sampling consistency and dominance constraints.
Calculation example
Example:
A user wants to check whether two position sensors can be fused for a medium-dynamics system with a required accuracy of 0.8 mm.
- σ1 — Sensor 1 accuracy = 1.0 mm
- t1 — Sensor 1 latency = 0.010 s
- f1 — Sensor 1 sampling rate = 100 Hz
- σ2 — Sensor 2 accuracy = 1.5 mm
- t2 — Sensor 2 latency = 0.015 s
- f2 — Sensor 2 sampling rate = 80 Hz
- ρ — Correlation coefficient = 0.2
- D — Dynamics class = Medium
- σreq — Required accuracy = 0.8 mm
Result: σfused = 0.930 mm, teff = 0.0116 s, G = 7.0%.
Assumptions and limitations
- The dynamics class is limited to Slow, Medium or Fast.
- Sensor accuracy values must be greater than 0.
- Sensor latency values must be greater than or equal to 0.
- Sensor sampling rates must be greater than 0.
- Required accuracy must be greater than 0.
- The correlation coefficient must be greater than -1 and less than 1.
- Sensor 1 latency must not exceed 5 times Sensor 1 sampling time.
- Sensor 2 latency must not exceed 5 times Sensor 2 sampling time.
- Fused accuracy is clamped to not go below 50% of the best individual sensor accuracy.
- Slow dynamics allow effective latency up to 2 times the shortest sampling time.
- Medium dynamics allow effective latency up to 1 times the shortest sampling time.
- Fast dynamics allow effective latency up to 0.5 times the shortest sampling time.