Mechatronics & Robotics

Sensor Fusion Uncertainty Calculator — Correlated Two-Sensor BLUE & Fusion Gain

Calculate 1σ fused uncertainty for two unbiased length sensors using correlated BLUE weighting, with fusion gain plus optional latency, paired sampling-rate and uncertainty-requirement checks.

Unit-aware inputs Deterministic calculation Engineering interpretation
Calculation workspace

Enter the known values and review the calculated result

Deterministic calculation
01
Parameters

Input parameters

Use consistent values and select the intended engineering units.

Sensor 1

Sensor 2

Fusion parameters

Performance requirement

02
Output

Results

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Engineering reference

Method, application and limitations

Review the calculation method, intended application and engineering assumptions before using the result in a design decision.

01
Method

Formula and calculation method

Correlated two-sensor BLUE model

The calculator combines the standard uncertainties of two sensors that estimate the same scalar measurand. Both sensor estimates are assumed to be unbiased. The correlation input describes correlation between the measurement errors, not correlation between the measured signal values.

Input quantities

Symbol Meaning
u1 Sensor 1 standard uncertainty, 1σ
u2 Sensor 2 standard uncertainty, 1σ
ρe Correlation coefficient of the two sensor measurement errors
ureq Optional maximum allowed fused standard uncertainty

The required uncertainty inputs must satisfy u1 > 0, u2 > 0 and −1 < ρe < 1.

Error covariance

C12 = ρe u1u2

where C12 is the covariance term associated with the two sensor errors.

BLUE denominator

D = u12 + u22 − 2C12

BLUE coefficients

w1 = (u22 − C12) / D

w2 = (u12 − C12) / D

The two coefficients sum to 1. With sufficiently strong positive error correlation and unequal sensor uncertainties, one BLUE coefficient can become negative. The calculator identifies this as a high-correlation weighting regime.

Fused standard uncertainty

ufused2 = (u12u22 − C122) / D

ufused = √[(u12u22 − C122) / D]

The result is a standard uncertainty (1σ). It is not an expanded uncertainty and no coverage factor is applied.

Fusion gain

First define the uncertainty of the better individual sensor:

ubest = min(u1, u2)

The percentage reduction in standard uncertainty relative to that sensor is:

G = [(ubest − ufused) / ubest] × 100%

Optional pair timing

When both sensor latencies are entered, the pair-ready latency is:

tpair = max(t1, t2)

When both sensor sampling rates are entered, the maximum paired update rate is:

fpair,max = min(f1, f2)

These timing relationships assume synchronized, time-aligned pairwise fusion in which every fused result requires one matching sample from each sensor.

Optional uncertainty requirement

If ureq is provided, the requirement is satisfied when:

ufused ≤ ureq

The corresponding requirement margin is:

Mu = [(ureq − ufused) / ureq] × 100%

A positive margin produces a PASS result; a negative margin produces a FAIL result.

02
Application

When to use this calculator

When to use this sensor fusion calculator

Use this calculator when two sensors provide unbiased estimates of the same scalar length-related measurand, such as position, displacement or distance, and you want to estimate the standard uncertainty achievable by combining them with a correlated Best Linear Unbiased Estimator (BLUE).

It is particularly useful when the sensors do not have independent errors. Instead of assuming that two measurements can be combined using a simple independent inverse-variance rule, the calculator includes an explicit error correlation coefficient ρe.

Typical engineering uses

  • compare the expected uncertainty of a two-sensor measurement architecture with the better individual sensor;
  • evaluate whether adding a second sensor provides meaningful uncertainty reduction;
  • study the effect of correlated sensor errors on fusion performance;
  • calculate the BLUE coefficients assigned to each sensor uncertainty model;
  • check whether the fused 1σ uncertainty remains below an entered requirement;
  • identify high-correlation cases that produce a negative BLUE coefficient;
  • screen pair-ready latency when both sensor latencies are known;
  • estimate the maximum paired update rate from the two sensor sampling rates.

Use standard uncertainty, not an unspecified accuracy limit

Enter u1 and u2 as 1σ standard uncertainties. A datasheet value stated only as ± accuracy, tolerance, maximum error or worst-case error is not automatically equivalent to one standard deviation. Convert or characterize the specification appropriately before using it as a standard uncertainty.

When this calculator is not appropriate

This is not a general measurement uncertainty budget calculator and it does not estimate the correlation coefficient from sample data. It also does not implement a Kalman filter, extended Kalman filter, asynchronous sensor fusion, state prediction, bias estimation, outlier rejection or sensor fault detection.

Do not use the model when the two sensors measure different physical quantities, when significant sensor bias has not been corrected, or when a static two-sensor covariance model cannot adequately represent the measurement errors.

03
Decision support

How to interpret the result

How to interpret the sensor fusion results

Fused standard uncertainty, ufused

This is the primary 1σ uncertainty result of the correlated two-sensor BLUE model. A lower value represents lower estimated dispersion of the fused measurement result under the entered uncertainty and correlation assumptions.

Compare ufused with u1 and u2 to determine whether the second sensor provides a practically useful uncertainty reduction.

Fusion gain, G

Fusion gain expresses the uncertainty reduction relative to the better individual sensor. A result close to 0% means the second sensor adds little uncertainty-reduction benefit under the assumed covariance model. A larger positive value means the fused 1σ uncertainty is correspondingly lower than that of the better standalone sensor.

BLUE coefficients, w1 and w2

The coefficients describe the linear BLUE weighting implied by the two sensor uncertainties and their error covariance. The coefficients sum to 1.

A smaller uncertainty does not automatically mean a sensor receives a coefficient of exactly 1. Both uncertainty magnitude and cross-correlation influence the optimum weighting.

High-correlation / negative-coefficient regime

If either BLUE coefficient becomes negative, the calculator reports a high-correlation weighting regime. This can occur with strong positive error correlation and unequal uncertainties.

The result is not automatically mathematically invalid, but it becomes more sensitive to errors in the assumed covariance model. Verify that ρe represents the correlation of the sensor errors and that both standard uncertainties are supported by appropriate measurement data.

Pair-ready latency

tpair is the larger entered end-to-end sensor latency. It represents the earliest time at which a synchronized pair can be ready when a fused result requires one corresponding measurement from each sensor.

The calculator also identifies which sensor is the latency bottleneck, or reports equal latency when both values are identical.

Maximum paired update rate

fpair,max is the lower of the two entered sampling rates. Under the implemented pairwise model, the slower sensor limits how frequently complete two-sensor pairs can be produced.

Uncertainty requirement and margin

PASS means ufused ≤ ureq. FAIL means the calculated fused standard uncertainty exceeds the entered requirement.

A positive uncertainty margin means the result is below the limit. A negative margin indicates by what percentage of the entered requirement the calculated uncertainty exceeds that limit.

A PASS requirement can still coexist with a high-correlation warning because the requirement status and the BLUE weighting regime evaluate different aspects of the result.

Meeting the entered uncertainty requirement does not confirm overall measurement-system performance. Sensor bias, calibration, synchronization, dynamic response, outliers, environmental effects and other error sources may still require separate verification.

04
Worked case

Calculation example

Worked example — two correlated position sensors

Two sensors estimate the same position. Their uncertainties are expressed as 1σ standard uncertainties. The design also has a maximum permitted fused uncertainty and optional timing information.

Inputs

Quantity Value
Sensor 1 standard uncertainty, u1 0.10 mm
Sensor 2 standard uncertainty, u2 0.15 mm
Error correlation coefficient, ρe 0.20
Sensor 1 latency, t1 8 ms
Sensor 2 latency, t2 12 ms
Sensor 1 sampling rate, f1 200 Hz
Sensor 2 sampling rate, f2 125 Hz
Maximum allowed fused uncertainty, ureq 0.095 mm

1. Calculate the covariance term

C12 = ρeu1u2 = 0.20 × 0.10 × 0.15 = 0.0030 mm2

2. Calculate the BLUE denominator

D = 0.102 + 0.152 − 2(0.0030) = 0.0265 mm2

3. Calculate the BLUE coefficients

w1 = (0.152 − 0.0030) / 0.0265 = 0.73585

w2 = (0.102 − 0.0030) / 0.0265 = 0.26415

Both coefficients are positive, so this case remains in the standard BLUE weighting regime.

4. Calculate fused standard uncertainty

ufused2 = [(0.102)(0.152) − 0.00302] / 0.0265 = 0.00815094 mm2

ufused = √0.00815094 = 0.09028 mm

5. Calculate fusion gain

The better individual sensor has ubest = 0.10 mm.

G = [(0.10 − 0.09028) / 0.10] × 100% = 9.72%

The BLUE combination therefore reduces the standard uncertainty by approximately 9.72% relative to the better individual sensor.

6. Evaluate timing

tpair = max(8, 12) = 12 ms

Sensor 2 is the pair-latency bottleneck.

fpair,max = min(200, 125) = 125 Hz

7. Check the uncertainty requirement

0.09028 mm ≤ 0.095 mm

The evaluated uncertainty requirement therefore passes.

Mu = [(0.095 − 0.09028) / 0.095] × 100% = 4.97%

The fused standard uncertainty is approximately 4.97% below the entered maximum. This PASS result verifies only the entered uncertainty criterion. It does not validate sensor bias, calibration, synchronization quality, dynamic tracking performance or the overall reliability of the measurement system.

05
Model boundaries

Assumptions and limitations

Assumptions and limitations

  • Exactly two sensors are modeled. The implemented equations do not perform an arbitrary multi-sensor covariance-matrix fusion.
  • Both sensors must estimate the same scalar measurand. The current inputs are length quantities, so the calculator is intended for length-related measurements such as position, displacement or distance.
  • The sensor estimates are assumed to be unbiased. Constant bias, calibration offset, drift and other systematic effects are not estimated or corrected by this model.
  • u1 and u2 are 1σ standard uncertainties. A datasheet ± accuracy, tolerance or maximum-error specification should not be entered directly unless it legitimately represents one standard deviation.
  • Error dependence is represented by one correlation coefficient. The model uses C12 = ρeu1u2 and requires −1 < ρe < 1.
  • The correlation applies to measurement errors, not signal values. Correlation observed between the two sensor outputs does not by itself establish the required error correlation coefficient.
  • The covariance model is treated as fixed for the calculation. Time-varying uncertainty, frequency-dependent noise, changing correlation and operating-condition-dependent covariance are not modeled.
  • No dynamic state estimator is implemented. The calculator is not a Kalman filter, extended Kalman filter, complementary filter or recursive Bayesian estimator.
  • No fused measurement value is calculated. The tool evaluates fused uncertainty and BLUE coefficients but does not accept the two actual sensor readings.
  • No expanded uncertainty is calculated. The result is a standard uncertainty; no coverage factor or confidence interval is applied.
  • Timing diagnostics require both values in a pair. Both sensor latencies must be supplied to evaluate pair-ready latency, and both sampling rates must be supplied to evaluate the maximum paired update rate.
  • Timing assumes synchronized, time-aligned pairwise fusion. Asynchronous measurements, interpolation, buffering strategies, timestamp uncertainty, out-of-sequence measurements and prediction are outside the implemented model.
  • Strong correlation requires careful interpretation. A negative BLUE coefficient can be mathematically valid, but the calculated benefit may be highly sensitive to uncertainty or correlation modeling errors.
  • The requirement check covers only fused standard uncertainty. PASS does not verify sensor reliability, functional safety, fault tolerance, dynamic accuracy or compliance with a specific standard.
06
Questions

Frequently asked questions

What does this sensor fusion uncertainty calculator calculate?
It calculates the 1σ standard uncertainty produced by combining two unbiased sensors that estimate the same scalar length-related measurand. It also reports the two BLUE coefficients and the percentage uncertainty reduction relative to the better individual sensor. Optional inputs add pair-ready latency, maximum paired update rate and a fused-uncertainty requirement check.
What is BLUE sensor fusion?
BLUE means Best Linear Unbiased Estimator. In this calculator, the uncertainties and error correlation of two unbiased sensor estimates determine the linear coefficients that minimize the variance of the fused estimate within the assumed two-sensor covariance model. The calculator reports those coefficients and the resulting standard uncertainty; it does not accept the actual sensor readings or calculate the fused measurand value.
What should I enter as sensor uncertainty?
Enter the standard uncertainty of each sensor as a 1σ quantity. Do not enter a datasheet ± accuracy limit, tolerance or maximum permissible error as 1σ unless its statistical interpretation is known and it has been converted appropriately. Both uncertainty values must describe the same measurand.
What does the error correlation coefficient mean?
ρe describes correlation between the measurement errors of the two sensors. It is not the correlation between the measured signal values themselves. A value of 0 means the error terms are treated as uncorrelated. Positive correlation means the errors tend to move together, while negative correlation means they tend to move in opposite directions.
Does the calculator determine the correlation coefficient from measurement data?
No. This is not a correlation coefficient calculator. The error correlation coefficient must be supplied as an input based on an appropriate error model, experimental characterization or other justified engineering evidence. The accepted numerical range is −1 < ρe < 1.
Why can a BLUE sensor weight become negative?
With unequal sensor uncertainties and sufficiently strong positive error correlation, the minimum-variance BLUE solution can assign a negative coefficient to one sensor. The calculator flags this as a high-correlation weighting regime. A negative coefficient is mathematically possible in an unconstrained BLUE model, but it makes the result especially dependent on the validity of the assumed uncertainty and correlation model.
How does error correlation affect sensor fusion gain?
Correlation changes both the BLUE coefficients and the fused variance. For comparable sensors, increasing positive error correlation generally reduces the independent information available from the second measurement, so the uncertainty-reduction benefit can become small. Negative correlation can produce a larger theoretical uncertainty reduction, provided the assumed covariance model is valid.
Is this a Kalman filter calculator?
No. The calculator performs a static two-sensor BLUE uncertainty calculation. It does not model state dynamics, prediction, process noise, time-varying covariance, nonlinear measurement equations or recursive filtering. Kalman, extended Kalman and other dynamic sensor-fusion methods require a different model.
How are sensor latency and sampling rate combined?
When both latencies are entered, pair-ready latency is the larger of the two sensor latencies. When both sampling rates are entered, the maximum paired update rate is the lower sampling rate. These diagnostics assume synchronized, time-aligned pairwise fusion in which each fused result waits for one matching sample from both sensors.
What does fusion gain mean?
Fusion gain is the percentage reduction in fused standard uncertainty relative to the individual sensor with the lower uncertainty. For example, a fusion gain of 10% means the calculated 1σ fused uncertainty is 10% lower than the standard uncertainty of the better standalone sensor under the assumed covariance model.
What does a PASS uncertainty status confirm?
PASS means only that the calculated fused standard uncertainty is less than or equal to the maximum value entered as ureq. It does not demonstrate overall sensor-system reliability, calibration validity, acceptable bias, fault tolerance, dynamic tracking performance or compliance with a particular engineering standard.
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Engineering notice. Verify input data, assumptions, applicable standards and required safety factors before using results for design release or safety-critical decisions.