Sensor Fusion Uncertainty Calculator

Calculate the fused position uncertainty when combining two sensor modalities — and see quantitatively why multi-sensor fusion always outperforms any single sensor alone.

⚙️ Input Parameters

Sensor A

Uncertainty σ_A (m)

GPS urban: 2–10m

Weight w_A

Relative confidence

Sensor B

Uncertainty σ_B (m)

LiDAR map-match: 0.02–0.1m

Weight w_B

Increase to trust B more

📊 Results

🔀

Enter the uncertainty (σ) of two sensors to see how fusion reduces total position uncertainty compared to either sensor alone.

Fused Uncertainty (1σ)

—m

Sensor A variance (σ²_A)—

Sensor B variance (σ²_B)—

Improvement vs Sensor A alone—

Improvement vs Sensor B alone—

Effective weight on Sensor B—

Why Sensor Fusion Reduces Uncertainty Below Any Single Sensor

What This Calculates

Weighted least squares fusion weights each sensor’s contribution inversely proportional to its variance — low uncertainty sensors get high weight. The fused result is always more accurate than any single sensor alone — mathematically guaranteed. This is the measurement update step inside every Kalman filter, in scalar form.

When to Use It

Quantifying benefit of adding a second sensor to your stack
Setting initial Kalman filter measurement noise matrices (R)
Justifying multi-sensor cost in safety case or project proposal
Evaluating what happens when one sensor degrades (raise its σ)
Understanding why BEV fusion outperforms late fusion

Formula

σ²_fused = 1/(w_A/σ²_A + w_B/σ²_B) σ_fused = √(σ²_fused)Effective weight on B: = (w_B/σ²_B) × σ²_fused × 100%Improvement vs A: = (1 – σ_fused/σ_A) × 100%

Built by Dr. Dilip Kumar Limbu · Co-Founder, MooVita · Former Principal Scientist, A*STAR · UDHY Engineering Tools

FAQs on Sensor Fusion Uncertainty

1. What is sensor fusion and why is it used in autonomous vehicles?

Sensor fusion is the mathematical combination of data from multiple sensors to produce a more accurate, reliable estimate than any individual sensor provides. AVs use fusion because no single sensor covers all scenarios: cameras fail in low light; LiDAR degrades in heavy rain; radar has coarse resolution. Fusing modalities exploits their complementary strengths while mitigating individual weaknesses — improving accuracy and providing redundancy when one sensor degrades or fails.

2. How does weighted least squares sensor fusion work?

Weighted least squares assigns each sensor influence inversely proportional to its variance (uncertainty squared). Low uncertainty sensors get high weight. The fused variance is computed as 1 / (w_A/σ²_A + w_B/σ²_B), and fused standard deviation is its square root. The result is always smaller than either individual σ — mathematically guaranteed regardless of sensor type or scenario. This is the same calculation performed in the measurement update step of every Kalman filter.

3. Why does fusion improve accuracy even when one sensor is much better?

Mathematically, adding any measurement with finite uncertainty always reduces combined uncertainty — even if the second sensor is much worse than the first. However, the improvement diminishes rapidly as the second sensor becomes worse. Combining GPS (5m) with LiDAR (0.05m) produces essentially the same result as LiDAR alone (≈0.05m fused) because GPS's large variance gives it near-zero weight. The largest proportional improvement occurs when both sensors have comparable, moderate uncertainty.

4. How do I set sensor weights in my fusion algorithm?

In a Kalman filter, weights are set implicitly through the measurement noise covariance matrix R (larger R = less weight on that measurement). Start with manufacturer-specified accuracy (1σ). Verify experimentally against ground truth in your operating environment — datasheet accuracy often doesn't reflect real-world urban canyon GPS degradation or adverse weather LiDAR performance. Update R dynamically based on sensor health signals (GPS satellite count, HDOP, LiDAR scan quality) for adaptive fusion that automatically reduces trust in degraded sensors.

5. What sensor combinations are most common in AV localisation fusion?

The most common AV localisation fusion combines: GNSS/GPS (absolute global position, 1–10m urban error); LiDAR map-matching (sub-10cm relative position within mapped area); IMU (high-frequency velocity and heading between LiDAR/GPS updates); wheel odometry (short-term distance estimates). These are fused in an EKF or Factor Graph Optimizer, with each sensor's R matrix set to reflect real-world uncertainty — which this calculator helps estimate for any two-sensor combination.

6. What is the difference between early, mid-level, and late sensor fusion?

Early fusion combines raw sensor data before processing (e.g. projecting LiDAR points onto camera image planes). Mid-level (feature-level) fusion processes each sensor independently to extract features, then fuses features — this is the BEV fusion approach where camera and LiDAR features are combined in a shared top-down grid. Late fusion runs each sensor through a complete pipeline independently, then combines final object detections. This calculator implements the uncertainty arithmetic applicable to late fusion uncertainty estimation.