Sensor Fusion: Combining Uncertain Measurements

Robotics and Automation · CHAPTER 5 / 5

Level: introductory undergraduate · Study session: 15–20 minutes including practice.

Learning outcome: Compute a weighted estimate under explicit assumptions.

Before you begin: Complete earlier chapters in this discipline; revise algebra and units as needed.

1. Understand the idea

Two independent unbiased measurements of the same static quantity can be combined using weights inversely proportional to variance. The estimate favours the more precise sensor. Shared bias, correlated errors and outliers invalidate the simple benefit claim. Calibration and timing must precede fusion.

2. Write the model

x̂ = (x1/σ1²+x2/σ2²)/(1/σ1²+1/σ2²)

Define the symbols from the worked case before substituting values. Keep units consistent, record the assumptions and round only after the calculation.

3. Follow a worked example

If x1=10 with σ1=1 and x2=12 with σ2=2, weights are 1 and 0.25. The fused estimate is 10.4 and its ideal standard deviation is √0.8 ≈0.894.

4. Explore the model

Figure: calculated teaching cases; straight segments connect sampled points, not experimental observations. Read exact values below.

Illustrative scenario sweep — synthetic educational data
σ2 (σ1=1)	x1=10, x2=12	Fused estimate
1	equal uncertainty	11
2	sensor 1 favoured	10.4
3	sensor 1 favoured more	10.2

Change sensor 2 uncertainty while holding measurements fixed. The estimate moves toward sensor 1 as σ2 grows.

5. Reproduce it yourself

Copy the input cases into Excel or a notebook and name each quantity and unit.
Recalculate each result from the stated equation or rule. For qualitative cases, explain the state or partition decision instead.
Change one input, predict the direction of change, then calculate again.
Write one sentence explaining where the model stops being valid.

6. Check your understanding

What is the fused estimate for equal uncertainties?

Reveal the answer and check your work

11.

Compare your result, units and reasoning. If they differ, return to the equation and test the worked case first.

7. Common mistake

Two biased sensors can produce a precise-looking but wrong result.

Further study and academic note

Explore the discipline reference and further learning resources. This is an original IIMEEE introductory teaching note with synthetic examples; it is not a peer-reviewed research result or a professional design specification. Figures were constructed from the listed teaching cases. Resource links support further study and do not imply endorsement.