Blend Weights and the Independence Assumption

Ada re-derives this chapter’s own numbers step by step, at full precision

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Ada ADA · CALCULATION AUDIT

Blend Weights and the Independence Assumption

The chapter makes two claims: the gyro owns a fast tilt update, its fused 10.352 deg barely leaving the pure-gyro 10.4 deg, and averaging two sensors reading 10.0 and 10.4 shrinks uncertainty from 0.5 toward 0.35. The second claim hides a condition. This audit checks the blend weights and the independence assumption to show fusion shrinks random noise but cannot remove a shared bias.

Companion to the chapter Sensor Fusion Foundations — every number here comes from that chapter.

Ada: This chapter makes two numeric claims — that the gyro carries most of one update, and that averaging can shrink uncertainty. Both are true, but the second one hides a condition. Let me check both with the chapter’s own values.

For the tilt update, the previous fused angle is 10.0 deg, the gyro rate is 20.0 deg/s, the time step is 0.020 s, the accelerometer estimate is 8.0 deg, and alpha is 0.98:

  • Gyro prediction: 10.0 + 20.0 x 0.020 = 10.0 + 0.4 = 10.400000 deg
  • Fused angle: 0.98 x 10.400000 + 0.02 x 8.0 = 10.192000 + 0.160000 = 10.352000 deg

The fused value sits 10.400 - 10.352 = 0.048 deg below the pure gyro prediction, so the accelerometer moved it only a hair toward gravity — the gyro owns this fast update.

Now the shared-bias example. Sensor A reads 10.0 and sensor B reads 10.4, each with random uncertainty of about 0.5:

  • Simple average: (10.0 + 10.4) / 2 = 10.200000
  • If the two errors are independent, the average’s uncertainty shrinks by 1 / sqrt(2): 0.5 / sqrt(2) = 0.5 / 1.414214 = 0.353553, about 0.35 — smaller than either sensor’s 0.5.
  • If instead both sensors share a +0.4 calibration offset, the average still carries the full +0.4, because averaging two copies of the same bias returns that bias unchanged.

The audit conclusion is narrow: the smaller 0.35 uncertainty band is real only under independence. Fusion arithmetic can shrink random noise, but it cannot remove a shared bias — so the honest fused record publishes the common-calibration risk right next to the tighter-looking number.

Every number above is taken from the chapter’s own material and re-derived step by step.