Gate and Exclusion Thresholds

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

ADA · CALCULATION AUDIT

Gate and Exclusion Thresholds

The chapter gates a 25.0 C sample against a 22.0 C prediction and rejects it — its normalized innovation of 18.0 sits far past the 3.841 chi-square limit — then excludes a sensor whose 0.72 m residual breaks a 0.30 m range gate. Every one of these is a single number compared to a threshold. This audit reproduces each gate and exclusion threshold to show one comparison decides accept, reject, exclude, and readmit.

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

Ada: Every gate in this chapter is a comparison between a number and a threshold, so let me confirm each number lands on the side the text claims, using only the stated values.

For the scalar innovation gate, the prediction is 22.0 C with variance P = 0.25, and the measurement is 25.0 C with variance R = 0.25:

• Innovation: y = 25.0 - 22.0 = 3.000000
• Innovation variance: S = P + R = 0.25 + 0.25 = 0.500000
• Normalized innovation squared: NIS = y^2 / S = 9.000000 / 0.500000 = 18.000000

The one-dimensional 95 percent chi-square gate is 3.841 (equivalently 1.96^2 = 3.8416). Since 18.0 > 3.841, the 25.0 C sample is rejected — not because 25 C is hot, but because it sits sqrt(18) = 4.242641, about 4.24 standard deviations, from the prediction.

The fault-exclusion example uses a fixed range gate of 0.30 m against three residuals:

• Sensor A residual 0.08 <= 0.30: keep.
• Sensor B residual 0.11 <= 0.30: keep.
• Sensor C residual 0.72 > 0.30: exclude — it is 0.72 / 0.30 = 2.4x the gate.

Recovery reads the same threshold forward. If C returns at 0.09, 0.07, 0.06 m, all three are <= 0.30, so the recovery rule can readmit it. If C instead returns at 0.09 m and then jumps to 0.41 m, that 0.41 > 0.30 keeps it quarantined on the second sample.

The audit conclusion is narrow: one one-line comparison decides accept, reject, exclude, and readmit. Recording the gate value beside the residual is what lets a later reviewer reproduce every one of those decisions instead of trusting a smooth output.

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