Error Budget and Airtime Load

Error Budget and Airtime Load

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

ADA · CALCULATION AUDIT

Error Budget and Airtime Load

A 30 m x 20 m work area places anchors at known coordinates and mounts a tag on a cart, where a geometry factor of 3 can turn a 0.20 m range error into about 0.60 m of position uncertainty. A fuller error budget of range 0.15 m, geometry factor 2.5, environment bias 0.20 m, and filter lag 0.10 m sums to about 0.675 m, which clears a room-level need but fails a 0.30 m robot-safety boundary, while a TWR design with 80 tags, 4 anchors, and 1 update per second asks for 320 ranging exchanges per second. This audit asks the question those sums invite: are the geometry-amplified error budget and the TWR airtime load really additive bookkeeping the way the chapter claims, and which decision do they actually clear?

Companion to the chapter UWB Indoor Positioning Systems — every number here comes from that chapter.

See the relationship before changing it

The figure reads from left to right. The blue card is range error. The middle card applies this page's rule. The green card is geometry-amplified error. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only range error, so the numeric fixture does not switch without explanation.

Walk the arrows. Poor anchor geometry multiplies the range error before the safety boundary is tested.

Derive the baseline in four named moves

1. 1

   Name the input. The chapter baseline is 0.2 m.

2. 2

   Name the relationship. position uncertainty = geometry factor 3 x range error

3. 3

   Substitute with units. 3 x 0.20 m = 0.60 m

4. 4

   Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change range error

Try Predict the direction of position uncertainty = geometry factor 3 x range error. Test another range error, then compare geometry-amplified error.

Range error

0.2 m

Chapter baseline

Geometry-amplified error

Observe Poor anchor geometry multiplies the range error before the safety boundary is tested. Reset range error to 0.2 and compare geometry-amplified error.

Explain Poor anchor geometry multiplies the range error before the safety boundary is tested.

Check yourself

What should you do before trusting a moved-control result?

Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only range error moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only range error. It is not a field prediction.

Try

Apply geometry factor 2.5 to a 0.15 m range error, add 0.20 m environment bias and 0.10 m filter lag, then total TWR load for 80 tags and 4 anchors at 1 Hz.

Observe

The position ledger reaches 0.675 m; TWR requires 320 exchanges/s while one-blink-per-tag TDoA uses 80 transmissions/s.

Explain

Geometry magnifies ranging uncertainty before other errors are added, and TWR airtime multiplies by anchor count because each tag performs a separate exchange with every anchor.

Ada: This chapter builds two kinds of number: a position error budget and an airtime load. Both are just sums, and both decide whether a design is honest. Let me total them with the chapter’s own values.

Geometry amplification, from the GDOP example: a 0.20 m range error under a geometry factor of 3 becomes 0.20 x 3 = 0.60 m of position uncertainty before any other term is added.

The full error budget (range 0.15 m, geometry factor 2.5, environment bias 0.20 m, filter lag 0.10 m):

• 0.15 x 2.5 = 0.375 m from range times geometry.
• 0.375 + 0.20 + 0.10 = 0.675 m total.
• That clears a room-level 1.0 m need but fails a 0.30 m robot-safety boundary – the same evidence is adequate or inadequate only relative to the decision.

Airtime load, TWR versus TDoA:

• TWR at 80 tags, 4 anchors, 1 Hz: 80 x 4 x 1 = 320 ranging exchanges per second.
• A 10-tool commissioning fleet at 2 Hz: 10 x 4 x 2 = 80 exchanges per second – fewer requests, stronger per-anchor evidence each.
• Relaxing 200 tags from 1 Hz to 0.2 Hz: 200 x 0.2 = 40 reports per second.
• TDoA at 80 tags, 1 Hz: one blink each, so 80 x 1 = 80 transmissions – there is no x anchors factor, which is exactly why it scales.

The audit shows both budgets are bookkeeping, not a rating: geometry multiplies the error term by term, and the anchor factor is the single line separating TWR airtime from TDoA – so the honest release states which sum was met, and under what geometry and load.

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

Technical boundaries: This additive budget omits covariance and error distributions, anchor survey error, clock synchronisation, RF collisions, protocol overhead, NLOS detection, scheduler capacity, and estimator dynamics.

Ready: work the ledger before checking it.