Timing Error to Range Error
Timing Error to Range Error
Ada re-derives this chapter’s own numbers step by step, at full precision
ADA · CALCULATION AUDIT
Timing Error to Range Error
A tag’s poll-and-response exchange runs from 0 ns to 220 ns, with the responder recording 10 ns and 210 ns, so the round trip reduces to a 20 ns flight time and a worked distance of 3.0 m at 0.30 m/ns. The same conversion shows a 2 ns round-trip timing error becomes about 0.30 m of range error, which the chapter says may be fine for a tool cabinet that only needs to know a device is within about 1.5 m, but not enough for a robot cell that needs a 0.25 m exclusion boundary. This audit asks the question that gap invites: does the chapter’s own arithmetic actually support the worked 3.0 m distance and the 0.30 m error warning, once calibration and clock behavior are accounted for?
Companion to the chapter UWB Ranging Techniques — every number here comes from that chapter.
See the relationship before changing it
The figure reads from left to right. The blue card is round-trip timing error. The middle card applies this page's rule. The green card is range 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 round-trip timing error, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 2 ns.
- 2
Name the relationship. range error = round-trip timing error x 0.30 m/ns / 2
- 3
Substitute with units. 2 ns x 0.30 m/ns / 2 = 0.30 m
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change round-trip timing error
Try Predict the direction of range error = round-trip timing error x 0.30 m/ns / 2. Test another round-trip timing error, then compare range error.
Observe Two-way ranging halves the round-trip distance, but nanoseconds still become large spatial errors. Reset round-trip timing error to 2 and compare range error.
Explain Two-way ranging halves the round-trip distance, but nanoseconds still become large spatial errors.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
From T1 = 0 ns, T4 = 220 ns, T2 = 10 ns, and T3 = 210 ns, subtract the responder delay and halve the remaining round trip.
The elapsed exchange is 220 ns, the programmed reply occupies 200 ns, and the residual 20 ns becomes 10 ns one-way or 3.0 m.
Two-way ranging removes the known turnaround interval from total elapsed time; the remaining propagation covers the path twice, so halving it yields one-way flight time.
Ada: UWB ranging is a physics calculation before it is a location claim. The chapter’s examples already give the values needed to check both the timing-error warning and the worked two-way range.
- The chapter states that light travels about
0.30 m/ns. - A
2 nsround-trip timing error becomes one-way range error after the two-way divide:0.30 m/ns x 2 ns / 2 = 0.30 m. - In the worked TWR example, the initiator elapsed time is
T4 - T1 = 220 ns - 0 ns = 220 ns. - The responder delay is
T3 - T2 = 210 ns - 10 ns = 200 ns. - The flight-time part of the round trip is therefore
220 ns - 200 ns = 20 ns. - One-way flight time is
20 ns / 2 = 10 ns. - Distance is
10 ns x 0.30 m/ns = 3.0 m.
The audit conclusion is bounded: the arithmetic supports the stated 3.0 m example and the 0.30 m error warning, but only for a measurement record that also proves calibration, clock behavior, and channel quality are good enough for the action using the range.
Every number above is taken from the chapter’s own material and re-derived step by step.
Technical boundaries: This four-timestamp example assumes calibrated, symmetric clocks and a known reply delay; it omits antenna delays, drift, timestamp quantisation, multipath, NLOS bias, packet detection error, and security checks.