Marvelmind IA-04-2D-Badge, Operating Manual

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11.10. System Accuracy Evaluation
Accuracy of distance measurement:
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Marvelmind navigation system can measure distances between beacons with
accuracy of +/- 2cm if correct ultrasound speed is used.
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The ultrasound speed depends on several factors: temperature, pressure, and
humidity. Other factors have an insignificant effect.
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The main factor is temperature. In temperature range of -
20…+50 °C the speed of
ultrasound changes on about 0.6 m/ (s* °C). It gives distance error about (0.6 / 340)
*100% ~ 0.17%/ °C. So caused by incorrect temperature setting absolute error of
distance measurement is 0.17% of real distance between beacons. For example, with
distance 30 meters and 5 °C error, this gives 0.85%*30 ~ 0.25
meters’ error.
Marvelmind system allows to setup temperature of air in the system settings.
Accuracy of position measurement:
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Marvelmind system uses trilateration algorithm to calculate position by distances. The
inaccuracy of position calculation is related to inaccuracy of distances measurement
and to geometry of relative location of stationary and mobile beacons
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Basic trilateration formulas are given in this article:
https://en.wikipedia.org/wiki/Trilateration
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As you see, the position of mobile beacons X , Y , Z is calculated from positions of 3
stationary beacons which are set by values of d , i , j . One of the beacons was shifted
to (0,0) position to simplify formulas in the article. In formulas for X , Y we see d and j
in denominators. This means that with low values of d and j small error of this value
can cause significant position error
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Please see the picture of the beacons in the article - in more simple words, in means
that if one of three beacons is close to line connecting other two beacons, it gives
increased inaccuracy of locating mobile beacon
For example:
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assume d= 10, i= 5, j= 0.1, r1= 7, r2= 7, r3= 4.8
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We get x= 5, y= 2.4375 , z = 4.25
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If we suppose that j=0.101 (0.1 cm error), we receive x= 5, y= -0.06 , z= 4.89
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You see very large Y error
Another example for Z. Assume mobile beacon is relative close to plane of stationary
beacons:
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d= 8, i= 4, j= 6, r1= 5.02, r2= 5.02, r3= 3.01
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This gives X=4, Y= 3.01169, Z= 0.36
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If we suppose r3= 3.0 (1 cm error), we receive X=4, Y= 3.016, Z= 0.44. Error
on Z is about 8 cm
Also, with r1= 5, r2= 5, r3= 3, Z will be 0. As you see, low change of distances causes
large change of Z value near the plane.
Learn more about accuracy here: Absolute vs. relative accuracy