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Cyclecomputer Math:
How Accurate?

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by John "Integers" Allen
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The question

The accuracy of processing in a cyclecomputer's internal computer chip plays into the overall accuracy of the cyclecomputer. But -- by enough to outweigh other issues such as tire inflation and tread wear? I don't have time to run a large-scale test, but a test I did run produced some interesting results.

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The test jig

I rigged up a crude but functional test jig using a phonograph turntable and a handlebar, as shown in the photo below. I mounted three cyclecomputers on the handlebar. The cyclecomputers at the right and left are Cateye Astrale -- same model, though with cosmetic differences. The cyclecomputer in the middle is a Cateye Micro. I ductaped the sensors to the tonearm and two magnets to the turntable platter opposite one another. I set the Astrale at the left to record kilometers; the other two cyclecomputers, to record miles. I set all three to the same wheel diameter, 2130mm -- or 213cm for the Micro, which takes only three digits of input. I zeroed the distance reading of all three computers before starting the turntable. In the photo, the platter is spinning at 78 RPM, the test has been running for about 4 minutes, and the computers have recorded 0.63 miles, 1.02 kilometers.

Cyclecomputer math test jig

The test

In case of a loose connection or misaligned sensor, I checked several times as readings increased, to make sure that the ratios of the distances recorded by the three computers remained consistent. That checked out. I left the test running until the computer on the right tallied up 156.25 miles. Why this mileage? It is also exactly 251.46 kilometers. This is the shortest distance at which an exact match in 1/100ths of a mile and kilometer occurs -- if the internal conversion factor of the computers is exact.

The video below shows the readings of the three cyclecomputers as the one on the right reaches 156.25 miles.

I counted frames in the video to interpolate and make the comparison more precise. Mileage should come out to 156.25, and kilometers, 251.46 -- but:

km Micro
mi
Astrale
mi
km/
Micro mi.
km/
Astrale mi
251.2974 155.7285 156.2500 1.613689 1.608304
Ratios as fractions --> 1391/862  10343/6431

I give only one entry for kilometers, because I was able to confirm with another test that the Astrale and Micro, when recording in kilometers, stay in step with one another with the wheel-size setting I used. (The Micro took a second or two longer to calculate each new reading. The delay varied from one reading to the next, but showed no trend.)

I calculated fractional ratios using Microsoft Excel's automated format converter, which rounds fractions to a specified number of digits. Though I allowed 5 digits of precision in the denominator, the fractions reduced to 3 or 4 digits. If I allowed more digits, I could not get a result within the range of 16-bit calculations (maximum denominator of 65,535). This result suggests that these computers are limited to 16-bit integer arithmetic, and that the fractions I obtained are the actual ratios which the computers use.

It is not surprising that the Astrale calculated faster and makes a more accurate conversion: it is a newer model, and must have a more elaborate processor. It is surprising, though, that for the Astrale, 10350/6431, not 10343/6431, and for the Micro, 1388/862, not 1391/862, would be closest to the exact ratio between miles and kilometers, 25146/15625-- which is still within the range of 16-bit arithmetic. A likely explanation is that the inaccuracy results from the initial conversion from kilometers to miles when the computer is set to record in miles.

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Conclusions

The conclusions below are in addition to the ones in our main article about cyclecomputer calibration.

The discrepancies found through my testing are most likely due to rounding errors of integer arithmetic. It is safe to assume that most cyclecomputers, for compactness, low power consumption and low cost, use simple processor chips capable only of integer math, and will show similar discrepancies..

(Having smashed the paradigm on all three of those counts, a GPS unit with a wheel sensor might apply more sophisticated math. A GPS unit could also use GPS readings to refine its calibration, too.)

The same cyclecomputer, or two of the same make and model, will almost certainly not record precisely the same distance in kilometers as in miles, though set for the same wheel circumference. For that reason, a roll-out test will result in slightly different distances when the computer is set for miles, or kilometers. Also, computers of different makes and models are likely to give different results.

For a cyclecomputer calibrated in millimeters of tire circumference, inaccuracy due to variation in air pressure in the tire is almost certainly larger than inaccuracy of internal math. Still, the km/ mile conversion inaccuracy for the Astrale, though small, 0.06%, is greater than the millimeter calibration steps, 0.04% at the wheel size used -- and there's no telling based on this experiment what the error might be for other wheel sizes. For the computer calibrated in centimeters, the Cateye Micro, at the wheel size used in the test, the discrepancy is 0.26% and the steps are 0.46%. Rounding the measured wheel size to the nearest calibration number -- higher or lower -- reduces the potential error by half, and so the km/mile discrepancy becomes more significant by comparison.

So, to achieve the greatest accuracy, do not expect the calibration to be exactly the same for miles and kilometers. Establish all the computer's settings, then run a roll-out test, and finally, adjust the calibration number based on GPS readings or a ride over a measured course of several miles or kilometers, or longer. Have the computer's sensor on the bicycle's front wheel, and keep the tire pumped up to a consistent pressure. Details of this procedure are described in our article on cyclecomputer calibration.

How much does accuracy matter? That's up to you to decide. I find that a cyclecomputer on a bicycle with a road-tread tire can give readings consistent within of 0.5% or better. This accuracy can be useful when following a cue sheet laid out using GPS. If you are running a cyclecomputer as well as GPS, as some cyclists do, you would like them to agree as well as possible. Clearly, on a cyclecomputer with wheel size settings in millimeters, calibration and conversion errors are smaller than wheel-size measurement error. With wheel-size settings in centimeters, calibration and conversion errors can dominate.

Further experimentation? Other applications?

My test setup could not measure the exact ratio between the number of turns of the wheel and the cyclecomputer readings. Checking this would require a rotations counter, which was not available for my experiment. Rotations-to-distance information would be useful when, for example, using cyclecomputers to record data in a scientific experiment which required high accuracy.

But, due to rounding, calibration steps are unlikely to correspond exactly to increments in wheel size. It would be practical to test for rotations-to-distance ratios at only a few wheel-size settings. I had my experiment running overnight. A test at every calibration step could take years! The only practical way to get complete documentation on a cyclecomputer's internal math would be from the manufacturer. Whether a manufacturer would provide this information is an open question. And then only a person skilled in computer assembly language would probably be able to interpret the documentation.

Still, cyclecomputers are inexpensive. Rotations-to-distance calibration followed by identical setup of identical cyclecomputers would allow them to be put into service at a large number of test stations, at low cost.

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Links:

Cyclecomputers and GPS, table of contents

Riding with RidewithGPS

Database of cyclecomputer and GPS instruction manuals

Installing cyclecomputers

Calibrating cyclecomputer wheel sensors

Printer-friendly cyclecomputer calibration chart

Troubleshooting cyclecomputers

Errors due to incorrect wheel magnet orientation

Accuracy limitations of cyclecomputers and GPS

Discrepancies due to the internal math of cyclecomputers

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