# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Cook's Incremental Reckoning**

**From:**Frank Reed

**Date:**2010 Apr 3, 18:14 -0700

I realize you dropped this thread almost two weeks ago, but there's one matter here that might be worth pursuing. You quoted me as saying:

"Round-off adds a small "random walk" to the data. But if the step size in that random walk is well below the normal uncertainties in the process of DR navigation itself, then the round-off is irrelevant."

And then you wrote:

"Ah! So we do agree that there is some roundoff! Yes, it will be irrelevant when we consider journeys of only a few days or weeks. Of course, this is also a function of the step size of record. Does Cook record to the nearest second, minute or 1/4 degree? Roundoff can and will pose a greater challenge when the period extends to years."

But here's the thing: the round-off error will be a random walk in the "sixth digit" of the calculation while the actual error in the dead reckoning occurs in the "fourth digit" if that makes sense. There were many navigators still doing pure dead reckoning for longitude even in the second half of the 19th century. Some of them were very good at this and their longitude estimates would only err by, let's say, five miles after a day of sailing (the standard deviation of the errors is 5 miles). Sometimes this error would be +7, sometimes -3, sometimes +2, etc. in a random fashion. So as a random walk, we get an expected error after "N" days of 5*sqrt(N). That is, the average error after 16 days would be 20 miles, and the average error after 100 days would be 50 miles. Now let's suppose we round the longitudes to the nearest minute of arc every day --very reasonable here. With perfect dead reckoning, this round-off error would accumulate like a separate random walk and spoil our results but since it's much smaller than our actual daily error, it's nearly undetectable. The standard deviation of round-off error is W/sqrt(12) where W is the width of the round-off band. So in this case even after 12 days it would only increase the error by 1 mile. The net standard deviation of the error combining the "real" error due to the limitations of dead reckoning with the round-off error would be given by the square root of the sum of the squares: sqrt(25+1/12) or 5.008 miles scarcely different from the real error alone and completely indistinguishable from it. That's what I meant by saying that the round-off is irrelevant. It's irrelevant because any navigator with a little sense always works his or her calculations a digit or two more accurately than the estimated accuracy of the data and the process itself.

You also worte:

"An analoguous situation occurs in robotic equipment. [...] However, over long term, those round off errors DO accumulate, and the calibration equipment will show a significant error. It is a well known problem with incremental positioning and robotics, requiring that the robot find the reference point periodically or face defects in manufacture."

That sounds like dead reckoning error to me. Could that be what you mean?

-FER

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