A second wheel cannot correct the first

What a wheel counts on a rough road found that an odometer’s wheel, rolling without slipping, counts the length of the path its own centre follows. It counts that exactly, because the contact is the wheel’s instantaneous centre and the centre moves at the wheel’s turning rate times its radius. On a road of 10 mm amplitude and 200 mm wavelength, a wheel smaller than the dips’ tightest radius rolls over all of the road and counts its full length, 2.4% more than the distance. A larger wheel bridges the dips, pivots over the crests and counts less, its excess falling as $\lambda^3/24(R + \rho)^2$ per wavelength, with ρ the dips’ radius.

A vehicle does not have one wheel, and its wheels do not meet the corrugation together. A front and a rear wheel a wheelbase apart are on crests and in dips at different times. That essay closed by asking whether the average of two wheels on one rigid frame is closer to the distance than either — whether a wheelbase of half a wavelength might cancel the corrugation’s error, as a drag link’s slow stretch cancelled a crank-rocker’s fast one in a quick return that cuts evenly.

It is not, at any wheelbase, and the reason takes one sentence. Something else does work.

Every wheel’s centre is on one curve #

A wheel on the road touches it at a point and has its centre one radius away along the road’s normal there — or, once it bridges a dip, one radius from each of the two crests it rests on. Either way the centre’s height at each position along the road is fixed by the road and the wheel’s radius and by nothing else. It is a curve, the same for every wheel of that radius on that road: the dashed line in the figure.

A frame on two such wheels does not change the curve. It fixes only where on the curve the second centre is: a wheelbase further along it, measured in a straight line between the two centres. The frame rises, falls and pitches as it goes, but each centre is on the curve at every instant, and each wheel counts its own centre’s path along it.

So consider the frame advancing one wavelength. Both centres advance one wavelength along the road, because the road repeats and the frame is rigid. Each therefore travels exactly one period of the curve, whatever the wheelbase and whatever the frame’s pitching did on the way. Each wheel counts one period of its centre curve per wavelength of travel, which is what a lone wheel counts. The average of two equal counts is the same count.

Measured — the frame advanced in 240 steps over a wavelength, each wheel’s count taken as its centre’s arc length — the two 50 mm wheels count 204.847 mm per 200 mm at every one of twelve wheelbases from a quarter of a wavelength to three, to $10^{-9}$ of the count. The 300 mm wheels count 202.024 mm per 200 mm, the same at every wheelbase. The 2.4% and the 1.0% are the lone wheels’ errors, unchanged.

What the wheelbase does change #

The count is not uniform within a wavelength. A wheel counts fast where its centre’s curve is steep and slow where it is flat, so its running total ripples about the steady rate. Two wheels a wheelbase apart do their fast and slow counting at different times, and the wheelbase decides how those ripples combine.

For the 300 mm wheels half a wavelength apart, each wheel’s running count wanders 0.40 mm either side of the steady rate, and their average wanders only 0.047 mm. At the end of the wavelength both have counted 202.024 mm, and the average is no closer to the 200 mm travelled than either. The wheelbase has smoothed the count’s approach to its error without touching the error.

How much it smooths depends on the wheel. A bridging wheel’s counting rate varies once a wavelength, fastest as it climbs a crest, and a half-wavelength wheelbase puts the two wheels’ fast stretches opposite each other: the average’s ripple falls to 0.12 of a lone wheel’s at one and a half and two and a half wavelengths as well as at a half. A small wheel, which rolls over the whole road, counts fast wherever the road is steep, going up or coming down. Its rate varies twice a wavelength, with the square of the road’s slope, and a half-wavelength wheelbase puts the two wheels in step on that second harmonic; the average’s ripple falls only to about half. At whole wavelengths the two wheels are in the same phase and the average ripples exactly as one wheel does, for both sizes.

The ripple matters for short trips. Over a distance of a few wavelengths, the count’s error is the steady excess times the distance plus a ripple term that depends on where the trip starts and stops, and a half-wavelength wheelbase shrinks that second term for a bridging wheel. Over a long trip the steady excess is everything, and no wheelbase reaches it.

Why the drag link’s trick does not carry over #

The hope the single-wheel essay attached to this one came from a quick return that cuts evenly, where a drag link placed ahead of a crank-rocker evened out the cutting speed: the drag link was slow where the crank-rocker was fast, and the two unevennesses cancelled. That worked because the thing being evened was a rate within a cycle — how fast the tool moved at each point of its stroke. Shifting one unevenness by half a cycle against another is exactly what cancels a rate.

An odometer’s error is not a rate within a cycle. It is the rate’s average over the cycle: how much the count exceeds the distance per wavelength. Averaging two copies of a periodic signal shifted against each other changes the signal’s harmonics — the first harmonic cancels at half a period, the second at a quarter — and never its mean. Every copy has the same mean, and so does any average of them. The wheelbase is a phase shift, and a phase shift cannot move a mean. The drag link could even out a stroke because it changed which crank angle the tool reached when; it could not have changed the stroke’s length, and the wheelbase cannot change the centre curve’s.

The same reasoning says where a second wheel’s phase is worth having. The ripple within a wavelength is a rate, and it can be cancelled, which is the half-wavelength result above. A system that cares about where it is within a few wavelengths — a positioning drive running over a toothed or corrugated track, say — can use a wheelbase to cancel its ripple. A system that cares about how far it has gone over many wavelengths cannot.

The price of smoothing: the frame rocks #

The wheelbase that smooths the count is the one that rocks the frame. At whole wavelengths the two centres are at the same height at every instant and the frame travels level: it does not pitch at all. At half a wavelength one centre is high where the other is low, and the frame pitches by 11.5° either way on 50 mm wheels and 7.1° on 300 mm wheels. The pitch falls at larger odd half-wavelengths — 3.8° at one and a half, 2.3° at two and a half for the small wheels — only because the same difference in height is spread over a longer frame.

So for a frame whose job is to measure distance, the wheelbase is a choice between a level frame whose count ripples and a pitching frame whose count is smooth, and neither choice touches the error per wavelength. That error belongs to each wheel’s centre curve, and every wheel of the same size has the same one.

What each wheel’s curve is made of #

The centre curve deserves a closer look, because it is the whole of what a wheel counts. For a wheel smaller than the dips’ radius it is the road’s parallel curve: the road pushed out along its normal by one radius. Its length per wavelength is the road’s own, because a parallel curve’s length differs from its base curve’s by the offset times the total turning of the tangent, and over one wavelength of a repeating road the tangent turns back to where it started. The road a wheel carries with it found the same rolling contact in a strand on a pulley, where the material at the tangency is at rest against the surface and the length that passes is the length rolled.

For a bridging wheel the curve is built from pieces. While the wheel rests on one crest it pivots about that crest, and its centre moves on an arc of radius $R$. While it rolls through the part of a dip it can reach, its centre runs on the parallel curve again. The arcs are shorter than the stretches of road they replace — the wheel’s centre cuts across what its rim does not follow — which is where the bridging wheel’s smaller count comes from. The frame changes none of these pieces, and two wheels of one size run over the same pieces in the same order, a wheelbase apart.

That is also why the count is a length and not a turning. A wheel that forbids nothing found a wheel’s rolling constraint to be one row of a matrix, a relation between the wheel’s turning and the vehicle’s travel, and on a flat road the relation is the radius. On a rough road the relation is still the radius, but the travel it relates the turning to is the centre’s, along its curve, and the vehicle’s horizontal progress is a different number.

Two sizes of wheel #

The one thing two wheels on one frame can do that one cannot is disagree, and they disagree only if their centre curves differ, which means their radii differ.

The single-wheel essay’s law says how they disagree. A wheel smaller than the dips’ radius, 101 mm on this road, counts the road’s full length: 2.424% over, the same at 20, 50 and 100 mm. Above it the error falls, following $\lambda^2/24(R + \rho)^2$ as a share of the distance, and for wheels well above the dips’ radius that is very nearly the inverse square of the wheel’s radius: 0.339% at 0.6 m, 0.099% at 1.2 m, 0.027% at 2.4 m, a slope of −2 on log axes.

An error that scales in a known way with a known size can be removed by extrapolation, the idea behind every measurement made at two step sizes and extrapolated to none. Two wheels of radii $R_1$ and $R_2$ on one frame count $D(1 + e_1)$ and $D(1 + e_2)$ over the same distance $D$, with $e_i \propto 1/R_i^2$. The count of an infinitely large wheel, which would count the distance exactly, is then

$$D \approx \frac{R_1^2\,c_1 - R_2^2\,c_2}{R_1^2 - R_2^2},$$

and nothing about the road appears in it.

On the 10 mm × 200 mm road, wheels of 0.6 m and 1.2 m overcount by 0.339% and 0.099%. Extrapolated on the inverse square, they leave 0.018%, five times better than the better wheel, and the extrapolation used neither the road’s amplitude nor its wavelength. Across three roads and three pairs of sizes the extrapolation always beats the better wheel. By how much depends on how far both wheels are above the dips’ radius: by a factor of 1.6 when the smaller wheel is one and a half times that radius, about five when it is six times it, and nine or ten when it is ten to twelve times it. Knowing the dips’ radius, and extrapolating on $1/(R + \rho)^2$ instead, removes nearly everything: $7 \times 10^{-7}$ of the distance for the 0.6 m and 1.2 m pair.

A large wheel is the obvious answer to a rough road, and the extrapolation says how much a second, different wheel buys on top of it. It also says what the second wheel must not be: the same size as the first.

What the argument used, and what it did not #

The argument that two equal wheels cannot help used three facts and nothing else. The road repeats. The frame is rigid. And each wheel counts its own centre’s path, which is a property of rolling without slip. It did not use the sine, the wheel’s size or the wheelbase, so it holds for any repeating road and any frame of equal wheels, and for any number of them: a vehicle averaging four equal wheels counts what one of them counts, over every whole period of the road.

It needs the road to repeat under both wheels in the same way. A road whose corrugation differs under the left and right wheels, or that does not repeat, is a different problem: then each wheel’s centre curve is its own, and averaging does average two different errors.

The extrapolation used one more fact, the law of the bridging wheel, and it is only as good as that law. Wheels near the dips’ radius, where the law is still turning from the road’s full length to the inverse square, extrapolate least well; that is the 1.6 in the table. And a wheel that cannot report its radius found that odometry cannot separate a wheel’s radius from the distance it measures, so the extrapolation needs both radii known — a calibrated radius, not a nominal one — or it extrapolates a calibration error along with the corrugation.

What the frame does not settle #

Towed wheels. A trailer’s wheel is a wheel on a frame whose other end is a hitch, and the path a towed wheel takes is decided by that hitch. On a rough road it too counts its own centre’s path, and nothing about being towed changes the curve that path runs on.

Suspension. The frame here is rigid and the wheels stay on the road. A sprung vehicle’s wheels follow the road through their springs, and each still counts its own centre’s path; what changes is only the frame’s motion, which the counts do not see. A wheel that leaves the road — at speed over a crest — counts nothing while it is in the air, and that is not modelled.

Slip. Every count assumes rolling without slip — the pure rolling that gear teeth manage at one point only and a wheel on a hard road manages everywhere. A driven wheel slips a little under load, and that error is proportional to the load, not to the road’s shape. A wheel that slips entirely, like one wheel on ice, counts turning that is not travel at all.

Tyres. A tyre deforms at its contact and rolls on an effective radius that changes with load and speed. The road’s corrugation also changes the load cyclically, so a real tyre’s effective radius ripples with the road — a coupling this rigid model leaves out.

Still open: a road that does not repeat #

Everything above rests on the road repeating. A real road is closer to a random surface, with a spectrum of wavelengths rather than one. Each wheel still counts its centre’s path, and the centre path of a wheel on a random road is the road seen through the wheel’s size: short wavelengths bridged, long ones followed.

The distinct argument would be that filter measured. For a road built from many sines of random phase with a realistic spectrum, the centre path’s excess over the distance as a function of the wheel’s radius, and whether it still falls as the inverse square — or whether the long wavelengths, which no wheel bridges, set a floor that no size of wheel and no extrapolation can get below. A second measurement would ask whether two equal wheels on a frame still count alike, which they need not once the road stops repeating, and by how much their average then beats either.