What a wheel counts on a rough road

A wheel that cannot report its radius found that rolling is the one place in these essays where an angle measurement carries a length: a wheel’s turning times its radius is how far it has gone, and an odometer is nothing more than an encoder and a radius. That essay’s wheels ran on flat ground, where “how far it has gone” has one meaning. On rough ground it has three: the horizontal distance travelled, the length of the road surface, and the length of the path the wheel’s centre follows. They are different numbers — on the road in the figures below, 200 mm of travel is 204.85 mm of road surface, and the axle’s path is somewhere between, depending on the wheel — and the question is which one the wheel counts.

The obvious answer is the road’s length, since rolling without slipping means the wheel’s rim and the road pass through the contact at the same rate. That answer is right for small wheels and wrong for large ones, and the reason it goes wrong is the reason there is a single answer that is always right.

A wheel sits on the road’s parallel curve #

The road here is a sine: 10 mm of amplitude and a 200 mm wavelength, the corrugation a gravel road develops. A wheel of radius $R$ above a point $X$ of the road sits as low as it can without passing through the surface, so its centre is at the greatest height $y(x) + \sqrt{R^2 - (x - X)^2}$ over the road points within $R$ of $X$, and it touches the road where that greatest height is attained.

Two cases follow from one number, the radius of curvature of the road at the bottom of each dip. For a sine it is $\lambda^2/4\pi^2 A$, here 101.3 mm. A wheel smaller than that fits into every dip: it touches the road at one point, the contact runs continuously along the whole surface, and the centre traces the road’s parallel curve at distance $R$. A wheel larger than that cannot get to the bottom of the dips. It touches two crests at once when it is over a dip, and as it moves on, its contact jumps from the crest behind to the crest ahead. Between jumps it rolls over the top of one crest; in the jump it touches no road at all.

Every wheel counts its own centre #

Rolling without slipping says that the wheel’s material at the contact is momentarily at rest — the same condition under which two curves rolled on each other lay equal arcs. So the contact is the wheel’s instantaneous centre of rotation, and every point of the wheel moves as if turning about it — in particular the wheel’s centre, which is at distance $R$ from the contact and therefore moves at the wheel’s angular speed times $R$. Turning times radius is the length of the centre’s path. That holds on any road, for any wheel, including at the jumps, where the centre does not move and the wheel does not turn.

That is the one answer that is always right: a wheel counts the path of its own centre. It is not the road’s length, and it is not the distance travelled; it is a third curve, the one the axle actually follows.

It is checked here by computing the two sides from nothing they share. The turning comes from the contact alone: wherever the contact moves continuously the wheel rolls over a road arc $ds$ while the line from the contact to the centre turns by $d\theta$, and the wheel’s turning is $ds/R - d\theta$ — the rolling itself, plus the pivoting that comes from the contact normal swinging round. The centre’s path comes from the centre alone, as the length of the curve its height traces. The travel is split recursively until the contact moves less than a millionth of a wavelength per piece, and a piece that will not converge is a jump.

For twelve wheels from 20 mm to 2.4 m, the two agree to $2 \times 10^{-9}$ mm per wavelength. The column beside them, the length of road the wheel actually touches, is something else once the wheel bridges the dips: the 2.4 m wheel touches 8 mm of each 205 mm of road and counts 200.05 mm. What makes up the difference is the second term in the turning, $-d\theta$: a wheel riding over a crest turns about it, the contact normal swinging from leaning back to leaning forward, and that turn covers no road.

Small wheels count the road #

Below the trough radius the contact runs over the whole road and never jumps. The centre’s path is then the road’s parallel curve, whose length over a whole wavelength is the road’s own length exactly: a parallel curve at distance $R$ is longer than its curve by $R$ times the curve’s total turning, and a periodic road turns back to where it started over each wavelength, so the excess is nought.

So every wheel smaller than 101.3 mm counts 204.847 mm per 200 mm travelled, to $4 \times 10^{-12}$ mm — the road’s length, 2.4% more than the distance. The size of the wheel makes no difference at all below the trough radius. That is the regime in which the textbook statement — a rolling wheel measures the road — is exactly true, and it is also the regime in which the odometer overcounts by the most.

How much it overcounts is the road’s own excess over its wavelength, and for a gentle sine that is $(\pi A/\lambda)^2$ to leading order: the square of the corrugation’s slope.

A 20 mm wheel, smaller than every trough on these roads, overcounts by 0.099% on a 2 mm corrugation, 0.39% on 4 mm, 2.4% on 10 mm and 6.0% on 16 mm, each equal to the road’s excess length to $10^{-9}$ and close to $(\pi A/\lambda)^2$, the last point falling below it by the expansion’s next term. A robot on small wheels crossing a corrugated floor gains that fraction on every metre, and a wheel-radius calibration of the kind the earlier essay described, done on a smooth floor, cannot remove it, because it belongs to the floor and not to the wheel.

Large wheels ride the crests #

Above the trough radius the count falls, and it falls towards the distance actually travelled. On the road here it is 204.76 mm at 120 mm radius, 204.31 at 150, 203.36 at 200, 202.02 at 300, 200.68 at 600, 200.20 at 1.2 m and 200.05 at 2.4 m.

A wheel much larger than the dips touches only the tops of the crests, and over each crest its centre moves on a circular arc: the parallel curve, at distance $R$, of the crest’s own curve, whose radius at the top is the same 101.3 mm, $\rho$. So the centre follows arcs of radius $R + \rho$, one per wavelength, meeting at a corner above each dip where the contact jumps. An arc of radius $r$ over a chord $\lambda$ is longer than the chord by $\lambda^3/24r^2$ to leading order, so

$$\text{count} \approx \lambda + \frac{\lambda^{3}}{24\,(R + \rho)^{2}}.$$

From three times the trough radius upward the measured excess is on that law to within 0.3%. Just above the trough radius it is not, and should not be: a 120 mm wheel still rolls some way down into each dip before it jumps, so its centre’s path is not a single arc per crest. The law says how quickly large wheels forgive a rough road — as the inverse square of their radius — and also that they never quite reach the true distance: on this road a 2.4 m wheel still counts 0.027% too much.

A large wheel hardly notices how deep the dips are #

The crest-riding law has a consequence that is easy to miss. Once $R$ is large against the crest radius $\rho$, the count’s excess over the distance is about $\lambda^2/24R^2$ of it — and the amplitude of the corrugation has dropped out. The depth enters only through $\rho$, which shrinks as the dips deepen, and a large wheel adds that to a much larger $R$.

Measured, with the 1.2 m wheel on the same 200 mm wavelength: corrugations 5, 10, 20 and 30 mm deep make the road 0.61%, 2.42%, 9.24% and 19.4% longer than its wavelength — thirty-two times as much from the shallowest to the deepest. The wheel’s count exceeds the distance by 0.084%, 0.099%, 0.107% and 0.110% — less than a third more. Each is on the crest law to better than 1%.

So the two regimes answer to different features of the ground. A small wheel’s error is the ground’s excess length, which grows as the square of the corrugation’s slope and does not care how big the wheel is. A large wheel’s error is a matter of the corrugation’s wavelength against the wheel’s size, and hardly cares how deep it is. A wheel whose radius is near the trough radius is in between, and there both matter.

Touching and counting are different #

The last figure puts the two quantities that the obvious answer confuses side by side.

Below the trough radius they are the same: the wheel touches the whole road and counts the whole road. Above it the touched fraction collapses — to 72% at 120 mm, 42% at 200 mm, 15% at 600 mm and 4% at 2.4 m — while the counted fraction hardly moves from the distance travelled. A wheel riding the crests covers almost no road and turns almost as much as it would on a flat one, because between crests it pivots rather than rolls.

That separates two things the rolling constraint is usually said to guarantee together. Equal arcs of rim and road is one of them, and it holds while the contact runs along a smooth stretch; the jumps, where the wheel spans a dip, are where it stops applying, much as a curve’s corners are where a traced path stops being smooth. A constraint that takes nothing away wrote the rolling constraint as a condition on velocities: the contact point of the wheel is at rest. That condition holds at every instant here, on the crests and across the jumps. What it does not say is that the wheel’s turning equals the road’s length. It says the wheel’s turning equals the motion of the wheel’s centre — which is what a velocity condition at an instantaneous centre can say, and all it can.

The threshold is where the dips get too tight #

The change between the two regimes happens at one radius, and it is worth seeing why that radius is the curvature of the dips and not their depth or their width.

A wheel sits in a dip by touching it at one point, with its rim tangent to the road there. For that to be the only contact, the rim must lie above the road on both sides of it, and near the contact that is a comparison of two curvatures: the rim’s, $1/R$, and the road’s. At the bottom of a dip the road bends up towards the wheel. If it bends up more gently than the rim, the rim lies above it and the wheel sits in the dip; if more sharply, the road rises into the rim on both sides before the rim reaches the bottom, and the wheel is held up by the dip’s two flanks instead. The borderline is a road curvature of exactly $1/R$ at the bottom of the dip, which for a sine is where its curvature is greatest, $4\pi^2A/\lambda^2$.

Just past that radius the wheel no longer reaches the bottom, but it reaches most of the way, and it rolls down each flank before it jumps across the small gap it cannot fill. That is why the count leaves the road’s length gently rather than suddenly — 204.76 mm at 120 mm radius against 204.85 below the threshold — and why the crest law does not apply there.

The wheel carries its own road #

The road a wheel carries with it found that a taut strand rolls on a pulley in the same way a wheel rolls on the ground, and that the two differ in whether the constraint integrates. The rough road adds a sharper version of the same contrast. A wheel on a smooth enough road carries, in effect, the whole road into its count; a wheel on a rougher one carries only the part of the road that its own size lets it reach, and replaces the rest by pivots. The same wheel on the same road can be in either regime depending only on how its radius compares with the road’s tightest dip.

For a designer of an odometer, the practical reading is a pair of statements. On surfaces rougher than the wheel’s radius can follow, the odometer counts the axle’s path, not the ground’s, and it overcounts the distance by the axle path’s excess — which falls as the inverse square of the wheel’s size once the wheel bridges the dips. On surfaces smooth enough for the wheel to follow, it counts the surface, and overcounts by the surface’s excess length, which no choice of wheel size below the trough radius changes.

What the road leaves out #

One road shape. A sine has the same curvature radius at its crests and its troughs. A road with sharp crests and wide troughs, or the reverse, would give different thresholds for bridging and different arcs for the centre; the claim that the wheel counts its centre’s path would not change, since it does not depend on the road.

One wavelength at a time. A real road’s roughness is a spectrum: long gentle undulations, shorter corrugations and fine texture together. By the argument here a wheel follows every component whose dips it fits, counting their full excess length, and rides the crests of every component it cannot, counting almost none of theirs. So a wheel filters the road — it counts the long wavelengths and ignores the short ones, with the crossover at the wavelength whose trough radius equals its own. Whether that filtering is as clean on a road where all the wavelengths are superposed is not measured here.

A rigid wheel and a rigid road. A tyre deforms into a dip and a gravel surface gives under a wheel; both shorten the centre’s path and move the count towards the distance travelled. A tyre that flattens is also a wheel whose effective radius changes, which is the separate error the earlier essay treated.

No slip. A wheel pivoting over a crest carries its full load on one point, and whether it slips there is a question of friction this essay does not ask. A wheel that slips counts neither its centre’s path nor anything else.

Still open: two wheels on one frame #

A vehicle does not have one wheel. Its odometry averages several, and on a rough road the wheels meet the corrugation at different phases: a front and rear wheel a wheelbase apart are on crests and in dips at different times.

The distinct argument would be what a pair of wheels on one rigid frame counts. Each wheel still counts its own centre’s path, but the two centres are held a fixed distance apart by the frame, so the frame pitches as it goes, and the two paths are not independent. Whether the average of two counts is closer to the distance travelled than either — whether a wheelbase that is a half-wavelength cancels the corrugation’s error the way a drag link’s slow stretch cancelled a crank-rocker’s fast one — is measurable, and the measurement would be the two wheels’ counts on the same sine as a function of wheelbase over wavelength, from a quarter to three.