Field

Wheels, and where they may not go

A rolling wheel forbids a velocity, not a position. It may not go sideways and it can still be parked anywhere, at any angle — and the gap between those two sentences is worth an exponent, a shortest path that is a whole circle, and a ball that comes back turned by the area it went round.

A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0.

A constraint that takes nothing away

The rolling constraint as a condition on velocities rather than on positions, why it removes a direction without removing a dimension, and the ledger of what each wheeled mechanism can reach.

$Which of these constraints is secretly about positions. How much of the bracket of two permitted directions lies outside the permitted directions, as a fraction of its own length. Frobenius' theorem says a distribution is the tangent field of a family of surfaces exactly when this is zero, so the test needs no integration and no recognition. The scale is logarithmic because the answers are seventeen orders apart: the rail returns nothing at all and everything else returns essentially the whole bracket. There is no mechanism in the middle.$

One character apart

Frobenius' theorem as a measurement — whether a velocity constraint is secretly a constraint on positions, tested on a wheel, a rail, a car and a gear train, with the answer seventeen orders apart and nothing in between.

Four legs that do not cancel. Drive forward, turn, drive back, turn back — each leg exactly as long as the one it is undoing. The mechanism does not come home. What is left over is 1149.9 mm at an amplitude of 1.10, and it points along the direction the wheel forbids. The gap and the computed bracket are 31.51° apart here and 3.15° apart at a tenth of this amplitude — the agreement is a leading-order statement and the departure is the third-order remainder, which falls with the manoeuvre rather than staying put. Every point on the path was reached by a permitted velocity, so nothing here cheats; the sideways motion is assembled out of motions that are not sideways.

The motion left over by going nowhere

The Lie bracket as a manoeuvre rather than a derivative: the four-leg wiggle, the displacement it leaves, its direction against the computed bracket, and the second-order law that says what a smaller manoeuvre wins.

How many wiggles

The growth vector of a wheeled mechanism, measured as a sequence of ranks, and the second route to it: the power of its own amplitude by which each direction moves.

The parking shuffle, three times over. Forward on left lock, forward on right lock, back on left lock, back on right lock — and repeat. The heading comes back to where it started and so does the position along the road, both exactly and at every leg length, because the four legs are a symmetric set. What is left is 255 mm of pure sideways translation per cycle at a leg of 0.80 m, which is exactly 4R sin(φ) tan(φ/2) with φ = s/R. Halve the leg and it quarters.

Parking is an exponent

The parking shuffle with a closed form, its second-order law measured rather than assumed, and what it costs in shuffles and metres to move a car half a metre sideways in a bay of a given length.

The path a towed wheel takes

The tractrix as the output of one heading equation, the rod length as a conserved quantity rather than an assumption, off-tracking in closed form, and the swept path of an articulated lorry against the regulation it is built to.

Which of these two tracks was made by the front wheel. The rear wheel of a bicycle is towed, so it points at the front wheel at every instant: the tangent to the rear track, extended forward by the wheelbase, lands on the front track. Done that way round the tangents land a mean of 0.13 mm off; done the other way round they land 461 mm off, a factor of 3625. It is a measurement rather than an eye for tracks, and it needs neither the wheelbase nor the direction of travel to be known in advance.

Which way did the bicycle go

Reading a mechanism from the marks it left: the tangent test that identifies the front track, the wheelbase recovered from the tracks themselves, and the one case in which the tracks genuinely cannot say.

Six words, and the shortest of them. Every path a car that may not reverse and may not turn tighter than R can take between two placements is one of six shapes: three arcs, or two arcs with a straight between. All six are drawn; the shortest is LSR at 2.2557 R and the longest is RSR at 14.788 R. A numerical shooting solve that shares no line of code with the closed forms returns 2.2557 R, which agrees to 4e-16.

A circle for the first millimetre

Dubins' six words as a classification rather than a search, checked against a numerical shooting solve, and the exact cost of a lateral offset for a vehicle with a curvature bound and no reverse gear.

A ball that remembers the area. Roll a ball round a closed loop on the plane, without ever twisting it about the vertical, and it comes back to the same place turned. The angle is the loop's area divided by r², and the dashed line is that law with nothing fitted to it. The departure at the top is not an integration error: it grows as the square of the angle — a fitted exponent of 1.99 — which is what a leading term's first correction does. The rotation is composed from exact exponentials, so the drawing carries no drift of its own.

The ball that remembers where it has been

Holonomy as the field's deepest bracket: a ball's orientation as a memory of the area it has enclosed, the growth vector that changes by one entry when twisting is forbidden, and the law checked against its own second-order correction.

How much each set of wheels forbids. Every wheel contributes the same row, and whether it is a constraint or a drive is one factor of sin γ in it — γ being the angle the rollers make with the wheel's own axle. At γ = 0 the row says the body may not move across the wheel and the wheel's speed drops out of the statement; at γ = 45° the row says nothing about the body at all and fixes the wheel's speed instead. The whole difference between a machine that shuffles and one that slides sideways is in that factor.

The wheel that forbids nothing

Mecanum and omni wheels written as the same row as an ordinary wheel, the rank that goes to zero, and why forty-five degrees is the only roller angle at which a platform is isotropic.

Every axis through one point

The instantaneous centre as a rank condition on the wheels' constraint rows, Ackermann's relation derived from it rather than assumed, and the scrub a violation forces, measured per metre travelled.

Forwards it settles, backwards it runs away. A trailer starting a hundredth of a radian out of line, with the steering held straight. Driving forwards the angle decays as e^(−s/d); reversing, the same equation runs the other way and it doubles every 4.16 m. Nothing about forces is involved and nothing about the driver: it is the sign of one exponent, and the length scale is the trailer's own length. The curve flattens at the top because the sine that generates it saturates — the runaway is exponential only while the angle is small.

The angle that doubles

The reversing instability as the sign of one exponent, the saturation that ends it, and the measurement that says a jackknife is not a kinematic singularity.

A wheel that cannot report its radius

What a scaling does to a rolling constraint, why a wheel's rotation is not a dimensionless reading, and what a vehicle can and cannot learn about itself from its own sensors.

An open belt, whose wraps make one turn. Two pulleys of radius 40 and 24 mm with their centres 160 mm apart, with the strand running the same way round both. The two tangency points on each pulley are marked; the run between them is computed from the signed radius difference and nothing else. The wrap angles are 191.48° and 168.52°, and they add to exactly one turn — the turning number of a simple loop. Length 522.6633 mm, against the textbook formula's 522.6633. positioned by solving, not by drawing.

The road a wheel carries with it

The strand as a rolling contact, the arms that replace the radii, and why a strand's no-slip condition is a length equation while a wheel's is not.

$Two pins, and what their bracket costs. The bracket, on the smallest case there is. Two revolute joints span a two-dimensional set of twists whichever way they are arranged, and no count in these essays can tell the two arrangements apart. Their brackets can. Two parallel pins bracket to a translation, which was not in the span, and the span closes at three — planar motion, which is the group the pair of them lives in. Two skew pins bracket to something that closes at six: nothing smaller than the whole of the rigid displacements contains them. The defect column is how far the bracket lies outside the original span as a fraction of its own length, and in both cases it is of order one — which is the ordinary case, and is why a mechanism confined to a subgroup is the exception.$

One bracket, two subjects

The Lie bracket in the rolling field and in the pairs field: a varying bracket of vector fields against a constant bracket on a six-dimensional algebra, and why non-closure liberates one subject and disqualifies the other.

A bearing is a planetary with no teeth

A deep-groove ball bearing treated as two rolling constraints and solved for its cage speed and ball spin; the result set against the planetary relation with pitch radii for tooth counts and against the gear-train solver's exact null space over seven ball sizes; the weighted mean when both races turn; and a tapered roller's slip along its contact line, nought everywhere only when the roller's axis meets the bearing's at the apex.

A wheel on a rough road, and the path of its centre. A road whose height varies as a sine of 10 mm amplitude and 200 mm wavelength, and a wheel of 150 mm radius rolling along it without slipping. The dashed curve is the path of the wheel's centre; the thick stretches of road are the parts the wheel touches. The road's tightest curve, at the bottom of each dip, has a radius of 101 mm; a wheel smaller than that touches all of it, and a larger one bridges the dips. Over one wavelength this wheel turns through 1.3621 radians, and that times its radius is 204.308 mm — the length of its centre's path, not the road's 204.847 mm and not the 200 mm it has travelled. Dragging changes the wheel.

What a wheel counts on a rough road

A wheel rolling without slipping on a sinusoidal road, with the path of its centre and the stretches of road it touches; what the wheel counts per wavelength against its radius, from 20 mm to 2.4 m; the count found from the contact alone against the centre's path found from the centre alone; the crest-riding law for large wheels; the small wheel's overcount against the road's own excess; and how much of the road a wheel touches against how much it counts.

Two wheels on one frame, on a rough road. A rigid frame on two wheels of 50 mm radius, 100 mm apart — 0.5 of the road's wavelength — on the road of 10 mm amplitude and 200 mm wavelength. The dashed curve is the path every wheel's centre follows on this road, whatever the frame does; the frame only fixes where on it the second wheel is. Over each wavelength of travel each wheel counts 204.847 and 204.847 mm, the length of one period of that path, against the 200 mm travelled; the frame pitches by up to 11.53°. Dragging changes the wheelbase.

A second wheel cannot correct the first

Two wheels on one rigid frame on a sinusoidal road: the frame and its wheels' shared centre path; each wheel's count and their average over one wavelength at a half-wavelength wheelbase; the average's ripple against wheelbase for a small wheel and a bridging one; the frame's pitch against wheelbase; each wheel's overcount against its radius, falling as the inverse square; and three roads and three pairs of wheel sizes, each extrapolated to an infinite wheel.

A wheel on a real road rides the long waves and bridges the short ones. Three metres of a class C road — the ISO 8608 average — built from 80 sines of random phase from 5 cm to 90 m wavelength, heights in millimetres and drawn far taller than they are. The second line is the lowest point of a wheel of 0.6 m radius rolled along it: where it lies on the road the wheel touches, and where it lifts off the wheel bridges a trough. On this stretch it touches 12% of the way. A wheel counts its centre's path, so the texture it bridges no longer adds to its count, and the waves it follows still do.

A real road sets its own power

Wheels rolled on class C roads built from ISO 8608 spectra with random phases: the centre path as the road dilated by a disc; the overcount against the wheel's radius over the ISO band and with texture down to 5 cm; the tightest trough below which a wheel counts the whole road; the large-wheel exponent against the spectrum's slope and the law 2(3 − w)/(5 − w); each wave's trough radius against its wavelength; and the one-sine extrapolation applied to pairs of wheels.

Three wheels read the road they measure

Three wheels of 0.3, 0.6 and 1.2 m rolled together over class A to E roads textured to 1 cm: their lowest points on one stretch; the excess against the radius with each road's three-wheel fit carried out to 6 m and the band floor marked; the distance error from one wheel, two wheels on the inverse square and three on a fitted power, road by road; the power read by four windows of radii on roads textured to 1 cm and to 5 cm; other spectral slopes and spectra with a bend; and the power and distance against an error in the counts.

One tyre pressed harder: its patch grows and averages the road it used to ride. 0.6 m of a class C road built from 110 sines from 1 cm to 90 m, heights in millimetres and drawn far taller than they are, under a 0.3 m tyre. The tyre is a circular tread on a bed of vertical springs; at each position its centre height is the one at which the springs carry the load. The grey line is a rigid wheel's lowest point, riding the peaks. The coloured line is the level the loaded tread's patch rests at, 9 mm deflection, a patch 146 mm long on a flat floor: it passes over the short texture as an average of it instead of following its tops.

A load reads the road a radius cannot

One 0.3 m tyre modelled as a tread on a bed of springs, rolled over class A to E roads textured to 1 cm at four loads: the level its patch rests at against a rigid wheel's lowest point; its overcount against the patch half-length on logarithmic axes; the distance error from one rigid wheel, three rigid wheels and one tyre at three loads, road by road; its overcount against the road's class, beside a rigid wheel's; the share of the patch in contact and the power read, class by class; and two windows of loads.

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