Machine Kinematics
solved, then drawn
Concept

Wheelbase — where it appears

The distance between a vehicle's front and rear axles, which sets its turning radius at a given steer angle and the rod length of its rear wheel's tow. It appears twice in the geometry — once in the turning radius and once as the rod length of the rear axle's tow — which is why a long vehicle turns wide and tracks tight.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

The path a towed wheel takes. The front wheel is given a path; the rear one obeys a single equation — roll along your own heading, and stay attached. The rod is drawn every twelfth sample and is never imposed: the integrator carries the axle's position and heading and nothing else, and the distance from hitch to axle comes out constant to 1.4e-13 m over the whole run. The rear track cuts every corner, which is off-tracking, and it is the reason a long vehicle needs a wide turn.

The path a towed wheel takes

A towed axle obeys one line: roll along your own heading, and stay attached. Nothing tells it to keep its distance from the hitch and it keeps it to 10⁻¹³ anyway, it settles onto a circle of exactly √(R² − L²), and the residual against that is not the integrator — it is the difference between a circle and the polygon it was sampled as, and it falls by four when the sampling doubles.

rolling · Rolling 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

Two tyre tracks in mud, and a question with a definite answer. The rear wheel is towed, so its tangent extended forward by the wheelbase must land on the front wheel's track — and it does, to 0.13 mm one way round and 447 mm the other. The test needs neither the wheelbase nor the direction of travel, and it returns both.

rolling · Rolling Every axis through one point. Four wheels on one rigid body, each rolling without sliding. Each turns about some point on its own axle line, and a rigid body has one such point, so every axle line has to pass through it. That is the whole of steering geometry, and it is a rank condition rather than a formula: here the four rows have rank 2 of 3, leaving a one-dimensional family of twists, and the centre they agree on is 12.000 m to the side. The scrub is 3.2e-17 m per metre — zero, to the last digit.

Every axis through one point

Bolt several rolling wheels to one rigid body and they impose one condition between them: every axle line must pass through a single point. The familiar steering formula falls out of it as a consequence rather than being quoted — cot δₒ − cot δᵢ = 0.574074 at a turn of six metres, of eight, of twelve and of twenty, on a track of 1.55 m and a wheelbase of 2.7.

rolling · Rolling 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 wheel that cannot report its radius

Rolling relates a wheel's turning to a vehicle's travelling, and the relation has a length in it. So a rolling constraint is the one place in these essays where an angle measurement does carry a size — and the size it carries is the one thing a vehicle's own odometry can never separate from its wheelbase.

rolling · Rolling 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

An odometer counts a wheel's turns and multiplies by its radius, and on a flat road that is the distance travelled. On a rough road it is not, and it is not the road's length either. It is the length of the path the wheel's centre follows — exactly, because the contact is the wheel's instantaneous centre. A wheel smaller than the road's tightest dip rolls over all of it and counts the road's full length; a larger one bridges the dips, pivots over the crests, and counts less, falling towards the true distance as λ³/24(R + ρ)². On a road 2.4% longer than its wavelength, a 50 mm wheel overcounts by 2.4% and a 2.4 m one by 0.03%.

rolling · Rolling 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

A wheel on a rough road counts the length of its centre's path, not the distance travelled. Put two on one rigid frame and average them, and the error does not shrink at any wheelbase: both centres run on the same curve, fixed by the road and the wheel's size, and each wavelength of travel carries each of them once round it. A half-wavelength wheelbase smooths the count's ripple and rocks the frame instead. What does work is two wheels of different sizes. A bridging wheel's error falls as the inverse square of its radius, so two radii measure their own error: wheels of 0.6 and 1.2 m that overcount by 0.339% and 0.099% extrapolate to 0.018%, knowing nothing about the road.

rolling · Rolling 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

On a road of one sine, a wheel's overcount falls as the inverse square of its radius, and two wheels extrapolate it away. A real road is a spectrum. Over the band road standards describe, every wheel smaller than the road's tightest trough — about a metre and a half on an average road — counts the road's whole length, and its size makes no difference at all. With the texture below that band, bigger wheels do bridge more, but the overcount falls as the two-thirds power of the radius, a power the road's spectrum sets: 2(3 − w)/(5 − w) for a spectrum falling as n^−w. The inverse-square recipe removes a sixth of the error.

rolling · Rolling

Named alongside it

The objects these essays reach for when they reach for this one.

Rolling constraintOff-trackingSecond orderTurning radiusApproximationNonholonomicScale invarianceTowed axleTractrixTwistAckermannClosed form

All concepts