Every axis through one point

A single rolling wheel turns about some point on its own axle line. Which point is not decided by the wheel — it depends on how fast the wheel is being turned and how fast it is being steered — but the point is on that line, always, because a wheel that rolled about a centre off its axle line would be sliding.

Bolt two wheels to one rigid body and the body has one instantaneous centre, not two. So the centre is on both axle lines, so the axle lines meet there. Bolt four wheels on and all four lines have to pass through the same point.

That is the whole of steering geometry, and everything anybody says about Ackermann is a consequence of it.

As a rank condition #

Written as arithmetic rather than as a picture, each wheel contributes a row saying the body may not move across it:

$$n_i \cdot (v + \omega \times p_i) = 0,$$

with three unknowns — two components of the body’s velocity and its turn rate. Four wheels give four rows for three unknowns, which is over-determined, and everything about steering geometry is the question of when an over-determined system has a solution.

Rank 3 — the generic case, four rows independent — means the only twist is zero. The vehicle cannot move at all without something sliding.

Rank 2 means a one-dimensional family of twists, and the vehicle can move. That is what the steer angles have to arrange, and it is arranged by making two of the four rows redundant.

Rank 1 happens when all the wheels are parallel: the vehicle can translate along them and also, if they are all on one line, turn about a point at infinity. It is the state of a four-wheeled trolley with its wheels welded straight.

Rank 0 is a set of wheels that constrain nothing, which is the mecanum platform and not a steering question at all.

So a steered vehicle is a mechanism kept deliberately at a rank deficiency. Its wheels impose four conditions of which only two are independent, and the two surplus conditions are made redundant by the geometry — which is exactly what an overconstrained linkage does, arrived at from wheels instead of from joints.

Ackermann, derived #

Take the centre on the rear axle’s line, a distance $R$ to the side, with a wheelbase $L$ and a track $t$. Each front wheel must be perpendicular to the line from the centre to that wheel:

$$\tan\delta_i = \frac{L}{R - t/2}, \qquad \tan\delta_o = \frac{L}{R + t/2}.$$

Take cotangents and subtract, and the $R$ cancels:

$$\cot\delta_o - \cot\delta_i = \frac{t}{L}.$$

That is Ackermann’s condition, and it fell out of the concurrency rather than being imposed. The measurement, computed from the centre with no formula involved:

against $t/L = 1.55/2.7 = 0.574074$. Six figures, at four radii, from a computation that never used the relation — the angles came from the perpendicularity condition at each wheel and the cotangent difference was taken afterwards.

That is a two-route agreement of the kind this site runs on, and the routes really are independent: one is point the wheel at right angles to the radius, applied wheel by wheel, and the other is a formula in the track and the wheelbase with no wheel in it.

What the derivation buys over the formula #

The formula is for two steered wheels with an unsteered rear axle. The construction is for any number of wheels anywhere.

Six wheels. A three-axle vehicle with the middle axle unsteered has no exact solution at all: the centre must lie on the middle axle’s line and on the rear axle’s line, and two parallel lines meet nowhere. Every three-axle rigid vehicle scrubs in every turn, and the amount is computable from the same rows.

Four steered wheels. A machine that steers all four has two conditions rather than one, and a centre anywhere in the plane rather than on a line. That is what a swerve platform is, and it is the reason such a machine can turn about a point outside itself, spin on its own centre, and crab — all of which are just choices of where the centre goes.

A wheel anywhere. The condition is per-wheel and knows nothing about axles. A castor is a wheel whose direction is a coordinate rather than a constraint, so it contributes no row and always agrees; that is what makes castors the standard fix for a vehicle that would otherwise be overconstrained.

What a violation costs #

When the axes do not meet there is no twist satisfying every wheel, so the wheels share the disagreement out. The vehicle takes up the twist that minimises the sliding — least squares, which is what tyres settle into once they have finished arguing — and every wheel is left with some.

Steer both front wheels to the same angle, which is what a simple parallel linkage does, and at a 12 m radius the worst wheel scrubs 1.71% of the distance travelled. Over one full circle of 75 m that is 1.3 m of sliding.

The number gets worse quickly as the turn tightens, because the two correct angles diverge:

That is why a car park is where tyres wear out and a motorway is not, and it is a statement about geometry with nothing about rubber in it.

The rig that cannot help it #

The three-axle case was named above as having no exact solution. The arithmetic is worth doing, because the size of the answer is not what “no exact solution” suggests.

Take the same track and wheelbase, steer the front axle, and add a second unsteered axle a distance $d$ behind the first — a tandem, which is what a tipper, a bus or a bogied trailer carries. The two unsteered axle lines are parallel and distinct, so no point lies on both, and the concurrency cannot be satisfied at any steer angle whatever. There are still six rows for three unknowns; what has changed is that no choice of the two free angles empties the residual.

Feeding those rows to the same least-squares step gives the sliding directly. At a tandem spacing of 1.35 m:

The comparison worth making is against the previous table. A two-axle car steered wrongly — both front wheels held parallel — scrubs 1.71% at a 12 m radius. A three-axle rig steered as well as it can possibly be steered scrubs 5.99% at the same radius, three and a half times as much. The exact geometry of a tandem is worse than the crude approximation of a car, and no steering linkage can close the gap, because the gap is not in the linkage.

The same rows say where the sliding goes. The six residuals come out at 2.25% and 1.88% across the front, 5.99% at both middle wheels and 4.01% at both rear ones. The unsteered wheels take nearly all of it, and the middle axle takes half again as much as the rear — which looks arbitrary until the effective centre is read off the fitted twist. It sits at 3.509 m behind the front axle, between the two rear axles but not at their midpoint, and each unsteered wheel’s scrub is then its own longitudinal distance from that centre divided by the radius actually turned. Distances of 0.809 m and 0.541 m over a radius of 13.497 m give 5.99% and 4.01% — the least-squares answer arrived at a second time from a length and a radius, with no minimisation anywhere in it.

That effective radius is the other thing the computation returns, and it is not the radius the rig was steered for. Steered for 12 m it turns 13.497 m, a quarter of a wheelbase wide, and the ratio settles towards 1.12 as the turn opens out. A tandem understeers geometrically, before any tyre has been asked to do anything, and the correction a driver makes for it is a steer angle the concurrency condition calls wrong.

The scrub is very nearly linear in the tandem spacing — 4.03%, 5.99%, 7.82% and 10.03% at spacings of 0.9, 1.35, 1.8 and 2.4 m. That is the trade a designer actually holds, because the spacing is chosen for load and for axle-weight limits and the tyre bill is what it costs. It is also why a lifting axle is fitted at all: raising one of a tandem deletes its two rows, the two lines that remain meet, and the residual goes to zero exactly rather than merely getting smaller.

Running the ordinary two-axle car through the identical code returns a worst-wheel scrub of 6.2 × 10⁻¹⁷. The step that reports 5.99% for the tandem reports nothing at all for the vehicle it cannot fault, which is what makes the 5.99% a statement about the rig rather than about the method.

The tandem is also the point at which this field’s rows stop being the whole story, and the path a towed wheel takes is where the rest of it is. Scrub is an instantaneous quantity: it says what is sliding now, at these angles, and says nothing about where the rig ends up. A trailer’s off-tracking is the integral of a hitch angle over a path, and the two questions have different answers on the same vehicle — a rig can scrub heavily while tracking a corner beautifully, and a long trailer can track badly with every one of its wheels rolling cleanly.

The condition is instantaneous, and that is the field’s whole point #

It would be easy to read the concurrency as a constraint on where the vehicle can be, and it is not one. It is a condition on the steer angles at each instant, and steer angles are things a driver sets.

That distinction is the field’s premise applied here. A vehicle whose four axle lines do not meet is not in a forbidden configuration; it is in a configuration from which no motion is possible without sliding. Change the steer angles — which is free, and costs no travel — and the same vehicle in the same place is free to move again. Nothing about where the car is has been restricted by any of this.

Compare that with a linkage. A four-bar whose lengths do not satisfy its closure equation cannot be assembled at all; the configuration does not exist. A car with badly-chosen steer angles exists perfectly well and simply cannot roll. The two failures look alike in a rank computation and they are entirely different mechanically, and the difference is exactly the one the field’s first essay is about.

What the condition does not decide #

Three things, and each of them is a reminder of where this field’s boundary is.

Which point on the line the vehicle turns about. The concurrency says the centre is where the lines meet; it does not say the vehicle is turning at all. A car with its wheels straight has a centre at infinity and may be stationary. The magnitude of the twist is set by how fast the wheels are being driven, and the geometry says nothing about it.

How the load is shared. Four wheels on a rigid body over uneven ground is a statics problem with one wheel too many, and which wheel carries what is decided by the suspension. Nothing in this essay’s rows knows that a wheel is touching the ground at all — the rows say what a wheel forbids if it is rolling, and a wheel in the air forbids nothing.

Whether the tyres actually hold. The scrub table above says how much sliding the geometry demands. Whether the tyre supplies it by squirming within its contact patch, by slipping in a series of small releases, or by breaking away entirely is a friction question, and the answer changes the wear and not the geometry.

Why real steering does not satisfy it #

Nothing above says how the two front wheels are made to take those angles, and that is the hard part, which an essay of its own is about. A steering rack and two track rods form a four-bar linkage, and a four-bar cannot produce the exact relation $\cot\delta_o - \cot\delta_i = t/L$ — it can only approximate it, exactly at a handful of angles and nearly everywhere else.

The two essays are two halves of one subject and the boundary is worth stating. This one is the condition: what the wheels require, derived from the rolling constraint, exact and independent of any mechanism. That one is the mechanism: the trapezoid that tries to deliver it, its error curve, and the one arm angle that minimises the error over a working range.

Reading them together gives the whole of it: a requirement that is a rank condition, and a linkage that meets it at three angles and misses everywhere else by a third of a degree.

The centre as an object #

The instantaneous centre is not a new object on this site. It is the same one a coupler curve’s centrode is built from, and reading a wheeled vehicle through it makes the two fields the same subject.

For a four-bar, the instant centre of the coupler is found by intersecting two lines — the two cranks extended — and the coupler’s motion is the rolling of one centrode on another. For a steered vehicle, the centre is found by intersecting the axle lines, and the vehicle’s motion is the rolling of its wheels on the ground. Both are the statement that a planar motion at any instant is a rotation about some point, and both compute that point by intersecting lines that the mechanism supplies.

The difference is which way the reasoning runs. In a linkage the lines are given and the centre follows. Here the centre is chosen — by the driver, through the steering wheel — and the lines have to follow, which is what a steering linkage is for.

What the redundancy is worth #

Four wheels for two independent conditions means two surplus rows, and a surplus row is either a nuisance or an instrument.

As a nuisance it is what makes a four-wheeled vehicle sensitive: the two rear wheels’ rows agree only because they are parallel and on a common axle, so a rear axle that is not square to the vehicle — from a bent mounting, an accident, or a badly-set tracking — puts the two rear rows in disagreement and the vehicle scrubs in a straight line. That is the fault a driver notices as a car that pulls, and it is geometric before it is anything else.

As an instrument it is what makes the disagreement measurable. Four rows for three unknowns leave a residual, and the residual is zero exactly when the four wheels agree. Nothing on a two-wheeled vehicle has that property, which is the same trade a redundant platform’s fourth wheel offers.

Where the centre goes to infinity #

Two cases are worth naming because they are where the arithmetic changes character rather than merely getting large.

Straight ahead. All four axle lines are parallel, they meet at infinity, and the twist has zero turn rate. The rank is still 2 and the vehicle still moves; nothing has degenerated. What has happened is that the centre — the point — is no longer a point, and any computation that reads the centre off by dividing by the turn rate will produce infinities on a perfectly ordinary configuration. The machinery here reads the centre off the twist rather than off an intersection for exactly that reason, and reports that there is no finite centre rather than a very large one.

Turning on the spot. A four-wheel-steered platform can put the centre at its own middle, at which point the vehicle rotates and does not translate. Every wheel is tangential and every axle line is radial, and the concurrency is satisfied as easily as anywhere else. A conventional car cannot reach this configuration, not because the condition fails but because its rear wheels are not steered and their axle line does not pass through the vehicle’s middle.