The steering that is never right

Turn a car and the two front wheels do not point the same way. The inner wheel is on a tighter circle than the outer one, so it has to be turned further, and the difference at full lock is several degrees.

The condition for the wheels to roll without scrubbing is geometric and exact: every wheel’s axis must pass through one point, the centre of the turn. For the two front wheels, with kingpins a distance w apart and a wheelbase l, that gives $\cot\delta_{\text{outer}} - \cot\delta_{\text{inner}} = w/l$, which is the relation Rudolph Ackermann’s name is attached to and which he did not invent. It is a statement about circles. It contains no linkage.

A four-bar generating a function #

This is the function-generation problem the synthesis field is built on, arriving under a car. One angle in, another angle out, and a required relation between them that the mechanism can only approximate.

The trapezoid has one free parameter once the track and wheelbase are fixed: the angle the steering arms are swept in by. The track rod’s length follows from it, since the arms must be parallel at straight ahead for both wheels to point forwards.

So the whole design space is one number, and the question is what to do with it.

The rule of thumb, and what it costs #

The classical answer is a construction: sweep the arms so that their lines, extended, meet at the middle of the rear axle. For the car here — 1,480 mm between kingpins, 2,650 mm wheelbase — that is an arm angle of 15.6°.

Zero at straight ahead and nowhere else in the lock range. At 20° of inner lock the outer wheel is 1.17° out; at 35° it is 2.06° out, and it is out in the direction that makes the outer wheel turn too much, which is towards parallel steering rather than away from it.

Two degrees at the wheel is not a subtle error. On a 2.65 m wheelbase at full lock the two front axes miss each other by a good fraction of a metre, and the tyres resolve the difference by scrubbing.

The arm angle that minimises the worst error #

The rule of thumb is a construction, not an optimisation, and there is no reason for it to be the best available. Scanning the arm angle and measuring the worst error over the whole lock range gives a different answer.

At 22.1° the worst error over the whole range is 0.34°, six times smaller. That is the same measure the rule of thumb was doing badly at, measured the same way over the same range.

The flatness of the curve near its minimum is worth as much as the minimum itself. It says the optimum is not a knife edge: a manufacturing error in the arm angle, or a decision to move it slightly for packaging, costs very little. That is a robustness statement of exactly the kind the synthesis field’s optimiser essays keep needing and rarely get for free.

What “percentage of Ackermann” means, and why it is not one number #

The phrase in the ledger is “100% Ackermann”, and it deserves unpacking because it is the form the claim usually takes.

The percentage is defined as how much of the required difference between the two wheel angles the linkage actually delivers: 100% is the geometric condition, 0% is parallel steering, and a number in between is a linkage that turns the inner wheel some fraction of the extra amount it should.

Written that way it is a ratio of two differences, both of which depend on the steer angle. So a linkage does not have a percentage of Ackermann; it has a curve of percentages, and quoting one number means quoting the curve at some chosen lock. On the rule-of-thumb trapezoid here that curve runs from 100% at straight ahead — where both differences are zero and the ratio is a limit — down through the mid-lock range and out to about two-thirds at full lock.

That is the same defect as the roll centre, one field over: a construction that returns a different answer at every position, compressed into a scalar by an unstated convention about where to evaluate it. The difference is that here the convention is not even fixed — some manufacturers quote at 20° of lock, some at full lock, and the two numbers are not close.

How the linkage is solved, and a sign that made the point #

The trapezoid is solved rather than expanded. The two arms and the track rod are a four-bar; the crank is one arm; the output is the other arm’s angle, read off the solved position by an arctangent.

That last step is where the phase’s own mistake lived and it is worth recording, because the symptom was diagnostic. The two arms are mirror images, so the first version of the code mirrored the output angle as well — negating it. The resulting error curve reached 55° at full lock.

Fifty-five degrees is not a linkage being imperfect. It is the size of number that says a sign is wrong rather than a mechanism, and it is worth having a sense of that scale: a real trapezoid’s error is a couple of degrees, so anything an order of magnitude beyond that is arithmetic rather than geometry. The site has made the same class of error before and caught it the same way — a velocity field that came out with a relative error of exactly 2, which is the signature of a flipped sign rather than an inaccuracy.

Both wheels turn the same way when a car steers. The mirror is in the arms, not in the motion, and writing the linkage down is a good way to notice.

Equal ripple, which is how the optimum announces itself #

The optimum has a signature, and finding it is the strongest evidence that the scan has done its job.

The error equioscillates: +0.344, then −0.336, with a zero between them. That is Chebyshev’s equioscillation criterion showing up in a linkage — the best approximation in the worst-case sense is the one whose error touches its extreme value the maximum number of times with alternating signs, and if it did not, the design could be nudged to reduce the biggest peak at the expense of a smaller one.

This site has met the criterion before, in where the precision points go: the Chebyshev spacing of the precision points is the spacing that equalises the ripple. There it was applied to choosing where a synthesis should be exact. Here nothing was chosen; a scan over one parameter found the equal ripple by itself, which is the same theorem arriving from the other direction.

How many angles can be exact #

Two, and the count is not an accident of this geometry.

Straight ahead is always exact: with both wheels pointing forwards the condition is satisfied trivially and any symmetric linkage meets it. That one is free.

Beyond that, the number of solutions of “linkage output = required output” is a question about how many times two curves can cross, and the required output is a cotangent relation while the linkage’s is algebraic in the crank angle. In the lock range they cross once more at best. Nothing in the family reaches three.

That is the real reason “100% Ackermann” is a quoted number rather than a wrong one. It is not that the linkage misses the target by a bit. It is that the target is a curve, the linkage’s output is a different curve, and two curves that cross twice are not the same curve. A percentage suggests a quantity that could in principle be 100, and there is no member of the family for which it is.

Why real cars do not want it anyway #

Here the essay has to stop, and it is worth saying exactly where.

Real cars are frequently set up with less Ackermann than the geometric condition, and racing cars sometimes with the opposite sign — parallel or “anti-Ackermann” steering. The reason is that the geometric condition assumes the wheels roll without slip, and a tyre generating a cornering force does not: it runs at a slip angle, and the outer tyre, more heavily loaded, wants a different slip angle from the inner one.

So the correct steering geometry for a car that is actually cornering is not the correct steering geometry for a car being pushed round a car park, and which is wanted depends on load transfer, tyre construction and how fast the corner is being taken. All of that is outside this site.

What is inside is the geometry, and the geometry has three definite statements: the condition is a cotangent relation, a trapezoid can meet it at two angles at most, and the arm angle that minimises the worst error over the lock range is 22.1° here rather than the 15.6° the construction gives.

What the four-bar is doing while all this happens #

One property worth reading off, since the mechanism is a four-bar and the site has machinery for it.

The transmission angle of a steering trapezoid stays healthy through the lock range — the track rod and the arms never approach being in line — which is why steering does not develop a dead spot at full lock. That is not automatic; it is a consequence of the arms being swept in rather than out, and a trapezoid with its arms swept outwards both gets the Ackermann sign wrong and approaches a toggle.

A flat optimum is a cheap tolerance #

The flatness near the minimum is worth as much as the minimum, and it is worth saying exactly what it is worth, because the argument is the same one this site has now made three times in three fields.

At a minimum the worst-case error depends quadratically on the design parameter, since the first derivative is zero. So an arm angle set half a degree away from 22.1° costs a small multiple of a quarter of a square degree, and the same half-degree error at the rule-of-thumb angle — which is not a minimum of anything — costs a term linear in the error, and therefore far more.

That doubles the case for optimising. The optimum is six times better at its own value, and it is also the value at which manufacturing error is cheapest, because the two effects multiply rather than trading off. A designer who scans for the minimum gets a smaller error and a wider tolerance band around it, and the second may well be worth more than the first on a part made in millions.

The same argument has appeared on this site with a rocker arm’s shim and with the choice of where to centre a working range, and the general form is worth stating because it applies to any design parameter at all. Design at a stationary point of the error, and the parameter’s own tolerance stops mattering to first order. Nothing about steering, cams or precision points is doing the work; it is the ordinary fact that a smooth function is flat at its extremum, put to a use it is not usually put to.

The corresponding warning is the one that makes it worth checking rather than assuming. A stationary point of the error is not the same as a stationary point of anything else, so an optimum found by a construction — a rule of thumb, a classical drawing method, an inherited value — has no reason to be flat. The rule of thumb here is a construction, and its sensitivity is the linear one; it is not merely worse, it is worse in a way that a tolerance cannot recover.

And the equioscillation is what says the flatness has been found rather than assumed. An error curve with two equal peaks of opposite sign either side of a zero is at a minimax optimum, and moving the parameter either way raises one of the two peaks. That is a check on the scan, and it is also the reason the flat region is flat: the curve of worst-case error against arm angle has its own minimum where the two peaks trade places, which is a corner in the underlying max and a smooth minimum in what a manufacturer experiences.

What this does not model #

Three things, all of them named rather than waved at.

The rack. Most cars steer through a rack and pinion, which replaces the track rod with a sliding bar and the trapezoid with two shorter links from the rack ends to the steering arms. It is a different mechanism — six links rather than four — and it has one more design parameter, the rack’s position fore and aft of the axle line. That extra parameter is exactly what makes rack steering able to do slightly better than a trapezoid, and it is not built here. Building it needs no new machinery, only a second closure, and it is the first thing a phase after this one should add.

Bump steer. The steering linkage and the suspension are two mechanisms attached to the same upright, and as the wheel rises the track rod’s end follows an arc that is not the arc the ball joint follows. The difference steers the wheel without anybody touching the wheel. Computing it is a genuinely spatial problem — the track rod is not in the plane of the suspension — and everything in this essay and in the suspension essay is planar. It is named in both and computed in neither.

Steering axis inclination and castor. A real kingpin is not vertical. It leans inwards and backwards, which makes steering lift the car slightly, gives the wheel a self-centring tendency, and — the part that matters here — means that steering the wheel also changes its camber. Every one of those is a spatial rotation about an inclined axis, and the planar model has no room for it.

What all three have in common is that they are more mechanism, not more physics. None needs a force or a tyre. That is a good sign for a boundary: the things left out are things this site could do rather than things it has ruled out, and saying which is which is most of what a boundary is for.

The ledger’s verdict #

The steering row is filed as quoted, and it is the clearest case in the field of a number that names a condition rather than a quantity: a percentage of a relation the mechanism meets at two points and misses everywhere else.

What makes this one different from the roll centre is that the target exists and is exact. There is a correct answer for the outer wheel’s angle at every inner angle; it simply is not producible by a four-bar. The roll centre has no correct answer at all — the construction returns a different number at every position and none of them is the right one, because there is no quantity for them to be right about.

Two failures, two different shapes. Both are recognisable from the fields that came before: the roll centre is a quantity that is not a number, and the steering is a linkage that is only nearly right, which is the synthesis field’s whole subject. What the applied lens adds is that both are on the same car, three feet apart, and neither is regarded as a problem by anybody who works on it — because the trade knows what its own numbers mean and the convention is invisible from outside.

There is a design lesson underneath the arithmetic, and it is the one this essay would defend. When a mechanism can only approximate its target, the free parameters should be spent on the measure that matters rather than on a construction that is easy to draw. The rule of thumb is a ruler-and-pencil answer from an era when scanning a parameter meant building forty linkages; scanning it now costs a second. Six times better on the worst-case error, from a number that was already in the drawing.

The next essay takes a mechanism where the failure is the feature: a latch built at exactly the configuration every other essay treats as a hazard.