A roll centre is not a point

The roll centre is the most quoted number in suspension design and one of the least defensible, and the interesting thing is that the construction behind it is completely sound. Nothing here is a case of a bad approximation. It is a case of a well-defined quantity that varies, quoted as though it did not.

The construction #

Two steps, both from the curves field.

Find the instantaneous centre of the upright. The wheel’s upright is the coupler of the suspension’s four-bar, so at any instant it is turning about one point relative to the body. Kennedy’s theorem puts it where the two arm lines cross.

Join it to the contact patch and extend to the centreline. The wheel touches the road at one point; if the body rolls, the wheel’s motion relative to the body is a rotation about the instantaneous centre, so the point of the body that has no sideways motion relative to the road lies on the line from the centre through the contact patch. Where that line crosses the car’s centreline is the roll centre.

Every step is exact and computable. The centre is found here from the solved velocity field rather than from the construction, and the two agree to 1.1 × 10⁻¹⁵ — Kennedy’s lines are drawn in the figure, and they are drawn to confirm the answer rather than to produce it.

Why it moves so much #

The instantaneous centre moves a long way for a small wheel movement, and that is not an accident of these dimensions.

The two arm lines are nearly parallel. Two nearly parallel lines cross a very long way away, and a small change in either angle moves the crossing enormously — which is exactly the ill-conditioning the site’s algebra field keeps meeting in a different guise. At ride height here the centre is 1,494 mm inboard; at full bump it is 659 mm; at full droop, 3,895 mm.

The roll centre moves much less than the instantaneous centre does, because the construction divides by the distance — a centre twice as far away subtends half the angle at the contact patch — and that partial cancellation is why the roll centre is a usable number at all. It moves 54 mm while the centre it is constructed from moves three metres.

But 54 mm is not nothing. It is comparable to the roll centre height itself, over a travel a road car uses going over a speed bump.

The strut, which is worse and more interesting #

A strut’s upright is carried by one arm and one sliding rod. There is no second arm line, so Kennedy’s construction has nothing to intersect and the velocity route is the only one available. The site’s insistence on two routes pays for itself exactly here: the route that works everywhere was checked against the route that works sometimes, on the mechanism where both exist.

The strut’s roll centre travels 131 mm — more than twice the wishbone’s — and crosses zero. A roll centre below ground level is not a paradox; it is what the construction gives when the line from the instantaneous centre to the contact patch slopes the other way. What it means physically is that the geometric coupling between lateral force and body motion has changed sign, which is a real effect and a real reason struts behave as they do.

The number in the specification is 60 mm.

What the number is used for #

It is worth being precise about why a moving roll centre matters, because the objection is not aesthetic.

The roll centre is used in two ways. As a height, it sets how much of a cornering force goes through the geometry rather than through the springs — the “jacking” term. As a point, its distance below the centre of mass sets the roll moment. Both are computed from a single number, and both are then used in a model where the number is constant.

The model is not wrong to want a constant; it is a model of a car in steady state at one attitude. What is wrong is taking the constant from a drawing at ride height and applying it to a car that is, by the time it needs the number, at forty millimetres of roll — which is precisely the position where the two wheels’ roll centres are furthest from their static values and, on a car with independent suspension, no longer even the same construction on both sides.

That is the standard criticism of roll-centre analysis in the vehicle dynamics literature, and it is not this site’s contribution. What this site can add is the number: on this geometry, 54 mm of movement on a wishbone and 131 on a strut, computed rather than asserted, with the construction visible at every position.

When the construction has no point in it #

There is a configuration where the arm lines are exactly parallel and the instantaneous centre is at infinity. The upright is translating: no rotation, no camber change, at that instant.

The roll centre is still perfectly well defined there. A centre at infinity means the line from it to the contact patch is parallel to the arms, and the roll centre is where that line crosses the centreline — a finite point, arrived at through an infinite one.

The site’s figures handle this the way the curves field learned to: the centre is drawn when it falls inside the frame and reported as off-frame when it does not, rather than being clamped to the edge or omitted silently. A quantity that goes to infinity in the middle of an ordinary sweep is not an error condition; it is the mechanism passing through a configuration, and the centrodes essay has the same behaviour drawn as a curve running off the page.

The construction assumes both wheels are where they are not #

There is a second problem with the number that is worse than its migration, and this site can point at it even though it cannot compute it.

The construction above is drawn on one wheel. The roll centre it produces is the point on the centreline that the body would rotate about if that wheel’s contact patch were the pivot — which presumes the other wheel is doing the mirror-image thing at the same time. In pure bump, it is: both wheels rise together and both constructions give the same point.

In roll, they do not. One wheel is in bump and the other in droop, and the two constructions give two different points — 52 mm and 106 mm on this geometry at the extremes — and the roll centre of the car is neither of them. It is found by the same construction applied to the two wheels together: the intersection of the lines from each wheel’s instantaneous centre through its own contact patch. That point is not on the centreline at all once the two sides are asymmetric, and it moves sideways as well as vertically.

So the number is a single sample of a curve that is itself a simplification of a construction that stops being symmetric in the very manoeuvre it exists to describe. None of that is a criticism of the geometry; the geometry is doing what it does. It is a criticism of compressing it into a scalar.

This site computes one wheel because it is planar and because the second wheel adds nothing kinematic — it is the mirror image, solved by the same solver. What it would take to do properly is a model where both suspensions and the body are one mechanism with the body’s roll as an input, which is a five-bar rather than a four-bar and is a perfectly ordinary thing to build. It is not built here, and it is the most obvious extension this essay leaves.

What the migration is worth as a rate #

If the height has to be quoted, the least useless companion to it is how fast it changes, and that is a number this sweep gives directly.

Near ride height the wishbone’s roll centre falls about 0.34 mm for every millimetre the wheel rises. The strut’s falls about 0.82 mm — two and a half times as fast — which is the compact way to say that a strut’s geometry is more position-dependent than a wishbone’s.

Both rates are themselves position-dependent, which is the same objection one level up, and at some point the honest thing is to publish the curve. The reason the trade does not is not laziness: a curve does not fit in a table, and every specification format ever written wants a number.

That is a recurring shape in this field. The scissor lift’s rise-per-ram, the rocker arm’s ratio, the camber gain, this. In every case the mechanism has a function, the format has a slot, and the slot wins.

Two routes, and what they cost #

Both routes to the instantaneous centre were built, and there is a real difference in what they can do.

Kennedy’s construction is exact, instant, and needs the mechanism to be a four-bar with two visible arm lines. Nothing has to be solved beyond the position itself.

The velocity route needs a solve and then a linear system: J·q̇ = −∂f/∂θ, one solve against a matrix that has already been formed for the Newton step. It works on the strut, on a five-bar, on anything whose Jacobian can be assembled — and it produces the whole velocity field, not just the centre.

The reason both exist is that the second is the one the site trusts and the first is the one that can check it. On the wishbone they agree to fifteen decimal places at every position of the travel. That agreement is the licence to use the velocity route on the strut, where there is nothing to compare it with.

This is the same pattern as the two routes to a sensitivity in the practice field, and the pattern is worth naming: build the general method, check it against the special one where the special one exists, then use the general method where it does not. The alternative — using the special method wherever possible and the general one elsewhere — leaves the general one unchecked in exactly the cases where nothing else can confirm it.

A number that survives, from the same sweep #

It would be unfair to leave the impression that nothing about a suspension is a number, so here is one that is.

The swing arm length — the horizontal distance from the contact patch to the instantaneous centre — decides the camber rate exactly: the wheel’s angular velocity is its vertical velocity divided by that distance. It varies through the travel like everything else here, but it has a property the roll centre does not: it is a length in the mechanism, so two suspensions with the same swing arm length at a given position have the same camber rate there, whatever else differs. It is a quantity rather than a construction on a construction.

The other survivor is the pair of travel limits. A suspension stops when its four-bar reaches a configuration it cannot pass — for the strut here, about a fifth of a radian of arm angle either side of ride height, past which the rod would have to pass through its own mount. That is a property of the linkage, it is the same at every load and every temperature, and it is exactly the kind of thing the limit-position machinery computes for a crank-rocker. A bump stop is a physical component put in front of a geometric limit.

The pattern is the same one the field’s opening essay set out: the numbers that survive are the ones that are properties of the mechanism rather than samples of a function. A count survives. A limit survives. A construction evaluated at a chosen position does not.

Four numbers, one point #

There is a stronger objection available than the roll centre migrates, and it follows from the construction rather than from the measurement. The roll centre is not an independent property of the suspension at all: it is one reading of the instantaneous centre, and the other quantities a suspension is specified by are other readings of the same point.

Follow the constructions. Camber gain is set by how far the wheel is from the instantaneous centre. Scrub is set by how far the contact patch is from it in the other direction. Swing arm length is the horizontal distance from the contact patch to it, quoted directly. And the roll centre is where the line from the contact patch through it crosses the centreline. Four numbers, one point, and the point has two coordinates.

So the four cannot vary independently, and that is a checkable claim about any published specification. Two coordinates fix all four; fix any two of the four and the other two are decided. A suspension advertised with a constant roll centre and a substantial camber gain curve is being advertised with two numbers that constrain the point in incompatible ways, and the arithmetic settles it without any measurement of the car.

That is a good deal more useful than the migration objection, because it does not require anybody to accept that a moving roll centre matters. It says the number is redundant: whatever a designer wants to control, there are two degrees of freedom in the geometry’s instantaneous behaviour and four quantities being quoted, so two of the quotations are consequences of the other two.

Which gives the honest specification its shape. Not four numbers with operating points attached, and not a roll centre with a travel range, but the path of the instantaneous centre through the travel — one curve, from which all four quantities are read at any position, and against which any claim about any of them can be checked. That is one object where there were four, it carries the position dependence that all four numbers were hiding, and it is exactly what the four-bar reading of the suspension produces without any extra work.

The reason it is not the industry’s specification is worth guessing at and not worth being confident about. A curve is harder to put in a table than a number, a number can be compared between cars and a curve cannot be at a glance, and the four quantities have each accumulated a literature of their own in which the others do not appear. None of those is a good reason and together they are a sufficient one.

What the site can say is the narrow thing. The four are readings of one point, the point moves, and any specification that treats them as four independent constants is asserting something the geometry does not permit — which is a stronger verdict than quoted and is available from the construction alone.

What would make the number defensible #

Not much, and that is the practical conclusion.

Quoting the roll centre with its travel — “73 mm static, 52 to 106 over ±80” — would be an entirely honest specification and is three numbers rather than one. Every suspension package computes the curve; it is one line of output.

Quoting the rate of migration is the next best thing, and is what a sensitivity would be: −0.3 mm of roll centre per millimetre of bump near ride height on this geometry. That has the same defect as camber gain — it is a derivative of a curve that bends — but it at least announces that there is a curve.

What is not defensible is the single number with no operating point, and the reason it survives is the reason most quoted numbers survive: everybody in the field knows it is quoted at static ride height, so nobody writes it down, so the convention is invisible to everybody else. That is the same mechanism by which a rocker ratio is quoted at zero lift and a chain drive at its mean — a convention that is knowledge inside a trade and a trap outside it.

The last of the four-bar essays takes a mechanism where the varying quantity is the product rather than a nuisance: an over-centre latch, which is deliberately parked at the one configuration every other essay treats as a hazard.