Six points and no more

Put a camera back on a telescope, take it off, put it back. Does it return to where it was?

That is a kinematics question and it has a kinematics answer, which is unusual for a question about precision. It has nothing to do with how tightly the bolts are done up or how good the machining is. It is decided by how many contacts there are and where they point.

The mechanisms in this field so far have been bars and pins. This one has neither. A part resting on a set of contacts is a mechanism whose joints are the contacts, and everything the spatial and screw fields built applies to it directly — because a contact is a joint and a contact force is a screw.

A contact is a joint #

A ball resting on a plane can move in every way except into the plane. It has five freedoms left out of six, so the contact takes one away — which makes it, in the vocabulary the pair table set up, a five-freedom pair, the loosest joint there is.

What it can transmit is a single force along the contact normal. Not a moment, because a point cannot carry one; not a force in any other direction, because nothing is holding the ball down sideways. One force, along one line.

In the language the screw field set up, that is a wrench of zero pitch — a pure force along a line, the same object the spatial field found in the constraint system of a planar four-bar. The whole apparatus for handling those already exists here: reciprocity, rank, the complement, the axis and pitch of whatever is left.

The ball’s centre is the whole geometry #

There is a simplification hiding in this that is worth being explicit about, because it is load-bearing and one radius away from being wrong.

A wrench is a force along a line, and a line needs a point on it. The natural point is where the ball touches the surface. But for a sphere, the contact normal passes through the centre of the sphere, whatever surface it is resting on — a flat, a groove, a cone, a hole. So the wrench built at the touching point and the wrench built at the ball’s centre are the same wrench, and a coupling’s entire constraint system can be written from the ball centres and the normal directions with the contact points never computed.

That is checked rather than assumed. Building both and comparing gives agreement to 1.1 × 10⁻¹⁶ in every component, over all six contacts of the clamp. It is the kind of statement that is obviously true once said and would be silently false if the balls were, say, cylinders.

Three arrangements that work #

Three ways of arranging six contacts are standard, and all three take all six freedoms.

Kelvin’s clamp — one ball in a trihedral socket (three contacts), one in a V-groove aimed at the centre (two), one on a flat (one). It is 3 + 2 + 1, and its virtue is that it is easy to make: a cone, a groove and a flat.

Maxwell’s — three balls, three V-grooves, each groove pointing at the centre. Two contacts each. Its virtue is symmetry, and the essay’s measurement is about what symmetry buys.

Three-two-one — the machinist’s version: three pads on the primary face, two on the secondary, one on the tertiary. Six normals along three orthogonal directions rather than six around a circle. It is what a machining fixture does, and it is the same theorem.

For each of the three, the six wrenches are assembled and their rank measured. All three give rank 6 — nothing left free, nothing constrained twice. That is what exactly constrained means, and it is a rank, which is why the ledger files it as exact.

Units, and a rank that would otherwise be a matter of opinion #

There is a technical trap here that this site has learned to look for. A wrench mixes a direction with a moment, so its six numbers do not have the same units, and a rank is a decision about which singular values count as zero.

Draw the coupling in millimetres and the moment rows are forty times the size of the force rows. Draw it in metres and they are forty times smaller. Neither is a property of the coupling, and a rank tolerance applied to the raw matrix is therefore a tolerance on the drawing units.

Every wrench here is built with its moment divided by the coupling’s own radius, which makes the matrix dimensionless. And that is checked: the same coupling built at 40 mm and at 0.04 m gives the same order, and so does the arrangement that fails. It is a one-line assertion that exists because getting it wrong produces a plausible answer.

The one that does not work #

Six contacts is not the condition. Six independent contacts is, and the difference is a mechanism.

Cut the three V-grooves parallel to each other instead of aiming them at the centre. There are still six contacts, still three balls, still two normals each, and every normal is still perpendicular to its own groove. Nothing about the parts has changed.

The rank comes out at five, and the freedom that is left is not described as “some looseness”. It is a screw, and the machinery names it: a twist of infinite pitch — a pure translation — along the direction (1, 0, 0), which is the groove direction. The part slides.

That is the same computation the spatial field runs on its overconstrained loops, in the opposite direction. There, the constraints were the complement of the joint screws and the question was what they were; here the freedoms are the complement of the contact wrenches and the question is the same. One bilinear form, ω·m + v·f, and everything else follows.

Fewer than six, on purpose #

Not everything wants six. A table on three legs is a part on three contacts, and its rank is three — the three freedoms gravity cares about, and no more.

The three freedoms that are left are named the same way the failure above was named: a rotation about the vertical axis, and two translations in the plane. Nothing about that is a defect — a stool is supposed to slide across the floor — and the arithmetic makes no distinction between a freedom that was wanted and one that was not. It reports what is there.

This is worth stating because “exact constraint” is often taught as a rule that a part needs six contacts. It does not. A part needs as many independent contacts as the freedoms it must not have, and the useful discipline is to say which those are before counting. A machining fixture that has to locate a casting in all six will use 3-2-1; a stool that has to stand still vertically and be draggable horizontally uses three; a hinge, which must keep exactly one, uses five.

What a coupling is worth, and why the neat one is the worse one #

All three good arrangements take six freedoms. That is where a textbook stops, and it is not where a designer can stop, because the three are not equally good.

Suppose a ball is a micron oversize, or a groove has been ground a micron deep. The part cannot stay where it was: it has to move to touch all six contacts again. How far it moves, and where, is a linear map from six contact errors to a rigid displacement — and it is exactly the sensitivity computation the practice field built for tolerances on link lengths, on a different mechanism.

The displacement of a point on the part is then three numbers from six, and the honest way to summarise it is a worst case over directions of error, which is the largest singular value of that map.

The reason is visible in the arrangement. A Kelvin clamp’s socket is a fixed point: three contacts meeting at one ball pin that ball completely, so every error anywhere else in the coupling turns the part about it, and a rotation about a point at the rim is amplified by the distance to the far side. Maxwell’s has no fixed point; an error in one groove moves the part a little and turns it a little, and the three grooves share the work.

That is a real design rule and it comes out of the arithmetic rather than out of a preference. It is also why the ledger’s exact verdict for a coupling carries a note: the exactness is a rank, and a rank does not rank couplings.

Two routes, and the one that is not a route #

The sensitivity above is a derivative, and this site does not trust a derivative that has never been compared with anything.

The first route differentiates the contact equations: the displacement of the material point at a contact, projected on the normal, is the reciprocal product of the part’s twist with that contact’s wrench, so the six equations are the wrench matrix and the answer is one linear solve against a matrix that has already been formed.

The second route moves the contacts by a finite amount and re-seats the part — with a real rotation, Rodrigues and all, not a small-angle matrix — and reports where the probe point ended up.

They agree to 2.7 × 10⁻⁴ relative at a contact error of one per cent of the coupling radius, and the disagreement is second order: halving the error halves what is left after dividing by it. The first version of that check asked for a factor of four and got two, which is not a defect in either route — it is the difference between a quantity and a quantity per unit error, and the check was rewritten to say what it was actually measuring rather than widened until it passed.

A linearised second route would have agreed exactly and proved nothing. That is the trap the synthesis-depth phase walked into with a solver seeded from a closed form, and it is worth saying plainly: two routes that share their algebra are one route.

Why this is the same subject as a linkage #

It would be reasonable to read this essay as a detour. The site is about linkages; a coupling has no links, no pins and nothing that moves once it is seated.

The connection is that a coupling is a mechanism at the moment of not moving, and everything that makes a mechanism interesting is visible in that moment. A mechanism’s mobility is the nullity of a constraint matrix; a coupling’s is the nullity of a smaller one. A mechanism’s overconstraint is a rank deficiency in the constraints; a coupling’s is the same deficiency and produces the same symptom — a part that will not go together unless something is made to give.

The spatial field’s headline was that the corrected mobility formula M = 6(n − j − 1) + Σf + ν needs ν, the number of redundant constraints, which is not in the joint graph. A coupling makes ν visible with no linkage in the way: count the contacts, measure the rank, and the difference is ν, exactly.

The same is true in the direction this field cares about. A machine tool’s slideway carries hundreds of contacts and is nowhere near exactly constrained; it works because the surfaces are made straight and flat to a tolerance that makes the redundancy harmless. That is the engineering answer to overconstraint the practice field arrived at from tolerance analysis, met here from the constraint side, and it is the subject of the next essay.

What is not here #

No preload. Nothing here says how hard the part is held down, and a coupling with no preload is a part resting on six points, which is a statement about where it can be rather than where it stays.

No friction, no Hertzian flattening, no wear. A real coupling’s repeatability is limited by all three and by the surface finish of the balls, and the numbers above are geometry — the ratio between an error in a contact and the resulting motion of the part. That ratio is what the design decides; how big the contact errors are is what the workshop decides.

And no seventh contact — which is not an omission but the next essay, because a seventh contact does something to a coupling that no amount of care in making it can undo, and the arithmetic that says so is the same rank, read from the other side.

Rank decides whether, conditioning decides how well #

The three good arrangements share a rank and differ by a factor in sensitivity, and it is worth naming the second quantity properly, because it is the one a designer chooses between them on and it is not in the exact-constraint argument at all.

The six wrenches form a matrix. Rank six says the six are independent and the part has one place to be — a binary property, true or false, with no degrees. The conditioning of that same matrix says how far the part moves when the contacts move, and it is a continuous quantity that varies widely among arrangements that all have rank six.

Those are the two questions this site keeps separating, arriving here in the cleanest form they take anywhere. Exact constraint is a rank; quality of location is a number; and one matrix answers both. A coupling that is exactly constrained is not thereby a good coupling, in exactly the way that a mechanism with the right mobility is not thereby a good mechanism.

Read as conditioning, the design rule has a shape rather than being a list of three named arrangements. The sensitivity is governed by the smallest singular value of the wrench matrix, so the arrangement to want is the one whose six wrenches are as far from dependent as possible — six directions spread as widely as the geometry allows, rather than six that happen to be independent. Two normals nearly parallel is two rows nearly equal, a small singular value, and a part that moves a long way for a small contact error even though the rank is perfect.

That is why the neat arrangement is the worse one, and it is a general statement rather than a fact about these three. Symmetry tends to cluster normals — a symmetric arrangement repeats a direction three times at three places — and clustering is what a small singular value is. The arrangement that spreads best is usually the one that looks least tidy, which is an uncomfortable design rule and a correct one.

It also says what to do when neither of the three named arrangements fits the part. Rather than reaching for a fourth named scheme, assemble the six wrenches for whatever the part’s geometry permits and read the smallest singular value; any arrangement with rank six is exactly constrained, and the best of them is the one that number likes. The named arrangements are good examples of a criterion rather than a list to choose from, and having the criterion is what lets a designer leave the list.

What to take from it #

Three things, in the order they are worth having.

A contact is a joint and a contact force is a screw. Once that is said, a coupling is inside the same machinery as a spatial linkage and there is nothing new to build. The site did not write a coupling library; it wrote thirty lines that turn contacts into wrenches and handed them to the screw machinery it already had.

Six is not the condition. Independence is, and independence is a rank — which is a computation, not an inspection. The parallel-groove arrangement looks entirely reasonable in a drawing and fails, and the failure is not a small looseness but a full translational freedom.

Exactness does not rank. Three arrangements are all exactly constrained and differ by a factor of nearly two in what a contact error costs at the rim. The verdict “exact” in the field’s ledger is about the count being a count, and a design decision needs the number underneath it — which, on this site, means a sensitivity, computed twice, by routes that share no code.