Four in the plane and seven in space

The most useful number in exact-constraint design is six. A rigid body in space has six freedoms, six independent constraints remove them, and a part located on three balls in three vee-grooves comes back to the same place every time it is put down. This site has an essay about it, and the arithmetic in it is right.

Take the same six contacts and ask a different question — not is the part located but can the part move — and the answer changes completely.

The arithmetic, which has no exceptions #

The whole of it is one observation about signs.

A set of vectors spans $\mathbb{R}^d$ when every vector is some combination of them. It positively spans $\mathbb{R}^d$ when every vector is a combination with non-negative coefficients. Spanning needs $d$ vectors; positive spanning needs at least $d+1$, and the proof is two lines.

Suppose $a_1, \dots, a_d$ positively span. Then $-a_1$ is a non-negative combination of them: $-a_1 = \sum \lambda_i a_i$ with every $\lambda_i \ge 0$. Rearranged, $(1 + \lambda_1) a_1 + \lambda_2 a_2 + \cdots = 0$, a vanishing combination with a strictly positive coefficient in it. So the vectors are dependent, and $d$ dependent vectors do not span $\mathbb{R}^d$ at all. Contradiction.

As the previous rung set out, a set of contacts leaves the part nothing exactly when its rows positively span the twist space. So the minimum count is $d + 1$: four in the plane and seven in space.

Three in the plane, checked forty ways #

The planar case is small enough to exhaust, and worth exhausting, because the arithmetic above is easy to believe and easy to believe for the wrong reason — a reader can come away with the impression that three contacts fail because they are badly placed.

Forty arrangements of three contacts on three edges of a square, at every spacing from a tenth of the way along to nine tenths, in every combination. The largest margin any of them reaches is exactly nought. Not small: nought, because the linear program that would have to find a positive combination cancelling three independent rows reports itself infeasible rather than returning a tiny number.

Put a fourth contact on the fourth edge and the margin goes to 0.211. That is the whole demonstration: the failure is not about placement, and the repair is not a better placement.

What six does instead #

If six contacts do not hold a box, it is fair to ask what the 3-2-1 scheme is for, since it is not a superstition — it is on the first page of every fixture-design text and it is the right answer to the problem it solves.

It locates. The six contacts are six independent constraints, and a part pushed against all six of them has exactly one pose. The word pushed is doing the work: a locating scheme assumes something holds the part against the pads — a clamp, gravity, a vice — and given that, six pads determine where the part is to within the accuracy of the pads.

So locate and hold are two different questions and the six-contact scheme answers one of them. Reading a locating scheme as a hold is the mistake this rung exists to prevent, and it is easy to make because the arithmetic that settles the first — count the constraints, check the rank — is exactly the arithmetic that fails to settle the second.

The distinction has a clean statement. Locating asks whether $A t = e$ has a solution, which is about the rank. Holding asks whether $A t \ge 0$ has a non-zero solution, which is about the signs. Same matrix, different question, and the answers are independent: an arrangement can be excellent at one and hopeless at the other.

Seven, and what it is worth #

Add a seventh contact and a box can be held. The version drawn here is the six above with a second contact on the top face, at the opposite corner from the first: nothing else changes, the rank stays at six, and the margin goes from nothing to 0.116.

That is the minimum being reached. What it is worth is a separate question, and it is the reason this rung has a census in it.

Four thousand arrangements of each size, drawn uniformly on the six faces of a box:

• six contacts hold in nought of four thousand — and that row is a proof rather than a measurement, since the arithmetic above forbids it;
• seven hold in twenty-one, which is 0.53 per cent;
• eight in seventy-seven, 1.9 per cent;
• nine in two hundred and twenty, 5.5 per cent;
• ten in four hundred and eighty-three, 12.1 per cent.

So the classical number answers a question about what is possible and says almost nothing about what a set of contacts put down without thinking will do. A designer who has counted to seven has established that the problem is not impossible. Half a per cent of the arrangements that reach the count are holds.

That gap between a necessary condition and a useful one is the same shape as Grübler’s count against a measured mobility: the formula is not wrong, it answers a question about numbers rather than about geometry, and the geometry is where the answer is. What is different here is the direction of the error. Grübler’s count is sometimes too small and sometimes too large; this one is never wrong about what it says and is simply about something else.

Two sevenths, and they are opposite #

This site already has an essay with a seven in the title, and the two sevens mean opposite things. It is worth putting them side by side, because a reader who has met the first will otherwise read this rung as contradicting it.

The seventh contact is about a part that is already exactly constrained on six bilateral contacts, and what a seventh does to it: nothing. It adds no rank, so it removes no freedom, and what it actually does is guarantee that one contact of the seven cannot touch — with contact errors of ten microns, the added pad under a Kelvin clamp is left with a gap of 3.5 µm. A seventh contact there is a redundancy and its cost is a rattle.

Here the seventh is the one that makes the hold exist at all. Six is not enough for a reason that has nothing to do with rank, seven can be enough, and the difference between the two arrangements drawn above is one pad.

Both statements are true of the same physical situation and they are answers to different questions, which is the whole burden of this rung. Counting constraints and counting contacts are not the same activity even when the contacts are the constraints, and the number six is the right answer to one and never the right answer to the other.

The Kelvin clamp itself makes the point better than any constructed example. It is the canonical exactly constrained coupling; its six contacts have rank six; and read as unilateral contacts it holds nothing at all — the escape it leaves is dominated by a straight lift, and a Maxwell coupling’s is a pure translation upward with nothing else in it. Every exact-constraint coupling on this site has that property, because every one of them is designed to be put down onto something, and what keeps it there is gravity.

Why seven and not eight #

It is worth checking that seven is genuinely attainable in space rather than merely permitted by the arithmetic, because the two are different and the plane is misleading about it.

In the plane, four contacts hold a generic convex part and it is easy to arrange: two pairs of roughly opposed faces. In space the count is seven and the naive constructions do not reach it. The 3-2-1 scheme plus one does not work. Six faces of a box, one contact each, plus one more, works only for some choices of the seventh — and the census says how few.

The reason is that a box’s faces supply only six distinct normals, and a positive combination of seven rows has to cancel three force components and three moment components with seven non-negative coefficients. That is six equations in seven unknowns: generically a one-dimensional solution space, which then has to happen to have every entry positive. There is no reason it should, and mostly it does not.

Which is also why a part with more distinct surface normals is easier to hold. A sphere has every normal and cannot be held at all, because they all pass through one point; a polyhedron with many faces in many orientations is easy. The count is about the number of contacts and the difficulty is about the directions, and the second is a property of the part.

The disc, which no count reaches #

The sharpest version of that is a part no number of contacts holds.

Every normal of a circular part passes through its centre. So every row’s moment entry — $(p \times n)_z$ with $p$ and $n$ antiparallel — is exactly nought, and the rows all lie in the plane $\omega = 0$ of the twist space. A set of vectors confined to a plane through the origin cannot have the origin in the interior of its hull, however many of them there are.

Three contacts, eight, twenty-four: rank two, margin nought, and a spin left over every time.

Mill one flat on the disc and four contacts hold it, with a margin of 0.213. That is not a metaphor for why shafts have flats on them; it is the reason, stated in the units of this field. A flat is a surface whose normal misses the centre, which is the only thing a round part is missing.

The same statement in the plane, which nobody quotes #

The planar number is four and it is quoted much less often than seven, which is worth noticing because the planar case is the one people can draw.

Part of the reason is that most planar holds people meet are not frictionless. A vice grips a part with two jaws and holds it, which contradicts nothing here: friction at a contact permits a cone of contact directions rather than one, and the counting changes completely. Every number in this field is a frictionless number and is therefore a lower bound on what a real fixture achieves. Two jaws with friction can hold what two frictionless contacts cannot.

That is the right side to be wrong on, and it is worth being explicit that it is the side this site is on. A frictionless hold is a hold under any friction whatever. An arrangement that passes the test here works with the surfaces greased; an arrangement that fails might still work, and whether it does needs a coefficient somebody measured, which is not a kinematic quantity.

What the number is for #

The count is worth carrying for three things, and none of them is design a fixture by counting.

It rules things out. Six pads is not a hold, ever, on anything, and no amount of care in placing them changes that. That is a fact a drawing cannot show and a count can, and it is the cheapest check there is.

It says what a part costs. A part needing seven contacts costs seven contacts, which for a machining fixture means seven located surfaces with tolerances on them — and the next rungs show that each of those tolerances buys a term in an inequality the part has to satisfy before it goes in. The count is the first term of a cost.

It says which parts are hard. A part whose normals cluster is hard: a disc is the limit and a shallow arc is nearly one. That is visible from the part before any fixture is drawn, and it is the same reading an exactly constrained coupling gets from its own wrenches — the arrangement’s quality is in the directions and the count is a preliminary.

What the count is not for is deciding whether an arrangement works, and the census is the argument. Between this has seven contacts and this holds there is a computation, and it is the subject of the next rung.

Three and six, or four and seven #

The pair of numbers is worth stating together, because the plane-to-space step is the one this site has already had a field about and the two steps are not analogous.

Going from the plane to space takes a body’s freedoms from three to six and a pin’s constraint from two to five, and the interesting consequence there is that a closed loop needs seven joints before it moves — with the famous exceptions, Sarrus and Bennett and the universal joint, that move with four and are counted immobile. That is a story about equations and about counts being wrong.

The step here is different and duller, which is why it is worth naming as different. Four to seven is exactly three-to-six plus one, with no exceptions anywhere and nothing to catch a count out. There is no part in any dimension that can be held by $d$ frictionless contacts, and no clever geometry that gets round it, because the obstruction is the sign argument at the top of this essay and it does not care what anything looks like.

So this is one of the few counts on this site that is safe. What is not safe is what happens above it, and the census is the measure of how unsafe: reaching the count establishes almost nothing, and the distance between possible and achieved is the field’s real subject.

One in two hundred #

The census numbers are the most useful thing in this essay and the ratio in them is worth pulling out, because it says exactly where the design work is.

Six contacts hold in none of four thousand random arrangements, which is a proof. Seven hold in twenty-one — one in a hundred and ninety. So a designer who places seven contacts on a box at random, with no thought about where, has about half a per cent chance of producing a hold.

That number is the measure of how little the count is worth on its own. The minimum is necessary and it is nowhere near sufficient, and the gap between the two is not a factor of two or three — it is a factor of two hundred. Essentially all of the design is in the placement and essentially none of it is in the count.

It also says what to do instead of counting, and the answer is what fixture practice already does. Adding contacts beyond the minimum raises the fraction, because more rows give the hull more chances to contain the origin, so the practical route is to use comfortably more than seven and to place them with the hull condition in view. A 3-2-1 locating scheme plus two or three clamps is eight or nine contacts, which is not extravagance — it is the count at which a reasonable arrangement is likely to work.

The right reading of the classical numbers, then, is as a lower bound with a very loose relationship to practice. Six locates and does not hold; seven can hold and usually does not; eight or nine placed sensibly does. Only the first of those three is a theorem, and it is the one most often quoted as though it settled the design.

Which is the same relationship Grübler’s count has to a working mechanism, arriving one field over. A necessary condition that is cheap and a sufficient one that is not, with the whole engineering in the gap — and a number that is quoted as though it were the second when it is only ever the first.

The number that was already on the site #

There is a version of $d+1$ this site has met before, and it is worth connecting because it makes the arithmetic feel less like a trick.

A network’s mobility is the nullity of its constraint matrix, and the network field spends a rung on the fact that a nullity is an upper bound: a direction the rank permits may not be a motion. That is the same asymmetry as here, arrived at from the equation side — the linear answer contains the true answer and may be larger.

Here the asymmetry runs the other way and is sharper. The rank is not an upper bound on anything in this field. It is an answer to a question nobody asked, and the reason the count of six survives in the literature as long as it does is that the question it answers — is the part located — is the one a fixture designer usually has.