The contact that is free not to touch

Four contacts hold a square. Five hold a hexagon. The fifth is doing nothing, and finding out which of the five it is takes five recomputations and no cleverness at all: remove each in turn and see whether the hold survives.

What removable means here #

The test is exactly as blunt as it sounds. For each contact, drop it, run the routine on what is left, and record whether the origin is still inside the hull of the remaining rows and by how far.

On the hexagon, four of the five removals drop the margin to nought and one leaves it at 0.0914. On the square at the minimum, all four removals drop it to nought — there is nothing spare, which is what being at the minimum means.

That the margin is unchanged rather than reduced is the interesting part, and it is not an accident of the arrangement. The margin is the origin’s distance to the nearest facet of the hull of the rows. Removing a point from a set of points can only shrink the hull, so the margin can only fall — and when it does not fall at all, the removed point was not on any facet the origin is nearest to. It was inside, or on a facet that was not binding.

So a removable contact is one whose row lies inside the hull of the others, which is a geometric statement about direction and has nothing to do with position or with how far it is from anything.

The unilateral version of an old problem #

This site has an essay about a redundant contact already, and the two answers are different in a way worth being precise about.

The seventh contact is about a part on six bilateral contacts — exactly constrained, rank six — with a seventh added. That seventh adds no rank, so it removes no freedom. What it does is leave one contact of the seven unable to touch: with contact errors of ten microns, the added pad under a Kelvin clamp is left with a gap of 3.5 µm, and the combination that constrains nothing is a left null vector of the constraint matrix, which under a four-legged table is the alternating sum of the four legs.

Here the combination is a positive one, and it cannot alternate. A hold means there is a $\lambda \ge 0$ with $\sum \lambda_i \hat a_i = 0$ and every entry strictly positive; the field’s whole test is that such a $\lambda$ exists. So the arrangement’s dependency is signed the same way throughout, which is not a detail — it is the difference between a table that rocks about a diagonal and a fixture a part will not go into.

The count of dependencies is the count of conditions #

Which is where the redundancy costs something rather than merely being idle.

For a hold at the minimum there is exactly one $\lambda$ up to scale. For the hexagon on five there are two independent ones, and they are enumerable by the same argument the rest of the field uses: an extreme ray of $\{\lambda \ge 0 : \lambda^{\mathsf T} A = 0\}$ lies on $N-1$ tight constraints, $d$ of which are the equations, so at most $d+1$ of its entries are non-zero. Take every subset of that size, take the null vector of its block, keep it when every entry has the same sign.

The two for the hexagon are

$$(0.500,\; 0,\; 0,\; 0.500,\; 0) \quad\text{and}\quad (0.144,\; 0.288,\; 0.144,\; 0,\; 0.424).$$

The first is a pair of directly opposed contacts, whose rows are exact negatives and cancel between themselves with nothing from the other three. The second uses four of the five and leaves out the spare.

Each of those is a condition the contact errors have to satisfy before the part goes in at all, which is the subject of a later rung. A hold at the minimum imposes one such condition. A hold with a spare contact imposes two, and the extra contact bought no restraint and sold a second inequality.

Free not to touch #

The phrase is deliberate and it is what makes the unilateral case worse rather than better than the bilateral one.

A redundant bilateral constraint has to be satisfied and cannot be. Six pads locate a part; a seventh demands that the part be somewhere it already is not, and since the pad is rigid and the part is rigid, something has to give — a gap opens, and the essay about the seventh contact computes how big.

A redundant contact has no such obligation. It says do not come closer and the part is not coming closer, so it is satisfied by not touching. There is no gap to compute, no stress to redistribute, and nothing to indicate that anything is wrong. The contact is simply idle, and whether it is touching on any particular occasion is decided by errors of a few microns.

That is a genuinely nastier situation than the bilateral one for anybody trying to make a repeatable fixture, and it is nastier in a specific way: the arrangement is not one arrangement. Sometimes five contacts are touching and sometimes four, and the two have different stiffnesses, different wear, and — since a hold with five contacts and a hold with four have different pose sets — different repeatability. A part that seats on a different subset each time will not come back to the same place, and every gate a fixture designer has says the arrangement is fine.

How to tell whether it matters #

There is a cheap way to decide whether a spare contact is a problem or a deliberate choice, and it is the same computation run twice.

Compute the margin with the spare contact and without. If the two are equal — as they are here, 0.0914 both ways — the contact is doing nothing at all and the arrangement is really the smaller one with a passenger. If the margin falls when the contact is removed, it was doing something, and the arrangement is not redundant even though it exceeds the minimum count.

That is a real distinction and the count cannot make it. Five contacts on a plane part is one more than the minimum, so a count says redundant; whether it is depends entirely on where the rows point, and an arrangement of five in which all five are on facets of the hull is not redundant in any useful sense.

Why anybody adds one #

The obvious question, given all of the above, is why a designer would ever exceed the minimum, and there are two honest answers and one bad one.

Stiffness, which is not this field’s. More contacts spread a load, and how far a part deflects between them is a question needing a material and a modulus. Every argument in this field survives with every force unknown and this one does not, so it is named and not developed. It is the standing reason for a sixth pad under a large panel and it is a good one.

Reach, which is geometric and is this field’s. A single contact holds only in its own half-space, and a part whose useful surfaces are all on one side may need contacts that are individually redundant in order for the set of them to surround it. That is the hexagon’s case read charitably: the fifth contact is inside the hull of the others, and it is where it is because the sixth face of the hexagon was not available.

And insurance, which is the bad one. Adding a contact because the arrangement might not be quite right is exactly what the seventh contact essay refutes on the bilateral side, and it fails here in a different way. It does not make the hold more secure — the margin is a distance and the extra point is inside the hull — and it does add a condition on the tolerances. It costs and does not buy.

Which one is spare is not obvious #

It is worth noting that the removal test is the only way to find out, because the answer is not readable off the drawing.

On the hexagon the spare contact is not the one furthest from the others, not the one on the shortest face, and not the one a reader would pick. It is the one at the middle of its own edge, and the four that are load-bearing are at 0.3 and 0.7 along theirs — which is the same distinction as the pinwheel against the working square, and for the same reason: a contact’s row is decided by its face’s line and by the sign of its moment, and the moment is what changes when a contact slides along a face toward the middle.

At exactly the middle of an edge of a regular polygon centred on the origin the moment is nought, so the row is a pure direction with no rotational content. A hold needs rows whose moments cancel with positive weights, and a row with no moment can contribute to that only through the other two components. So a contact at the middle of a face is the most likely one to be spare, which is a rule of thumb the test confirms and does not need.

The margin does not measure this #

One caution, because the field now has a number that a reader will reasonably want to use for everything.

The margin says how far the origin is from the boundary of the hull, and it is the right measure of how close an arrangement is to letting go. It is not a measure of how much redundancy there is, and the two move in different directions: the square on four has a margin of 0.211 and no spare contact, and the hexagon on five has 0.091 and one spare. The arrangement with the spare is the weaker one.

That is not a paradox. The hexagon’s contacts are worse placed — its rows cluster more — and adding a fifth in the middle of the cluster does not fix the placement. The margin is reporting the placement and the removal test is reporting the redundancy, and no single number does both.

Two contacts on one face, which is the commonest case #

The arrangement a workshop most often builds with a spare in it is not the hexagon; it is a part resting on two pads on the same flat.

Two contacts on one face contribute rows with the same normal and different moments, so they are not identical — the pair genuinely constrains rotation where one alone does not. But the pair spans only a two-dimensional set of directions, and a third contact on the same face is inside the hull of the first two for certain: its row is a convex combination of theirs, since its moment lies between theirs and its normal is the same.

So three contacts on one flat face always contain a spare, whatever the spacing, and it is always the middle one. That is the three-legged-stool argument arriving from the direction of a hold rather than of a location, and it explains why the the four-legged table is the standard example on the bilateral side: four legs on a floor is four contacts on one face, three of which do everything.

Three kinds of saying the same thing twice #

This site now has three fields in which a constraint repeats one that is already there, and the three cost different things. They are worth setting side by side, because the word redundant has been doing three jobs.

In a network a redundant constraint is a dependency among the rows — a combination that comes to nothing — and its cost is that the count of freedoms stops being computable from the numbers of bodies and joints. The deployable ring’s forty-four constraints have four dependencies and the count declares it immobile. What a dependency does not do there is anything physical: it is a statement about a matrix.

In an exactly constrained seating a redundant constraint is the same left null vector, and its cost is a gap: something has to not touch, and which thing is decided by the errors. The count of freedoms is still right.

Here a redundant contact is a row inside the hull of the others, and its cost is a condition: the count of freedoms was never the question, the margin is unchanged, and what is added is one more inequality the tolerances have to satisfy before the part fits.

Three fields, three instruments, one word — and the only thing the three share is that a count cannot see any of them.

An equality becomes an inequality, and that changes the odds #

The unilateral case is described above as nastier than the bilateral one, and the arithmetic behind that comparison is worth doing, because it turns nastier into a probability.

A redundant bilateral constraint produces a gap functional of the contact errors that has to be exactly zero for every contact to touch. It never is: the functional is a continuous function of errors that are not controlled, and its zero set has measure zero. So a bilateral redundancy is essentially never satisfied, which is the seventh contact’s whole finding — one contact is always left with a gap.

A redundant contact produces the same functional and the condition is an inequality rather than an equality: the part seats on all of them when the functional falls on the right side of zero, and on all but one when it does not. That is not a measure-zero event. If the errors are as likely to go one way as the other, the condition is satisfied about half the time.

Run that over the hexagon and the number is concrete. It has two dependencies, so two such conditions; if they are independent and the errors are symmetric, all five contacts touch on roughly a quarter of assemblies, and on the other three quarters the part seats on four and one contact stands off with a gap. Not always and not never — sometimes, with the outcome decided by errors nobody controls and nothing reports.

That is exactly why the unilateral case is worse for repeatability. A bilateral redundancy has one predictable outcome: the same contact is always short, by an amount that varies. A unilateral one has several outcomes, each with its own seating and its own location for the part, and which one occurs is settled at assembly time by a coin the errors toss. The measured position of the part is then multi-valued, and the values are what the different seatings give.

It also says what to measure if the behaviour is suspected. Not the scatter, which is what a metrologist would reach for, but the distribution’s shape: a part seating in one of a few configurations gives clusters, and the number of clusters is bounded by the number of subsets of contacts that can carry it. Continuous scatter is a tolerance problem; clusters are a redundancy problem; and the two have entirely different remedies, only one of which involves making anything more accurately.

What a spare contact says about the part #

Reading the removal test across a whole family of arrangements gives one thing that is worth having and is not obvious.

A contact is removable when its row is inside the hull of the others. The rows are unit vectors built from normals and moments, so the hull is a picture of which directions the part offers. A part with a spare contact under one arrangement will tend to have one under most arrangements, because the redundancy is coming from the part’s own surfaces rather than from where anybody put the pads.

The extreme case is the disc, where every row is inside the hull of every other pair, because they all lie in a plane and the hull has no interior at all. Every contact on a disc is removable in the sense of this rung, which is a way of saying the same thing the earlier rung said and it is worth having both: a part whose normals do not spread cannot be held, and a part whose normals spread only just is a part on which most contacts are spare.