Six things a hold is not

Six things that get said about parts held by contact, each of them reasonable, each of them the natural extension of something that is true of a linkage, and each answered here with a number. The arrangements are the seven on the field’s ledger, and the instrument is the same throughout: one row per contact, take the hull, ask where the origin is.

One: six constraints leave nothing #

True of equations and false of inequalities.

A part on six pins has one configuration because six equations in six unknowns have one solution. A part on six pads has a set of configurations, because six inequalities bound a cone and a cone has an interior. The step from a rank to a mobility is legitimate for the first and not for the second.

The arithmetic that forbids it is two lines. If $d$ vectors positively span $\mathbb{R}^d$ then $-a_1$ is a non-negative combination of them, which rearranges to a vanishing combination with a strictly positive coefficient — so they are dependent, and $d$ dependent vectors do not span at all. The minimum is $d+1$: four in the plane and seven in space.

Every exact-constraint coupling on this site has rank six and an escape, and the escapes are clean rather than marginal — a Maxwell coupling’s is a pure vertical translation with nothing else in it. That is not a defect: it is what a coupling is for, and gravity is the seventh contact nobody draws.

The confusion is easy to make because the two questions share a matrix. Locating asks whether $At = e$ has a solution, which is about the rank; holding asks whether $At \ge 0$ has a non-zero one, which is about the signs. Same rows, different question, and an arrangement can be excellent at one and hopeless at the other.

Two: reaching the minimum is most of the way there #

It is almost none of the way there.

Four thousand arrangements of contacts drawn uniformly on the faces of a box: six hold in nought of four thousand, which is a proof rather than a measurement; seven hold in twenty-one, which is 0.53 per cent; ten hold in 12.1 per cent.

So a designer who has counted to seven has established that a hold is possible with that many. Half a per cent of the arrangements reaching the count are holds, and the difference between a necessary condition and a useful one is the whole of what the rest of the field measures.

That gap has the same shape as Grübler’s count against a measured mobility and one difference worth naming. 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. A count that cannot be wrong and cannot decide anything is a strange instrument to be handed, and it is what the classical numbers are.

Three: a part touched on all sides cannot move #

The pinwheel is the counter-example and it is the sharpest thing in the field.

Four contacts, one on each edge of a square, each a third of the way along, taken the same way round. Every row’s moment about the centre has the same sign, so no positive combination can cancel it, and the part turns clockwise about any of a large region of centres.

Move two of the four to the other ends of their own edges — the same four edges, the same distance along each — and the margin goes from nought to 0.211.

Nothing about either drawing distinguishes them. The count is the same, the rank is the same at three, and the singular values are identical to every figure, because the two matrices differ by the sign of two rows. A rank is invariant under negating a row and a cone is not, which is why no function of the row space can decide this.

Four: a null direction is a motion #

It is a candidate.

An ellipse of semi-axes 1.4 and 0.9 in a pocket the size of its own bounding box has four contacts whose normals all point at its centre, so every row’s moment entry is exactly nought, the rank is two, and the cone of permitted twists is a whole line — a free spin, both ways, at first order.

It cannot turn by any amount whatever. The reach toward the top and bottom walls is $\sqrt{a^2\sin^2\theta + b^2\cos^2\theta}$, which is smallest at $\theta = 0$, so the penetration grows as $(a^2-b^2)\theta^2/2b$ — measured at $6.4 \times 10^{-5}$ for a hundredth of a radian, with a fitted exponent of 1.9944 against 2.

A circle in the same pocket has the same rank, the same nullity, the same escape cone and the same margin, and turns for ever. The entire difference is a second derivative, and it is the same asymmetry the network field found on a flex arrived at from a different direction.

The same claim in its usual form #

Four contacts is enough in the plane is how this one is normally met, and it is the counting claim and the pinwheel claim run together. Four is enough for some arrangements of four and not for others, and both of those sentences are needed.

The vee makes the second half concrete in a way the pinwheel does not. Four contacts, two on each of two faces: the count is right, the directions repeat — two contacts on one flat contribute rows that differ only in their moments — and the part slides out of the open side. So an arrangement can fail for two independent reasons, one about signs and one about repeated directions, and a count sees neither.

Five: a part free in every direction is not restrained #

A unit disc among three points on a circle of 1.1 radii is free at every instant, in every direction, at every configuration it can reach — rank two, margin nought, both regions of centres shaded everywhere — and it cannot get out.

The area its centre can reach is 0.060. Push the obstacles to 1.25 radii and it is 26.9 and unbounded. The threshold is exactly $1/\sin(\pi/3) = 1.154701$, where the gap between neighbouring obstacles is the part’s own diameter — checked against a flood fill of the free space that is told nothing about circles and agrees at every radius sampled.

Everything in this field before that rung is a statement about an instant, and whether a part can get out is not one. Neither implies the other in either direction: a held part is caged trivially, and a caged part need not be held anywhere in its cage.

Six: tightening the tolerances will pin it down #

Not if the arrangement is not a hold, and no amount of tightening changes whether it is.

With a clearance $c$ on every contact the permitted poses are $\{p : Ap \ge -c\}$, a polyhedron whose every dimension scales exactly with $c$ — asserted at seven clearances four orders of magnitude apart, with the ratios constant to $10^{-9}$. And it is bounded exactly when the arrangement holds, because an unbounded direction of it is a ray of the escape cone.

So halving the clearance on an arrangement that is not a hold narrows the pose set in two coordinates and leaves the third exactly as infinite. Taking the clearance to zero collapses the set to a line rather than to a point.

The order the two questions have to be settled in is therefore fixed: the arrangement first, the tolerance second, and nothing about the second bears on the first. Every part in tolerance, every pad where the drawing says, the count of contacts right, and the part not held — with no gate a workshop has that would catch it.

A seventh, which gets said less and is also wrong #

Worth adding because it is the one a reader who has followed the first six will reach for: more contacts is at least never worse.

It is worse. A contact beyond the minimum whose row lies inside the hull of the others adds no restraint at all — the margin without it is unchanged, and on the hexagon held by five it is unchanged to every figure at 0.0914 — and it adds a second condition on the contact tolerances, because the cone of positive combinations that cancel gains an extreme ray.

Measured as a yield: on the square held by four, 202 of 400 random error vectors admit a pose; on the hexagon held by five, 280 of 400 do not. The extra contact takes the fraction of parts that fit from about a half to about three tenths, on an arrangement whose margin is worse and whose pose set is larger.

And it does one more thing, which is the reason it is worse rather than merely useless. A contact that adds no restraint is the one that is free not to touch — it is satisfied by not touching, since it says only do not come closer — so the arrangement is not one arrangement. Sometimes five contacts are in play and sometimes four, decided by errors of a few microns, and the two have different stiffnesses and different repeatability.

And the version of it a workshop says #

If it is not tight enough, add a clamp is the same claim in the form it is actually met in, and it is right when the clamp’s line cuts the escape region and wrong otherwise.

That is a condition a reader can check by eye once the escape region is drawn: the added contact’s own surface line has to pass on the far side of every corner of the region, with the right sense. A clamp pressing into a face the part is already located against duplicates a row that is already there, changes the count and changes no answer — which is the workshop complaint that a clamp “does not do anything”, stated as a geometric fact.

And two things a hold is #

For balance, because a list of refutations is a poor way to learn what something is.

A hold is the statement that the origin is inside the hull of the contact rows. That is the whole definition, it is a distance rather than a decision once the hull is drawn, and the distance falls continuously to nought as an arrangement approaches one that lets go. Every other quantity in the field is derived from it.

And a hold is a relation between a set and its complement rather than between a part and the ground. Two interlocked pieces hold each other, mutually, with the frame contributing nothing to either — each is blocked in every direction by the other alone, and the pair lifts straight out. So the ground is one of the parts, and which part comes out first is a question that can have no answer.

Three, and the one it is easiest to get right #

Of the two things a hold is, the first has a consequence worth stating on its own: the test is cheap.

One row per contact — the moment of its normal about the origin, then the normal itself. The hull of at most a dozen unit vectors in three dimensions, which is 220 small determinants. The origin’s distance to the nearest facet. That is the whole computation, it is linear throughout, there is no root-finding anywhere, and it runs in less time than reading the drawing takes.

So none of the six failures above is a case of a hard question being answered badly. Each is a case of an easy question not being asked, with a different question’s answer used in its place.

Four questions, one matrix #

The observation that the two questions share a matrix is made above about the first misconception, and it is the whole diagnosis of five of the six. The matrix of contact rows answers several different questions, the answers are all numbers, and nothing about a number says which question it came from.

There are four questions in this field and it is worth setting them out with what each one cannot answer, because that is the part the failures exploit.

How many independent conditions are there? The rank answers it. It cannot say whether the part is held, because a rank counts rows and a hold is about which side of them the part is on — which is the first misconception exactly, and the second, since a minimum row count is still a row count.

Which directions are candidates for motion? The nullity and the null space answer it. They cannot say whether any candidate is realisable, because a candidate direction may be forbidden at second order by the same contacts that permitted it at first — the vee block and the ellipse, which is the fourth.

Is the part held at all? The hull condition answers it: the origin inside the convex hull of the rows. Nothing else does, and it is the question the other three get mistaken for. The pinwheel is not a failure of this test; it is a case where the test is the only thing that separates two drawings nothing else separates.

And where can the part actually get to? The escape region answers it, and it is the only one of the four that is not a statement about an instant. A part free in every direction at every instant may still have a bounded reachable set, which is the fifth misconception and is the one that cannot even be posed in the vocabulary of the other three.

Read that way, the sixth is a question about none of them. Tolerance narrows the polyhedron of permitted poses and does not move the origin relative to the hull, so it operates on a quantity that only exists once the hold question has been answered in the affirmative. That is why the order is fixed rather than merely advisable: three of the four questions have answers whatever the arrangement, and the tolerance question does not have one at all until the third is settled.

The practical remedy is unglamorous and it is the same one every list of this kind ends at. Name the question before reading the number. Every quantity here is cheap, correct and computed from the same rows, so nothing is gained by choosing carefully between them — all four can be had at once, and the failure is never a wrong computation but a right one asked to mean something it does not.

What the six have in common #

Five of the six are a quantity that is right about what it measures being read as an answer to a different question, which is the standing shape of everything this site’s wrong field collects. The rank measures the row space; the count measures the number of contacts; the nullity measures a linearisation; the clearance measures a gap. Each of them is exactly correct and none of them is the answer to can this part move.

The sixth — the pinwheel — is different and is worth separating out, because it is not a misread quantity. It is two arrangements that are identical in every number anybody would compute except the one that decides, and the one that decides is a sign. That is a failure a more careful reading of the usual instruments could never have caught, because the usual instruments do not carry the information.

Which is the reason the field exists at the size it does rather than as a footnote to the constraint field. The routine is short — one row per contact, take the hull, find the origin — and there is no way to get to it from the machinery of the first twenty fields, because those are built on equations and this one is not.