Held is not located

Every contact in this field so far has been exact: the obstacle is at a place and the part touches it. Nothing is like that. Give every obstacle a clearance and the part stops having a pose and starts having a set of them.

The pose set #

With a clearance $c$ on every contact, the non-penetration conditions become $a_i \cdot p \ge -c$: the part may move toward each obstacle by up to $c$ before touching. So the permitted poses are

$$P \;=\; \{\, p : A p \ge -c\,\},$$

a polyhedron in the three-dimensional pose space — two translations and a rotation — with one facet per contact.

And there is an immediate consequence, which is the rung.

$P$ is bounded exactly when the arrangement is a hold. If $P$ were unbounded there would be a direction $u$ with $A(p + tu) \ge -c$ for all $t > 0$, which forces $Au \ge 0$: a ray of the escape cone. And conversely a ray of the escape cone gives an unbounded direction. So the set is finite precisely when the cone is trivial.

That is worth sitting with, because it collapses two questions this field has been keeping apart. Is the part held and does the part have a finite set of possible positions are the same question, asked at two scales, and the clearance is what turns the second one from a point into a set with a size.

Measured with six programs #

The size is what a designer wants and it is what the simplex is for. The extreme value of each pose coordinate over $P$ is a linear program — maximise $e_j \cdot p$ subject to $-Ap \le c$ — and there are six of them, two per coordinate.

For the square held on four contacts, at $c = 0.01$:

For the hexagon on five, at the same clearance: 0.1770, 0.0526, 0.0236 — ratios of 17.70, 5.26 and 2.36.

Two things in those numbers.

The ratios are not one. A clearance of a hundredth buys 0.0856 radians of rotation on the square, which is nearly five degrees. Nothing about the arrangement suggests a factor of 8.56, and it is not a factor anybody chose.

And they are not equal to each other, which is the number an error budget would have to carry per coordinate rather than as a single figure. The hexagon leaves more than twice the rotation the square does per unit of clearance, and less than half the translation in one coordinate. So a clearance does not “loosen the part by $c$”; it loosens it by $c$ times a number per coordinate, and the numbers are properties of where the contacts are.

Linear, exactly #

The scaling is exact rather than approximate and it is worth saying why, because it is what makes the ratio a property of the arrangement.

$P$ is defined by $-Ap \le c\mathbf{1}$, and scaling $c$ scales the whole polyhedron about the origin: if $p \in P(c)$ then $\lambda p \in P(\lambda c)$. So every extent is exactly proportional to $c$ and the ratio is constant.

It is asserted at seven clearances from $10^{-4}$ to $10^{-1}$ — four orders of magnitude — with the ratios agreeing to $10^{-9}$. That is not a numerical check of a plausible claim; it is a check that the routine implements the claim, since the claim itself is a triviality once written down.

Which means the useful number is the ratio rather than the extent, and it can be quoted for an arrangement before anybody has decided what the clearances will be.

The arrangement with the better margin is the looser one #

Here is the finding that makes the rung worth its place, and it is the kind of thing this site exists to catch.

The square on four contacts has a margin of 0.211 and leaves 8.56 radians of rotation per unit clearance. The hexagon on five has a margin of 0.091 — less than half — and leaves 17.70, more than twice as much.

So the arrangement with the better margin has the tighter pose set, which is the direction anybody would guess. But the hexagon has more contacts, and adding a contact is what everybody does when a fixture is not tight enough.

The two numbers are not the same quantity and neither determines the other. The margin is the origin’s distance to the boundary of the hull of the rows. The pose extent is a support function of the polar of that hull, which is a different functional of the same object — sensitive to the shape of the hull rather than to its inradius.

A designer wanting a tight pose set has to compute the pose set. There is no shortcut through the margin and there is certainly no shortcut through the contact count.

Where this leaves a clearance #

The site has an essay arguing that a clearance is a link — a pin in a hole with play in it is not a pin, it is a short link of unknown orientation, and the mechanism has more freedoms than the drawing shows. That is the same observation in the language of mechanisms.

Here it becomes a statement about a set rather than about a linkage, and it is sharper in one respect: the extra freedoms a clearance introduces are bounded if the arrangement holds and unbounded if it does not, and there is nothing in between. A mechanism with clearances is a mechanism whose configuration is a small blob rather than a point; whether that blob is small is a property of the joints and whether it is a blob at all is a property of the arrangement.

Three coordinates, and the one that has no units #

A caution about the table above, because it puts a rotation and two lengths in one column and that is a units question the field has been careful about elsewhere.

The rows of $A$ are built with the moment divided by the part’s own size, so a pose vector’s rotation entry is a rotation and its translation entries are lengths divided by that same size. The extents in the table are therefore radians and dimensionless ratios, and comparing 0.0856 against 0.0411 is comparing a rotation to a translation-in-part-radii — which is a fair comparison in the sense that both are “how much of the part’s own size does this move a point at unit radius”, and is not a comparison of a radian to a millimetre.

That is the same convention the seating essays adopted and for the same reason, and it is checked the same way: build the arrangement at a thousandfold difference in scale and require the same ratios. Without it, halving the drawing’s units would halve the rotational extent and leave the translations alone, and every number in this rung would be a statement about a draughtsman.

A tolerance cannot buy a hold #

The negative statement is the one to carry, and the picture is the whole argument.

A square in a vee has four contacts and does not hold. Give it a clearance and its pose set is unbounded along the direction it slides out. Halve the clearance and the set narrows in two coordinates and stays exactly as unbounded. Take the clearance to zero and the set collapses to a line — the escape direction — rather than to a point.

So tighten the tolerances is not a remedy for an arrangement that is not a hold. It reduces two of the three dimensions of a set whose third dimension is infinite, which is not an improvement in any sense a designer cares about. The arrangement has to be fixed first, and only then does a tolerance mean anything.

That is the practical content of the whole field in one sentence, and it is worth noting that no gate a workshop has would catch the failure. Every part is in tolerance, every pad is where the drawing says, the count of contacts is right, and the part is not held. It is the same silence a stack-up analysis keeps about a mechanism that has stopped assembling: every dimension inside its limits and the thing does not go together.

The pose set and the escape cone are the same object #

There is a tidier way to say the whole rung, and it is worth having because it explains why the boundedness statement is not a coincidence.

The recession cone of a polyhedron is the set of directions in which it is unbounded, and for $P = \{p : Ap \ge -c\}$ that recession cone is exactly $\{u : Au \ge 0\}$ — the escape cone, unchanged, with the clearance dropping out because a recession direction is about behaviour at infinity and a constant does not reach infinity.

So the escape cone is not merely related to the pose set; it is the pose set’s recession cone, at every clearance including zero. The five rungs of first-order analysis before this one were computing the asymptotic shape of an object nobody had drawn yet.

That reading also settles a question the field has been avoiding. A margin of nought is reported for every arrangement that does not hold, with no gradation, and it looked like an honest admission that failure has no size. It is more than that: the thing that fails to have a size is a recession cone, and a recession cone genuinely has no scale in it — it is the same set at every clearance and at none.

Which coordinate is worst #

The three extents are not equal and which one is largest is decided by the arrangement, so it is worth knowing what makes the rotation large.

Rotation is the coordinate whose row entries are moments, and a moment is a length times a direction. An arrangement whose contacts are close together has small moments, so its rows are nearly horizontal in the twist space, so the hull is flat in the $\omega$ direction — and a flat hull has a wide polar in that direction, which is a large rotational extent.

Concretely: contacts near the middles of their faces have small moments and leave more rotation. The hexagon’s five include one at the middle of its face, which is the spare one, and it is the same fact showing up twice — a contact that is redundant for holding is also one that is not helping the rotational extent.

The design rule that falls out is simple and is not the one people use. Spread the contacts along their faces, not around the part — which is close to the opposite of what the counting rung suggests, since spreading the directions is what makes an arrangement hold at all. Two contacts far apart on one face contribute a large moment difference and cut the rotational extent hard; two contacts on adjacent faces contribute directions and do less for the rotation.

What a repeatability actually is #

The word this rung is circling is repeatability, and it is worth pinning down because it is the specification a fixture is bought against and it is not the pose set.

Repeatability is how far a part moves between one placement and the next. The pose set is where it may be. The second bounds the first and is not equal to it: a part that seats in the same corner of its pose set every time is perfectly repeatable inside a large set, and a part that seats in a different corner each time is not repeatable inside a small one.

Which corner a part seats in is decided by what pushes it — a clamp, gravity, the direction the operator loaded it from — and that is outside this field. What is inside it is the bound, and the bound is worth having on its own terms: a fixture cannot be more repeatable than its pose set is wide, whatever else is true.

And it explains why exact-constraint couplings are built the way they are. A Kelvin clamp has six contacts with no clearance — every ball is in contact with every surface it is meant to be — so its pose set is a point and its repeatability is limited by the pads’ own errors rather than by any play. What it gives up in exchange is holding: it is not held at all, and gravity does that.

The extent is at a vertex, and the vertex names the contacts #

There is one more thing the polyhedron gives that a bounding box does not, and it is the answer to the question the field’s neighbouring rung asks: which contact to make accurately.

A linear program’s optimum is attained at a vertex, and a vertex of this polyhedron is a pose at which three of the constraints are tight — three contacts simultaneously at their clearance limits, with the rest slack. So each of the six extents comes with a list of exactly three contacts, and those three are the only ones that decided it. The others could have been made looser without moving that extent at all.

That turns a global number into a local instruction. The extent in rotation is decided by one triple of contacts; the extent in $x$ by another, possibly overlapping, possibly not. Tightening a contact that appears in no active triple buys nothing anywhere, and tightening one that appears in all six buys everything at once.

So the error budget is not a matter of dividing a tolerance among the contacts in proportion to anything. It is a matter of reading the active sets off the six programs, counting how often each contact appears, and spending the accuracy on the ones that appear — which is a list the simplex already produced on its way to the numbers in the table and that costs nothing extra to record.

The arrangement also explains a phenomenon a fixture designer meets and usually attributes to luck. Tightening one contact often improves nothing measurable, and then tightening a second one improves several coordinates at once. That is the active set changing: while the first contact was slack at the optimum, its clearance was irrelevant, and the improvement arrives only when it becomes one of the three that bind.

And it says what to do when the answer is unsatisfactory. If a coordinate’s extent is too large and its active triple is already at the tightest clearance available, the remedy is not a tighter clearance — it is to move a contact, because the extent depends on the triple’s geometry through the moments in those three rows. Changing where a contact sits changes which triple is active and how far the vertex is from the origin, and that is a design change rather than a tolerance one.

Which is the same ordering the rest of the rung insists on, one level down. The arrangement decides whether the set is bounded; the arrangement also decides which contacts bind and how far the vertices sit; and the clearance is the last factor, scaling a shape that was already fixed.

What is left over #

Two limits worth naming.

Every clearance here is the same, which is a strong assumption. The pose set for unequal clearances is $\{p : Ap \ge -c_i\}$ with a different bound per contact, which is the same linear program with a different right-hand side and no harder — but the scaling argument fails, since the set is no longer a dilate of itself, and the ratios stop being constants. That is the realistic case and this rung does not compute it, which is a scope decision rather than a difficulty.

And the pose set is not a probability. It says where the part may be, not where it is likely to be, and nothing here says a part is uniformly distributed over its pose set — it is not, and the difference between a worst case and a distribution is a whole rung of the practice field. What this computes is the support.

The support is the right thing for the question the rung asks, which is whether the set is finite at all. Once it is known to be finite, and how large, what a part does inside it is a different subject with a different instrument.