A constraint that only pushes

Twenty fields of this site have been about mechanisms whose parts are fastened together. A pin holds two points at the same place, a bar holds two points at a fixed distance, a mesh holds one angle proportional to another, and a rolling wheel forbids a velocity. Every one of those is an equation, and everything the site does with them follows from that: a configuration is a root, the set of configurations has a tangent space, and mobility is the dimension of that space.

A part resting on a table is none of it.

The table says do not come below this height. It says nothing whatever about going up. So the constraint is an inequality, and three things change at once — not as refinements of the picture the earlier fields drew, but as replacements for it.

A freedom stops being two-sided #

Take the simplest case there is. One flat contact under a part, with the contact normal pointing up into it.

A velocity of the part is a twist $t = (\omega, v_x, v_y)$ — an angular rate and the velocity of whichever material point is at the origin — and the rate at which the part separates from a contact at $p$ with inward normal $n$ is

$$\dot s \;=\; n \cdot v(p) \;=\; \omega\,(p \times n)_z + n_x v_x + n_y v_y .$$

That is linear in the twist, so each contact contributes one row $a = [(p \times n)_z,\, n_x,\, n_y]$, and the condition that the part does not drive into the obstacle is

$$a \cdot t \;\ge\; 0 .$$

An equation would have been $a \cdot t = 0$, and the set of twists satisfying a stack of those is a subspace — closed under negation, with a dimension, and nothing else to say about it. The set satisfying a stack of inequalities is a cone. It has a dimension too, and it also has a shape, and it need not contain the negative of anything in it.

That is not a technicality about signs. It means the sentence the part has one degree of freedom has stopped being well formed. A part on a table can go up and cannot go down; whatever that is, it is not one freedom, and it is not half of one either.

The question stops being a rank #

Mobility on this site has been the nullity of a constraint Jacobian since the first field, and everything about how that number behaves — that it is an upper bound, that a count can disagree with it, that it is a decision about where a spectrum stops being signal — has been argued about that quantity.

Ask the same question here and it does not parse. The cone $\{t : At \ge 0\}$ is not determined by the rank of $A$. The rank is a fact about the rows and the cone is a fact about their signs, and two arrangements can have identical ranks and opposite answers.

The right question is whether the cone is trivial — whether the only twist every contact permits is the zero twist. When it is, the part is held, and the word for that condition is form closure: the shape of the contacts closes around the part, with nothing said about how hard anything is pressed.

There is a clean way to see when it happens. $\{t : At \ge 0\} = \{0\}$ exactly when the rows of $A$ positively span the whole twist space — when every direction is refused by somebody — and that is the same statement as: the origin is in the interior of the convex hull of the rows, taken as unit vectors.

So the routine for every arrangement in this field is three steps. Write one row per contact. Take the convex hull of those rows. Ask whether the origin is inside it.

The row, and the two conventions inside it #

The row is the whole of the machinery, so it is worth being explicit about what goes into it, because two conventions are buried there and each of them is invisible when it is wrong.

The normal points into the part. It is the direction the part is free to move at that contact, not the direction anything pushes — there is nothing pushing. Reverse every normal in an arrangement and every row is negated, so the cone $\{t : At \ge 0\}$ becomes its own negative: the same dimension, the same number of extreme rays, the same answer to whether anything is left. Every count printed in this field is unchanged and every escape direction is reversed. That is a defect no count can see, and the only test that catches it builds both and requires the escapes to be negatives of one another.

And the moment is divided by something. A row mixes a moment with a direction, so its three numbers do not share units, and a hull taken from a stack of them is a statement about numbers whose relative size depends on whether the part was drawn in millimetres or metres. Every row here is built with the moment divided by the part’s own size, which makes the matrix dimensionless — and the check is to build the same arrangement at a thousandfold difference and require the same margin, which is the discipline the seating essays earned for the same reason and by meeting the same trap.

Neither of those is a subtlety about the subject. Both are places where a figure would have been drawn, published and believed, and where the number underneath it would have been about the draughtsman’s choice of units or the order in which somebody happened to list the contacts.

The counts change #

Six independent constraints fix a body in space, and the whole of exact-constraint design is built on it: three balls in three vee-grooves, six contacts, six freedoms removed, and a part that returns to the same place every time it is put down.

Six unilateral contacts fix nothing at all.

The reason is arithmetic and it has no exceptions. $d$ vectors can span $d$ dimensions and they cannot positively span them: a positive combination of $d$ independent vectors is never zero, because the only combination that is zero is the trivial one. Positively spanning $\mathbb{R}^d$ takes at least $d+1$ vectors. So the minimum is four in the plane and seven in space, and the classical numbers are not empirical.

The pair of pictures above is worth reading twice, because it separates two things this site has been treating as one. Both arrangements have rank six. Both are, read as bilateral constraints, complete: a version of either with the contacts glued down would fix the box exactly. One of them holds the box and the other does not, and nothing in the rank says which.

The seven arrangements #

The ledger at the top of this essay is seven parts and their contacts, chosen so that between them they produce every answer the routine can give. It is worth going through them, because five of the seven are cases a reader would have called held.

A square on four contacts is the minimum, and it works: one contact on each edge, opposite edges taken at opposite ends. Margin 0.211, no escape, and no contact that can be removed.

The same four, placed pinwheel is the same square, the same four edges, the same distance along each — and taken the same way round rather than alternately. Every row’s moment then has the same sign, no positive combination can cancel it, and the part turns.

A square in a vee has four contacts on two edges. The count is right and the directions repeat, so the part slides out of the open side. A square on three contacts cannot hold whatever is done with it. A hexagon on five holds, with one contact spare — which turns out to matter, and has an essay of its own.

And then two whose rank is not three at all.

A disc cannot be held by any number of frictionless contacts. Every normal of a circular part passes through its centre, so every row’s moment entry is exactly nought, the rows lie in a plane through the origin, and the origin can never be interior to their hull. Three contacts, eight, twenty-four: the answer is the same and it is the same for a reason about the shape rather than the count.

An ellipse in its own bounding box has the same rank deficiency and the opposite behaviour, which is the subject of the rung about second order: the first-order analysis says it spins, and it cannot turn by any amount whatever.

What this field does not take #

The dual of everything above is a statement about forces — a set of contacts is form-closed exactly when it can resist any wrench — and that statement is structural-engineering-statics.com’s rather than this site’s. The boundary is worth drawing precisely, because this is the closest this site has come to crossing it, and because the temptation to reach for the force reading is strong: the phrase the contacts take the load is how the subject is usually explained.

The separating test is the one the network field applied to a state of self-stress, and it is the same test. Every number in this field survives with every force in the assembly unknown. A margin is a distance between a point and a hull. An escape is a solution of a system of inequalities in velocities. Neither changes if the part is made of something else, neither needs a stiffness, and neither has a unit of force anywhere in it.

Friction is out for the same reason and more strongly. A contact with friction permits a cone of contact forces rather than a single direction, which is a different object from the one above and belongs to the same neighbours; every contact here is frictionless, and that is a restriction rather than an approximation. A frictionless hold is a hold under any friction, so everything this field measures is a lower bound on what a real fixture does, which is the right side to be wrong on.

What the site already had, unnamed #

Reading back, the inequality has been in this site twice before and both times it was allowed to collapse into an equation as quickly as possible.

A strand is slack or taut, and taut is an equation. The field made real use of the switching — a mechanism with a belt has different mobility in different places — but each individual configuration was solved with the strand either doing nothing or holding a distance exactly.

A pawl on a ratchet is the other one, and it is closer: whether the pawl holds is decided by two lines through the contact, and the argument is entirely about which side of them things fall on. What that essay did not do is ask what the set of permitted motions is, because with one contact the answer is a half-plane and there is nothing to compute.

The difference here is that there are several contacts at once, and the interesting object is what they leave between them. One inequality is a half-space and a reader can see it. Four of them intersect in something that has to be computed, and the answer is not the intersection of four intuitions.

How the rest of the field goes #

The ladder from here has a shape, and it is worth stating so a reader can see where each rung is going.

The next two rungs are the object itself: what one contact rules out, and why the minimum is $d+1$. Then two on how the question is answered — the test is a linear program rather than a rank, and an escape is best drawn as a place rather than a direction, which is Reuleaux’s own picture and is what every figure in this family is built on.

Then two rungs on where the first-order answer stops: an ellipse that is free at first order and cannot move at all, and the contact that is free not to touch. Then the boundary of what any of it measures — a part that can move in every direction and cannot get out. Then three on assemblies rather than parts: which way something comes out, why neither of two parts comes out first, and where a chuck’s jaws put a bar.

And last, the two rungs about tolerances, which are where the field turns out to say something a workshop can use. A hold makes the set of poses a part can occupy finite, and a clearance gives that set a size; and a hold turns a set of contact tolerances into a single inequality whose weights say which contact is worth making accurately.

Convexity is what makes any of it computable #

Replacing an equation by an inequality sounds as though it should make everything harder, and on most problems it would. It does not here, and the reason is worth naming because the whole field’s tractability rests on it.

The separation rate at a contact is linear in the twist — one row, one dot product, one sign. So each contact contributes a half-space, the permitted set is an intersection of half-spaces, and an intersection of half-spaces is a convex polyhedral cone. Not a general set, not a curved region, not something that has to be sampled: a convex cone with flat faces and finitely many extreme rays.

That single property buys everything the field does. Whether the cone is trivial is a linear program. Its extreme rays are enumerated by taking subsets of the rows. The margin is a distance from a point to a convex hull, which is another linear program. Whether one row lies inside the hull of the others is a third. Every question the field asks has an exact answer from a routine that terminates, and none of them needs a search or a tolerance chosen by hand.

Compare what would happen if the rate were not linear. A contact between two curved bodies at second order, a contact with friction, a contact that can also slide along a rough surface — each of those turns a half-space into a curved region, the intersection stops being convex, and the questions become optimisations over a set that can have several local answers. That is the situation the second-order rungs step into, and it is exactly why they are a separate and much smaller part of the field.

So the honest account of what changed when the equations became inequalities is not the problem got harder. A subspace became a cone, both are convex, and the whole apparatus of exact computation survived the change intact — which is why a field about parts resting on other parts has more closed-form answers in it than most of the fields about parts pinned together.

Why this is a field and not a footnote #

The honest objection to all of this is that a fixture is a specialised thing, and that a site about linkages does not need a field about it.

What answers it is the ledger. Five of those seven arrangements are things somebody would draw and call held, and one of them — the pinwheel — differs from a working hold only in which end of each edge the contacts sit at. Nothing about the drawing distinguishes them. The count is the same, the rank is the same, the parts are the same, and the answers are opposite.

That is the same shape as every argument this site has made from its first field: a count of freedoms can be confidently wrong, and the thing that catches it is a computation on the geometry rather than on the numbers of parts. What is new is which computation. For a linkage it is the rank of a Jacobian; for a set of contacts it is where the origin sits inside a hull, and no amount of familiarity with the first prepares anybody for the second.

There is one more reason, and it is about what a mechanism is. The site has spent twenty fields on the position of parts that are joined, and every joint in every one of them is an idealisation of surfaces in contact: a pin is a shaft in a hole, a slider is a block in a way, a cam pair is two profiles touching. The equation is what those surfaces become once somebody has decided that they never separate. That decision is usually right and it is a decision, and the field below the decision is this one — which is why the first thing it finds is that a part with more contacts than a hinge has is not necessarily held by any of them.

The rest of the field is what that computation says.