Two bars that have to cross

Every figure on this site is drawn on a page, and for twenty-three fields that has been free.

A planar mechanism, in the sense the first field sets up, is one whose bodies have three freedoms each rather than six: two translations and a rotation, all in one plane, with every joint axis perpendicular to it. That definition says nothing about where along those axes anything sits. It is a statement about the motion, and the motion is the same whether the crank is a millimetre above the coupler or a metre.

Give the links bodies and the freedom that definition leaves becomes a decision that has to be made, and made correctly, or there is no machine.

The reading that starts it #

Sweep the crank rocker of the opening essay — ground 4, crank 1, coupler 3.5, rocker 3 — at 0.16 width, through 180 solved positions of a full turn. The smallest gap over the tested pairs is −0.432 at every one of them.

That is not a near miss and it is not a corner case. It is the coupler and the frame bar fully crossed, by the whole width of the coupler, for the entire turn. And it is the same reading on Watt’s linkage, on the drag link and on Peaucellier’s cell — the last of them between a long arm and the crank, two links at opposite ends of the chain with no joint and no equation in common.

The mechanisms are correct. Every position solves to 10⁻¹⁴, every mobility is one by formula and by rank, every classification is what it was. What has been discovered is not an error in any of them but a cost that has never been on the bill: to make these of material, some of their links have to be somewhere else along the joint axes.

Two kinds of conflict, and only one of them is measured #

Two parts cannot share a plane for either of two reasons.

They are joined at a pin. Two links pinned together both surround the same pin, and both have to have material there — that is what a boss is for. This one is decided before the mechanism moves: it depends only on the graph of links and joints, and it holds at every configuration, so it is not measured but read off. A four-bar has four such pairs: crank to coupler, coupler to rocker, crank to frame, rocker to frame.

They overlap somewhere on the drive. Two links with no pin in common whose material is in the same place at some position. This one is a measurement: sweep, and test the pair at every configuration. On the four-bar there is exactly one such pair, the crank against the rocker, and it holds for part of the turn rather than all of it.

The two kinds are drawn differently in every figure in this field — solid for joined, dashed for overlapping — because they mean different things to a designer. A joined conflict is structural and cannot be designed away by changing lengths. An overlapping conflict is a fact about these particular dimensions, and moving a pivot or shortening a link can remove it.

What the overlap costs, measured as a share #

An overlapping pair is not overlapping for the whole drive, and how much of the drive it takes is worth having, because it is the difference between a conflict that could be removed by a small change of dimensions and one that could not.

On the four-bar the crank and the rocker are in the same place for a fraction of the turn — the part where the crank swings towards the rocker’s side of the frame. On Peaucellier’s cell there are two such pairs and both are brief. Chebyshev’s linkage, over the arc it has, has none at all: its two long arms cross in the drawing and never occupy the same place at the same instant, which is why it is the only machine in the catalogue that could be built flat.

The distinction is exactly the one the field’s opening essay draws about coupler curves: a picture of a whole motion has thrown the time away. Two bars that cross on the page have crossed paths; two bars that conflict have been in the same place at the same configuration. Chebyshev’s is the machine that separates the two readings, and without it the field could plausibly claim that any drawing with crossing lines needs another plane.

Why the joined pairs are not measured #

There is a temptation to treat both kinds of conflict the same way — sweep, test every pair, and let the joined ones fall out of the measurement like the others. It does not work, and the reason is worth a paragraph because it decides how the whole field is coded.

Two links pinned together have material at the same pin at every configuration, so a measured test on them returns overlapping at every sample, always, on every mechanism. That is not a finding: it is the definition of a pin, arriving as though it were a discovery. Worse, it would swamp the collision test — the smallest gap over all pairs would be the boss overlap at whichever pin happened to be tightest, at every angle, and the number a designer wants would be buried under a constant.

So the rule is stated once and applied everywhere: joined pairs are excluded from the collision test and included in the conflict graph. The first half is what makes the gap function about something; the second half is what makes the plane count right. Getting the second half wrong is the more interesting failure, and it happened in a first version: with joined pairs left out of the graph as well, a four-bar’s conflict graph has one edge and the answer comes out as two planes — a machine whose crank and coupler are in the same plate, sharing a pin, occupying the same material.

What an offset changes #

Nothing, and that is the point.

Take the four-bar, put the crank in one plane, the coupler in a second, the rocker in a third, and the frame wherever it likes. Every link length is unchanged: the pins are still at the same (x, y), and the offset is along the joint axis, which is perpendicular to everything the length is measured in. Every angle is unchanged. Every position the solver returns is unchanged, because the closure equations never mentioned the third coordinate.

The mobility is unchanged too, and this is where a reader who has done the spatial field may hesitate, because that field’s whole lesson is that a planar mechanism built in space is overconstrained — counted at −2 by Kutzbach and moving anyway, with a rank of 3. That is still true and it is still the same mechanism. Offsetting the links along their parallel axes changes none of it: the joint screws are the same four parallel axes, the screw system they span is the same three-dimensional one, and the mechanism is overconstrained by exactly as much as it was.

So the offsets are free, kinematically. What they are not is arbitrary.

What an offset does change #

Three things, all of them about the material rather than the motion.

A pin becomes a shaft with a length. Two links in planes 1 and 3 are joined by a pin that has to span both, which means it passes through plane 2 — and anything in plane 2 whose material covers that pin is pierced by it. That is a condition involving three links at once, and it is the subject of the next essay.

A ground pivot becomes a bearing pedestal. A fixed pivot has to be carried by something standing up from the base, and that something is in every plane at once — so unlike a link’s conflict with another link, it cannot be resolved by choosing heights. A moving link that passes over the pivot in plan passes through the pedestal in fact — which is the one obstacle a planar linkage cannot design around by choosing heights, and it turns out to be what really limits how wide the links of a four-bar can be. That is two essays further on.

And the machine gets thicker. Every plane is a plate thickness plus a running clearance, so a mechanism needing three planes is three plates deep whatever its plan dimensions are. On a scissor stack that number is the whole design question, because the point of the machine is to fold flat.

What decides a pair, when it is not obvious #

Two of the catalogue’s machines are worth reading in detail, because their conflicts are not what a glance at the drawing suggests.

The drag link — ground 1, crank 2, coupler 1.6, rocker 2.2 — is the four-bar in which both the input and the output turn all the way round. Its frame is the shortest link, a stubby unit between two pivots, and its crank and rocker are twice as long. The overlapping pair is the crank against the rocker, as on the crank rocker, but the geometry is different: they are long links on short centres, and they sweep through each other’s territory twice per turn rather than brushing past once. It still needs three planes, and it is a machine where a designer would notice the need without any of this machinery, because the drawing is visibly a tangle.

Watt’s linkage is the opposite case. It looks like the tidiest mechanism on the site: two arms and a short coupler, everything symmetric, nothing anywhere near anything. Its conflict is the coupler against the frame bar — and the frame bar is the one link a reader is least likely to think of as a link at all, because on this site it is drawn as a dashed line and in a real machine it is a plate. Give the frame a body and it is in the way; leave the frame out, and the machine reads as needing fewer planes than it does.

That second case is a modelling decision with consequences, and it is made explicitly: the frame is a part like any other in the machines that carry one. Some of the catalogue’s machines do not, because their frames are represented instead by an obstacle bolted to them — a post, a pedestal, a guide rail — and those are tested against every moving part in the same way.

The count, over the catalogue #

Every machine in the catalogue, swept at ninety positions, with its conflicts counted and the fewest planes that resolve them found exactly:

The four-bars — crank rocker, drag link, Watt, the bell-crank variant — all need three. Peaucellier’s cell, with eight links and fourteen joined pairs, needs four. The slider crank needs three. Chebyshev’s linkage and the two machines whose only conflicts are with a fixed obstacle need two.

Not one of them fits in a single plane, and the number for a four-bar is three rather than two, which is the mildly surprising part. Four links pinned in a ring make a four-cycle, and a four-cycle is two-colourable: alternate the planes round the loop and every joined pair is resolved. What breaks it is the one overlapping pair — the crank and the rocker, which sit at opposite corners of that cycle and would have been given the same plane by the alternating scheme. One measured conflict, and the machine gains a plate.

That is a small result with a general shape. The structural conflicts are decided by the graph and give an answer anybody could work out on paper; the measured conflicts are decided by the dimensions and change it. A count from the graph alone would have said two, and a four-bar built to it would have its crank and rocker in the same plate.

Why this has not come up before #

Twenty-three fields is a long time for a question this basic to go unasked, and the reason is instructive rather than accidental.

Every one of those fields is answering a question about motion, and motion is exactly what the offsets do not affect. A mobility is a rank. A coupler curve is a locus. A transmission angle is an angle between two lines. A tolerance band is a set of loci. Not one of them changes if a link moves along its own joint axes, so not one of them has ever had to know which plane anything is in — and the site could go on drawing everything on one page indefinitely without a single number becoming wrong.

The moment the drawing becomes a claim about material, the page stops being free. And it stops being free in a way that is invisible to every gate: nothing on this site could tell a figure of a buildable four-bar from a figure of an unbuildable one, because the two figures are identical.

The offsets are not what a spatial mechanism is #

One confusion to head off, because the words are the same.

Offsetting a planar mechanism’s links along their parallel axes does not make it a spatial mechanism. A spatial mechanism is one whose joint axes are not all parallel: the motion leaves the plane, the bodies use all six freedoms, and the mobility formula changes. Here every axis is still parallel to every other, the motion is still planar, and the mechanism is the same one it was.

What has changed is only where along those axes the material sits, and the reason it needs saying is that the fix for a conflict looks like a spatial construction: bars at different heights, pins spanning several plates, a shaft standing through the stack. It is a planar mechanism assembled in three dimensions, which is what every planar mechanism ever built has been.

The genuinely spatial version of this question — what happens when the axes are not parallel and the bodies sweep volumes rather than areas — is not in this field. It is a real subject, it needs a solid model rather than a polygon, and this field’s whole machinery is two-dimensional.

What the conflict graph is, as an object #

It is worth naming what has been produced, because it is a new kind of answer for this site.

The conflict graph has one node per link and one edge per pair that may not share a plane. It is derived from the mechanism — half of it from the chain’s graph and half from a swept measurement — and it is not itself a mechanism, a motion, or a configuration. It has no dimensions and no units. Two mechanisms with different lengths and the same conflicts have the same graph.

That puts it beside the topology field’s objects rather than beside the loop fields’. The difference is that a kinematic chain’s graph is given by the designer as an input, and this one is measured out of a mechanism that already has all its lengths: it is a graph that depends on the dimensions, which is a thing the topology field deliberately does not have.

And what is asked of it — the fewest planes, which is the graph’s chromatic number — is a colouring, which is the subject of the next essay, where it turns out not quite to be one.

The bill, stated plainly #

For twenty-three fields this site has drawn mechanisms on a page and computed everything about them correctly. The page was not a simplification of anything: the mechanisms really are planar, and their motion really is what the figures show.

What the page hid is that a drawing of a planar mechanism is not a drawing of one object. It is a drawing of a family of machines, differing in which link is at which height, all with identical motion — and most members of that family cannot be assembled. Choosing one is a design decision with a right answer and a wrong one, it is forced by the material rather than by the kinematics, and until a link had a width there was no way even to state it.