Which link to bolt down

A kinematic chain has no frame. Nothing in the graph says which link is bolted to the floor, and until something does, the thing is not a machine — it is a set of parts that can move relative to one another, with no reference for the word moves.

Choosing the frame is therefore a separate decision from choosing the chain, and it is a decision with consequences a reader of this site has met twice already. The four-bar’s four inversions — crank-rocker, double-crank, double-rocker, and the one obtained by grounding the coupler — are one chain, one set of lengths, and four machines. A robot arm’s eight ways of holding the same tool is the same idea one level along.

What this field adds is the count. How many genuinely different machines does a chain give?

Not one per link #

The obvious answer is $n$: bolt down link 0, bolt down link 1, and so on. It is wrong on the very first case.

The four-link chain has four links and gives one mechanism. Every link is carried to every other by a symmetry of the chain — the graph is a four-cycle, and a four-cycle can be rotated onto itself — so grounding link 0 and grounding link 2 produce the same machine. Not a similar machine: the same one, with the page turned round.

The reason the four-bar’s four inversions are four is that the four-bar has lengths. Once the links are 1, 3.5, 3 and 4 units long, grounding the shortest link and grounding the longest are different machines — that is what Grashof’s condition is about. Strip the lengths away and the distinction goes with them.

So there are two different things called inversion and this site now has both. Dimensional inversion is grounding different links of a mechanism whose lengths are fixed; it is what the linkages field means and it gives four machines from a four-bar. Topological inversion is grounding different links of a chain, with no lengths at all, and it gives one. The second is a lower bound on the first and it is the one a census can count.

Two links are the same when a symmetry says so #

The rule is exact and it is short. Two links give the same mechanism exactly when some automorphism of the chain carries one to the other — a relabelling of all the links that leaves every pin between the same two links it was between before.

That is an equivalence, so the links fall into orbits, and the number of mechanisms is the number of orbits.

Two plus three is five, and those five are what the subject calls Watt I, Watt II, Stephenson I, Stephenson II and Stephenson III.

A reader who has met those names as a list has met the answer without the question. The question is a count of orbits, and it has an answer for every chain in the census rather than only for the two anybody named.

The numbers #

Across the four censuses the mechanism counts are 1, 5, 71 and 1,834.

Seventy-one is the classical number of eight-link mechanisms and it comes out here as a sum of sixteen orbit counts. It is worth seeing how unevenly it is made up: the sixteen chains give between two and eight mechanisms each, and the spread is entirely explained by symmetry.

The chain with the largest automorphism group — sixteen relabellings leave it unchanged — gives two mechanisms from its eight links. The two chains with no symmetry at all beyond the identity give eight, one for every link. Everything else is between.

The average rises with size: 1.00, 2.50, 4.44, 7.97. It approaches the link count because most large graphs have no symmetry, and a chain with no symmetry gives exactly one mechanism per link. That is why the mechanism count grows faster than the chain count, and it is the same statement as the observation that a randomly chosen large graph is rigid under relabelling.

The group, and what it is doing #

The word automorphism is doing real work above, so it is worth unpacking on the smallest interesting case.

Watt’s chain has four of them. There is the identity, which moves nothing. There is the swap of the two ternary links, which carries each binary path onto the other. There is the reflection that keeps the ternary links where they are and reverses both binary paths. And there is the composition of the last two. Four relabellings, and every one of them leaves all seven pins between the same two links.

Apply those four to link 1 and it lands on links 1, 5, 2 and 4 — all four binary links, so they are one orbit. Apply them to link 0 and it lands on 0 or 3, so the ternary links are the other. Two orbits, two mechanisms, and the arithmetic checks: the orbits partition the six links as 2 + 4.

Stephenson’s chain also has four automorphisms and three orbits, which is the useful demonstration that the size of the group does not decide the answer. Its four relabellings move the two ternary links between each other and swap the two short binary paths, and they never touch the long path — so the two links on it form an orbit by themselves.

That is why the count has to be computed rather than looked up from the symmetry: a group of order four gives two orbits on one six-link chain and three on the other, and nothing about the number four says which.

Counting the orbits without listing them #

The observation that a group of order four gives two orbits on one chain and three on the other invites the obvious question — what does decide it — and there is an exact answer, which gives the whole census a second route.

The number of orbits is the average number of links each automorphism leaves where it is:

$$\text{orbits} = \frac{1}{|G|}\sum_{g \in G} \operatorname{fix}(g).$$

Not the order of the group, then, but how much of the chain its elements hold still. A group whose relabellings move nearly everything gives few orbits; a group of the same size whose relabellings each pin several links gives more.

The four-link chain is short enough to do entirely by hand, and it is the case the classical answer is most surprising for. Its graph is a four-cycle, its automorphisms are the eight symmetries of a square, and their fixed links are: the identity holds all four; the two quarter-turns hold none; the half-turn holds none; the two reflections through opposite links hold two each; the two reflections through opposite pins hold none. The sum is $4 + 0 + 0 + 0 + 2 + 2 + 0 + 0 = 8$, and $8/8 = 1$. One orbit, one mechanism, from a chain of four links — the count this essay opened with, arrived at without grounding anything.

Run the same arithmetic backwards on the six-link pair and it says exactly where the difference between them lives. Watt’s chain has four automorphisms and two orbits, so its fixed-point total must be eight; the identity contributes six, so the other three automorphisms fix two links between them. Stephenson’s has four automorphisms and three orbits, a total of twelve; the identity contributes six again, so its other three fix six links between them, three times as many.

That is the answer to the puzzle. The two groups are the same size and are doing entirely different amounts of work: Watt’s non-identity relabellings move almost the whole chain, and Stephenson’s leave a substantial part of it standing. The orbit count reads the fixed points, and the order of the group is only the denominator.

It is also the census’s cheapest self-check, and the reason it is worth having rather than merely elegant. The orbit count and the fixed-point average are computed from the same automorphism group by two different summations — one partitions the links and the other tallies them — so a bug in the partitioning shows up as a disagreement rather than as a plausible wrong number. Seventy-one is a sum of sixteen orbit counts, and a sum of sixteen numbers is exactly the sort of quantity that can be wrong by one without looking wrong at all.

Inversion is already everywhere on this site #

It is worth pointing at how much of the collection is inversion in disguise, because the idea reads as a technicality and is not one.

An epicyclic gear train has three shafts and the ratio depends on which one is held: holding a member chooses the ratio is the transmission field’s whole opening move, and it is inversion with the frame chosen among the sun, the annulus and the carrier. The quick-return mechanism is an inversion of the slider-crank — the same chain, a different link bolted down, and a stroke that takes longer one way than the other.

A Geneva mechanism and a ratchet are inversions of arrangements the timing field builds from the same few chains. A scissor lift grounded at a different link is a different machine with the same members. And every robot arm’s eight ways of holding a tool is the same structure applied to configurations rather than to frames.

What none of those has is a count. Each is an observation that this arrangement inverts into those arrangements, arrived at by working through the cases. The orbit count is the general form, and it applies to every chain in the census without anybody working anything through.

What grounding actually changes #

It is worth being concrete about why the choice matters, because “the same parts, differently bolted” sounds like it should not.

Grounding decides which links are known at the start of a position solve, and therefore what has to be solved together. On Stephenson’s chain, grounding a ternary link leaves a problem that a draughtsman could do with a compass and grounding a binary one does not. That is a difference in kind, and it is a consequence of the frame alone.

It also decides which link can carry the input, what the output looks like, and which of the mechanism’s singular configurations are reachable. None of that is visible in the chain, and all of it follows from a choice with a small number of distinct options — which is exactly the situation a census is useful in.

The naive answer becomes right #

There is a pattern in the averages worth reading, because it says the obvious answer this essay opened by refuting is not wrong so much as premature.

Grounding gives $n$ mechanisms if no two links are equivalent, so $n$ is the ceiling and the orbit count is what symmetry takes off it. Against link counts of four, six, eight and ten the averages are 1.00, 2.50, 4.44 and 7.97 — which is 25%, 42%, 55% and 80% of the ceiling. The fraction climbs steadily, and the reason is that symmetry gets rarer as chains get bigger: there are far more ways to wire ten links than eight, almost all of them lopsided, and a chain with no symmetry at all has one orbit per link and hits the ceiling exactly.

So the four-link chain is the extreme case masquerading as the typical one. It is the smallest chain, the most symmetric object in the census, and the one every reader has already met — and it is the single worst place to form an intuition about how many mechanisms a chain gives. At ten links the naive count is out by a fifth; extrapolating the trend, it stops being out by much at all.

That is a mildly deflating thing for a census to establish about its own headline, and it is the useful form of the result. Symmetry is a small-chain phenomenon. The orbit count is indispensable at four, six and eight links, which is where all the named mechanisms are and where the classical lists were compiled; it is a correction of diminishing size everywhere beyond, and the correction has to be computed to know that.

Why the count is not the whole story #

Two warnings, because an orbit count is a clean number and clean numbers get over-read.

A chain’s mechanisms are not equally useful. Grounding a link with many pins gives a machine whose frame carries many bearings; grounding a binary link gives one where most of the mechanism swings. Both are counted once here. The census says how many distinct choices exist, not how many are worth making, and nothing in a graph could say the second.

And the count is a lower bound on what dimensions give. Two links in the same orbit give the same mechanism as chains. Give the chain lengths and the two may well be different machines, because the symmetry that identified them does not survive unequal links — which is precisely the four-bar’s one topological inversion becoming four dimensional ones.

So the right reading of “seventy-one eight-link mechanisms” is: seventy-one is how many distinct problems there are to give dimensions to. It is a floor on the design space and a ceiling on the topological one.

Where the classical lists came from #

The five six-bars have names because two people found two chains and later authors worked through the groundings. The seventy-one eight-link mechanisms have no names at all, and that asymmetry is itself informative: the point at which a subject stops naming things is the point at which the list stops being memorable, which is between five and seventy-one.

That is a good argument for the census as a reference rather than as a result. Nobody is going to remember which of the sixteen gives seven mechanisms. What matters is that the question has an answer, that the answer is computable in a second, and that a designer choosing an eight-link arrangement is choosing from a known seventy-one rather than from whatever they have seen before.

And it is worth noticing what the naming convention concealed. Watt I and Watt II are two orbits of one graph; Stephenson I, II and III are three orbits of another. The Roman numerals are orbit indices and the surnames are chain names, so the classical vocabulary is exactly this field’s structure — chain, then inversion — written down by people who had the structure without the count.

Drawing the frame #

One convention, since every figure in this field that shows a mechanism rather than a chain has to make the choice visible.

The frame is drawn dark and its pins are drawn solid; every other link is drawn light and its pins hollow. That is the same convention the rest of the site uses for a ground link, so a reader arriving from the linkages field needs no key — and it matters here more than there, because in this field the frame is the only difference between two pictures that are otherwise identical.

The two pictures above are the whole essay in two frames. Nothing about the parts changed. What changed is which pins are bolted down, and the second one is the other of Watt’s two mechanisms.

What is left to choose #

It is worth closing on where this rung sits in the sequence of decisions a designer actually makes, because the sequence is longer than it looks and this field covers the first two steps of it.

Choose a link count. That is the census row, and it is usually chosen for reasons about part count and cost rather than about kinematics.

Choose a chain from that row. There are 1, 2, 16 or 230 of them, and the choice is what the catalogue rung is about — a search over a finite list, with every filter reading the graph.

Choose a frame. That is this rung: between one and $n$ options, decided by the orbits, and it changes the position problem, the input, the output and the workspace.

Then choose dimensions, which is everything else on this site and is where the search stops being finite.

The useful observation is that the first three are discrete and small and the fourth is continuous and enormous — and that almost all published attention goes to the fourth. A synthesis routine started on the wrong chain will optimise beautifully within it and never find what a different topology would have given for nothing. The three cheap decisions are made first, made by habit, and never revisited — which is a good description of where a census earns its keep.