Same links, same pins, different machines

Two six-link chains exist. Every textbook that names them names them separately — Watt’s and Stephenson’s — and every one of them draws the two pictures side by side and moves on. What almost none of them says is what the difference is, in a form that could be checked.

Here is what it is not.

Both have six links. Both have seven pins — forced, since one degree of freedom needs $j = (3n-4)/2$ and that is seven. Both have four links carrying two pins and two links carrying three, which is forced as well: the degrees must sum to fourteen across six links with none below two, and the only sequences that do it are $(3,3,2,2,2,2)$ and $(4,2,2,2,2,2)$ — and the second, as it happens, admits no chain at all.

So the two chains agree on every count that can be made without looking at where a pin goes. That is the entire content of the link assortment, and the assortment is the first thing anybody reaches for.

What separates them #

Number the ternary links and ask one question: is there a pin between them?

In Watt’s chain there is. The two ternary links share a pin directly, and the remaining four binary links form two paths of two, each running from one ternary link back to the other. In Stephenson’s chain there is not. The two ternary links are joined by three separate paths of binary links — one path of one link, one of one, and one of two.

That is a fact about adjacency and not about counting, and it is the smallest possible instance of the field’s central claim. The arithmetic is a filter and the graph is the answer.

What the difference is worth #

It would be a thin result if the two chains behaved alike. They do not, and the differences are the ones a designer cares about.

Watt’s chain gives two mechanisms and Stephenson’s gives three. A mechanism is a chain plus a choice of which link is held still, and two links give the same machine when a relabelling of the chain carries one to the other. Watt’s two ternary links are interchangeable and so are all four of its binary links, so it has two orbits. Stephenson’s ternary links are interchangeable, but its binary links split into two pairs that are not — the pair on the single-link paths and the pair on the double-link path — so it has three.

Two plus three is five, and those five are Watt I, Watt II, Stephenson I, Stephenson II and Stephenson III. A reader who has met those as a list of five things has met an answer with the question removed; the question is a count of orbits, and it is the subject of the rung on inversions.

Their position problems are different. Hold a link still, drive a neighbour of it, and ask what has to be solved. Watt’s chain comes apart into two pairs of links from every one of its fourteen frame-and-input choices — and a pair of links is two circles meeting, which is a quadratic. Stephenson’s comes apart that way from only eight of its fourteen; the other six leave all four remaining links as a single group that has to be solved together.

That is a difference in kind rather than degree, and it is the reason the two names have survived in a subject that has otherwise forgotten most of its nineteenth-century vocabulary. A draughtsman with a compass could always position a Watt six-bar and could not always position a Stephenson one, and the rung on Assur groups is about why.

Their loop structure is different. Both carry two independent loops — that is $j - n + 1$, and it is fixed by the same two totals the assortment is. But the shortest pair of independent loops is not the same length. Watt’s is two loops of four links each, total eight. Stephenson’s is a loop of four and a loop of five, total nine.

That is the smallest number of link dimensions any tolerance stack-up on the mechanism can involve, and it is decided by the graph before a single dimension has been chosen.

What the loops share #

There is a second way to say the difference, and it is the one that makes the behavioural differences above look inevitable rather than surprising.

Both chains carry two independent loops. In Watt’s chain the two loops share exactly one pin — the pin between the ternary links — and nothing else. Each loop is a four-bar in its own right, and the two four-bars are hinged together at a single point.

In Stephenson’s chain the two loops share two pins and the whole binary link between them. Cut the chain along its longest binary path and what is left is not two four-bars hinged at a point; it is two loops with a common side.

That is why Watt’s chain always comes apart into pairs and Stephenson’s does not. A loop that shares one pin with what is already known has two unknown links and three usable pins — the arithmetic of a pair, exactly. A loop that shares a whole link with a loop still unsolved has nothing to hold on to, so the two must be solved as one.

It is also why the coupler curves differ in degree, why the two chains appear in different machines, and why every treatment that lists five six-bars lists them in two groups. All of it comes from one pin.

Every count agrees and the chains differ #

It is worth listing the counts explicitly, because the list is the argument.

Links: six and six. Pins: seven and seven. Binary links: four and four. Ternary links: two and two. Degrees of freedom by Grübler: one and one. Degrees of freedom by the rank of the constraint Jacobian at a generic placement: one and one. Independent loops: two and two. Sum of the link degrees: fourteen and fourteen. Number of pins on the two ternary links together: six and six.

Nine counts, nine agreements, two different chains. There is no eleventh count that would have worked either, and the reason is structural rather than a lack of imagination: a count is a function of the multiset of degrees, and the two chains have identical degree multisets by construction. Anything computed from how many links have how many pins is going to agree.

What is needed is something that reads which pins go where. The cheapest such thing turns out to be the characteristic polynomial of the adjacency matrix, and on these two it works.

Watt’s is $\lambda^6 - 7\lambda^4 + 7\lambda^2 - 1$ and Stephenson’s is $\lambda^6 - 7\lambda^4 + 9\lambda^2 - 4\lambda$. The $\lambda^4$ coefficients agree at $-7$, and they must: for a graph with no triangles that coefficient is minus the number of edges, and both have seven. The first disagreement is at $\lambda^2$, which for a triangle-free graph counts four-cycles and pairs of disjoint edges — the first coefficient that notices which links the pins run between rather than how many there are.

The $\lambda^1$ term is more striking still. Watt’s is nought and Stephenson’s is $-4$, and a nonzero odd-degree coefficient means the graph has an odd cycle. Stephenson’s five-link loop is one; Watt’s chain has none, so its polynomial is even and its eigenvalues come in $\pm$ pairs.

Why this is the right first rung #

The temptation with a two-element census is to treat it as a curiosity. It is not, and there are two reasons.

The first is that this pair is the whole field in miniature. Everything the field does later — the census that admits eight graphs for every mechanism, the spectral test that works until it does not, the four chains no compass can position — is a larger version of the counts agree and the objects differ. At six links a reader can hold both pictures in their head and see it. At ten links there are 230 chains and nobody can.

The second is that Watt and Stephenson were not doing topology. Watt’s linkage of 1784 was an attempt to guide a piston rod in a straight line and his six-bar came from adding to it; Stephenson’s is a different arrangement that arrived from a different problem. Neither man enumerated anything. The census says, after the fact, that between them they found all of it: there are no other six-link chains, and there was never going to be a third six-bar for someone else to discover.

That is the kind of statement only a census can make, and it is worth having. The synthesis field chooses a topology by habit before it chooses any dimension, and the habit is drawn from a list of the arrangements somebody happened to name. Knowing the list is complete — and knowing it is five rather than “the usual ones” — changes a search from open-ended to finite.

Telling them apart on a real machine #

Everything above is a statement about graphs, and it reduces to a question anybody can answer while looking at a mechanism: do the two plates share a pin? If they do it is Watt’s; if they do not it is Stephenson’s; and there is no third answer, because there is no third chain.

That is worth spelling out because a five-second inspection replacing a classification is unusual, and because two things make the inspection harder than it sounds.

The first is that a ternary link does not have to look like a plate. It is a rigid body carrying three pins, and it can be a triangular plate, a bent bar, a casting, a welded bracket or a machined block with three holes in it. What matters is the count of pins on one rigid part, not its shape, and a bar with a third pin part-way along it is a ternary link exactly as much as a triangle is. Counting pins per part rather than looking for plates is the reliable version of the inspection.

The second is the one that actually causes misclassification: the frame is a link too. A six-bar’s ground is one of its six, and if the ground is one of the two ternary links then one of the plates is the machine’s own baseplate — three bearings in a casting that reads as structure rather than as a member. Somebody counting the moving parts sees one plate and four bars, concludes there is only one ternary link, and stops. The question then cannot be answered at all, and the mechanism gets called a six-bar with no further classification.

So the inspection has an order to it. Identify the frame and count its bearings; then count the pins on every moving part; then, having found two parts with three pins each, ask whether a pin joins them directly. Three steps, no measurement, and the answer names the machine and settles which of the two position problems it has — whether it comes apart into pairs of links that can be placed with a compass, or whether four of them have to be solved together.

That last consequence is why the classification is worth making rather than merely being possible. Watt’s chain and Stephenson’s differ in a way that decides how a position analysis is written, so somebody about to write one has a practical reason to know which is in front of them, and the reason arrives before any dimension is measured. The names are usually presented as historical labels on two arrangements. They are labels on two different computations.

Where each of them is met #

Neither chain is a museum piece, and it is worth naming where a reader will have seen them, with the caution that a machine is a chain with dimensions and the census has nothing to say about the dimensions.

Watt’s chain is the one that arrives by adding to a four-bar. Take a four-bar, take a point on its coupler, and hang a second dyad from it: the coupler becomes ternary, the frame becomes ternary, and the result is Watt’s chain. Every mechanism built by the standard move of drive something from a point on a coupler curve is this graph — a dwell made from a curve, a stroke amplified, an output taken from a point that traces something a four-bar cannot.

Stephenson’s chain is the one that arrives when the added dyad is grounded rather than hung — when the second loop is closed to the frame instead of to the first loop’s coupler. Both are reasonable things to do, both are done constantly, and which chain results is usually not noticed. That is the practical reason to know there are two: a designer who adds a dyad has made a topological choice whether or not they were thinking about topology, and the choice decides whether the resulting mechanism can be positioned in closed form.

The third six-bar that was never going to be found #

One more consequence, and it is the kind of statement only an enumeration can make.

There is no third six-link chain. Not none has been found — none exists, and the search that establishes it is small enough to describe in a sentence: two admissible degree sequences, five graphs between them that are connected with no link below two pins, three of those carrying a triangle, two left.

That is worth contrasting with how the subject actually proceeded. Watt’s linkage of 1784 came from a problem about guiding a piston rod; the six-bar that carries his name came from adding to it. Stephenson’s arrived from a different problem in a different decade. Neither man enumerated anything, and there was no reason for either to believe his arrangement was one of a complete pair rather than one of many.

The census settles it after the fact, and it settles it in the direction that is rarely available: the two of them, between them, found all of it. Every planar six-link mechanism of one degree of freedom that anybody will ever build is Watt’s chain or Stephenson’s with one of five links held still. A search for a better six-bar arrangement is not a search at all — it is a choice among five, and the only remaining freedom is dimensional.

And it does not extend #

One warning, because the six-link case is unusually clean and it is tempting to generalise from it.

The characteristic polynomial separates the two six-link chains and all sixteen eight-link ones, so at those sizes it is a complete isomorphism test as well as a cheap one. It is exact on every case anybody ever checked by hand. It stops being exact at ten links, where two pairs of genuinely different chains share a polynomial — and one of those pairs does not even share an assortment, so counting the ternary links separates them and the spectrum does not.

So the lesson of this rung is not use the polynomial. It is that separating two chains requires reading the adjacency, that several things read the adjacency, and that which of them is sufficient is a question about the size of the census rather than about the test. The next rungs are about the one that is always sufficient and what it costs.

There is one honest caveat on the inspection, and it is the same caveat the whole field carries. What the three steps identify is the chain, and a chain is not a machine: the same Watt chain grounded at a different link is a different mechanism with different behaviour, and the inspection above does not distinguish those. Naming the chain settles the position problem and the loop structure and stops there. Which of the chain’s inversions is in front of the observer is a second question with its own short answer — which link is bolted down — and the two together are what actually names a six-bar.