Choosing the chain before the lengths

Three problems are called synthesis and all three are dimensional: given a motion, given a path, given a function, find the lengths. Every method the field carries — Burmester’s construction, the five-position root count, an optimiser started somewhere — begins after a topology has been assumed.

The assumption is rarely examined. Three positions of what? A four-bar, if the answer is a dyad; a six-bar, if a four-bar will not do; and which six-bar is usually not asked at all.

This rung is about that decision, and about the one quantity the graph fixes before any dimension exists.

How many numbers a chain has #

A mechanism’s shape is decided by where its pins sit. Put them all down at one reference configuration and that is $2j$ numbers — two coordinates each.

Not all of them are free. Sliding the whole mechanism about and turning it gives the same machine in a different place, which is three numbers; and moving it along its own motion gives the same machine in a different configuration, which is one more. So the number of independent dimensional parameters is

$$2j - 4.$$

The four-bar has $2 \times 4 - 4 = 4$, which is exactly the four link lengths anybody would have listed. A six-bar has $2 \times 7 - 4 = 10$. An eight-link chain has $2 \times 10 - 4 = 16$.

Two things follow immediately, and the second is the interesting one.

More links buys parameters through the pins #

The first is that design freedom scales with the pin count rather than with the link count, and the two are locked together by the mobility arithmetic: $j = (3n-4)/2$, so each pair of extra links buys three extra pins and six extra parameters.

That is a large step. A four-bar has four numbers to choose and a six-bar has ten, so moving up one row of the census more than doubles the dimension of the search space. It is why a six-bar can hit prescribed conditions a four-bar cannot, and it is why a dwell made from a curve needs the extra dyad. Four numbers is a small space and it is exhausted quickly by any demanding requirement.

It is also why the search gets harder. Six more parameters means six more dimensions for an optimiser, more local minima, and a defect rate that does not improve — 1,176 exactly correct three-position syntheses of which 176 are usable is a ratio per topology, and adding parameters does not raise it.

Both six-bars have ten #

The second consequence is that the parameter count cannot distinguish two chains with the same pin count. Watt’s chain and Stephenson’s both have seven pins and both have ten free numbers.

So how much freedom does this topology give is answered identically for both, and every published comparison of the two six-bars is about something else. What differs is which conditions those ten numbers can be made to satisfy, and that is not a count at all.

The clearest instance is the one the topology field measures. Watt’s chain comes apart into two dyads from every way of driving it; Stephenson’s from eight of its fourteen. A synthesis method built on placing dyads in sequence — which is what Burmester’s construction is — applies directly to a Watt six-bar and applies to a Stephenson six-bar only for some choices of frame.

That is a genuine methodological difference between two topologies with identical parameter counts, and it is decided before any of the ten numbers is chosen.

Checking the parameter count #

The formula $2j - 4$ is a derivation and derivations on this site get checked, so it is worth doing on the case where the answer is known independently.

A four-bar is described by four link lengths and everybody agrees about that. The formula gives $2 \times 4 - 4 = 4$. Good.

But four link lengths is not obviously the same list as eight pin coordinates less four, so the correspondence is worth making explicit. Put the four pins down: eight numbers. Fix the frame by putting one ground pin at the origin and the other on the positive $x$ axis: that is the three numbers of a planar displacement, and it leaves five. Then note that turning the crank moves the two moving pins without changing the mechanism: that is the fourth subtraction, and four remain. Those four are the ground distance, the crank, the coupler and the rocker.

The correspondence breaks down as a list the moment a link has three pins, which is why the pin-coordinate form is the right one. A ternary link is not described by lengths at all — it is a triangle, three numbers, and calling them “lengths” hides that two of them are not link lengths in any useful sense. The pin count handles it without a special case.

There is one thing the formula quietly assumes and it is worth naming: that the mechanism has mobility one. The subtraction of one for moving along its own motion is a subtraction of $M$, so a two-freedom chain has $2j - 5$ parameters. Every chain in this field has $M = 1$, so the formula is stated for that case and would need the general form elsewhere.

What the four is #

The formula subtracts four and the essay above has so far treated that four as a fact about the four-bar checking out. It is worth saying what is actually being removed, because naming it turns the quiet assumption at the end of the last section into an exact statement.

Start from a reference placement of the chain: every pin somewhere in the plane, $2j$ coordinates. Those coordinates are not the mechanism’s dimensions, because two different placements can describe the same mechanism. There are exactly two ways that happens. The whole placement can be moved — two translations and a rotation, three numbers — and the mechanism can be advanced along its own motion, which is one number because the mobility is one. Four numbers of placement that carry no dimensional information, and $2j - 4$ that do.

So the four is three rigid-body motions plus one degree of freedom, and the caveat about mobility one stops being a caveat and becomes the fourth term. A chain of mobility $M$ has $2j - 3 - M$ dimensional parameters, and the general form says something faintly counterintuitive: a chain with more freedom has fewer dimensions to choose, because a reference placement of it is that much less pinned down.

Overall size is not among the four, which is worth noticing because the similarity group of the plane is four-dimensional and the coincidence invites the wrong reading. Scale is a genuine dimensional parameter here: the four numbers of a four-bar are four lengths, and doubling all of them gives a different mechanism that happens to trace a doubled curve. Nothing in the count treats that as the same design. A synthesis problem stated in a scale-free way — this shape of coupler curve, with no size attached — is therefore one condition short of what the parameter count would suggest, and the missing condition is recovered as soon as any absolute length is prescribed.

The count as a budget, and where it goes negative #

Reading $2j - 4$ as a budget makes one thing immediately checkable that is otherwise a matter of experience: whether a synthesis problem is even the right shape before any method is chosen.

A four-bar has four parameters. Prescribing five positions of the coupler as a rigid body is Burmester’s problem, and it is famously solvable — which looks like five conditions against four parameters and ought to be impossible. The resolution is that the problem is not posed on the chain’s parameters at all. It is posed on the two dyads separately: each dyad has its own small set of unknowns, each is solved through the same five positions, and the four-bar is what the two solutions assemble into. The chain’s parameter count is the budget for the assembled mechanism and the synthesis is run one Assur group at a time, which is a second and quite different use of the decomposition the topology field computes for position analysis.

That is why the parameter count is a poor predictor of what can be prescribed and a good predictor of what a search costs. Sixteen numbers for an eight-link chain is a sixteen-dimensional space to look in, and that statement survives whatever method does the looking. Sixteen conditions may be imposed does not survive anything.

The budget does have one unambiguous use, and it is a negative one in the field’s usual style. A problem asking for more independent conditions than the chain has parameters has no solution for any dimensions whatever, and that is decidable from the graph alone, before a single length is guessed at. A four-bar cannot be made to pass through nine arbitrary coupler points; the ten numbers of a six-bar cannot satisfy eleven independent conditions. Those are not statements about the difficulty of a search — they are statements that the search space does not contain the answer, and they are the cheapest thing this rung produces.

What a topology fixes about a synthesis problem #

Four things, and each is checkable from the graph alone.

The number of free parameters, which is $2j - 4$.

Whether the problem decomposes. If the chain comes apart into dyads, the synthesis can too: place the first dyad through the prescribed positions, then the second, each with its own small closed-form problem. If it does not, the whole thing is one system.

How many assembly branches the result will have. A chain solved as $k$ dyads has $2^k$ assemblies at a given input, so a Watt six-bar has four — and branch and circuit defects, which are the reason 111 of 1,176 syntheses are usable, are defects between those branches. How many there are to be defective about is a topological number.

And which links can be the frame. A synthesis problem is usually stated with respect to a fixed frame — these positions of this body, relative to ground — so the choice of which link is grounded is part of the problem statement, and the number of genuinely different choices is an orbit count.

What it does not fix #

Everything else, and it is worth being blunt because a parameter count invites the wrong kind of confidence.

Ten free numbers does not mean ten conditions can be satisfied. The relationship between parameters and prescribable conditions is the subject of its own rung and it is not one-to-one: some conditions consume more than one parameter, some are consumed by the requirement that the mechanism assemble at all, and the count of usable solutions is smaller again.

Nor does it say anything about the quality of the result. Two topologies with ten parameters each can produce solutions with wildly different transmission angles, different defect rates and different sensitivity to manufacturing error, and none of that is visible in the graph.

What the extra parameters are actually spent on #

It is worth asking what a designer gets for the six extra numbers a six-bar buys, because more freedom is a vague answer and the field has a precise one.

A four-bar’s coupler traces a sextic curve, and the shape of that curve is decided by six numbers once a coupler point is included. There are curves it cannot trace — a curve with a long straight portion followed by a genuine dwell, for instance — and no choice of the six produces one.

Adding a dyad does two things at once. It adds six parameters, and — more importantly — it changes the kind of output: the second dyad’s output link is driven by a coupler point rather than by a crank, so its motion is a function of a function. That is why a dwell becomes available: a dwell needs a portion of the input’s motion to produce no output motion at all, which a single four-bar cannot do and a curve with a circular arc in it can.

So the six extra numbers are not simply six more dimensions of the same search. They are the parameters of a second mechanism whose input is the first one’s output, and the compositional structure is what makes the new behaviours reachable rather than the count.

Where a coupler point sits in the count #

One omission in the parameter count is worth addressing, because it is the parameter a path-synthesis problem cares about most.

A coupler point — the point whose path is prescribed — is not a pin. It is a point rigidly attached to some link, and it costs two more numbers: its coordinates in that link’s own frame. So a four-bar with a prescribed coupler path has six parameters rather than four, and a six-bar has twelve rather than ten.

Which link the point is attached to is a topological choice, and it is one this field does not enumerate. Any link will do, so at eight links there are eight choices per mechanism — but the point’s coordinates are continuous, so the choice is really which link followed by a two-dimensional search, and only the first half is discrete.

That is the general shape of what a census can and cannot enumerate. Discrete choices that change the structure — which chain, which frame, which input, which link carries the output point — are countable. Everything downstream of them is a continuum, and the boundary between the two is exactly where this field stops.

The two-stage split, honestly #

The standard organisation of the subject is type synthesis then dimensional synthesis, and the second half has all the methods. That imbalance is old, it is visible in every textbook, and it is worth understanding rather than complaining about.

The reason is not neglect. Dimensional synthesis has a continuum to search and a rich algebra behind it: Burmester’s circle and centre point curves, resultants, root counts, homotopy continuation. Type synthesis had, until a census existed, nothing to search — the space was a list of arrangements somebody remembered, and there is no method for consulting a memory. A subject develops methods where there is something for a method to act on, and a list of five names is not that.

What changes with a census is modest and specific. The first stage becomes a filter over a known space rather than a recollection, so a design can state what it needs from the topology — a quaternary link, a closed-form position solve, a particular assortment — and get a shortlist with a size attached.

And the shortlist is then handed to the second stage one member at a time. That is more work, not less: a synthesis run per topology rather than one. What it buys is the answer to the question a design review always asks and a habit never answers — what else was there? — which at ten links is 230, and at six links is five and always was. The second number is the reassuring one: at the size where the subject formed its habits, the habit and the census were the same thing, which is exactly why the habit felt complete.

That is a smaller claim than a reader might want from a whole rung, and it is the honest one. The census does not tell a designer which topology to use, and nothing in it ranks anything. What it supplies is the first stage’s missing ingredient — a space — and the observation that the space is small enough to search and large enough that nobody was searching it.

The one place it does more than that is negative, and it is the same shape as everything else in the topology field. A requirement no topology in a row can satisfy is best discovered before an optimiser runs, and there are such requirements: a ten-link mechanism with a six-bearing plate does not exist, and ninety of the 230 ten-link chains cannot be positioned in closed form from any way of driving them. Both are microseconds to establish, and both would otherwise be discovered by a search failing slowly inside a topology that was never going to work.