A catalogue is a search space

The synthesis field solves the designer’s problem: given the motion, find the lengths. Burmester’s construction places a dyad through three prescribed positions; a five-position problem has a finite root count; an optimiser searches a continuum for a best fit.

Every one of those starts by assuming a topology. Three positions of what — a four-bar, a six-bar, which six-bar? The question is answered before the synthesis begins, and it is almost always answered from memory.

This rung is about answering it from a list instead.

The discrete part of the problem #

A design decision here has four steps and only the last is continuous.

Link count. Four, six, eight, ten. Usually decided by cost and part count rather than by kinematics, and decided first because everything else is conditioned on it.

Chain. One, two, sixteen or 230 of them, depending on the row.

Frame and input. Between one and $2j$ choices, reduced by symmetry — nine distinct ones at six links, 153 at eight.

Dimensions. A continuum, with no count at all, and where every published method spends its time.

The first three are finite and small. That is the whole argument for a census: a small finite set can be searched rather than chosen, and a search that is exhaustive answers a question a judgement cannot — namely, what was not considered.

A requirement that mentions no length #

The filters worth applying at this stage are exactly the ones that read the graph. Three examples, applied in turn to the 230 ten-link chains.

At least one link with four pins. A quaternary link is a plate carrying four bearings, and a designer might want one because it is where three separate connections can be taken off a single machined part. It is also where accuracy is expensive, so the requirement runs both ways; either way it is a statement about degrees.

A link none of whose neighbours is binary. This is the graph’s way of saying a plate bolted directly to other plates — a stiff core with the bars hung off it — and it is not something the assortment can express, because it is about adjacency rather than about counts.

A driving choice that comes apart into pairs. The closed-form condition: the mechanism can be positioned with a compass, its branches are enumerable, and its position derivatives with respect to the link lengths come out in closed form rather than through an implicit-function argument. Ninety of the 230 fail this outright.

Each one is a microsecond. Applied together they leave a specific subset, countable, which can then be handed to dimensional synthesis one topology at a time.

What a filter cannot be #

The discipline is worth stating explicitly, because the temptation is to filter on things the graph does not know.

Nothing about reach. Whether a mechanism’s output covers a required region is entirely dimensional. The same chain with different lengths reaches different places, and there is no graph condition that bounds it.

Nothing about transmission. The transmission angle is an angle between two links in a configuration; a chain has neither.

Nothing about whether the crank turns. Grashof’s condition compares four lengths. A chain has none, so is this a crank-rocker is not a question about a chain — which the arc chart makes uncomfortably plain.

And nothing about accuracy, except structurally. Which link lengths matter most is a dimensional question the tolerance field answers. What the graph does decide is how many lengths any loop equation involves, which is the shortest-cycle-basis question and is a real constraint on how bad a stack-up can get.

Filtering the ten-link census, step by step #

It is worth watching the numbers fall, because the shape of the fall is the argument.

Start with 230. Every ten-link chain there is: every planar mechanism of ten links and one degree of freedom is one of these with a link held still.

Require at least one quaternary link and 163 remain. That is a mild condition and it removes about a quarter, which is a useful calibration — most conditions of this kind are mild, and a designer who expects a filter to be decisive will be disappointed by nearly all of them.

Require additionally that some link has no binary neighbour, and the count drops sharply. That is a condition about adjacency rather than about degrees, it cannot be expressed in the assortment table at all, and it is the first one that does real work.

Require finally a closed-form driving choice and what survives is the shortlist.

Two things are worth reading off that sequence. The filters are not independent — chains with a quaternary link are somewhat more likely to have a link with no binary neighbour — so the counts do not multiply, and there is no way to predict the shortlist size without running it. And the order does not matter to the answer, only to the cost: every condition is a predicate on a chain, so the intersection is the same however it is reached.

What makes a good filter at this stage #

Three properties, and they are not the ones a dimensional criterion has.

Cheap. Every condition above is microseconds on a ten-vertex graph, so applying twenty of them to all 230 costs less than drawing one figure. That means filters can be exploratory: try a condition, see how many survive, change it.

Monotone in nothing. A dimensional criterion usually has a direction — longer is better up to a point, more accurate is better always — and an optimiser exploits it. A graph condition is a predicate: a chain has a quaternary link or it does not. There is no gradient, nothing to relax, and no partial credit, which makes the search a filter rather than an optimisation.

And permanent. A chain that fails a graph condition fails it for every set of dimensions, so the filter never has to be re-run. That is the property that makes the two stages separable at all: everything decided here stays decided while the dimensional search runs.

Two ways this is normally done instead #

Worth naming, because both are reasonable and both have a specific failure.

Working from a named repertoire. A designer knows the four-bar, the slider-crank, the two Watt six-bars, the three Stephenson ones, and perhaps a handful of eight-link arrangements met in practice. That is between five and fifteen topologies and it is enough for most work.

Its failure is invisible: nothing in the process says how large the unexplored remainder is. At six links it is nothing at all — the five are the census — and the habit is therefore reinforced by being exactly right at the size where a person learns it. At eight links the remainder is 56 of 71 mechanisms, and at ten links it is essentially all of 1,834.

Adding a dyad to something that works. Take a four-bar, hang a dyad off a coupler point, and the result is a six-bar. Repeat and it is an eight-bar. This is constructive, it always produces a mechanism, and it produces a mechanism that is dyadic by construction and therefore easy to position.

Its failure is that it only reaches the chains that decompose that way. Every chain built by repeated dyad addition has an all-dyadic driving choice, so the four hard eight-link chains and the ninety hard ten-link ones are unreachable by the method — not rejected, unreachable — and a designer using it will never see them.

Both failures have the same shape: a method that cannot reach part of a space also cannot report that the part exists.

A filter deletes and a preference ranks #

The permanence of a graph condition was listed above as a virtue, and it is also the property that makes filtering dangerous. It is worth separating the two, because the discipline that follows is the difference between a shortlist and a mistake.

A dimensional criterion ranks. Transmission angle, accuracy, reach, the size of the swept region — each produces a number for every candidate, so a candidate that scores badly is still present and can be reconsidered when a requirement turns out to have been over-stated. Nothing is lost by evaluating it, and the ordering is available at the end for a designer to argue with.

A graph filter deletes. A chain that fails is gone, it does not appear in the shortlist, and there is no residue of it to reconsider. That is exactly what makes the filter cheap and permanent, and it means every graph condition applied is a claim that the requirement it encodes is hard. Requiring a quaternary link because a plate would be convenient discards 67 of the 230 on a preference, and nothing downstream will ever say that it happened.

So the rule for this stage is narrower than filter on anything the graph knows. Filter only on conditions that would make a mechanism unacceptable however well its dimensions turned out. A requirement that some link is bolted to three others because three things have to be mounted on it is that kind of condition. A requirement that the chain be solvable in closed form is not — it is a preference for a cheaper position analysis, the site’s own solver handles the alternative, and encoding it as a filter throws away the four hard chains for a reason that stopped being binding a century ago.

The ordering of the filters is free, which is worth saying because it removes a temptation. Set intersection is commutative, so the shortlist does not depend on the sequence and there is no need to reason about which condition to apply first — the only thing sequence changes is how many graphs the later conditions are evaluated on, and at microseconds each across 230 candidates that is not a cost anybody can measure. Every filter may as well be applied to the full census and the results intersected, which also makes the individual pass rates visible.

Those pass rates are the part worth keeping. A condition that removes two graphs of 230 has told the designer almost nothing and should be viewed with suspicion, since it is doing no work and may be encoding an assumption rather than a requirement. A condition that removes 200 has either found a strong structural requirement or been stated too strongly, and which of those it is deserves a moment’s thought before the remaining thirty are taken as the answer. The informative filters are the ones in the middle, and reading the fall of the counts is a check on the requirements rather than merely a report of progress.

What the census does not settle #

An enumeration answers what are the options and never which one. The second question needs a criterion, and every useful criterion at this stage is a filter of the kind above — necessary conditions, cheaply checkable, stated about the graph.

So the honest workflow is: filter to a shortlist, then run dimensional synthesis on each member of the shortlist and compare the results. The census makes the outer loop finite. It does not make the inner loop cheaper, and the inner loop is where the time goes.

There is one genuine efficiency, though, and it is worth stating. A requirement that no member of a topology can satisfy is best discovered before the optimiser runs. If a design needs a mechanism whose position problem is differentiable in closed form, ninety of the 230 are out before anything is optimised; if it needs a link with six bearings, all 230 are out and the arithmetic says why. Both answers cost microseconds and both would otherwise be discovered by an optimiser failing slowly.

The two-stage split is older than the census #

It is worth noting that the division this rung relies on — choose a type, then choose dimensions — is not new and is not this field’s idea. It is the standard organisation of the subject, and the two halves have had names for a century: type synthesis and dimensional synthesis.

What the standard organisation lacked was a complete list for the first half. Type synthesis in practice meant think of an arrangement, and the literature’s contribution was a set of worked examples and a body of judgement about which arrangements suit which jobs. That is a perfectly reasonable engineering discipline and it is what every textbook teaches.

The census does not change what type synthesis is. It changes it from a generative process into a selective one, and the difference matters in exactly one respect: a selective process can report what it rejected, and a generative one cannot. A Watt six-bar and a Stephenson six-bar were considered and all five six-bar mechanisms were considered are the same sentence at six links and very different sentences at ten.

What is still judgement #

Nothing above removes the need for it, and it is worth being clear about where it moves to.

The requirements are judgement. Whether a design wants a quaternary link, whether closed-form positioning is worth restricting the space by two fifths, whether a stiff core with bars hung off it is the right architecture — none of that is decided by anything here, and each is a preference about manufacture and stiffness rather than about kinematics.

The link count is judgement. Ten links is a lot of bearings; four is usually not enough freedom in the coupler curve. That trade is about cost and lost motion rather than about topology.

And the choice within the shortlist is judgement, informed by whatever the dimensional synthesis produces on each member. The census narrows a space; it ranks nothing, and a ranking would need a criterion that reads dimensions.

Using it as a reference #

The practical form of all this is not a filtered list; it is a browsable one.

Each entry carries the things a designer would otherwise have to work out: the assortment, how many mechanisms the chain gives, how symmetric it is, and how many of its driving choices come apart into pairs. None of those requires a dimension and all of them are decided before the interesting work starts.

That is the modest claim this rung makes. Not that a census designs anything, but that it converts what else is there from a question of experience into a question with a number attached, and that the number at ten links is 230 rather than five.

The one thing it settles absolutely #

There is a single question the census answers with no judgement in it at all, and it is worth ending on because it is the one a design review actually asks.

Is there a mechanism of this many links that does this? — where this is any condition on the graph at all — has a plain yes-or-no answer, and when the answer is no it is a proof rather than a failure to find one. A ten-link linkage with a six-bearing plate does not exist. A ten-link chain with no closed-form driving choice and no quaternary link does or does not exist, and the search takes a millisecond either way.

That is the difference between a catalogue and a search space. A catalogue says what somebody has built; a search space says what there is, and occasionally says that the thing asked for is not in it. Both are useful and only the second can be exhausted, which is why the subject spent a century assembling the first, and why half a second of arithmetic now produces the whole of the other.

There is a second-order use of the same counts that costs nothing and is worth taking. The pass rates of two filters applied together say whether the conditions are independent: if requiring a quaternary link leaves 163 of 230 and requiring a link with no binary neighbour leaves some fraction of those, the fraction can be compared against what independence would predict. A pair of conditions that turn out to be nearly the same condition is a pair worth collapsing, and a pair that interacts strongly is a signal that the requirements are pulling against each other structurally rather than merely narrowing the field. Neither observation costs more than the filtering already done, and both are invisible to anybody applying the conditions one at a time to a remembered shortlist.