Machine Kinematics
solved, then drawn
Drawn wrongly

Six things a chain is not

A count read as a verdict, a rank trusted where it is blind, a fingerprint used as a proof, a list of five taken for a complete one, a solver treated as a convenience, and a census read as a catalogue of machines. Six claims, each with the number that kills it.

Assumes The mechanism is the graph.

13 min read 8 figures Two routes to the same numberExact and approximateThe singularities are the point

Six things that get said about kinematic chains, each of them reasonable, each of them the natural extension of something that is true at the sizes anybody checked, and each answered here with a number. The instrument throughout is the census: every planar chain of one degree of freedom to ten links, enumerated rather than quoted.

None of the six is a careless claim and none is refuted by a subtlety. Every one is exactly correct on the four-link, six-link and eight-link cases, which is to say on everything a person can hold in their head, and every one fails at ten — which is one row past where the subject’s evidence stops, and is where every number in the rest of this essay comes from. Two of the six fail at eight links as well, and one of those two — the count read as a verdict — fails at six, where five graphs pass it and two are mechanisms. That is the one instance on this list small enough to check with a finger, and it is the reason the others are worth believing.

One, two, sixteen, two hundred and thirty. Every planar chain of mobility one, up to ten links, counted by enumeration rather than quoted. The pins column is forced: a chain of 10 links has one degree of freedom only if it has exactly (3n − 4)/2 pins, which is why no odd link count appears. Pass the count is how many graphs satisfy Grübler's rule, are connected, are simple and give every link at least two pins. Are chains is how many of those survive the fourth condition, that no proper subchain is already a structure — and the gap between the two columns is the whole of this field's first argument: at ten links 1,878 graphs pass a rule that 230 of them deserve. Mechanisms is larger again, because a chain is not a mechanism until a link is held still, and how many different mechanisms that gives is a question about the chain's own symmetry.
Fig. 1 The census the rest of this essay argues from. One, two, sixteen, two hundred and thirty.

One: the count decides whether it is a mechanism

It decides nothing of the kind, and at ten links the gap is eight to one.

At ten links, 1,878 graphs satisfy Grübler’s rule with every link carrying at least two pins, connected and simple. 230 of them are mechanisms with ten links. The other 1,648 contain a subchain whose own count is nought or less, so they move as mechanisms with fewer links, one of which is welded out of several pieces.

The reason the rule cannot decide it is structural rather than a matter of precision. Grübler’s rule takes two integers as input — how many links, how many pins — and the question is about which links the pins run between. No refinement of a formula with that input can answer it.

The ratio does not improve with size either — it worsens. Graphs admitted per mechanism run 1.00, 2.50, 4.44 and 8.17 across the four census rows, roughly doubling each time, so the fourth condition does more of the work the larger the census gets.

Nor is the rejection a rare accident of arrangement. Four of the eleven arithmetically admissible ten-link assortments contain no mechanism at all — seventy-eight graphs between them, every one satisfying every count, every one carrying a structure — so the failure is systematic rather than incidental. Those four are exactly the assortments needing a link with six pins or more, and the consequence is a rule a designer could use directly: a ten-link mechanism of one degree of freedom has no link carrying more than five pins, which no arithmetic produces and an exhaustion does.

Two: if the count is unsure, take the rank

The rank is unsure about a different thing.

The site’s standing habit is that a count and a Jacobian rank are two routes to one number, and the habit has caught real failures — a mechanism the count says cannot move is caught by the rank immediately.

On a graph with a rigid triangle in it the rank returns one, which is what the count returns and is the truth about the assembly. Three bodies and three pins is nine coordinates and six independent equations, so the triangle removes exactly the three freedoms it is supposed to. There is no rank defect to detect.

Across the eight-link census the correspondence is exact and it is the useful form of the statement: all 46 graphs whose worst subchain sits at nought measure mobility one; all nine whose worst subchain sits below nought measure two. The rank is a perfect detector of over-constraint and a perfectly blind one for rigidity.

So the two routes this site has checked everything with are unanimous about 1,165 of the 1,878 ten-link graphs, and they are unanimous and correct about a number that answers a different question from the one being asked. What answers it is a third instrument the site did not have: a mobility count run over every subset of the links rather than over the whole — 1,024 small pieces of arithmetic on a ten-link graph, exhaustive, with nothing to tune. It also returns a set of links rather than a number, which is why it can be shaded in a picture and a rank cannot.

3 of these 6 links never move relative to one another. A graph that passes every arithmetic test and is not a mechanism of 6 links. The shaded links are a subchain that is already a structure: 3 links held by 3 pins, whose own count is 3 × 2 − 2 × 3 = 0. Grübler cannot see it, because the formula reads two totals for the whole graph and this is a statement about a subset. Neither can the rank. A rigid triangle removes exactly the freedoms the count says it does, so the constraint Jacobian is not deficient, the measured mobility is 1, and the two routes this site checks everything with agree — with each other and with the wrong answer. The assembly moves, and it moves as a mechanism with 4 links, one of which happens to be welded out of 3 pieces.
Fig. 2 The third instrument’s output. Not a number, but the particular links that never move relative to one another.

Three: matching characteristic polynomials means the same chain

At six and eight links it does. At ten links there are two pairs where it does not.

The polynomial is a genuinely good fingerprint and the argument for one direction is airtight: relabelling permutes rows and columns alike, which is a similarity, and a similarity does not move an eigenvalue. So a difference is a proof of difference, always.

A fingerprint that costs nothing and is almost always right. The characteristic polynomial of each chain's adjacency matrix, coefficient by coefficient. Isomorphic chains have identical polynomials — relabelling the links is a permutation similarity and a similarity does not move an eigenvalue — so a difference anywhere in this column is proof that two chains are different, obtained without searching over a single relabelling. Watt's and Stephenson's first differ at λ^2, which is where a triangle-free graph's polynomial first notices how many four-cycles it has. The coefficients are computed on the integers by the Faddeev–LeVerrier recursion and checked against the eigenvalues of the same matrix from an unrelated Jacobi routine, which agree to 3 × 10⁻¹³.
Fig. 3 The two six-link chains, coefficient by coefficient. They differ at λ², which is proof enough and costs a determinant.

The converse is what a test needs and it was never established, which is the whole of that rung. Among the 230 ten-link chains, two pairs share a polynomial in all eleven coefficients. On the first of those pairs, the two chains do not even have the same assortment — six binary, two ternary and two quaternary against four binary and six ternary — so counting the ternary links separates them and the spectrum does not.

Different numbers of ternary links, and the same spectrum. Two of the 230 ten-link chains whose adjacency matrices have identical characteristic polynomials — identical in every one of the eleven coefficients — and which are not the same chain. They do not even share their assortment — 6×2 + 2×3 + 2×4 on the left and 4×2 + 6×3 on the right. Counting the ternary links tells them apart and the spectrum does not. That is worth pausing on: the spectrum is the more sophisticated invariant, it is the one that got written into the literature as a test, and here it is beaten by the first thing anybody would try. The polynomial both of them have is λ^10 − 13λ^8 + 52λ^6 − 4λ^5 − 76λ^4 + 8λ^3 + 32λ^2.
Fig. 4 The pair where the cheapest invariant in the subject beats the sophisticated one.

The instructive part is the size at which it broke. Six links: two chains, checkable on a page. Eight links: sixteen chains, 120 pairs, checkable by somebody patient. Ten links: 230 chains, 26,335 pairs, and the first counterexample.

Four: there are five six-bar mechanisms, and a longer list further on

The five is exactly right and it is the last size at which a remembered list is a complete one.

Two six-link chains, five orbits of links between them, five six-bar mechanisms — the two named for Watt and the three for Stephenson. Nothing is missing and nothing ever will be.

The five six-bar mechanisms, and there are only two chains. Two chains and five machines. Watt's chain has two orbits of links, so grounding it gives two mechanisms; Stephenson's has three. The frame is drawn dark in each. This is the whole of what "Watt I", "Watt II", "Stephenson I, II and III" name — not five linkages somebody invented, but two graphs and the five genuinely different links there are to bolt down. Anyone who has met the names as a list of five things has met the answer without the question, and the question is a count of orbits.
Fig. 5 The five. Two chains, five choices of frame, and every planar six-link mechanism of one degree of freedom.

At eight links there are 16 chains and 71 mechanisms, and none of them has a name. At ten there are 230 chains and 1,834 mechanisms. A designer working from a remembered repertoire is working from perhaps fifteen arrangements, which is complete at six links and is 20 per cent of eight links and 1 per cent of ten.

The failure this produces is invisible from inside, which is why it is on this list. Nothing about consulting a memory reports how large the unexamined remainder is, and the habit is reinforced by being exactly correct at the size where it is learned.

There is a second and quieter version of the same thing. The constructive habit — take a four-bar, hang a dyad off a coupler point, repeat — always produces a mechanism and always produces a dyadic one, so the four hard eight-link chains and the ninety hard ten-link ones are not merely unfamiliar to a designer using it. They are unreachable, and a method that cannot reach part of a space cannot report that the part is there.

Five: the solver is a convenience

On four of the sixteen eight-link chains it is the only route, and at ten links on ninety of the 230.

Hold a link, drive a neighbour, and the rest of a mechanism comes apart into the smallest sets that can be positioned one after another. A set of two links is two circles meeting — a quadratic, a closed form, two branches, and a compass. A set of four or more is a system that must be solved as a whole.

A dyad is two circles, and that is why it has two answers. The whole of what makes a dyad easy. Two links, three pins: one pin onto something already placed at each end, one pin between them. The free pin is at a fixed distance from each of the placed ones, so it lies on both circles — and two circles meet in two points, in nought, or in one. That is a quadratic with a closed form, and its two roots are the two assembly branches every four-bar on this site has. A group of four links has no such picture: its unknowns do not separate into one circle each, the system does not factor, and what replaces the compass is Newton's method from a seed. positioned by solving, not by drawing.
Fig. 6 The construction the whole distinction turns on: two links, three pins, and the free pin on both circles.

Twelve of the sixteen eight-link chains have at least one way of being driven that comes apart entirely into pairs. Four have none — twenty frame-and-input choices each, eighty between them, and not one of the eighty works.

That is not a statement about difficulty. It is a statement that no construction exists, permanently and for reasons no set of dimensions repairs — so a site that had chosen constructions over a solver would have had nothing to draw for those four.

It also comes with a prediction the graph makes about a count of configurations, which is the pleasant half of the same argument. A chain that does come apart entirely into pairs assembles in two to the power of the number of them — four for either six-bar — and the count from 240 random seeds agrees. That prediction is how a defect in the solver was found: Watt’s chain came back at eight, and the four extra solutions had its ternary link mirrored, because three distances fix a triangle only up to reflection.

Six: a census of chains is a catalogue of machines

It is a catalogue of the things machines are made of, and it decides nothing about any machine.

Every number in this field survives multiplying every link by a different scale factor, because there are no lengths to scale. So the census says nothing about whether a crank turns fully, what the coupler traces, whether the mechanism assembles at all, where the singularities are, or whether it is any good.

How far the driven link turns is a fact about the lengths, not the chain. Each of the sixteen eight-link chains, given the arbitrary placement its own layout produces, driven from its first available choice of frame and input, and swept until a frame stops closing. 10 of the sixteen reach every angle and the rest rock through between 107° and 244°. Nothing in this chart is a property of the chains. Change the placement and the bars change; the census above them does not. It is here because it is the sharpest way to say what this field does and does not decide, and because the temptation to read a topology census as a catalogue of machines is exactly the mistake it prevents.
Fig. 7 The sixteen chains, each given one arbitrary set of dimensions and driven until a frame stops closing. Ten reach a full turn and six do not — and none of that is a fact about the chains.

That chart is in the field precisely because it looks like a result about the census and is a result about one layout algorithm. Change any pin position and every bar in it moves; the census above it does not.

The separating test is a clean one. Multiply every link by a different scale factor — not the whole mechanism by one factor, each link by its own. Every quantity in this field is unchanged, because there was nothing to multiply; every quantity a designer cares about is different, which is the boundary that rung is about. That is the same shape of boundary the holding field draws against force, where every number survives with every force in the assembly unknown.

What the graph does decide is a real list — whether it is a mechanism at all, how many mechanisms it gives, how many ways there are to drive it, whether it has a closed form, how many assemblies a dyadic decomposition has, how many independent loops there are and how short they can be. That is more than a reader would guess and it is not a machine.

Two of those are worth marking as surprising, because they are things a designer would assume are dimensional and are not. The number of assembly branches is topological: a Watt six-bar has four at a given input angle whatever its lengths are, provided it assembles. And the smallest number of link dimensions any tolerance stack-up can involve is topological too: twelve to fifteen across the sixteen eight-link chains, decided before a single dimension is chosen.

A repertoire is not a sample

The remedy is worth stating as precisely as the failures were, because test rules on harder inputs is advice nobody can act on and there is a specific thing that was done here instead.

Every one of the five shared failures is a sampling fault, and the sample has a name. A repertoire is the set of mechanisms somebody has built: four-bars, slider-cranks, the five six-bars, a Geneva, a scissor lift. It is not a random draw from the space of chains and it is not a small draw either — it is the image of that space under a filter, and the filter is somebody found this useful and it worked. A rule tested on a repertoire is tested on inputs pre-selected to satisfy it, because a graph with a rigid triangle hidden in it never became a mechanism anybody built, and so never became an input to anybody’s test.

A census is the opposite object. It is the set of chains that satisfy a stated arithmetic condition, generated exhaustively, with no filter for usefulness and none for having ever been built. Almost everything in it is useless — that is not a defect, it is the definition — and the useless members are precisely where a rule shaped by practice has never been asked anything.

That is the whole of the method, and it is why five separate confident claims fell to the same instrument within one field. Nothing cleverer was applied to any of them. The census was generated, the old rules were run over it, and the disagreements were counted.

The limit is worth stating in the same breath, because it is not comfortable. The census exists up to ten links, and every number on this list is a number about chains of ten links or fewer. The six claims are refuted at those sizes and untested at twelve, and there is no reason to expect the ratios above to hold there — the one trend that can be read, the eight-to-one gap between graphs that pass the count and graphs that are mechanisms, was worsening with size rather than settling. So the honest position is that this list is a list of six things that are false, and not a list of six things whose falsity is understood well enough to extrapolate.

There is a second limit, and it is the one that would matter most to somebody wanting to use this. A census refutes; it does not construct. Knowing that 1,648 of 1,878 ten-link graphs contain a structure says nothing about how to generate the 230 that do not, other than by generating all of them and testing. Every result on this list is of the form this rule admits things it should not, and none is of the form here is a rule that admits exactly the right things. The subset scan comes closest and it is a test rather than a construction.

Which is the right note to end a field of wrong things on. The instrument that found all six is exhaustive enumeration, exhaustive enumeration is available at these sizes and not at larger ones, and what it produces is a reliable supply of counterexamples rather than a replacement theory. That is less than a reader might want and it is a great deal more than the five claims had behind them.

What the six have in common

Five of the six are the same failure wearing different clothes, and it is worth naming because it is the one this whole field is about.

A rule was tested only on inputs already known to satisfy it. Grübler’s count was applied to declared topologies, which come from mechanisms somebody built and therefore have no hidden triangles. The spectral test was checked on the six- and eight-link censuses, which have no cospectral pairs. The five six-bars were checked against everything anybody could name, and everything anybody could name is one of the five. The compass construction was checked on four-bars and six-bars, all of which are dyadic.

In each case the rule is correct on everything it was shown, and the set it was shown was chosen by the same process that guarantees the rule works. A necessary condition applied only to things known to satisfy the sufficient one never fails; a test checked on a census with no counterexamples in it never collides; a repertoire compared against itself is always complete.

The way out is not better rules. It is generating the inputs rather than choosing them, which is what a census does and is the only reason any of the six is visible here. Nothing above required a new idea about mechanisms; it required handing existing ideas a set of cases nobody selected.

The search generates 3,000 candidates for 1,878 answers. How much work the enumeration does, against how much it has to show for it. The upper line is the number of complete labelled graphs the search reaches and the lower is the number of distinct graphs they turn out to be, so the vertical gap is waste — every candidate above the lower line is a graph the search had already found under a different labelling. At eight links the unpruned version of this search generated 8,494 candidates for the same 71 answers, and at ten links it did not finish at all; with the pruning it generates 3,000 for 1,878 in 442 milliseconds. The rule that does it is one line long: when two links carry the same number of pins, reject the labelling that would be lexicographically smaller if they were swapped. It cannot reject a labelling that is the largest in its class, so nothing is lost, and it is not a complete test, which is why the canonical form is still taken at the end.
Fig. 8 Where the unselected cases come from: a search that reaches 3,000 candidates and returns 1,878 graphs, of which 1,648 are cases nobody would ever have drawn.

The sixth item is different in kind and belongs on the list for the opposite reason. It is not a rule that fails on unfamiliar input; it is a boundary that the field’s own figures make it easy to forget, since twenty pictures of driven mechanisms invite the reading that the subject is mechanisms. The correction there is repetition rather than a measurement, and every caption in the field carries it.

Where this goes next

One, two, sixteen, two hundred and thirty. Every planar chain of mobility one, up to ten links, counted by enumeration rather than quoted. The pins column is forced: a chain of 10 links has one degree of freedom only if it has exactly (3n − 4)/2 pins, which is why no odd link count appears. Pass the count is how many graphs satisfy Grübler's rule, are connected, are simple and give every link at least two pins. Are chains is how many of those survive the fourth condition, that no proper subchain is already a structure — and the gap between the two columns is the whole of this field's first argument: at ten links 1,878 graphs pass a rule that 230 of them deserve. Mechanisms is larger again, because a chain is not a mechanism until a link is held still, and how many different mechanisms that gives is a question about the chain's own symmetry. The mechanism is the graph Twenty-one fields of this site have been handed a mechanism and asked what it does. Take the mechanism away and keep only which link is pinned to which, and there is still a finite list of answers: one chain of four links, two of six, sixteen of eight, two hundred and thirty of ten — and 1,878 graphs at ten links that pass every count and are not among them. Three verdicts, and only one instrument can give all three. Every 10-link graph that satisfies Grübler's count, split by what is actually true of it. 230 are mechanisms with 10 links. 1,165 carry a subchain whose own count is exactly nought — and neither of the two standing routes can see them: the count returns one and the rank returns one, and both are right, because a rigid subchain removes exactly the freedoms it is supposed to. What is false is the description. 483 carry a subchain whose count is below nought, and those the rank does catch: the surplus pins repeat a constraint already imposed, the Jacobian loses rank, and the measured mobility comes out above the count. The third instrument — a count run over every subset of the links — is the only one that answers the question at all. What a count cannot see At ten links, 1,878 graphs satisfy Grübler's rule and 230 are mechanisms. The other 1,648 contain a subchain that is already a structure — and on 1,165 of them the count says one degree of freedom, the rank of the constraint Jacobian says one degree of freedom, and both are right about a mechanism that does not have ten links. Different numbers of ternary links, and the same spectrum. Two of the 230 ten-link chains whose adjacency matrices have identical characteristic polynomials — identical in every one of the eleven coefficients — and which are not the same chain. They do not even share their assortment — 6×2 + 2×3 + 2×4 on the left and 4×2 + 6×3 on the right. Counting the ternary links tells them apart and the spectrum does not. That is worth pausing on: the spectrum is the more sophisticated invariant, it is the one that got written into the literature as a test, and here it is beaten by the first thing anybody would try. The polynomial both of them have is λ^10 − 13λ^8 + 52λ^6 − 4λ^5 − 76λ^4 + 8λ^3 + 32λ^2. Right until the size nobody checked The characteristic polynomial of a chain's adjacency matrix is a fingerprint that costs nothing and separates every six-link chain and every eight-link one. At ten links it fails on two pairs — and on one of them, counting the ternary links tells the two chains apart while the polynomial does not.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Assur groupCanonical formCospectralDegenerate chainDesign ruleGraph isomorphismGrübler's criterionInversionKinematic chainType synthesis