Machine Kinematics
solved, then drawn
Concept

Canonical form — where it appears

A labelling of a graph chosen by a rule that reads the graph and never the labels it arrived with, together with the adjacency string that labelling produces. Two chains are the same chain exactly when their canonical forms match, which turns a search over relabellings into a string comparison and has no counterexamples by construction.

Named by 15 essays across 6 fields — each of them below, with the objects they name alongside it.

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.

topology · Topology Colour by degree, recolour by neighbours' colours, stop when nothing changes. The cheap half of every isomorphism routine there is, and the half that does most of the work. Start by colouring each link with how many pins it carries. Then repeatedly recolour it with its own colour plus the multiset of its neighbours', until a pass changes nothing. On this chain the process ends with 3 classes of sizes 2, 2, 2, and two links of different colours are certainly different links — no relabelling can carry one to the other. What refinement cannot do is separate links that are alike to every local measurement, and that residue is what the backtracking search is for. It is also, exactly, why a spectral test fails: an eigenvalue is a global average over walks and has no more to say about two locally identical links than the refinement does.

Deciding that two chains are one

Two chains are the same chain when a relabelling of the links carries one to the other. Ten links admit 3,628,800 relabellings, and the census asks the question 26,335 times — so the answer is not a search but a rule that picks one labelling out of the graph itself, and asking whether the two strings match.

topology · Topology 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.

Which link to bolt down

A chain is not a machine until one of its links is held still, and which one is a decision. Two links give the same machine exactly when a relabelling of the whole chain carries one to the other — so the number of mechanisms a chain gives is a count of orbits, and the classical five six-bars and seventy-one eight-bars are that count.

topology · Topology Four of the sixteen cannot be positioned without a solver. For every chain, every way of choosing a frame and a driven link pinned to it, and for each the decomposition into Assur groups. All dyads counts the choices whose groups are all two links — those are the mechanisms a draughtsman can position with a compass, two circles at a time. The last column is the one that matters: chains for which no choice of frame and input is all dyads, so every way of driving them leaves a group of four or more links that has to be solved as a single system. At eight links there are 4 of them and at ten there are 90. This site has run a Newton solve on every mechanism it has ever drawn, and it has always been possible to read that as convenience. On these it is not.

Eight ways to drive it, and one machine

Bolting a link down is half the decision; the other half is which link carries the input. A four-bar has eight frame-and-input pairs and exactly one of them is a distinct machine — and across the eight-link census 320 listed pairs collapse to 153.

topology · Topology Two circles each, so two to the power of the dyads. A dyad has two solutions, and a chain whose groups are all dyads is solved one dyad at a time, so the number of ways it can be assembled at a given input angle is 2 raised to the number of them. That is a prediction made from the graph about a count of configurations, and it is checked here against a count: the same chain solved from 240 random seeds, with the distinct converged configurations counted. The two agree in every row. Watt's did not at first — it came back at eight — and every one of the four extra answers had its ternary link mirrored: three distances fix a triangle only up to reflection, so the distance equations admit a part that has been turned inside out. A reflected link is a different part rather than a different pose, and the solver refuses those frames now.

Two to the power of the dyads

How many ways a mechanism can be assembled at a given input angle is a count of configurations, and it is predicted here by a graph: two circles per pair of links, so two to the power of the number of pairs. The prediction came back four for Watt's chain and the count came back eight, and the four extra had a link turned inside out.

algebra · Algebra The four eight-link chains a compass cannot position. Every one of the twenty ways of choosing a frame and a driven link, on each of these four chains, leaves a group of four links or more that has to be solved as one system. There is no order in which they come apart two at a time, so there is no ruler-and-compass construction for any of them and no closed form for their positions. They are numbers 1, 3, 4, 10 of the sixteen, and they do not share an assortment: 4×2 + 4×3 and 5×2 + 2×3 + 1×4 both appear. Three of the four are among the most symmetric chains in the census — automorphism groups of 16, 8, 8 against a median of three across the sixteen — which is the direction one would guess, since a symmetric chain has few genuinely different places to attach a driven link. The fourth has an automorphism group of 2, so symmetry is a tendency here and not the reason.

Four that a compass cannot reach

Twelve of the sixteen eight-link chains can be positioned two links at a time, from at least one choice of frame and input. Four cannot be positioned that way from any of their twenty choices — and at ten links ninety of the two hundred and thirty are in the same position.

topology · Topology 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.

The candidates a search throws away

The obvious enumeration generates every labelling of every chain and keeps one. At eight links that is 8,494 complete graphs for 71 answers; at ten it does not finish. One rule — reject the labelling that a swap of two equal links would improve — takes it to 3,000 candidates for 1,878 answers in half a second, and twelve links is still out of reach.

topology · Topology A catalogue is a search space, and a requirement is a filter on it. What a census is for. Four requirements applied in turn to the 230 ten-link chains, each of them a statement about the graph alone: a link carrying four pins, a link none of whose neighbours is binary, and a way of driving it that comes apart into dyads. 26 chains survive all of them. None of this is dimensional synthesis and none of it can be — no requirement here mentions a length, an angle or a position, and every one of them can be checked before a single dimension is chosen. That is the argument for having the census at all: the design problem is a search over shapes within a topology, and knowing which topologies there are turns an open question into 26 closed ones.

Choosing the chain before the lengths

Every synthesis method on this site starts by assuming a topology, and the assumption is usually a habit. What the graph fixes before any dimension is chosen is the number of free parameters — two per pin less four — and therefore how many positions can be prescribed at all.

synthesis · Synthesis 11 assortments are arithmetically possible and 7 contain a mechanism. The 10-link census organised the way every published table organises it: by how many links carry two pins, three, four and more. The assortments themselves are a small piece of arithmetic — the degrees must sum to twice the pin count and none may be below two — and it admits 11 of them. 4 contain no chain at all. Each of those 4 needs a link carrying six, seven or eight pins, and a link with that many pins in a chain this small always drags a structure in with it: the graphs exist, they satisfy Grübler exactly, and every one of them has a rigid subchain. That is a result the arithmetic cannot reach, because the arithmetic never looks at where a pin goes.

Eleven assortments and four that are empty

How many links carry two pins, how many carry three, how many carry four: two lines of arithmetic admit eleven answers at ten links. Seventy-eight graphs have degrees the last four of them describe, every one of those graphs satisfies Grübler's rule exactly, and not one of them is a mechanism.

topology · Topology A catalogue is a search space, and a requirement is a filter on it. What a census is for. Four requirements applied in turn to the 230 ten-link chains, each of them a statement about the graph alone: a link carrying four pins, a link none of whose neighbours is binary, and a way of driving it that comes apart into dyads. 26 chains survive all of them. None of this is dimensional synthesis and none of it can be — no requirement here mentions a length, an angle or a position, and every one of them can be checked before a single dimension is chosen. That is the argument for having the census at all: the design problem is a search over shapes within a topology, and knowing which topologies there are turns an open question into 26 closed ones.

A catalogue is a search space

Dimensional synthesis searches over lengths within a topology, and the topology is chosen first — usually from memory, usually from a list of five. With a census the list is two hundred and thirty, every requirement that reads only the graph is a filter on it, and the choice stops being a habit.

topology · Topology 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.

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.

wrong · Misconception 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.

The chain has no lengths

Every number in this field survives multiplying every link by a different scale factor, because there are no lengths to scale. Which is also the statement of what a census cannot decide — and the sixteen eight-link chains, each given one arbitrary set of dimensions and driven, produce a chart in which nothing belongs to the chains.

topology · Topology Watt chain: 7 pins, and nothing else. A kinematic chain drawn as what it is — a graph. Each disc is a link and carries its number; each line is a pin joining two links. There are no lengths here, no angles and no positions, and every quantity this field computes survives moving any disc anywhere: the picture is a way of reading the graph and not a picture of a machine. The fill says how many pins a link carries — 4 binary, 2 ternary — which is the coarsest thing that can tell two chains apart and the first column of every census table. The count reads two numbers off this picture and nothing else: 6 links and 7 pins give 3 × 5 − 2 × 7 = 1.

A graph has no numbers at all

Every other field on this site has parameters a measurement could try to recover. This one has none. A chain is a graph, a graph is a set of links and a set of joints, and there is nothing about it that a scaling touches, a tolerance perturbs or an instrument determines.

topology · Topology The chains that never close, counted by the same routine. An open chain has one joint fewer than it has links, so its graph is a tree and its mobility is the joint count rather than one. Feeding the same enumerator that produced the closed-chain census — with the minimum-degree condition relaxed to one, because an arm's base and its end each carry a single joint — gives 1, 2, 3, 6, 11, 23, 47, 106 for three links up to ten. That is the number of unlabelled trees, a sequence anybody can look up, and reproducing it is the strongest check the enumerator gets: it was written for a different problem, tested against three mechanism counts, and asked here for a number from a different subject entirely.

An arm is a tree

The serial field's chains are the ones that never close, and as graphs they are trees. There are 106 distinct arrangements of ten links joined that way, and exactly one of them is the straight arm every essay in the field has drawn — the other 105 branch.

serial · Serial The same links and pins, and between 12 and 15 link lengths in the stack-up. Every closed loop in a mechanism is one equation a tolerance analysis has to satisfy, and the equation involves every link the loop passes through. All 16 chains here have the same number of independent loops — 3, which is pins minus links plus one and is fixed by the two totals — but not the same shortest set of them. The bars are the total length of a minimum cycle basis, and they run from 12 to 15. So the smallest number of link dimensions that any stack-up on this mechanism can involve is decided by the graph, before a single dimension has been chosen, and two topologies a count cannot tell apart differ by 3 of them.

Where the shortest loops are

A tolerance stack-up goes round a loop, and every link the loop passes through is a dimension in it. Two eight-link chains with the same links, the same pins and the same number of loops can need twelve link lengths in their shortest independent set or fifteen — decided by the graph, before any dimension is chosen.

practice · Tolerance

Named alongside it

The objects these essays reach for when they reach for this one.

Kinematic chainType synthesisInversionMobilityAssur groupGraph isomorphismLink assortmentAutomorphismDegenerate chainLoop closureOrbitAssembly branch

All concepts