Machine Kinematics
solved, then drawn
Concept

Automorphism — where it appears

A relabelling of a chain's links that leaves every joint between the same two links, so that the chain is carried to itself. Its importance is that two links related by one give the same mechanism when grounded, so the number of mechanisms a chain gives is a count of orbits rather than a count of links.

Named by 8 essays across one field — each of them below, with the objects they name alongside it.

Same links, same pins, different chains. Watt chain on the left and Stephenson chain on the right. They have the same number of links, the same number of pins and the same assortment — 4×2 + 2×3 — so no count of anything can tell them apart. What differs is where the pins go: on the left the two ternary links share a pin, on the right they do not, and that single fact makes two mechanisms with different coupler curves, different numbers of inversions and different position problems. It is the smallest case in the subject of the thing this field exists to say: the arithmetic is a filter and the graph is the answer.

Same links, same pins, different machines

Watt's six-bar and Stephenson's have six links, seven pins, four binary links and two ternary ones. Every count anybody can make on them agrees. They are different chains, they give two mechanisms and three, and the difference is whether the two ternary links share a pin.

topology · Topology 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 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 Watt chain with 1 slide: 3 chains, 11 mechanisms. The same 6 links and 7 joints with 1 of the joints made a slide instead of a pin, drawn as a block astride the line. There are 7 ways to choose the joint, and the chain's 4 symmetries fold them into 3 that are genuinely different: with the slide at 0–3, 2 mechanisms; with the slide at 0–1, 6 mechanisms; with the slide at 1–2, 3 mechanisms. The mechanism count is the number of orbits of a held link and the slide set together, so a slide breaks symmetry the pin-only chain had, and links that gave one machine between them give two. The pin-only chain gave 2; one slide gives 11.

A slide turns nothing

Make one joint of a chain a slide instead of a pin and the graph has a second decision in it before any length exists. The symmetries that counted mechanisms count these too — Watt's chain with one slide is three chains and eleven machines — and two facts read off the graph say which placements still work: a loop of slides alone is freer than the count, and a pin in a group of links the slides hold at one orientation cannot turn. Across 102 placements on the three smallest chains, both agree with the rank of the constraint Jacobian.

topology · Topology

Named alongside it

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

Kinematic chainInversionType synthesisCanonical formGraph isomorphismLink assortmentOrbitMobilityAssur groupDegenerate chainFrameLoop closure

All concepts