Machine Kinematics
solved, then drawn
Series

Topology — the series

21 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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 collection 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.

    part 1 · topology
  2. 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.

    part 2 · topology
  3. 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.

    part 2 · topology
  4. 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.

    part 3 · topology
  5. 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.

    part 3 · topology
  6. 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.

    part 4 · topology
  7. 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.

    part 4 · topology
  8. 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 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.

    What has to be solved together

    Hold a link, turn a neighbour, and the rest of a mechanism comes apart into the smallest sets that can be positioned one after another. Every set of two links is two circles meeting — a quadratic, two branches, no solver. A set of four is a system, and the Newton solve stops being a convenience.

    part 5 · topology
  9. 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.

    part 5 · topology
  10. 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 469 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.

    part 6 · topology
  11. 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.

    part 6 · topology
  12. 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.

    part 7 · topology
  13. 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.

    part 7 · topology
  14. A machine with one thing in it that must be solved at once. Each compiled machine's joints, walked in the order they can be placed: a joint goes down as soon as two things already placed decide where it is. Every one of these machines unwinds completely, and each contains exactly one pair that has to be solved together — the arm and the parallelogram that carries its second angle back to the pivot. Nothing larger than a dyad appears in a machine of two hundred and forty joints. The topology field's four-bar, at four, has no such decomposition at all; a compiled linkage is enormous and structurally trivial, which are not the same axis.

    A machine with one dyad in it

    Two hundred and forty joints, and two hundred and thirty-eight of them can be placed one at a time from parts already positioned. The whole of what has to be solved simultaneously is a single pair — the arm and the parallelogram carrying its second angle home. Size and structural depth are different axes, and a compiled machine is extreme on one and trivial on the other.

    part 8 · topology
  15. 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.

    part 9 · topology
  16. A stop on the coupler, and the one direction in which it stops nothing. A four-bar — frame 4, crank 1.2, coupler 3.4, rocker 2.8, crank at 70° — whose coupler is stopped by a contact with the frame at (1.59, 2.16). The dashed line through the stop is its normal; the ringed point is the coupler's pole, where the crank's and rocker's lines meet, at (2.505, 6.883). With the normal 10° off the line to the pole the stop removes the four-bar's one freedom; through the pole it removes nothing, because the contact point is already moving along the stop's face. The graph of the stopped four-bar is the same in every case.

    A contact turns and slides at once

    A cam on its follower, a gear tooth or a pin in a slot takes one freedom away where a pin or a slide takes two, and it turns as well as slides. Give every joint of the four-bar, Watt's and Stephenson's chains a third label and there are 1,317 different chains. The rule that read slides off the graph extends to all of them, agreeing with the constraint rank on every one, and the two obvious shortcuts miss 36 and 769. What the graph cannot record is which way a contact's normal points — and on these chains that never matters, while a single stop added to a four-bar holds everywhere except through one point.

    part 10 · topology
  17. A seat round the stop's nose holds the four-bar if it is tighter than the nose centre's path allows. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at a crank angle of 70°, its coupler carrying a round nose of radius 0.3 centred at (1.593, 2.160). The nose sits in a hollow seat on the frame of radius 0.8, touching it at (1.650, 2.454), on the line from the nose's centre to the coupler's pole at (2.505, 6.883), off the top of the drawing. The dashed circle is where the nose's centre may go without the nose entering the seat: a disc of radius 0.50. The curve through the nose's centre is the path the four-bar gives it, with a radius of curvature of 0.893. The seat's disc is tighter than the path, so the path leaves it either way and the four-bar cannot turn: the stop holds. The threshold is a seat of radius 1.193, the path's radius plus the nose's.

    A stop aimed at its pole holds by curvature

    A stop whose normal passes through its pole forbids nothing at first order, and whether it holds at all is decided by curvature. For a round nose the deciding curve is not the one it touches: the seat that just holds is the path of the nose's centre moved out by the nose, 1.193 on a four-bar where the contact point's own circle says 0.776. Past it, one contact holds both ways — loosely, as the square root of its clearance.

    part 11 · topology
  18. The seat at the pole, and the friction cone at its contact. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at 70° of crank, its coupler's round nose of radius 0.3 in a seat of radius 0.8 on the frame, aimed through the coupler's pole. At the contact the shaded wedge is the friction cone for a coefficient of 0.1: every force the seat can press with lies within 5.7° of the normal. The normal itself runs through the pole, so it has no lever about it; the cone's edges do. Under a moment of 0.001 on the crank against a contact of stiffness 10⁶ the crank turns 1.70e-4 radians, against 1.48e-3 with no friction; released, it stays at 1.70e-4.

    A seat with friction stays where it is pushed

    A stop aimed through its pole holds a four-bar by curvature, because its normal has no lever about the pole. Friction has one, and the obvious guess is that it lets such a stop hold at first order. It does something narrower and stranger. It cannot hold what curvature releases, since a gap that opens carries no force for friction to be a share of. It makes a seat that holds up to twenty-six times stiffer, turning its cube-root give into a square root. And it makes any stop whose pole lies inside its friction cone stay where a load pushed it: released, the seat at the pole does not come back.

    part 12 · topology
  19. A V of two faces, each turned off the line to the pole. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at 70° of crank, its coupler's round nose of radius 0.3 resting in a V of two flat faces on the frame. Each face's normal is turned off the line from the nose to the coupler's pole so that at the contact it misses the pole by 12.79°, one each way, so that each face holds the crank one way at first order. The shaded wedges are the friction cones for a coefficient of 0.2, half-angle 11.3°: the pole lies outside each, so released from either side the crank springs back. Each face's moment per radian, through a contact of stiffness 10⁶, is 72 thousand.

    Return and stiffness come from the same aim

    A stop with friction looked like a trade: aim it near its pole and friction multiplies its stiffness seventeen times, aim it outside the friction cone and it springs back when released. The trade is an artefact of how the gain was counted. Friction's factor multiplies a stiffness that vanishes at the pole, and measured outright a stop aimed further away is stiffer at every friction — 271 per radian at 1.1° and 104,000 at 21.3° with none, 4,631 and 184,000 at a coefficient of 0.3. Return and stiffness are bought by the same aim. What the aim costs is the other side: a face aimed off its pole holds nothing the other way, so a stop that must return both ways is a V of two faces, and across a friction band from 0.1 to 0.2 a V aimed 12.3° each way returns from both and is 8,800 times stiffer than the seat at the pole.

    part 13 · topology
  20. A V carried on a slide and pressed onto the nose. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at 70° of crank, its coupler's round nose of radius 0.3 held in a V of two flat faces. The V rides on a slide along its own axis and a spring of force F presses it onto the nose, so it can be pushed back as the crank turns. Each face's normal is turned 45° off the V's axis, and misses the line to the coupler's pole at its contact by 47.6°. The shaded wedges are the friction cones for a coefficient of 0.2. Turning the crank past the V takes a moment of 1.39 F; released, the V pushes the crank back with 0.64 F.

    A V pressed onto its nose has a window

    A V of two faces holds a four-bar's coupler both ways and returns it, as long as each face is aimed off the coupler's pole by more than the friction angle. Squeeze the nose between both faces and two different things can happen. A rigid V cut a hundred-thousandth tight cannot lock, because the crank moves the nose across the V and never into it; it only stops short of home, inside a dead band that is μ/tan ψ of the play an equally loose fit would leave. A V pressed on by a spring, as a detent is, rides back along its own axis, and that is the direction a wedge locks in. It returns from an aim of arctan μ at the contact and locks at 90° less arctan μ about its axis, so its aim has a window, and the V that is most nearly reversible inside it is a right angle.

    part 14 · topology
  21. 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 of this collection 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.

    part 15 · topology

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