Machine Kinematics
solved, then drawn
Field

The chain before the lengths

Every field before this one is handed a mechanism and asked what it does. Here the connections are the unknown: which graphs of links and pins are mechanisms at all, how many there are, and which of them are the same mechanism drawn twice. The search space is finite and every quantity is a count — one four-link chain, two six-link, sixteen eight-link, two hundred and thirty at ten — and the counting rule everything else on the site rests on turns out to admit eight graphs for every one that deserves it.
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

Kinematic chains as graphs: the four conditions that make one, the census from four links to ten, and the gap between the graphs a count admits and the ones that are mechanisms.

 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

Why a count of anything cannot separate two chains: the six-link census, the assortment both of them have, and the first invariant that tells them apart.

 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

Degenerate chains: the subchain whose own count is nought, why neither the mobility formula nor the Jacobian rank can see it, and the third instrument that can.

 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 spectral isomorphism test for kinematic chains: why it is exact on every census small enough to check by hand, the two cospectral pairs at ten links, and what an eigenvalue cannot see.

 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

Canonical form for kinematic chains: refinement by neighbours' colours, backtracking on what refinement cannot split, and why the drawing of a chain has to be a function of its canonical form and nothing else.

 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

Inversion as an orbit count: why grounding different links gives different mechanisms, how many each chain gives, and where the classical numbers come from.

 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

Choosing the frame and the driven link as one decision: the orbit of an ordered pair, why the count is the denominator for every solvability statement, and the nine driven six-bars.

 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

Assur groups: the arithmetic that decides which links must be positioned together, why a pair of links is a compass construction, and how the decomposition is read off the graph.

 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

The chains with no closed-form driving choice: what the census says about them, why symmetry is a tendency rather than the reason, and what it means for a solver to be the only route.

 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

What a census costs: labelled graphs against distinct ones, the pruning rule that makes ten links possible, and an honest account of where the routine stops.

 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

The link assortment as an invariant: the arithmetic that produces it, how the census distributes across it, and the four ten-link assortments that contain no chain at all.

 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

What a complete census is for: filtering topologies by conditions that mention no dimension, and where the discrete part of a design problem ends.

 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

The boundary of the topology field: what a graph decides, what dimensions decide, and how the two are kept apart in a figure that has to draw both.

 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

Walking the graph of a compiled linkage: every joint placed in order except one dyad, and what that says about a mechanism whose size is enormous and whose topology is not.

 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

The one field on this site with no continuous parameters, what that means for measuring it, and where its counts get their evidence from instead.

 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

Slides placed on the graphs of the four-bar, Watt and Stephenson chains: the placements counted as orbits under the chain's symmetries, the double-slider's three classical inversions recovered and the slider-crank's four not, and a graph rule — pinless loops and orientation-locked groups — that predicts mobility and locked pins on all 102 distinct placements, checked against the Jacobian rank of each chain built at random.

 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

The four-bar, Watt and Stephenson chains with every joint a pin, a slide or a contact: the stopped four-bar and the one normal direction that stops nothing; the census of labellings, chains and mechanisms with the graph rule against the Jacobian rank and two shortcuts' misses; one joint as a slide, a contact and a pin; the relative freedoms the rest of a chain leaves each contact's two links, never one; four stops on one four-bar with their Kennedy poles; and how much a stop holds against the angle its normal misses the pole by.

 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 four-bar's coupler stopped against the frame by a round nose aimed through the coupler's pole: seats of several radii and the gap's second derivative; the threshold seat against the nose centre's path and the contact point's osculating circle, at seven nose radii, both sides and six crank angles; the cubic at the threshold; play against clearance and deflection against load, against a stop aimed off its pole.

 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

The pole-aimed seat and stops aimed off their pole, with Coulomb friction at the contact: the friction cone at the seat's contact; the seat's turn under load at four frictions, measured and against a cubic; load-and-release cycles for the seat and a stop just off its pole at frictions either side of its cone; which of seven off-pole stops lock at five frictions; and the turn of seats either side of the threshold with and without friction.

 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 four-bar's coupler nose in a V of two flat faces turned off the line to the pole, with their friction cones; the moment per radian of a flat stop against its aim at four frictions, measured and in closed form; the share of a pushed turn each stop keeps when released, against its aim at three frictions; the free turn the other way of a seat aimed off its pole; the turn under load either way for the V and the seat at the pole; and a V designed to return across four friction bands.

 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 four-bar's coupler nose in a V of two flat faces, rigid and cut tight, or carried on a slide and pressed on by a spring: the sprung V on the four-bar with its friction cones; the moment to carry the crank past it and the moment that brings it back, against the aim, at three frictions; the window of aims against the friction coefficient, with the limits measured by bisection; the ratio of returning to holding moment inside the window; the tight rigid V's loop of moment against turn; and the free play of a loose V against the dead band of a tight one.

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