Machine Kinematics
solved, then drawn
Theme

The thread: The crank is the argument

A mechanism is not a state, it is a relation between an input and an output. The figures are draggable because turning the crank is the explanation rather than a decoration on it.
Six assemblies, one routine, three disagreements and one accident. Every row is the same three steps: write down the constraint Jacobian, take its rank, and subtract it from the number of unknowns. The representations differ — bars between points, bodies joined by pins, panels joined by creases, one cell of a pattern that repeats for ever — and the routine does not. The counted column is the arithmetic on the numbers of bodies and joints; the measured column is the nullity of the matrix. They agree on the lazy tong and on the kagome cell and disagree on the other four, most sharply on the deployable ring, which the count declares immobile and which is sold as a mechanism that opens. The right-hand column is the reason: constraints that repeat what another constraint has already said, which the count has no way of seeing and the rank cannot help seeing. The fourth row is worth reading twice: the count says nothing can move and nothing can, so the two agree — and they agree for the wrong reason, because that pattern's flat state shows four freedoms and not one of them is a motion. Many of one thing

Many loops, one freedom

A scissor lift, a folded sheet and a deployable ring are one small unit repeated thirty times, and three things change at once: the count of bodies becomes a parameter, mobility becomes the rank of a matrix, and a unit that moves can be rigid the moment it is joined to another of itself.

 elbow arm at a posture. elbow arm, drawn from 6 joint values through a product of 6 exponentials — no equation is solved anywhere in this picture, because an open chain has none to solve. The thin lines are the joint axes at this configuration, which are also the columns of the arm's Jacobian: the tool's velocity is a sum of turns about exactly those lines. Here the smallest singular value of that Jacobian is 0.3674 and the largest is 2.407, so the arm is comfortably away from a configuration where a direction of motion is lost. Drag θ₂ shoulder. One path to the tool

The chain that does not close

Every mechanism on this site so far has been a loop, and a loop is why a configuration here is a solve. An arm has no loop. Its pose is a product of six transforms, evaluated, with nothing to converge and nothing to refuse — and the difficulty does not disappear, it moves to the other end of the problem.

 A four-bar at 60°, solved. Ground 4, crank 1, coupler 3.5, rocker 3. Every joint position here is the output of a Newton–Raphson solve on the loop-closure equations, converged to 0.0e+0 — not a placement that looked right. Grashof's condition classifies these lengths as a crank rocker, and sweeping the crank through 360° confirms it: 120 of 120 positions assemble. The transmission angle at this instant is 66.9°. Linkages

Four bars and four pins

The smallest interesting machine there is. Four lengths decide everything about it — which link can turn all the way round, how hard it pushes, where it stops and whether it can be assembled at all — and every one of those is a number that falls out of a solve rather than a judgement about a drawing.

 The coupler's motion at 66°. Every point drawn as a stub is a point of the coupler's own plane, and the stub is that point's velocity — solved, not sketched. They all point different ways and they are all consistent with one statement: at this instant the whole plane is turning about a single point, the pole, marked with a cross. It is off this frame at 1.5 coupler lengths from the crank pin, which happens whenever the coupler is close to translating. The arrow through it is the pole's own velocity, which is a quantity about the motion rather than about any point of it, and half of everything in this field follows from its direction and its size. positioned by solving, not by drawing. The motion, not the mechanism

The mechanism drops out

Every other field here is about a machine. This one is about the motion a machine makes — a plane sliding over a plane — and near any instant that motion is a handful of numbers with no linkage in them. Two mechanisms that agree on those numbers make the same motion, and one of them can always be thrown away.

 One contact, and the point the normal has to pass through. Two wheels on fixed centres, turning in the ratio 24 : 36, with one flank of each drawn. The contact is found by solving n·(v₁ − v₂) = 0 along the first flank — the two velocities are formed from the two rotations and subtracted, and nothing in that calculation knows where the pitch point is. The pitch point, marked with a cross, is computed separately as the one place where the two bodies' material points have the same velocity. The common normal misses it by 8.44e-15 of a millimetre, which is the law of gearing arriving as a measurement rather than as an assumption. positioned by solving, not by drawing. The shape is the unknown

The second shape is not a choice

Two bodies on fixed centres, told to stay in contact. Give one of them a shape and the other one's shape is no longer available to be designed — it is the envelope of the first one's positions, there is exactly one of it, and one routine computes it for a gear, a cam and a rotary engine alike.

 A bevel differential, turning. cage in, hold left, with every member's speed taken from the train's null space and every angular position that speed integrated. The teeth are marked at the pitch points rather than cut as involutes — the flank is the teeth field's subject — but the count is the tooth count and the positions are the solved ones, so what turns and how fast is real. Drag it and watch which way each member goes: left 0.000 · right 2.000 · cage 1.000. More than one input

Two inputs and one output

Every mechanism in this collection so far has had one input, and its output has been a function of that input. A differential does not. Its cage turns at the mean of two wheels, so knowing one of them tells you nothing at all about where the third shaft is going — and that is not a complication of the mechanism, it is a different kind of object.

 Four legs that do not cancel. Drive forward, turn, drive back, turn back — each leg exactly as long as the one it is undoing. The mechanism does not come home. What is left over is 1149.9 mm at an amplitude of 1.10, and it points along the direction the wheel forbids. The gap and the computed bracket are 31.51° apart here and 3.15° apart at a tenth of this amplitude — the agreement is a leading-order statement and the departure is the third-order remainder, which falls with the manoeuvre rather than staying put. Every point on the path was reached by a permitted velocity, so nothing here cheats; the sideways motion is assembled out of motions that are not sideways. Wheels, and where they may not go

The motion left over by going nowhere

Drive forward, turn, drive back the same distance, turn back the same angle. Every leg is undone by another leg and the mechanism does not come home — it has moved sideways, in the one direction it is forbidden to move in. The leftover has a name, a formula, and a measured exponent of 1.997.

 What a Simpson gearset can be made to do. Every distinct ratio the gearset offers, as a reduction. Each one is the same mechanism with one more constraint imposed — a brake holding a member to the case or a clutch locking two members together — and the input is a choice as well, which is why a four-speed needs an input clutch rather than four brakes. The shaded rows are the 4 a real transmission on this gearset is sold with; the rest are ratios the mechanism has and the gearbox does not buy the elements to reach. Every value is exact: the reductions are ratios of integers and are printed as such. More than one input

Holding a member chooses the ratio

A gearset offers a plane of motions and a shift element is one linear condition, so a gear is a line in that plane. Enumerate every brake and every clutch and a Ravigneaux's twenty-seven combinations collapse to seven ratios — eighteen of them being the same gear, because locking any two members at all locks the whole gearset solid. Neutral is not one of the seven, because neutral is not a gear.

 Six sizes, one freedom, and a count going the other way. Creases and constraints both grow as the square of the sheet's side, and they grow at different rates: two per panel against three per interior vertex. So the counted column runs (n − 1)(3 − n) and is positive at two, nought at three and increasingly negative after that, while the measured mobility is one on every row. The redundant column is the difference and it is exactly (n − 2)² — one at three, four at four, nine at five, thirty-six at eight. A twelve-by-twelve sheet of a hundred and forty-four panels is counted at minus ninety-nine and has a hundred repeated constraints, and it is the same mechanism as the smallest one on this table. Many of one thing

The freedom that survives repetition

A Miura sheet has one freedom at four panels and one at a hundred and forty-four, and the count runs the other way: plus one, then nought, then minus three, minus fifteen, minus ninety-nine. The gap between them is exactly (n − 2) squared, which is a hundred repeated constraints on a sheet with one degree of freedom.

 Slider-crank at 50°. Crank 1, connecting rod 3. The slider's travel is 2.0000 — exactly twice the crank throw, which is the one thing about this mechanism that does not depend on the rod length. Everything else does: the rod length decides how far the piston's motion departs from a sine wave, and that departure is the second harmonic every engine balancer has to deal with. Linkages

The slider-crank

Replace one pin of a four-bar with a slide and you get the mechanism in every reciprocating engine ever built. Its stroke is exactly twice the crank throw and does not depend on the connecting rod at all. Everything else about the motion depends on the rod, including the part that is always described as a sine wave and is not.

 How much of a compiled machine is computing anything. Each machine's bars split two ways: the ones that build an angle — reflectors, means, rigid offsets, the arm — and the ones that carry a direction from where it was computed to where it is needed. On the smallest machines the arithmetic is nearly all of it. By the quintic the carrying is 80 per cent, and it goes on rising, because the arithmetic grows with the number of terms and the carrying grows with the number of pairs of them. That is the answer to why a universality construction is enormous, and it is not about the algebra being hard. The curve as an equation

A parallelogram carries an angle, and only so far

A direction computed at the frame is no use where it is needed. A parallelogram carries one from point to point — but only between two points a fixed distance apart, and that single proviso is what makes a compiled machine quadratic in the number of terms and turns most of it into transport.

 The parking shuffle, three times over. Forward on left lock, forward on right lock, back on left lock, back on right lock — and repeat. The heading comes back to where it started and so does the position along the road, both exactly and at every leg length, because the four legs are a symmetric set. What is left is 255 mm of pure sideways translation per cycle at a leg of 0.80 m, which is exactly 4R sin(φ) tan(φ/2) with φ = s/R. Halve the leg and it quarters. Wheels, and where they may not go

Parking is an exponent

Four legs — forward on left lock, forward on right lock, back on left lock, back on right lock — return a car to its own heading and to its own place along the road, exactly, and move it sideways by 4R sin φ tan(φ/2). Halve the room and the gain quarters, so the number of shuffles goes up by four and the distance driven doubles.

 Where the output stops, and why the return is quicker. A crank-rocker's output reaches an extreme exactly when the crank and coupler line up — stretched out, so O₂ to B is 4.50, or folded back, so it is 2.50. Nothing about the rocker enters the condition, which is why the limits can be written down rather than searched for. Those two crank angles are 40.8° and 228.5°, so the crank spends 187.7° going one way and 172.3° coming back while the rocker covers the same 40.0° both times. The ratio is 1.0894 predicted and 1.0894 measured over 7200 swept positions — a shaper cuts on the slow stroke and returns on the fast one, and this number is what the proportions are chosen to get. Linkages

The return stroke is quicker

A crank-rocker's output stops at two definite places, and the crank angles at which it does are calculable without touching a solver. The interesting number is not where they are but how far apart — because the crank turns at a constant speed and the output covers the same swing twice in unequal times.

 The wheel is the coupler — double wishbone. The suspension solved at 0 mm of bump, with the whole travel ghosted behind it. The two arms are the cranks and the upright between them is the coupler; the wheel is bolted to that coupler, so camber is the coupler's rotation and nothing else. Camber here is 0.00° and the contact patch has moved 0.0 mm across the road. The cross is the instantaneous centre of the upright, found from the solved velocity field; the roll centre is where the line from it to the contact patch crosses the car's centreline, and it is at 73 mm here. Machines you have met

The wheel is the coupler

A double wishbone is a four-bar standing on end whose coupler carries a wheel, so camber is coupler rotation and scrub is a coupler point's path. Both are computable, and the second one comes out with the opposite sign from the model every suspension book uses — by more than the whole scrub.

 Three bars, and no rotation left. Each of a delta robot's legs ends in a parallelogram, which keeps the bar on the platform parallel to the bar on the arm. That leg therefore permits the platform no turn about either direction perpendicular to its bar: it imposes two couple constraints, drawn here as rings about the directions they act on. Three legs impose 6; together they span only 3, so 3 are redundant. What is reciprocal to them is three couples — three pure translations, and the platform cannot turn at all. Several legs, one platform

Why the platform stays flat

A delta robot has three legs and three freedoms, and there is no obvious reason those freedoms should be the three translations rather than some mixture. The reason is a parallelogram in each leg, and the argument from there to "the platform cannot turn at all" is a constraint computation that takes six wrenches and a rank.

 What the count is right about. Freedoms minus dependencies, against what the count predicts, on seven assemblies from three different representations. The difference is exact every time and it is exact for a reason that has nothing to do with mechanisms: the count is unknowns minus constraints, the rank is a number no larger than either, and the two nullities are what each of them has left over. So a count is not wrong in the way a mismeasurement is wrong. It is a statement about a difference being read as a statement about one of the terms — and on four of these seven rows both terms are large and the difference is nearly meaningless. Many of one thing

A constraint that has been said already

Every constraint matrix leaves two null spaces, and a mechanism only lives in one of them. The other is the set of combinations of constraints that come to nothing — and its dimension is exactly the amount by which the count is wrong, on a deployable ring, a Miura sheet and a framework with twelve bars and six joints.

 What it costs to multiply an angle. Multiplying an angle by n is done by doubling and adding — 2k is one reflector applied to k, and 2k+1 is one addition of k's result and the original — so the cost follows the binary expansion of n and not n itself. Eight costs three doublings; seven costs six gadgets, twice as much, for a smaller number. The other line is what a chain of n−1 additions would cost. The difference matters because the highest multiple a curve of degree d needs is d, so the arithmetic in a compiled machine grows like d log d while the carrying grows like d⁴ — which is why the arithmetic is not what makes these machines large. The curve as an equation

Doubling is cheaper than adding

Multiplying an angle by eight costs three gadgets and multiplying it by seven costs six. The cost of an integer multiple follows the binary expansion of the integer and not its size — which is why the arithmetic in a compiled machine grows like d log d while everything else grows like the fourth power.

 One chain, four mechanisms. The same four bars and the same four pins in every panel. What changes is which link is bolted to the bench, and that is not a property of the chain — it is a decision about where the bench is. The four mechanisms are crank rocker, double crank, crank rocker, double rocker: one input turns fully in some and rocks in others, and what each one is for is different. What cannot change is the shape of the closed loop, and the two diagonals measure that without reference to which link is held still: swept independently, all four visit the same locus of diagonal pairs to within 2.9e-3, half the sampling resolution. This is why the Whitworth quick-return and the oscillating-cylinder engine are not merely similar to a slider-crank; they are one. Linkages

One chain, four mechanisms

Which link of a four-bar is bolted to the bench is not a property of the chain. It is a decision about where the bench is, and making a different one gives a mechanism that looks and behaves completely differently while being, as a chain, the same object — which is why the Whitworth quick-return and the oscillating-cylinder engine are both a slider-crank.

 crank rocker: the closest pair at one position. The site's standard four-bar: ground 4, crank 1, coupler 3.5, rocker 3. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.5438 here, between coupler · frame. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour. Links with a width

Two bars that have to cross

The site's own four-bar has its coupler inside its frame by a full link width for the whole of a turn. It is not an impossible mechanism; it is a mechanism that cannot be built in one plane — and the plane it has been drawn in for twenty-three fields was a convenience nobody had to pay for.

 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. The chain before the lengths

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.

 A colouring is not an assignment. A pin joining links in planes 1 and 3 has to pass through plane 2, and anything in plane 2 whose material covers that pin is pierced by it. That condition involves three links at once where a colouring's conditions involve two, and it is invisible to a colouring because permuting colours preserves a colouring and destroys betweenness. It does not change the number of planes on any machine here. What it removes is arrangements: of the 6 three-plane colourings of a crank rocker, 2 can be built, and of Peaucellier's 192 exactly 96 can. Links with a width

A plane is a colour

Assigning links to parallel planes so that no two conflicting parts share one is a graph colouring, and the answer for a four-bar is three. Then the pins have to get through, and the problem stops being a colouring: of the six proper three-plane colourings, two can be built.

 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. The chain before the lengths

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.

 Every curve in the catalogue, compiled and counted. The bar count is not a formula: it is the number of bar constraints the compiled mechanism actually carries, asked of the object rather than predicted. A line costs five bars and a quintic four hundred and thirteen, which is the field's central number — exactness is paid for in size. The all-revolute column replaces the one prismatic pair that closes the chain with a Peaucellier cell, which costs seven bars whatever the curve, so it is more than half the machine on a line and a rounding error on a quintic. The last column is the polynomial evaluated at the traced point, worst over each machine's working arc. The curve as an equation

Five bars for a line, four hundred for a quintic

Nine curves compiled and counted: a line at five bars, a circle at eleven, a lemniscate at fifty, a general quintic at four hundred and thirteen. The growth is a fourth power of the degree, and three quarters of the largest machine is not computing anything at all.

 A spherical four-bar: 90°, 40°, 100°, 80°. Four revolute axes, all through one point, so the linkage lives on a sphere. Its link "lengths" are the angles between consecutive axes — 90°, 40°, 100°, 80° — and the arcs drawn between the axis directions measure 40.00°, 100.00°, 80.00°, 90.00°. The whole planar four-bar theory carries over with sines where lengths were, including Grashof's condition: sorted, the arcs give s + l = 140° against p + q = 170°, so the shortest arc does turn all the way round — and swept, the input reaches 36 of 36 positions over a driveable range of 360°. This is the mechanism a universal joint is a special case of, with two of the four arcs at 90°. Out of the plane

When the link lengths are angles

Put every axis of a four-bar through one point and the mechanism lives on a sphere. Its bars become arcs, its lengths become angles, and every planar result carries over with a sine where a length used to be — including Grashof's condition, which still predicts exactly which link goes all the way round.

 The slider turns round at the dead centre. The slider's position against the crank angle, through the dead centre. The curve has a maximum there — that is what a dead centre is — so the slider retreats on both sides of it, and the latch works because a mechanism resting past the top of this curve has to be pushed back up it before anything can move. Set 4° past, the slider is 71 µm below the peak, and because the curve is quadratic there, doubling the setting quadruples the depth. Machines you have met

Locked on purpose

A toggle clamp, a landing-gear downlock and the catch on a folding table are all the same mechanism parked a few degrees past its dead centre, where the slider's motion is second order in the crank's. Seventy-one microns of slider travel undoes a latch set four degrees over — and setting it eight degrees over does not double that, it quadruples it.

 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. The chain before the lengths

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 this site's Newton solve stops being a convenience.

 Turning all the way round, against being made of something. Seven four-bars, classified by Grashof's inequality on their four lengths and then asked a question Grashof cannot answer: with a bearing pedestal at each ground pivot, how wide may the links be? The two instruments have nothing in common — one is an inequality on four numbers, the other counts sign changes of (B − A) × (G − A) over a sweep — and they agree about something Grashof was not for. Every four-bar that turns all the way round sweeps a link straight over a ground pivot, so its closest approach is exactly zero and no positive width is admissible; not one of the rockers does, and they take widths up to 0.20 of their shortest link. Links with a width

The crank that cannot turn all the way

Grashof's inequality says which four-bars turn fully. Ask instead how wide their links may be with a bearing at each ground pivot, and the same inequality answers the opposite question: every four-bar that turns all the way round sweeps a link straight over one of its own pivots.

 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. The problem backwards

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.

 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. The chain before the lengths

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.

 How close a synthesis puts two pins. The distribution of the smallest pin-to-pin distance on any one link, over all 1,176 exact syntheses. Burmester's construction returns points, and points can be arbitrarily close together: the shortest here is 0.103, on a mechanism whose poses span more than two units. The shaded band is what a boss of radius 0.25 forbids — 148 of them, 12.6%. The link that is worst is most often the crank, which is not where a designer looks: the frame's two ground pivots are the pair everybody checks by eye. Links with a width

A pin is not a point

A joint in the fields before this one is a name and two coordinates. A pin is a cylinder with material round it, a length through the stack of plates, and a head — and every one of those turns some construction that returns points into a construction that may return nothing buildable.

 What the valve does, against what the ratio promises. The solved valve lift, and the cam's lift multiplied by the nominal ratio. They are the same curve to the eye and they are not the same curve: the peak is 12.777 mm against a promised 12.842 mm, short by 0.51%. The shortfall is not an error in either number. It is what happens when a ratio measured at one position is applied across a movement, and it is why cam cards and rocker ratios are quoted together. Machines you have met

The cam is not the valve

A rocker arm's ratio is the ratio of two moment arms measured at one position, and the rocker swings twelve degrees while the valve opens. The instantaneous ratio runs 1.588 to 1.605, so the peak valve lift is 12.78 mm where the number on the box promises 12.84 — and the shortfall depends on how the rocker was set up, not on the cam.

 What region each four-bar's two cognates land in. One row per region of length space. Against each, the regions of the two four-bars Roberts's construction gives for the same coupler curve, and how many of the 24000 chains in the census landed in that region. Every row has one entry: across the whole census, and at each of 5 tracing points, the original's region decides its cognates' regions with nothing left over. The four Grashof regions are above the rule and the four triple rockers below it, and no row crosses it — a crank-rocker has a double rocker and a rocker-crank, a double crank has a double crank and a double crank, a rocker-crank has a rocker-crank and a double rocker, a double rocker has a crank-rocker and a crank-rocker, a 0–π rocker has a 0–π rocker and a 0–π rocker, a π–π rocker has a π–π rocker and a π–0 rocker, a π–0 rocker has a 0–0 rocker and a 0–0 rocker, a 0–0 rocker has a π–0 rocker and a π–π rocker. The paths points trace

The kind is decided before the lengths are

Roberts's construction hands a four-bar two others that draw its curve, and which of the eight kinds those two are is settled by the kind of the first — not by its lengths within that kind, and not by where the tracing point sits. Twenty-four thousand chains at five tracing points produce no exception, and the reason is one line: the tracing point enters the construction only as a scale, and a region is scale-blind.

 The chain's whole configuration space, drawn on its two angles. A slider-crank asks one thing of its two angles: a cos θ + b cos φ = e, with θ the crank's angle and φ the rod's. Each panel is the square of those two angles from −π to π, with the curve that equation cuts out, followed by arclength on the equation alone. The number under each panel is how many whole turns θ, φ and φ − θ make around a circuit. A curve that crosses the square from side to side carries a turn in that angle; a closed loop inside the square carries none. The two regions where a member turns have two circuits each and the two where none does have one, which is what the four-bar regions these came from predict. Linkages

Three rotations, and four benches

The slider-crank chain's four inversions are four famous machines, and they are not four classifications. Four links make six pairs, one of those pairs cannot rotate at all because a slide is a rotation of nought, and the five that are left are three quantities between them. So an inversion chooses which two of the chain's three rotations sit at its bench, and the four regions of length space already say what all three do.

 The gap is a straight line in the width. Three machines, four widths each, every width a fraction of that machine's own limit. Each set of points is collinear to the last bit of a double — the slopes wander by less than 10⁻¹³ across the range — because a feature-to-feature distance is linear in the corners of the two bodies and a bar's corners are linear in its width. The slope reads the contact: Chebyshev's linkage at -2.692, a crank rocker with a post at -1.350, a crank passing a stud at -1.350. A bar's boss grows 1.35 times as fast as its side, so −1.35 is a boss against something that is not growing and −2.70 is boss against boss. Extending each line to zero gives the widest link the machine will take, and the bisection that finds it the hard way agrees — 0.30113 against 0.30112, inside the bisection's own residual. Links with a width

The gap is a straight line in the metal

Thickening every link by the same amount subtracts the same amount from every clearance, exactly, and moves the angle at which the worst one occurs by nothing at all. So a whole swept check can be done once on bars of any width and every other width read off by subtraction — until the closest pair changes hands, and never past zero.

 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. One path to the tool

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.

 One link, five bodies. Five bars with the same two pins, offset by -0.24, -0.12, 0, 0.12, 0.24 of the link's length, drawn to a common scale. Every one of them holds its two pins exactly the same distance apart, so every one of them is the same link: put any of them into a mechanism and the mechanism solves to the same joint positions at every configuration. Nothing in the kinematics of this collection — no loop equation, no velocity, no coupler curve, no mobility count — can tell them apart. What they do not have in common is which ground they occupy on the way from one pin to the other. Links with a width

A link may be bent

A link is two pins at a fixed distance and the metal between them is a free choice. Bending it moves no joint of the mechanism by more than 10⁻¹³ and moves the clearance by a tenth of a link length — enough to build a machine that a straight bar refuses, and worth exactly nothing against a bearing pedestal the link sweeps over.

 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. The chain before the lengths

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.

 Eight contacts at once, and the sense each turns the disc. The 12-pin drive at 40° of its eccentric. 8 pins are in contact with the disc at this instant, and each one's line is the common normal, which passes through the pitch point on the pin circle. A pin can only push, so the sense in which it turns the disc is decided by which side of the disc's own centre its normal passes: 4 of the contacts turn it one way and 4 the other, with the largest arm in each sense 42.2 and 45.0 on a pitch offset of 55.0. A pair with one contact has no such choice, which is the whole of why two identical rotors cannot drive each other. The shape is the unknown

What a second contact is for

Two identical rotors that are exactly each other's conjugates cannot drive each other, and the reason has nothing to do with conjugacy. A ring of pins and the disc they generate is just as exactly conjugate, has eight to eleven contacts at once instead of one, and never loses more than thirty per cent of the arm its geometry allows.

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