Machine Kinematics
solved, then drawn
Concept

Mobility — where it appears

How many independent motions a mechanism has, counted from its joints by a formula and measured from its constraints as a rank. The two routes are independent — one counts links and joints, the other measures the rank of the constraint equations — so where they disagree the formula is the one that is wrong.

Named by 87 essays across 21 fields — each of them below, with the objects they name alongside it.

Seven arrangements, one routine, and the two that hold. Every row is the same three steps: write down one row per contact — the moment of its normal about the origin, then the normal itself — take the convex hull of those rows, and ask whether the origin is inside it. The parts differ, the numbers of contacts differ, and the routine does not. Two of the seven hold. The other five leave the part something, and the interesting column is what: four rays of rotation for the pinwheel, a translation straight out of the vee, and for the last two a whole line rather than any number of rays, which is what a rank below three means and is the case a reader has to be warned about. Note that the four contacts of the second row are the four of the first row, on the same four edges of the same square, at the same distance along each. positioned by solving, not by drawing.

A constraint that only pushes

Every constraint on this site so far has been an equation: a pin holds two points together, a bar holds two apart, a mesh holds a ratio. A part resting against another part says only *do not come closer* — so what it may do is a cone rather than a subspace, and whether it can move at all stops being a rank.

holding · Restraint A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0.

A constraint that takes nothing away

A rolling wheel forbids one direction of motion and removes no coordinate from the mechanism's description. It cannot slide sideways and it can still be brought to any position at any heading — and the gap between those two sentences is the whole of this field, because in every mechanism built of pins and slides the two agree.

rolling · Rolling a crank rocker with a post: the closest pair at one position. The same four-bar with a post bolted to the frame, just clear of the coupler's path. 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: 0.2561 here, between rocker · post. 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.

A link that takes up room

For twenty-three fields a link on this site has been a distance between two points, and a distance cannot collide with anything, because it is not anywhere. Give every link a body and a question arrives that none of the constraint equations can ask.

bodies · Body 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 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.

networks · Network Kutzbach's count against the measured mobility. Five closed loops of revolute joints. The count is 6(L − 1) − 5j, a statement about how many links and joints there are; the measurement is the number of joints minus the rank of the loop's screw system, which knows only where the axes point. They disagree for four of the five, and the one they agree on is the generic seven-joint loop — so the formula is not broken, it is blind to the special geometry that makes the other four work. The universal joint is counted at -2 degrees of freedom and is in every car built.

Six freedoms, not three

Every mechanism on this site so far has been flat, and flatness is not a simplification made for teaching — it is a special case that hides the most interesting thing constraint counting does, which is get the answer wrong about mechanisms that are in daily use.

spatial · Spatial 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.

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.

serial · Serial 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 site 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.

topology · Topology Every way of stopping, on the same four questions. Six mechanisms that all turn a continuous input into an output that moves and then waits. Index is how far the output steps. Moving is the fraction of the input's turn the output is actually going for; the rest is dwell. From rest says whether the output starts and stops at zero velocity, and acceleration whether its acceleration is a number at all. Every entry is computed from the mechanism's own library, which matters for two of them: a Geneva's moving fraction is (n − 2)/2n and not 1/n, and its entry rate is zero in closed form rather than to the accuracy of a sampled sweep. The three rows whose acceleration is not a number are not badly made — they are mechanisms whose output velocity has a step, and no tolerance improves that.

The mechanism that waits

Every mechanism in this collection so far moves whenever its input moves. A ratchet, an indexer and an escapement do not: they are still for most of a turn and moving for the rest, and what decides which is not an equation running out of answers but a tooth arriving at a face.

timing · Intermittent What each machine is sold with. Every mechanism in this field arrives with a number. The middle column is that number; the right-hand column is what the mechanism was measured to have, by the library named under the row. The coloured bar and the word under each machine are the verdict: exact means the mechanism has the number at every position, mean that it is the average of something that varies within every cycle, bounded that it is false by a stated and negligible amount, point that it is the value at one position, and quoted that the mechanism has no such quantity at all. Of the 14 rows, 6 are quoted and 3 are exact.

The number on the box

Fourteen machines, fourteen quoted numbers, and five different things a quoted number can be. Six of the fourteen name a quantity the mechanism does not have at all; three are exactly right, and all three of them are counts. The field is built on the difference.

applied · Misconception 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.

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.

transmission · Transmission Three bars and four bars. On the left, two bars to a common point: three links, three joints, and Grübler gives 3(3−1) − 2(3) = 0. The Jacobian agrees — two free coordinates, rank 2, nothing left over — and the shape cannot change without a bar changing length. On the right, one more bar and one more joint gives mobility 1, and the whole of this site follows from that difference. The triangle is why bridges are triangulated and the quadrilateral is why machines are not.

What decides whether it moves

Before a mechanism does anything it has to be able to. Two bars pinned to a point cannot move; three can. The count that separates them is one subtraction, it is the first thing anybody computes about a machine, and it can be wrong in a way that no amount of care with the arithmetic will catch.

constraint · Mobility A simple planetary, as a graph. Members are vertices and meshes are edges. Each edge carries the two tooth counts and the body the two axes are stationary in — its carrier — and that third label is the whole of what makes an epicyclic different from an ordinary train. Write the mesh relation relative to the carrier and one formula covers both: an ordinary train is the case where every carrier is the frame. This train has 2 meshes across 5 members and 2 freedoms.

A ratio is a null space

Write a gear train as a graph — bodies for vertices, meshes for edges, and on every edge the body the two axes are stationary in — and one formula covers a countershaft gearbox, a planetary, a harmonic drive and a car's differential. The ratio is the null space of a matrix whose entries are tooth counts, so it comes out as a fraction and not as a number that is nearly one.

transmission · Transmission A 3-stage stack at 35°. Every bar is the same length and every stage is at the same angle, which is a consequence of the solve rather than an assumption in the drawing: the pins at the crossings and the roller on the ground leave one freedom, and it has been used. The platform is at 1547 mm and the ram — the heavy line from the ground pin — is 428 mm long. Moving the ram one millimetre from here raises the platform 5.0 mm; at the bottom of the travel the same millimetre is worth 17.6 mm.

A roller is not a slider

A scissor lift has one degree of freedom, at every height and for any number of stages. Grübler's criterion agrees — if the rollers under it are counted as pins in slots. Count them as slider blocks, which is how every textbook draws a slider, and the same formula declares a machine holding a car in the air to be a structure with minus one.

applied · Mobility Mobility, counted and measured. Grübler's criterion counts links and joints and knows nothing about the dimensions; the rank of the constraint Jacobian measures the dimensions and knows nothing about the topology. They agree for four of these five. The parallelogram with a redundant third bar is the exception: the formula declares it a structure with zero degrees of freedom, and it moves. The formula is the one that is wrong, because it cannot see that the third bar's constraint equations are already implied by the other two.

Counting and measuring mobility

Grübler's criterion counts links and joints and never asks how long anything is. The rank of the constraint Jacobian measures the lengths and never asks what a joint is. Two calculations with no inputs in common, producing one number — which is the only arrangement under which agreement is evidence.

constraint · Mobility 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 The mechanism, and the graph that decides how many loops it has. A lazy tong of 5 scissor units drawn over its own joint graph: a node for every body — 10 of them — and an edge for every pin, 13 of those. The number of independent loops is e − v + 1 = 13 − 10 + 1 = 4, which is how many closure equations somebody writing this mechanism out by hand would have to find and is the one quantity in the field that can be read straight off a drawing. It is also all the count knows: Grübler's 4 is 3(n − 1) − 2j and contains no geometry at all, which is why it is right here and wrong four rows further down the ledger. positioned by solving, not by drawing.

The loops are in the graph

Before a network is a mechanism it is a graph, and the one quantity that can be read straight off a drawing is how many independent loops it has: edges less nodes plus one. Grübler's count is that arithmetic and nothing else — which is why it is right about a tong at every size and says a deployable ring cannot open.

networks · Network Two things that are not the same configuration. The mechanical advantage of a four-bar, from the velocity solution: for a lossless mechanism it is the reciprocal of the output-to-input speed ratio, so wherever the output momentarily stops the force ratio diverges. It reaches 1673 at 222° — a toggle, where crank and coupler line up, and the reason a knee-joint clamp holds with almost no effort. The transmission angle is at its worst somewhere else entirely: 46.5° at 0°, 222° away, where coupler and rocker line up instead. Both get called "the mechanism jamming" and they are opposite situations — one is enormous force output, the other is force going into the bearings.

Two things called jamming

A four-bar's mechanical advantage peaks at 1,673 in one configuration and its transmission angle collapses in another, 222° away. Both get described as the mechanism jamming. One is enormous force output and the other is force disappearing into the bearings, and they are opposite situations.

wrong · Misconception What a pattern that cannot fold leaves behind. The best a least-squares solve can do with the vertex closures, against the fold it is asked for. The Miura pattern closes at every angle, at the arithmetic's own floor — the line along the bottom is 10⁻¹⁵ and below. The same grid with its vertices moved by a tenth of a panel does not close at any angle at all: its residual starts at 5.9e-6 at the smallest fold and grows with it, and no seed and no number of iterations moves it. Both patterns have the same panels, the same creases, the same graph and the same developable vertices, and every one of those vertices folds perfectly well on its own.

Each one moves, and together they do not

Take the pattern a Miura sheet folds along and move every interior vertex by a tenth of a panel. Every vertex still folds on its own — each is a spherical four-bar with a freedom of its own — and the four of them together fold to no angle at all, with a residual that starts at six millionths and never falls.

networks · Network 6 contacts, and the box lifts straight off. 3-2-1, six contacts. Each pad is a contact and each arrow the direction the box is free to move there — the inward normal, and the whole of what the contact contributes. The rank is 6, which is full: these 6 contacts are 6 independent constraints and a bilateral version of them would fix the box completely. The margin is nought, so the box is free to leave, and no amount of tightening the tolerances on where the pads are would change that. Rank and hold are different questions and this is the pair of pictures that separates them. positioned by solving, not by drawing.

Four in the plane and seven in space

Six independent constraints fix a body in space and six contacts fix nothing, because d vectors can span d dimensions and can never positively span them. The minimum is one more than the dimension — and it is a floor rather than an answer: six contacts on a box held it in none of four thousand random arrangements and seven held it in twenty-one.

holding · Restraint Peaucellier's cell: exact straight-line motion from pin joints. The rhombus and the two long arms hold |OP| · |OQ| constant at 16 = 5² − 3², which is inversion in a circle about O. Inversion carries circles through the centre to straight lines, and the link CQ makes Q run on exactly such a circle — so P travels on a line, with no approximation anywhere. Measured over 160 solved positions the deviation is 6.5e-16 of the span, which is arithmetic noise rather than a small error.

Peaucellier and the exact answer

Eighty years after Watt settled for an approximation, a French army officer found a linkage that draws an exactly straight line from pin joints alone. It works by inversion in a circle, the product it holds constant is measurable, and on this site it comes out straight to 10⁻¹⁶ of its span.

curves · Coupler One point, six groups, six shapes. A group has no picture, so here is the next best thing: fix one point of the moving body — the marked one — and draw everywhere the group can send it. A prismatic pair sends it along a line, a revolute round a circle, a helical along a helix, a cylindrical over a cylinder, a spherical over a sphere, a planar over a plane. Those six shapes are the six surfaces the previous figures drew, which is not a coincidence and is the field's first argument read backwards: a lower pair's surface is an orbit of its own group, which is exactly why the surface can slide on itself.

The count cannot tell a pin from a slide

A revolute, a prismatic and a helical pair all take five freedoms away in space and leave one. Grübler adds the same number for each, the constraint rank measures the same number for each, and the three joints have nothing whatever in common — one sends a point round a circle, one along a line, and one along a helix at a rate the joint decides.

pairs · Pair The lever of a simple planetary. Each member sits at a position on the lever fixed by the tooth counts alone, and its speed is the height of one straight line over that position. The line here is drawn through sun and ring; every other member is plotted where the train's null space puts it, and lands on the line exactly — the residual is zero in rationals, and 2.2e-16 once the coordinates have been rounded to doubles for the drawing. Where the line crosses the axis is the member that is standing still, and that is what a brake does: it pins the line to the axis at one position and leaves it free to pivot there. The planets are on the lever too, off the end of it, which is where they belong — they are members of the train and are not shafts anybody can reach.

The lever that is the gearset

The lever diagram of an epicyclic is usually offered as a mnemonic. It is exact, and the reason is a fact about the null space: a gearset whose frame carries no teeth can turn as a block, and that one motion supplies the coordinate every member is plotted at. Where the line crosses the axis is the member standing still, and the ordering of the members on the lever settles which gears are reductions and which run backwards, without a formula anywhere.

transmission · Transmission Three parallel bars, and a formula that says this cannot move. Five links and six pin joints, so Grübler's criterion gives 3(5−1) − 2(6) = 0 and calls it a structure. The Jacobian has rank 5 against 6 free coordinates, so it measures one degree of freedom — and the sweep assembles 59 of 60 positions, which settles the matter. The third bar removes no freedom because its constraint is already implied by the other two, and a formula that counts joints cannot notice that they happen to be parallel. Mechanisms of exactly this kind carry drafting machines, anglepoise lamps and locomotive coupling rods, where the redundant bar is there for load sharing and for keeping the linkage out of its change point.

The mechanism Grübler says cannot move

Three parallel bars between two frames. Five links, six pins, and the criterion every engineering course teaches gives zero degrees of freedom — a structure. It is a mechanism, it is in drafting machines and locomotive coupling rods, and the formula cannot see why.

constraint · Mobility 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.

topology · Topology Bennett's four-bar at 40°. Four bars, four revolute joints, and axes that are not parallel — a spatial four-bar, which Kutzbach counts at -2 degrees of freedom. Bennett's condition, sin α / a = sin β / b, makes the screw system rank 3 instead of 4, so the mechanism has 1. Driving the first joint through a full turn, 48 of 48 positions assemble. Orthographic projection, viewed from 40° azimuth and 24° elevation; dashed stubs mark the joint axes.

Bennett, and the condition that moves it

A spatial four-bar is immobile by every count there is, and generically it cannot even be assembled at more than isolated configurations. Bennett found the one relation between four lengths and two twists that makes it turn through a full revolution — and break the relation by two parts in a thousand and most of the travel is gone.

spatial · Spatial Sun 24, ring 72, planet 24. An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. With the ring held, a sun input turns the carrier at 1 + 72/24 = 4.000 to one. Willis's equation and the tabular superposition method are independent derivations and both are computed here; the site requires them to agree, because planetary ratios are the most commonly mis-stated numbers in mechanism work and the sign errors are notorious.

Epicyclic ratios, two ways

An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. The sign errors are notorious, so every ratio here is computed by Willis's equation and by the tabular method, and the two are required to agree.

gears · Tooth 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.

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.

transmission · Transmission 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.

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.

networks · Network 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.

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.

linkages · Fourbar What each kind of joint takes away. Grübler's formula is M = 3(n − 1) − 2j₁ − j₂, and the 2 and the 1 in it are not conventions. A lower pair — a pin or a slide — holds two bodies together over a surface and leaves one relative freedom, so it costs 2. A higher pair — a cam against a follower, a wheel on a rail — touches at a point, the contact travels along both surfaces, and it costs 1. Five chains, each built and each measured from the rank of its constraint Jacobian, which has never heard of the formula. The last row is the one worth having: count that cam contact as a pin, as is very easily done, and the formula returns 0 where the mechanism has 1. The Jacobian does not move.

What each joint takes away

Grübler's formula has a 2 in it for pins and a 1 for cam contacts, and those numbers are not conventions to be memorised. They are the number of constraints each kind of joint imposes, they are measurable as the rank of a matrix, and miscounting one of them is the commonest way the formula is got wrong.

constraint · Mobility A vertex is a spherical linkage, and the sectors are its link lengths. The four creases of one folded vertex, drawn as directions from the vertex itself, with the great-circle arcs between consecutive ones. Those arcs are the sector angles of the flat pattern — 80°, 60°, 100°, 120° — and they are those angles at every fold, to 8.9e-16 radians. That is the whole of the claim in the title: four axes through a point at fixed arcs from each other is a spherical four-bar, the object this site's spatial field built two phases ago, and a crease pattern's vertex is one of them with the arcs printed on the paper. The dihedral angle of the sheet at each crease is a half turn less that crease's fold angle, which here run 68.8°, 14.4°, 68.8°, 14.4°. positioned by solving, not by drawing.

Every vertex is a spherical linkage

Four creases through a point at fixed arcs from one another is a spherical four-bar — the object the spatial field is built on — with its link lengths printed on the paper as sector angles. The arcs hold to four parts in ten thousand million million at every fold, and on a flat-foldable vertex the half-angle tangents keep a ratio constant to nine figures.

networks · Network Where a point held by three strands may be. Three anchors, three strands of 130, 130, 120 mm, and a point tied to all three. A rigid link of those lengths would leave nothing to decide — three distance equations in two unknowns have no solution at all — and three strands leave a region, because each of them says no further than rather than exactly. The region is the intersection of the three discs; its area here is 2721.0 mm² and it has 3 corners. Inside it nothing is taut and the point has both its freedoms; on an arc one strand is taut and it has one; at a corner two are taut and it has none. positioned by solving, not by drawing.

The strand that is slack

A rigid link removes a freedom wherever the mechanism stands. A strand removes one only where it is taut — so a point held by three of them has two freedoms in the middle of its region, one on an arc, none at a corner, and no single mobility count describes it at all.

strands · Strand A pin in a hole is a short link. Left: a pin of radius 0.86 in a hole of radius 1, so the clearance is 0.14. The pin's centre may sit anywhere within that of the hole's centre. Right: the same joint as it enters the kinematics — a binary link of fixed length 0.14 and free direction, with a revolute at each end. That is not an analogy. It is the same set of relative positions, so every count, every Jacobian and every solve on this site applies to it unchanged, and a four-bar with play at each pin is a mechanism with eight links and eight joints.

A clearance is a link

A pin in a hole is not a joint at a point. Its centre may sit anywhere within the difference of the two radii, so the two links it joins are connected by a body of fixed length and free direction — a binary link with a revolute at each end. That is not an analogy, and taking it literally makes a four-bar a mechanism with eight links, eight joints and five degrees of freedom.

practice · Clearance 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.

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.

networks · Network 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.

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.

linkages · Fourbar What each loop's constraint system is made of. For each mechanism: the order of the screw system its joints span, the order of the reciprocal system — the wrenches it carries without moving, which is always six minus the first — and what those wrenches are. A planar four-bar carries one force and two couples; a mechanism whose motion lies in no subgroup carries screws of finite pitch instead, and 2 of these 6 do.

What a mechanism cannot do

A mechanism's freedoms are a subspace of screw space. Everything orthogonal to that subspace under the reciprocal product is a force the mechanism carries without moving — so the constraints are not a separate thing to be worked out, they are what is left, and one matrix gives both.

spatial · Spatial Four pulleys, one idler, and one equation. A closed strand over four fixed pulleys and an idler carried on an arm. The idler sits outside the loop the other four make and bulges the strand out to reach it, so its wrap is signed the other way — -25.34° against the four positive ones — and the five still add to exactly one turn. The arm angle is not a free choice: the strand has a length, and matching it is one scalar equation, whatever the number of pulleys. At this position the run is 1018.046 mm and the idler takes up 0.4387 mm of strand for every millimetre it moves along the bisector of its two spans. positioned by solving, not by drawing.

The tensioner is the unknown

A linkage closes when a vector comes back to where it started: two equations, two unknowns. A strand closes when a number does — its length — however many bodies it runs over. So a run with one free body is determined, a run with two is not, and the arm angle that takes up 1,020 mm of belt is the root of one scalar equation solved to 1·10⁻¹³ mm.

strands · Strand 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 What each of them can reach. Nine hundred control histories of four legs each, from the same starting configuration, with the resulting position plotted. The wheel's cloud is two-dimensional and fills the region; the trolley's is one-dimensional and lies exactly on its rail — the same number of coordinates, the same number of constraints, the same count of freedoms, and a reachable set of a different dimension. Nothing here is a matter of degree.

Not unreachable, only expensive

The sentence is false of every wheeled mechanism in this field and true of exactly one — the trolley bolted to a rail. A rolling constraint forbids a direction and reaches everywhere; the mistake is reading a statement about instants as a statement about intervals, and it is made in both directions.

wrong · Misconception 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.

One wheel on ice

A differential with one wheel stopped turns the other at exactly twice the cage, and the relation it imposes is satisfied the whole time — nothing has failed, nothing is confused, and the reason the car does not move is not in this site. What is here is the other half: a locked axle is an overconstrained mechanism, and the sliding it produces is 2π times the track per circle driven, whatever the radius.

transmission · Transmission Watt's chain. Six links and 7 pin joints, so Grübler counts 3(6 − 1) − 2(7) = 1 and the rank of the Jacobian measures 1. Two of the six links must carry three joints; they are shaded. Here they share a joint, which makes this a Watt chain. That adjacency is the entire classification, and it is read off the graph rather than off the picture.

Six bars, and what the extra dyad buys

Every linkage so far has had four bars, because four is the smallest closed chain that moves. The next one up is six, not five, and six is where the subject stops being one family and becomes a taxonomy — two chains, distinguished entirely by whether the two links carrying three joints happen to touch.

linkages · Synthesis Kutzbach, plus the constraints counted twice. Each row is 6(n − j − 1) + Σf, then the redundant constraints ν measured from the two legs' constraint systems, then the mobility measured from the rank of the whole loop's screw system. The first column is wrong for 5 of 6 of these mechanisms; adding ν repairs every one. The catch is that ν is not a property of the joint graph — Bennett's linkage and a spatial four-bar with one twist changed have the same graph and different ν — so the corrected formula needs the measurement it was supposed to replace.

The formula is repaired by the thing it replaced

Kutzbach's count is wrong about most of the mechanisms worth building, and every textbook gives the same repair — add back the constraints that were imposed twice. The repair works on every loop this site has. It is also not a formula, because the number it adds cannot be read off the joint graph.

constraint · Spatial The ring a count says cannot move. 8 pairs of angulated elements, each pair two mirror-image bent bars pinned at their kinks, with each pair joined to the next at two pins on a common radius. 16 bodies and 24 pins give Grübler's 3(n − 1) − 2j = 0: no freedom at all, a structure. The rank of the constraint Jacobian is 44 of 48, which leaves 4 — the three rigid motions of the whole ring and one deployment — and 4 constraints that repeat what the others have already said. The kink angle is not a style: it is 135.0000°, a half turn less the 45.0000° the ring subtends per pair, and any other value gives a ring that will not deploy. Inner radius 0.6876, outer 2.2688. positioned by solving, not by drawing.

The ring that closes at every size

Two bent bars pinned at their kinks hold the angle between their connection lines at 135.000000° whatever you do to them, and two straight ones hold it at nothing. That is the whole difference between a scissor chain that grows in a line and a ring of eight that opens and shuts — and the count says the ring cannot move.

networks · Network What every pair of groups meets in. The intersection of two subgroups is always a subgroup — that needs no computation — and which one is the useful part. This is the design rule behind every parallel machine on this site: choose legs whose groups meet in the motion the platform is wanted to have, and it has that motion whatever the leg lengths are, with no synthesis and no tolerance. The row and column are built about different axes, at right angles, because an intersection is a statement about particular subgroups rather than about their kinds: two planar groups with the same normal meet in the whole of themselves, and two with different normals meet in a line.

Three legs and one plane

The parallel field argued that a platform stays flat from the structure of its legs, one leg at a time. The same fact is one line of linear algebra: each leg confines the platform to a group, the platform gets the intersection, and an intersection of groups is a group whatever the leg lengths are. The motion type is decided before a single dimension is chosen.

parallel · Parallel Whose constraints stay put. The largest principal angle between a mechanism's screw system at the start of its motion and at each later position. The 3 mechanisms whose motion lies in a subgroup of the rigid displacements — planar, spherical, translational — never leave the same subspace, and read between 1.9e-6 and 3.0e-6 degrees, which is the precision of an arccosine near one rather than a movement. The paradoxical ones turn through 49° and 57°. Every curve runs over its own range of motion, because Bricard's linkage assembles over 120 degrees and a shared axis would hide it.

Two ways to be overconstrained

A planar four-bar and Bennett's four-bar report the same redundancy, the same rank and the same wrong count. One of them is overconstrained at every set of link lengths; the other at exactly one ratio and nowhere near it. The difference is not in any of the numbers so far — but it is measurable, and the measurement is an angle.

spatial · Spatial Four joints that give a group, and four that do not. Two chains of four revolute-and-slide joints, each drawn at its home position with its joint axes dashed, and each with a cloud of the tool positions it reaches. The counts are identical: four joints, four freedoms, the same Jacobian rank everywhere off a singularity. On the left the three pins are parallel and the slide is along them, and the displacement set is the Schoenflies group — every translation and one rotation direction, four dimensions, closed. On the right the axes are at random and the set is four-dimensional too, and it is inside no group smaller than all the rigid displacements. The instrument is in the caption of each panel: take the logarithms of the displacements the chain reaches and count the dimensions they occupy. Four means a group. Six means there is nothing to be inside.

Four joints that give a group, and four that do not

Two chains of four joints. Same joint types, same count, same mobility, same Jacobian rank, same everything this site has measured for twenty-two fields. One of them reaches a four-dimensional set of displacements that closes under composition; the other reaches a four-dimensional set whose logarithms fill all six dimensions. The difference is six hundredths of a radian in where two axes point.

pairs · Pair The same error, twice, in two directions. A Sarrus linkage with one axis of one chain tilted off true, and the motion range that survives. Tilted within the plane the chain works in, it does not care: at 0.2 radians — eleven and a half degrees, which is not a manufacturing error by any standard — it still drives through a full turn. Tilted out of that plane, 0.001 radians stops it dead. Two hundred times the error, in the other direction, for no cost at all. What separates them is whether the perturbation lies in the screw system the mechanism leaves unconstrained — so "an overconstrained mechanism must be exact" is not merely crude, it is wrong about the case it is usually said of.

Fragility has a direction

Tilt one axis of a Sarrus linkage out of true by a thousandth of a radian and it stops dead. Tilt the same axis of the same mechanism by two hundred times as much, in the other direction, and it drives through a full turn with nothing measurably wrong. Three orders of magnitude between two errors of the same size — and the direction that matters is the one the reciprocal screw system names.

practice · Overconstraint The network where counting works. 5 scissor units, each two bars pinned at their middles, with consecutive units sharing their end pins. Sixteen bodies and twenty-two pins at eight units, and Grübler's 3(n − 1) − 2j gives 4 — three rigid motions and one internal freedom — against a measured nullity of 4 and no redundant constraints at all. That holds at one unit and at sixteen. The tong is here to make the field's point in the direction nobody expects it: a network is not a place where counting fails, it is a place where counting stops being checkable by hand, and one of the two assemblies that spans this plate is counted perfectly. Span 8.1388 mm at this opening, which is 5 times one unit's 1.6278. positioned by solving, not by drawing.

One input at one end

A lazy tong's reach is 2nL cos θ — exactly, to a part in a million million, over eight sizes and sixty openings — so it multiplies its input by the number of units. It multiplies everything else by the same number, including the part of the drawing nobody wanted multiplied: a unit cut a hundredth of a radian out puts a tong of thirty-two units 0.374 out at the far end.

networks · Network 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.

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.

topology · Topology Two numbers, and where they differ. Every mechanism in this field, with what its constraints leave and what its brackets fill. On every mechanism without a rolling contact the two columns are the same number, which is why nobody had to say which one mobility meant. Here only the rail agrees with itself — and the rail is the one mechanism in the table that cannot go anywhere new.

The count that counts the wrong thing

Mobility has meant one number for six fields, because until a wheel appeared no mechanism could tell two questions apart. A rolling wheel has two velocity freedoms and a three-dimensional reachable set, and the formula that gives 2 is not wrong — it is answering the question about instants when the question anybody asks is about intervals.

constraint · Mobility The smallest mechanism that is not one. Two bars from a free joint to two pinned ones, with the three points in line. The constraint matrix has two rows and two columns, both rows are horizontal, and its rank is 1: one freedom left over, pointing straight up, and one dependency among the two bars. Move the joint up and neither bar changes length to first order, which is what the freedom says. To second order both bars get longer, by the same amount and in the same direction, and there is nothing to trade off against — which is what the dependency says. The obstruction is the dependency applied to the second-order stretch and comes to 1.414214; anything but nought there and the freedom is not the beginning of a motion. Lifted by 0.34 the bars are 0.05622 longer, which is the whole argument drawn to scale.

It moves to first order and not at all

Two bars from one joint to two pinned ones, all three in line: the rank leaves a freedom pointing straight up, and lifting the joint stretches both bars. The obstruction is 1.414214, the walk travels a millionth of what it is asked to, and how far it gets is a property of the tolerance rather than of the mechanism.

networks · Network What every pair of groups meets in. The intersection of two subgroups is always a subgroup — that needs no computation — and which one is the useful part. This is the design rule behind every parallel machine on this site: choose legs whose groups meet in the motion the platform is wanted to have, and it has that motion whatever the leg lengths are, with no synthesis and no tolerance. The row and column are built about different axes, at right angles, because an intersection is a statement about particular subgroups rather than about their kinds: two planar groups with the same normal meet in the whole of themselves, and two with different normals meet in a line.

Legs intersect

A serial chain multiplies its joints' groups and the product is almost never a group. A parallel machine's platform gets the intersection of what its legs permit — and an intersection of groups is a group, always, with no coincidence required. That is the only construction in this field that produces closure for free, and it is why a platform can be designed for a motion type instead of discovered to have one.

pairs · Pair What each instrument returns, on each kind of graph. The 8-link census, three rows, and the same three questions asked of every graph in it. Grübler returns 1 in every row — it has to, because that is what the census selected on. The rank returns 1 in the first two rows and 2 in the third. Only the third column changes across all three rows, and it is the one this site did not have before this field: a mobility computed for every subset of the links rather than for the whole. Read down the middle two columns and the site's standing pair of routes is unanimous about 62 graphs, of which only 16 are what it says they are.

The count was right and the name was wrong

The constraint field has checked Grübler's count against a Jacobian rank since the foundation, and the two disagree only where the geometry is special. Here is an assembly where they agree, where both are correct, and where the mechanism does not have the number of links it is described as having.

constraint · Mobility What the clearance has to swallow. Bennett's linkage with its second length multiplied by 1 + δ, and the closure error the solver drives down to and then cannot improve on. The loop does not close at any δ tried, including one part in a million. But the gap is exactly proportional to δ — the ratio varies by 0.07% across four decades — with a measured constant of 0.507. Shared over 4 joints that is 0.127 δ of play per pin, so a linkage machined to one part in a thousand needs about 0.20 mm of clearance in a link of 1.6 m, or a hundredth of a millimetre in a link of 1.6 cm. That is an ordinary running fit, and it is why a mechanism that cannot be built is in every folding table.

Why a hinge works

A door hinge with three knuckles is overconstrained — three axes imposed where one would do, and exactly parallel is a condition no bored hole has ever met. It works because the misfit is 0.507 times the error and the play in each knuckle is larger than that. The mechanisms this site called unbuildable are built every day, and the thing that builds them is the clearance that was already there.

practice · Overconstraint A brace is one redundant equation, on purpose. The compiled machine, counted and measured, with and without 4 braces. The count says the braced machine has -3 degrees of freedom — it cannot move — and the rank of the constraint Jacobian says it has 1, the same as before. Every brace contributes exactly one equation the others already imply, which is what overconstraint is, and here it is being added deliberately: the redundancy is what removes the assemblies the count knows nothing about. This is Grübler being wrong for the useful reason rather than the embarrassing one.

A bar between two midpoints

In a parallelogram the midpoints of two opposite sides are exactly one side apart, and in the crossed assembly they are not. One bar between them admits the first and refuses the second — and it is one redundant equation per parallelogram, added on purpose, on a site whose constraint field is otherwise about overconstraint arriving by accident.

computing · Compute One tool pose, a curve of postures. 41 postures of S-R-S arm, every one of them holding the tool at the same pose. The tool moved 1.0e-13 of a metre over the whole sweep, which is the arithmetic's noise floor rather than a tolerance. The joint that moves most is the elbow, and it runs on a circle of radius 0.192 m about the line from shoulder to wrist — a curve computed here by walking the Jacobian's null space, and known independently as the intersection of two spheres. The two agree to 10⁻¹³.

The freedom that does nothing

Give an arm a seventh joint and one tool pose stops having eight answers and starts having infinitely many. The arm changes shape while the tool stands still — measured at 3.3 × 10⁻¹² of a metre over forty-one postures — and the elbow runs on a circle that two entirely different computations agree about to a tenth of a picometre.

serial · Serial What it takes to build each of the twelve. The same twelve, read as a bill of materials. Six of them are one joint, because a lower pair permits the whole symmetry group of its surface and those six groups are exactly the symmetry groups surfaces have. The other five with a dimension take a chain: two slides for planar translation, three for Cartesian motion, a thread and two slides for the screw-in-a-plane group, and three parallel pins with a slide along them for Schoenflies motion — which is a SCARA arm, and is why a pick-and-place machine has four joints and not one. The group each chain produces is measured from four hundred sampled poses rather than declared, and every row agrees.

The freedom that is a set

Grübler's rule has been on this site since its first essay, and it adds up numbers. Each of those numbers is the dimension of a group of displacements, and the group has eleven siblings the number cannot distinguish. The count is not wrong; it is a projection, and this is what the projection discards.

constraint · Mobility The framework Maxwell's count calls a structure. Six joints and twelve bars in space. Three coordinates each gives eighteen unknowns, six rigid motions come off, and twelve bars is exactly twelve constraints — Maxwell's count is 6 against six rigid motions, which is the definition of isostatic: no mechanism, no redundancy, every bar carrying its own share and nothing spare. The rank is 11, not twelve. There is one dependency among the bars and one freedom left over, and the freedom is a genuine finite motion: walked here with every bar held to 4.4e-16 of its own length. The reason is a symmetry — three pairs of joints exchanged by a half turn about one line — and it is built into the coordinates rather than asserted about the result. positioned by solving, not by drawing.

Twelve bars and a symmetry

Six joints and twelve bars in space is Maxwell's count exactly: no mechanism, no redundancy, nothing spare. Place three pairs of the joints so that a half turn about one line exchanges them and it moves — a finite motion, walked with every bar held to five ten-thousand-billionths of its own length, on a framework the arithmetic calls a structure.

spatial · Spatial Sarrus, as two planes meeting in a line. Each arm of Sarrus's linkage is three pins with parallel axes, so each arm holds the platform inside a planar group — the one whose normal is that arm's axis direction. The platform has to satisfy both, so what it may do is the intersection, and the intersection of two planar groups whose normals are not parallel is the one-dimensional group of translations along their common perpendicular. The platform goes up and down and does nothing else, and that is the exact straight line the spatial field measured to 10⁻¹⁶ of its span — arrived at here with no mechanism solved and no tolerance anywhere. Drag the arms towards each other: the answer is a translation at every angle but zero, where the two groups become one group and the intersection jumps to three dimensions. That is the linkage built flat, and it is the configuration in which it stops being a straight-line mechanism.

Two planes meeting in a line

Sarrus's linkage draws an exact straight line out of six pin joints, and the spatial field proved it by solving the mechanism sixty times and measuring a departure of 9.8 × 10⁻¹⁶. Here the same fact comes out of two planes and a cross product, with no mechanism solved anywhere — and the two routes are not redundant, because only one of them can tell you the linkage as built delivers it.

pairs · Pair Two branches, meeting where the sheet is flat. The same vertex's two folding modes, plotted as one crease's fold angle against another's. Both curves pass through the origin, which is the flat sheet, and they cross there and nowhere else. That crossing is what a rank cannot see: at the origin the tangent directions of both branches are available to the constraint matrix at once, so the nullity there counts them all and the mechanism has only one of them once it has left. Every branch argument this site has made — a four-bar's assembly configurations, an arm's eight postures, the components a solve's paths turned out to run between — is this picture with different axes.

Where the branches meet

A flat sheet is the one configuration every folding of a pattern passes through, and it is the one configuration where the rank is wrong about all of them. Three of a three-by-three Miura sheet's four apparent freedoms are not motions — and a grid that folds to no angle at all reports exactly the same four.

networks · Network How much of Bennett's turn survives a bar being wrong. The same four bars and the same four twists, with one bar's length changed by the amount on the left and nothing else touched. The bar shows the fraction of 48 sampled positions of the first joint at which the loop closes to within 10⁻⁹. Two parts in a thousand already costs most of the travel. This is what it means for a mechanism to work only on a condition rather than approximately near one — and it is why Bennett's linkage was a curiosity for eighty years before anyone could machine to it.

Bennett's condition is a ratio

A spatial loop's parameters are lengths and angles together, so a scaling touches only half of them. Bennett's condition — a over sine alpha equals b over sine beta — is a relation between the two halves, and what it demands of a machine is a relation between its lengths and its twists rather than a property of either.

spatial · Spatial Four instruments, and only the last one names the group. Every instrument this site has for an overconstrained loop, on the same six mechanisms. Kutzbach's count gives −2 for a planar four-bar and −2 for Bennett's. The rank of the constraint Jacobian gives three and three. Both are right and neither separates them. The last two columns are this field's: the span is how many dimensions the logarithms of the displacements the moving link actually reaches occupy, and closes at is the dimension after those are closed under the bracket. A planar four-bar closes at three and the three are planar motion; Sarrus closes at one, a translation, which is the exact straight line the spatial field measured by solving the mechanism sixty times. Bennett closes at six: its displacements occupy four dimensions and no group smaller than all of them contains those four. That is what "paradoxical" has meant on this site for six phases, stated as an integer.

Compose two positions and see where you land

Take two configurations a mechanism actually reaches, compose the displacements that got it there, and ask what kind of thing the result is. A planar four-bar lands inside planar motion, to 4 × 10⁻¹⁶. Sarrus lands on its own line. Bennett's linkage lands three tenths of a radian outside the four dimensions its own displacements occupy — a one-freedom motion that generates all six.

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

Eleven assortments and four that are empty

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

topology · Topology In space the arithmetic allows almost nothing. A body in space has six freedoms and a revolute joint takes five, so a mobility of one needs (6n − 7)/5 joints — and that is an integer only when the link count leaves a remainder of two on division by five. The whole table is this: 7, 12, 17, 22 links, and nothing else. At seven links the degrees must sum to fourteen across seven links with none below two, so every link is binary and the graph is a single seven-cycle: there is exactly one spatial chain, and it is a loop. That is the census explanation for something the spatial field has lived with since it was written — every spatial mechanism on this site is one closed loop — and it had never been stated as a count. The next admissible size is twelve links and thirteen joints, where two assortments are arithmetically possible, 157 candidates give 33 graphs, and 5 of them are chains — every one with ten binary links and two ternary, so the assortment with a quaternary link is empty exactly as four of the planar ones are.

In space there is one chain

A body in space has six freedoms and a revolute joint takes five, so a mobility of one needs (6n−7)/5 joints — an integer only when the link count leaves a remainder of two on division by five. At seven links every link is binary, the graph is a single seven-cycle, and there is exactly one spatial chain.

spatial · Spatial One freedom, whatever the count says. Every one of these machines has exactly one degree of freedom, measured as the number of unknowns minus the rank of the constraint Jacobian. Unbraced, the count agrees. Braced, the count says the largest machine has -153 — that it cannot move, by a wide margin — and the rank says it still turns exactly as it did. The gap is one equation per brace and every one of those equations is implied by the others. This is the constraint field's oldest example, at a scale nobody would try by hand: a count that is wrong by a hundred and fifty-three about a mechanism that works.

One freedom and four hundred links

Braced, the machine compiled from a quintic has 1,249 equations in 1,096 unknowns and a Grübler count of minus a hundred and fifty-three. It turns. The rank of its constraint Jacobian is 1,095, so its mobility is one — and every one of the hundred and fifty-four surplus equations was added deliberately.

constraint · Mobility Arms with their tools pinned down. Pin an arm's tool to the ground and the open chain is a closed loop, which the first field of this site knows how to count. Kutzbach gives 6(n − 1) − 5n = n − 6 for a loop of n revolutes, and the measurement is n minus the rank of its screw system — the columns of the arm's own Jacobian, read as constraints rather than as velocities. The two agree on every row but one, and the one is the arm at a wrist singularity: the formula says the pinned arm is a structure and the mechanism has a freedom. That is the finding this site opened with, arrived at from the far end of its subject.

Pin the tool and it is a loop

Hold an arm's tool still and the open chain becomes a closed one, which this site has known how to count since its first field. Kutzbach's criterion says a pinned six-joint arm is a structure. At each of its three singularities the measurement says it can still move — the site's founding finding, arrived at from the far end of its own subject.

serial · Spatial 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.

Six things a network is not

A count that is right about a difference and read as an answer, a nullity taken for a mobility, a flat state that cannot tell a mechanism from a structure, a scissor ring that closes nowhere, a vertex that folds while its sheet does not, and a null space computed with an instrument whose floor is above the answer. Six claims, each with the number that kills it.

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

A catalogue is a search space

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

topology · Topology Four instruments, and only the last one names the group. Every instrument this site has for an overconstrained loop, on the same six mechanisms. Kutzbach's count gives −2 for a planar four-bar and −2 for Bennett's. The rank of the constraint Jacobian gives three and three. Both are right and neither separates them. The last two columns are this field's: the span is how many dimensions the logarithms of the displacements the moving link actually reaches occupy, and closes at is the dimension after those are closed under the bracket. A planar four-bar closes at three and the three are planar motion; Sarrus closes at one, a translation, which is the exact straight line the spatial field measured by solving the mechanism sixty times. Bennett closes at six: its displacements occupy four dimensions and no group smaller than all of them contains those four. That is what "paradoxical" has meant on this site for six phases, stated as an integer.

A name for each overconstraint

The spatial field separated subgroup overconstraint from paradoxical by measuring how far a mechanism's screw system turns: 2 × 10⁻⁶ degrees against 89. That is a verdict without a name. Closing the logarithms of the reached displacements under the bracket gives the same verdict and says which group — planar, spherical, a translation — and for Bennett's linkage it says six.

spatial · Spatial A network with no boundary at all. The kagome lattice, drawn out to 5 cells across and continuing for ever. The measurement is made on one cell: 3 joints, 6 bars, and a bar that leaves the cell comes back into it, written against the far end's position in the neighbouring cell. There is no boundary anywhere in the arithmetic, and the size of the network has gone from being a parameter to not existing. The highlighted triangle is the cell; every other line on the page is a copy of one of its six bars. positioned by solving, not by drawing.

The cell that repeats for ever

Take the size of a network to infinity and it stops being a parameter. What is left is one cell, six bars, and a question nobody has to ask about a finite assembly: does the pattern's period count as a body? A square grid is rigid if it does not and shears if it does, and so does the kagome.

networks · Network A geared five-bar at 3 : 2, its coupler pin traced over 2 input turns. Two cranks of length 1 on pivots 3 apart, meshed through gears whose pitch circles, dashed, have radii 1.800 and 1.200, so the second crank turns 1.5000 times for every turn of the first and the other way. Coupler bars of 3 and 2.5 join them at the pin C, and the thin curve is where C goes over 2 turns of the input. The two crank pins are always between 1 and 5 apart and the bars can span 0.5 to 5.5, so the chain never meets a dead position. After 2 turns the machine is exactly where it began and the curve is closed.

One freedom, and a motion that never repeats

Mesh a gear on each crank of a five-bar and the count and the rank agree that one freedom is left, at every gear ratio. Whether the machine ever comes back to where it started is a different question, and neither instrument can see it: at 3 to 2 it is home after two turns, at 37 to 23 after twenty-three, and at the golden ratio never, with its nearest returns at the Fibonacci numbers.

constraint · Mobility Two permitted motions, and a composite that lifts off. A disc resting on a straight edge — a roller follower on a flat-faced cam, and the simplest higher pair there is. Two bodies touching at a point rather than over a surface, two freedoms: slide along the edge, and turn, because a disc is its own symmetry group about its centre. Both are permitted and both keep the contact exactly. Their composite does not. The faint discs are the two permitted displacements taken separately; the solid one is one followed by the other, and its centre sits 1.049 radii off the dashed line where a tangent disc's centre has to be. The excursion is exactly |t₂ sin φ₁| — the second displacement's slide times the sine of the first one's turn — derived from the two displacements rather than from the composition and agreeing to 10⁻¹⁶. A lower pair's freedoms compose and a higher pair's do not, which is why a joint's freedom count is the dimension of a group in one case and the dimension of nothing in the other.

A higher pair has no group

A disc resting on a straight edge may slide along it and may turn about its own centre. Both keep the contact exactly. Do one and then the other and the contact lifts off the edge by |t sin φ| — up to 1.2 radii over an ordinary range — so the two freedoms are real and the pair of them is not closed. The count is still right and there is nothing for it to be the dimension of.

pairs · Pair Seven arrangements, one routine, and the two that hold. Every row is the same three steps: write down one row per contact — the moment of its normal about the origin, then the normal itself — take the convex hull of those rows, and ask whether the origin is inside it. The parts differ, the numbers of contacts differ, and the routine does not. Two of the seven hold. The other five leave the part something, and the interesting column is what: four rays of rotation for the pinwheel, a translation straight out of the vee, and for the last two a whole line rather than any number of rays, which is what a rank below three means and is the case a reader has to be warned about. Note that the four contacts of the second row are the four of the first row, on the same four edges of the same square, at the same distance along each. positioned by solving, not by drawing.

Six hold nothing

Every exact-constraint coupling on this site — Kelvin, Maxwell, three-two-one, and a Kelvin clamp with a seventh pad added — has rank six and holds the part not at all. The escape a Maxwell coupling leaves is a pure vertical translation with nothing else in it, which is not a defect: it is what a coupling is, and gravity is the seventh contact nobody draws.

holding · Mobility How far the driven link turns is a fact about the lengths, not the chain. Each of the sixteen eight-link chains, given the arbitrary placement its own layout produces, driven from its first available choice of frame and input, and swept until a frame stops closing. 10 of the sixteen reach every angle and the rest rock through between 107° and 244°. Nothing in this chart is a property of the chains. Change the placement and the bars change; the census above them does not. It is here because it is the sharpest way to say what this field does and does not decide, and because the temptation to read a topology census as a catalogue of machines is exactly the mistake it prevents.

The chain has no lengths

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

topology · Topology What one point can and cannot see of a group. For each of the twelve: the group's dimension, the dimension of one point's orbit under it, and the difference — the stabiliser, the motions that leave that particular point exactly where it is. The orbit is the only picture a group has, and this table is the honest caption on it. Planar motion and spherical motion are both three-dimensional and both sweep a point over a two-dimensional surface, so each leaves one motion doing nothing at all: a turn about the plane's normal in one case, a turn about the radius in the other. A point does not see the whole group, and no drawing of one trajectory can be a complete picture of what a joint permits.

Six things a joint is not

A freedom count read as a description, a screw system read as a group, a pair list read as a convention, a trajectory read as a determination, a nominal alignment read as a delivered one, and a higher pair read as a larger joint. Six claims, each of them what a careful person would say, each answered with a number.

wrong · Misconception Nine bars joining two sets of three joints, at three placements of one motion. Joints B₁, B₂ and B₃ lie on the horizontal line at -2, 1, 3, joints W₁, W₂ and W₃ on the vertical line through the same point at -1.5, 1, 2.5, and every joint of one set is barred to every joint of the other. Counted, nine bars on six joints leave no freedom. Drawn here at three placements, the joints have slid along their lines — B₂ at 0.632 and W₂ at 1.265; B₂ at 1.000 and W₂ at 1.000; B₂ at 1.265 and W₂ at 0.632 — and every one of the nine bars has the same length in all three, to 4.4e-16.

Nine bars that ought to be rigid

Join each of three joints to each of three others and the nine bars leave no freedom, by the count and by the rank, wherever the joints are. Put one set on a line and the other on a line at right angles and the framework moves, all the way round a loop, with no bar repeating any other: take away any one of the nine and the motion is unchanged, take away any two and it gains a freedom. Tilt the lines by a degree and it still has a freedom by rank and cannot move at all.

constraint · Mobility 5 braces, and it is rigid. A 3×3 grid of squares with 5 of its cells braced by a diagonal, and no freedom left. The bipartite graph on the 3 columns and 3 rows, with one edge per braced cell, has 1 component — and the number of freedoms is one less than that, at every bracing there is. Nothing in the rank computation knows about columns, rows or graphs.

Which diagonal rigidifies a grid

A three-by-three grid of squares needs five diagonals and eighty-one of the hundred and twenty-six ways of placing five will do. Which ones is not a rank question at all: it is whether a graph on the grid's columns and rows is connected, and eighty-one is the number of that graph's spanning trees.

networks · Network What each machine is sold with. Every mechanism in this field arrives with a number. The middle column is that number; the right-hand column is what the mechanism was measured to have, by the library named under the row. The coloured bar and the word under each machine are the verdict: exact means the mechanism has the number at every position, mean that it is the average of something that varies within every cycle, bounded that it is false by a stated and negligible amount, point that it is the value at one position, and quoted that the mechanism has no such quantity at all. The lit row is the one this essay is about. Of the 14 rows, 6 are quoted and 3 are exact.

Which numbers survived

Fourteen machines were measured against the numbers they are sold with. Three survived exactly, and all three are counts. One is false by a stated bound, two are exact means of things that vary, three are honest values at a stated position, and six name quantities their mechanisms do not have. The pattern is not about honesty; it is about what kind of thing a number is.

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

A graph has no numbers at all

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

topology · Topology The same patch twisted 17.2°: 35 mechanisms, all at the edge. A rhombus of 8 × 8 kagome cells — 192 joints, 346 bars — with every up-pointing triangle turned by 17.2° about its own centre. Each joint is drawn with an area proportional to its weight: its share of the diagonal of the projector onto the patch's mechanisms, which does not depend on how the mechanisms are written down and adds up over the joints to the number of mechanisms, 35. That number is Maxwell's count, 2 × 192 − 346 − 3, and the rank agrees with no redundant bar. The mean weight is 0.341 on the outermost ring of cells and 0.015 on the innermost. The twist kinks every line of bars at every joint, and the mechanisms fall away from the edge 23-fold in 3 cells.

The count says how many and not where

A kagome lattice has three joints and six bars in every cell and counts to exactly nothing, so a patch cut from it has as many mechanisms as its edge has lost bars: 5L − 5 for a rhombus of L cells a side, which the rank confirms at every size with no bar redundant. Straight or twisted, the number is the same. Where the mechanisms are is not: a straight patch keeps nearly half its edge weight in the middle, and a patch whose triangles are turned by 17° keeps a twentieth.

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

An arm is a tree

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

serial · Serial A framework built off square, and the play it asks for to move at all. The nine bars with their two lines 4° from perpendicular, as built and after the first joint has been pushed 0.2 along its line. The framework has a freedom by rank and no motion, so the push cannot be taken with the bars at their lengths. With every bar allowed to be wrong by 1.509e-4 — which is a radial clearance of 7.547e-5 at each end — a placement exists, and the worst bar in it is out by 1.509e-4. The same push on the perpendicular framework needs no allowance at all, because there it is a motion.

The right angle as a tolerance

Dixon's nine bars move only when their two lines are exactly perpendicular, and a framework built a degree off square has a freedom by rank and no motion at all. Give its joints clearance and it moves a bounded distance: the play each bar needs is proportional to the tilt and to the square of the travel, one constant serves every tilt, and all nine bars end up at that play exactly. Then the framework reaches its first crossing and the law is left three hundred times behind.

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

Where the shortest loops are

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

practice · Tolerance Every surface tried, and the group it permits. The census the six lower pairs come out of. Each row is a surface, sampled at 240 points; the freedoms column is six minus the rank of a matrix with one row per point, saying that the velocity a twist gives that point is tangent to the surface. Nothing is fitted and no shape is recognised — the surface's own normals write the matrix down. Three different surfaces of revolution give the same group, which is the content of the classification; two surfaces give nothing, which is what almost every surface gives. Eleven surfaces, six groups. The last column is the ratio of the smallest singular value kept to the largest discarded, so a row reading 10¹⁵ is not near being reclassified by anybody's tolerance.

The pair a catalogue sells

A plain bearing is a cylindrical pair and a catalogue calls it a bearing. Add two thrust faces and it is a revolute pair, which is a different joint and changes every mobility count downstream. The kinematic identity of a bought part is decided by which surfaces touch, and the catalogue's word for it is not the same information.

practice · Clearance The bores, and the one line that has to pass through all of them. A hinge of 6 knuckles, its bores drawn at the distance each was made from the nominal axis in units of the bore tolerance. The leaf is a rigid body, so its pins are on one straight line — two parameters of position and two of direction — and it assembles when some line passes within the clearance of every bore. The line drawn is the one whose largest miss is smallest, and that miss is 0.875 of the tolerance. 2 of the 6 bores are at that distance and hold the fit; the rest are slack and could have been bored anywhere inside it without changing the answer.

A piano hinge is not forty door hinges

A three-knuckle hinge works because the misfit its bore errors create is smaller than the play already in its pins. A piano hinge has forty knuckles and thirty-nine of them are redundant, so the obvious reading is that it needs thirteen times the play. It needs two and a half times, and it can never need more than the bore tolerance itself — because a rigid leaf has one axis and a line through the middle of the errors misses every bore by at most the largest of them.

practice · Overconstraint Which numbers have a size. Twelve quantities from eight fields of mechanism kinematics, each scaled by taking every length in its mechanism up and down together and fitting the power its value follows. A shape lands on zero, a length on one, a curvature on minus one, an area on two, and every row lands on an integer to 3.3e-11. The sorting is not a units argument — it is measured from each field's own library, by asking that library the same question. What it says is that most of what kinematics computes is a shape: a mobility, a Grashof class, a transmission angle, a contact ratio and a velocity ratio are all unchanged by making the machine bigger, so all of them are recoverable from a measurement that cannot recover a single length. The row worth stopping at is the tolerance band: held to a fixed ±0.01 it lands on minus one, because ±0.01 is a length and a bigger machine held to the same absolute tolerance is a better machine. Written as a percentage the same band lands on zero. One quantity, two drawing conventions, two different kinds of number.

A count is neither

Every quantity in the scaling survey lands on an integer power — zero for a shape, one for a length, two for an area. A mobility lands nowhere. It has no dimension at all, it does not move under any perturbation, and the probe that sorts the rest of the site's numbers returns nothing for it.

metrology · Parameter

Named alongside it

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

RankConstraintOverconstraintGrübler's criterionRedundant constraintDegrees of freedomKinematic chainNetworkToleranceConstraint jacobianDisplacement subgroupInversion

All concepts