Machine Kinematics
solved, then drawn
Drawn wrongly

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.

Assumes Many loops, one freedom.

15 min read 8 figures Two routes to the same numberExact and approximateThe singularities are the point

Six things that get said about assemblies of repeated units, each of them reasonable, each of them the natural generalisation of something that is true of a four-bar, and each of them answered here with a number. The mechanisms they are about are the six on the field’s ledger, and the instrument is the same one throughout: write the constraint Jacobian, take its rank, subtract.

One: the count says whether it moves

It gives the difference between two nullities, exactly, on every mechanism there is:

m−s=(unknowns)−(constraints).m - s = (\text{unknowns}) - (\text{constraints}).

The rank is a number no larger than either dimension of the matrix and the two nullities are what each of them has left over, so subtracting cancels the rank and leaves an arithmetic with no geometry in it. That is a true statement about a difference being read as a statement about one of the terms.

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.
Fig. 1 Six assemblies, one routine. Three of the counts are wrong and a fourth is right by accident.

The reading is safe when s=0s = 0 and only then. A deployable ring of eight pairs counts at nought — a structure — and has four freedoms and four dependencies. A Miura sheet of thirty-six panels counts at minus fifteen and has one freedom and sixteen dependencies. A Bricard octahedron counts at exactly isostatic and has one of each — no mechanism and no redundancy, on a framework that moves.

The count cannot certify itself. Whether it can be trusted is decided by ss, and finding ss means taking the rank, at which point the count is not needed.

Two: the nullity is the mobility

It is an upper bound. The tangent space to the configuration set is inside the null space of the Jacobian, so nothing is missed — and things are added.

The smallest mechanism that is not oneTwo 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.the whole travelrank 1 · one freedom · one dependencyobstruction 1.4142
Fig. 2 Two bars from a free joint to two pinned ones, all in line. The rank leaves a freedom pointing straight up; lifting the joint lengthens both bars.

Two bars in line have a rank of 1, a mobility of 1 and a dependency of 1. The freedom points upward, and lifting the joint stretches both bars quadratically with nothing to trade against. The second-order obstruction comes to 1.414214, and the walk travels 6.1×10−76.1 \times 10^{-7} of the 0.6 it is asked for — against a four-bar’s 0.384 of 0.4 and the octahedron’s 0.159982 of 0.160.

A framework with no dependencies has no obstruction and every flex extends, which is why this never came up in eighteen fields of chains. In a field of networks it is the ordinary situation.

Three: the flat state says whether a pattern folds

It cannot, and this is the most confidently wrong of the six because it is where anybody would measure.

At the flat state every crease lies in the plane, so each interior vertex’s three constraint rows span two dimensions, the rank falls by one per vertex, and the nullity rises. A three-by-three Miura sheet gives rank 8 and nullity 4, on a mechanism with one freedom.

What the flat state does and does not know. At the flat state every crease lies in the plane, so the three rows each vertex contributes span two dimensions rather than three and the rank is exactly twice the interior vertex count on every row here. The nullity is correspondingly larger than the mechanism's — four freedoms on a three-by-three sheet with one — and the last row is the point: a grid whose vertices have been moved and which cannot fold at all has the same rank and the same nullity at the flat state as the Miura does. The flat state is where the branches meet, so the tangent space there is the union of all of them and belongs to none; and whether any of those directions is the start of a motion is a second-order question that no rank answers.
Fig. 3 Three sizes of Miura sheet at the flat state, and one grid that folds to no angle at all.

The same grid with its interior vertices moved a tenth of a panel — which does not fold, at any angle, with a best residual of 5.9×10−65.9 \times 10^{-6} at a fold of 0.02 radians — gives rank 8 and nullity 4. Identical. The flat state’s first-order behaviour does not distinguish a mechanism from a structure, and the difference is entirely second order.

The honest instrument is to push the pattern off the flat state and see whether anything closes — and the flat state is where every branch of every folding meets, which is why its nullity counts them all.

Four: enough scissors will make a ring

No number of ordinary scissors closes into a ring, at any size, and the reason is one measurement.

Two straight bars pinned at their middles join their neighbours along two lines — the line through the two left-hand end pins and the line through the two right-hand ones. Those lines are parallel at every opening, to 2.5×10−142.5 \times 10^{-14} radians.

Two straight bars, whose connection lines never meet. The same measurement made on an ordinary scissor. Two straight bars pinned at their middles, opened four different amounts; the dashed lines through the two left-hand pins and through the two right-hand ones are parallel every time, to 0.0e+0 radians. A ring needs those lines to be its radii and needs them to turn by 2π/n from one pair to the next. Parallel lines turn by nothing, so an ordinary scissor ring closes at n = ∞ and nowhere else. This is the same fact the tong figure states from the other end.
Fig. 4 Two straight bars at four openings, with their two connection lines drawn extended. Parallel every time.

A ring needs those lines to be its radii, and radii turn by 2π/n2\pi/n from one unit to the next. Parallel lines turn by nothing. So an ordinary scissor ring closes at n=∞n = \infty and nowhere else, which is a lazy tong.

Bending each bar turns the lines by the bend, exactly, at every opening — 135.000000000° for a 135° kink — and a ring of nn closes when n(180°−β)=360°n(180° - \beta) = 360°.

Five: if every unit moves, the assembly moves

Take a Miura pattern and displace each interior vertex by about a tenth of a panel. Every vertex is still developable — its sectors sum to a full turn to nine decimal places, because a vertex drawn on a flat sheet has no choice about that. Every vertex is still a spherical four-bar and folds perfectly well on its own, with two branches and a large range.

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.
Fig. 5 Four vertices that each fold, assembled into a patch that folds to no angle at all.

The assembly of four of them folds to nothing. 5.9×10−65.9 \times 10^{-6} at 0.02 radians and 1.6×10−31.6 \times 10^{-3} at half a radian, growing monotonically, with no seed and no number of iterations moving it.

A unit’s freedom and the assembly’s are different objects, and the condition that connects them is a property of the tiling. On this family there are (n−2)2(n-2)^2 such conditions — one at nine panels, a hundred at a hundred and forty-four.

Six: a null space is a null space

The sixth is about the instrument rather than about mechanisms, and it is the one that would have quietly falsified the other five.

The usual way to get a null space out of a small matrix on this site is to accumulate ΣvvT\Sigma vv^{T} and take its symmetric eigenbasis. Forming ATAA^{T}A squares the condition number, so the smallest singular value the route can distinguish from nought is ε\sqrt{\varepsilon} — about 1.5×10−81.5 \times 10^{-8} of the largest — whatever tolerance the caller passes.

The instrument, and the floor nobody had written down. The usual way to get a null space out of a small matrix in this fleet is to accumulate Σvvᵀ and take its eigenbasis. That squares the condition number, so the smallest singular value it can distinguish from nought is √ε — about 1.5 × 10⁻⁸ of the largest — whatever tolerance it is handed. Asked at 10⁻¹⁰ it over-states the rank of every constraint matrix in this field: a triangle's three bars come back as five independent constraints, and the deployable ring's forty-four as forty-five, which reports the ring as a rigid body with no deployment. At 10⁻⁷, which is what its own callers pass and what a six-by-six screw system wants, it is right every time — which is exactly why the floor had never been reached. The rank column is one-sided Jacobi on the matrix itself, which resolves a ratio of 10⁻¹⁴.
Fig. 6 Four of this field’s constraint matrices, ranked two ways.

Asked at 10−1010^{-10} it over-states the rank of every constraint matrix in the field. A triangle’s three bars come back as five independent constraints. The octahedron’s twelve come back as fifteen — a rank larger than the number of rows. The deployable ring’s forty-four come back as forty-five, which reports the ring as a rigid body with three freedoms, all of them rigid motions, and no deployment at all.

Nothing says so. The null vector is returned as part of the row space and the nullity is simply smaller. At 10−710^{-7}, which is what every existing caller passes and what a six-by-six screw system wants, the same route is right every time, which is exactly why the floor had never been reached on any figure this site had drawn.

Where each of them comes from

It is worth saying why these six in particular, because none of them is a careless belief and five of them are the correct thing to believe about the mechanisms in the rest of this collection.

A four-bar’s count is right, so counts are right. A four-bar’s nullity is its mobility, so nullities are mobilities. A four-bar’s configuration space is a manifold near every configuration a designer cares about, so a rank taken anywhere settles the question. Four bars that each move give a mechanism, so units that each move give an assembly that moves.

Each of those inferences is sound for exactly one reason: the constraints do not repeat one another. That condition is invisible because it is universal in eighteen of this site’s twenty fields, so it never had to be stated and never was. It fails everywhere in the nineteenth, and the fifth belief — that a unit’s freedom is the assembly’s — fails there for the same reason and not for a different one.

The fourth item is the exception on the list, and it is the one that costs nothing. A scissor ring does not close, so nobody builds one and then wonders why; the geometry announces itself at the drawing board. It is here because it is the case where the same arithmetic is visible without any rank being taken at all.

What each of them costs

It is worth pricing the six, because they are not equally expensive to believe.

The count costs a design. Believing a deployable ring cannot move means not building it; believing a Miura sheet cannot fold means the same. The two most useful mechanisms in this field are both ruled out by an arithmetic that is free and wrong.

The nullity costs a mechanism that does not work. A drawing whose rank leaves a freedom and whose freedom is blocked is a mechanism somebody will build and find rigid, and nothing in the analysis will have said so.

The flat state costs a whole pattern. It is the measurement anybody would make, it takes no solving, and it gives the same answer for a sheet that folds and a sheet that does not.

The scissor ring costs nothing, because the mechanism refuses to be assembled and the mistake is discovered at once. It is on the list because it is the one case where the geometry announces itself.

The unit-implies-assembly claim costs a manufacturing run. Every vertex checks out, every panel is right, and the thing does not move.

And the instrument costs everything else on the list, silently, which is why it is the one worth writing down.

A seventh, which is a modelling mistake rather than a claim

A network is a big mechanism. It is not, and treating it as one produces the wrong instrument rather than the wrong answer.

A chain of a dozen bodies is analysed by finding its loops, writing their closures and solving. A network of three hundred is analysed by assembling a matrix and taking a rank, because there is no order in which its loops can be solved one at a time — a crease between two interior vertices appears in two closures, and neither can be solved before the other.

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.
Fig. 7 The step that is invisible on a chain: the loops are the graph’s independent cycles, and there are as many as edges less nodes plus one.

The practical form is that every quantity in this field is per unit or is a rank, and no quantity is a description of a particular body. A picture of sixty panels with every pin labelled is a diagram of nothing.

The measurements, in one place

Each of the six is answered by a number and it is worth having them together, because a refutation without a measurement is only a contrary opinion.

For the count: a ring of eight pairs at 0 against 4 measured, a sheet of thirty-six panels at −15-15 against 1, an octahedron at 6 against 7, and a tong of any size at 4 against 4. Four assemblies, three disagreements.

For the nullity: an obstruction of 1.414214 on the two bars, and a walk that travels 6.1×10−76.1 \times 10^{-7} of 0.6 — against 0.999885 of what was asked on the octahedron and 0.959 on a four-bar’s coupler. Six orders of magnitude between the two cases.

For the flat state: rank 8 and nullity 4 for both a three-by-three Miura sheet and the grid that will not fold, against a folded rank of 11 and mobility 1.

For the scissor ring: 2.5×10−142.5 \times 10^{-14} radians between connection lines that should differ by 2π/n2\pi/n, and 135.000000000° when the bars are bent.

For the units: sectors summing to 360.000000000° at every one of four vertices, each folding on its own, and an assembly whose best residual is 5.9×10−65.9 \times 10^{-6}.

And for the instrument: ranks of 5, 15, 46 and 45 where the answers are 3, 11, 44 and 44, at a tolerance of 10−1010^{-10} — and 3, 11, 44 and 44 at 10−710^{-7}.

There is one more number that belongs with them and is easy to overlook. Every rank in this field is reported with the gap it was decided across, and the gaps run from 101210^{12} on the largest sheet to effectively infinite on the small assemblies. Without that column the sixth item on this list would be unanswerable, because a rank quoted without its margin is a rank that cannot be argued with.

What these six have in common

Five of the six are the same mistake: something true of a chain, generalised.

A four-bar’s count is right, so counts are right. A four-bar’s nullity is its mobility, so nullities are mobilities. A four-bar’s configuration space is a manifold, so a rank is enough. Four bars that each move give a mechanism, so units that each move give an assembly that moves. Each of those inferences is correct for exactly the reason the four-bar case is: no dependencies among the constraints. That condition is invisible because it is universal in the first eighteen fields of this site, and it fails everywhere in the nineteenth.

The sixth is different in kind and is the one worth carrying furthest. An instrument that has been right everywhere it was pointed has been right within its own range, and nobody had written the range down.

The pattern behind all of them

There is a single sentence that generates five of the six and it is worth stating on its own, because it is what a reader should carry out of this field.

A constraint matrix has two null spaces, and every mechanism in this site’s first eighteen fields has only one of them.

Where the second is empty, the count is the answer, the nullity is the mobility, a rank at any configuration settles the question, and an assembly’s behaviour is its units’ behaviour added up. Where it is not, none of those hold, and how badly they fail is measured by the same number: the dimension of the second null space.

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.
Fig. 8 That number, in its own column, next to the two it explains.

Nought on a lazy tong at any size, which is why a tong behaves like a chain. Four on a deployable ring. One on a Bricard octahedron. Sixteen on a Miura sheet of thirty-six panels and a hundred at a hundred and forty-four. Read that column first, and it says what the rest of the table is worth.

Two more that are nearly true

“A network’s mobility is a property of the network.” Nearly. For a finite assembly it is, once it is said what is held; for a pattern that repeats for ever it is not, and a square grid is rigid if its period is held and shears if it is not. The same six lattices measured both ways disagree on two of the six.

“Redundant constraints could be removed.” Nearly. Four dependencies do not mean four particular pins are spare: they mean the forty-eight constraint rows span a forty-four-dimensional space, and there is no distinguished set of four to delete. Removing four arbitrary pins gives a different mechanism with more freedoms, not the same one.

What this essay does not claim

It does not claim that counts are useless. Grübler’s arithmetic is exact about what it is about, is free, and is right on every mechanism in eighteen of this site’s twenty fields — including a lazy tong at any size, which is a network by every other measure.

It does not claim that a rank is the last word either. A rank is an upper bound and a candidate list, and the walk is what settles the matter.

And it does not claim that any of these six is an unusual belief. Five of them are the correct thing to believe about every other mechanism on this site, which is why they are worth a page.

The six share a shape and it is worth naming as sharply as the list allows: each of them is an instrument answering correctly and being asked the wrong question. A count is right about m−sm - s and is asked for mm. A nullity is right about a tangent space and is asked about a motion. A flat state’s rank is right about that configuration and is asked about the pattern. A unit’s mobility is right about the unit and is asked about the assembly. A null space computed at a threshold is right about that threshold and is asked about the matrix. In not one of the six is anything computed incorrectly, and in all six the number is used to answer something it does not address. That is why the remedy is never compute more carefully and always compute the other thing. The field’s own arrangement follows from it: every table here carries the count, the mobility, the dependency count and the singular-value gap side by side, precisely so that no one of them has to stand in for another — and the cost of carrying four numbers is one decomposition, which was being taken anyway.

Where this goes next

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. 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. 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.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

DeployableDesign ruleFlat stateGrübler's criterionMobilityNetworkNull spaceRankRedundant constraint