Machine Kinematics
solved, then drawn
What can move

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.

Assumes Counting and measuring mobility.

14 min read 8 figures Two routes to the same numberSolved, not drawn

This field’s founding comparison is Grübler’s count against the rank of the constraint Jacobian. Two instruments, both computing the mobility of an assembly, and the interesting cases are the ones where they disagree.

Almost every disagreement the site has drawn is an accident: a mechanism whose axes happen to satisfy a condition, whose constraints therefore repeat one another, and whose count comes out short. Grübler says the mechanism cannot move and it moves; a Sarrus linkage is counted at zero and has one; a planar four-bar treated as a spatial loop is counted at minus two.

This rung is the first case on the site where the disagreement was built in, and it is the largest by two orders of magnitude.

Three parallel bars, and a formula that says this cannot moveFive 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 redundant oneGrübler: 3(5−1) − 2(6) = 0Jacobian: 6 − rank 5 = 1
Fig. 1 The field’s founding disagreement at its usual scale: a mechanism the count says cannot move.

Why the field needed a case this large

The comparison at the head of this field has been made on small mechanisms for seventeen phases, and there is a limitation in that which is worth naming before the numbers arrive.

On a four-bar, a count of one and a rank of one agree, and on Sarrus a count of zero and a rank of one disagree by one. A disagreement of one is real and it is also the smallest possible disagreement, which means every earlier example is consistent with the count being nearly right — off by a constant, off by the number of loops, off by something small enough that a rule of thumb could absorb it.

A disagreement of a hundred and fifty-four cannot be absorbed. It rules out any reading in which the count is approximately the mobility, and it makes the relation between the two instruments visible as what it is: the count is an upper bound on the number of independent conditions, and how far it overshoots depends entirely on the mechanism.

A large case is not a bigger version of a small one when the quantity in question is a difference. That is why this rung exists at a scale nobody would build, and it is the same reason the topology field’s census runs to three thousand candidates rather than to thirty.

The numbers

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.
Fig. 2 Four compiled machines, counted and measured, braced and unbraced. The count is right about the unbraced ones and badly wrong about the braced ones, and the rank says one every time.

Unbraced, the quintic’s machine: four hundred and eighty unknowns, four hundred and seventy-nine equations. Counted mobility one; rank four hundred and seventy-nine; measured mobility one. The two instruments agree.

Braced: one thousand and ninety-six unknowns, one thousand two hundred and forty-nine equations. Counted mobility minus a hundred and fifty-three. Rank one thousand and ninety-five; measured mobility one.

A hundred and fifty-four surplus equations, and a hundred and fifty-four braces.

The machine compiled from an ellipse. x^2 + 2.2y^2 − 1.6, compiled: 34 bars and 29 joints, painted by what each part is for. The two-link arm at the pivot carries the tracing point; the reflectors and means build each term's angle; the rigid offsets fix the constant φₖ and the amplitude; the translators carry those directions out along the summing chain, whose last vertex is held on a line. That last constraint is the equation. Every joint drawn is the output of a Newton–Raphson solve on 54 equations, converged to 7.4e-16, and the polynomial at the tracing point is 2.2e-16.
Fig. 3 An unbraced compiled machine. Every gadget in it contributes exactly as many equations as unknowns, which is why the count is exactly right about it.

Why the unbraced agreement is worth noticing first

It is easy to walk past the top half of that table, and it is the more surprising half.

Grübler is wrong about a great many of the mechanisms this site draws, and the reason is always the same: constraints that are not independent. A ring of hinges closes at every size because its conditions repeat, and a network of loops has fewer independent conditions than loops.

An unbraced compiled machine has no redundancy at all, at any size. Every gadget contributes exactly as many equations as it contributes unknowns — a mean two and two, a reflector four and four, an offset two and two, a translator two and two — and none of them repeats a condition another has already stated.

That is a consequence of the construction being acyclic. A redundant constraint is a statement that two routes through the mechanism reach the same conclusion, and a mechanism with no loops in its construction has one route to everything.

So a four-hundred-bar machine can be a case where the count is exactly right, and the reason is structural rather than lucky.

The third instrument the field has, and why it is silent here

This field carries three ways of asking what a mechanism can do, not two, and the third one is worth bringing in because it has nothing to say about a compiled machine and the reason is informative.

Grübler counts. The Jacobian’s rank measures. And a mobility counted over every subset of the links is the only one of the three that can find a rigid subchain — a part of a mechanism that cannot move even though the whole can. The topology field needed it because a count over the whole assembly is unanimous about a great many ten-link graphs that are not the mechanisms they are described as.

Run over a compiled machine it finds nothing, and it could not. Every gadget is determinate, so no subset of a compiled machine is rigid while the whole moves — the walk that places every joint in order is the same fact from the other side.

An instrument that reports nothing is reporting something, provided it is known to report things elsewhere. On these machines it says the structure is as simple as the count and the rank between them suggest, and that there is no third phenomenon hiding.

The same four bars, and the bar that tells them apart. Left, a parallelogram; right, the crossed assembly of exactly the same four lengths. In the parallelogram the midpoints of the two opposite sides are one side-length apart — here 1.000, and it stays that as the linkage moves — and in the crossed assembly they are 0.115. So a bar between those two midpoints admits the first and refuses the second. That is the brace: two rigid attachments and one bar, and it is the whole repair of Kempe's argument.
Fig. 4 The brace and what it asserts: in a parallelogram the two midpoints are already one side apart, so a bar between them adds an equation and no information.

What a brace does to the count

A brace is one bar and two rigid attachments: four unknowns, five equations. One equation more than it pays for.

That equation is not new information. In a parallelogram the two midpoints are already one side-length apart, as a consequence of the four bars; the brace asserts something the mechanism was doing anyway. So the machine’s constraints on the intended branch are unchanged in content and increased in number, which is exactly what redundancy is.

The count cannot see that. It counts equations and unknowns and subtracts. A hundred and fifty-four redundant equations subtract a hundred and fifty-four, and the answer goes from one to minus a hundred and fifty-three.

The rank can see it. The Jacobian’s rank is the number of independent conditions, and the redundant rows contribute nothing to it, so the rank comes out at one thousand and ninety-five and the mobility at one.

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.
Fig. 5 The same numbers as a table: unknowns, equations, the count, the rank and the two mobilities.

What a builder would see

The count and the rank are both abstractions, and it is worth saying what the braced machine is like as an object, because that is where the disagreement stops being a curiosity.

It is a mechanism with one input and one output, and it turns. Driven at its crank, every joint moves, and the tracing point runs along the curve. Nothing about handling it would suggest a structure.

What a builder would notice instead is that it is fussy. A hundred and fifty-four of its bars assert something the rest of the machine already asserts, so a hundred and fifty-four of them have to be made to the length the others imply, or the machine will not assemble at all. An unbraced machine tolerates a mis-made bar by taking up a slightly different configuration; a braced one has nowhere to put the error.

That is the practical face of overconstraint and it is the same face a three-knuckle hinge has: the redundancy that buys the good property is also what makes the thing require a jig. The practice field measures the cost in general and the tolerance rung beside this one measures it for these machines specifically.

Why this is not the same finding as the field’s earlier ones

The distinction matters and it is easy to blur.

The mechanism Grübler says cannot move is a case where the count is misleading about the mechanism. Somebody looking at the count would build the wrong thing, or would fail to build the right thing, and the site’s whole point in that essay is that the count is not to be trusted alone.

Here the count is not misleading anybody. It is measuring the machine correctly and reporting a quantity that is not the one wanted: the number of equations minus the number of unknowns really is minus a hundred and fifty-three, and it really does have the interpretation the formula assigns it only when the equations are independent.

The difference between an instrument that is wrong and an instrument being read past its assumptions is worth keeping, because the repair is different. The first needs a better instrument. The second needs the assumption stated, and Grübler’s assumption — that the constraints are independent — is stated in every textbook and forgotten in most applications.

What a brace takes away. The same census, run again with every parallelogram braced — a bar between the midpoints of two opposite sides, which is a length that does not change in a parallelogram and does not hold in the crossed one. 4 spurious assemblies become none, the machine still assembles, and every assembly that survives puts the tracing point in the same place. The brace is one redundant equation per parallelogram: it is deliberate overconstraint, and the mobility measured from the Jacobian's rank does not change by it.
Fig. 6 What the hundred and fifty-four surplus equations are for: the spurious assemblies stop existing, and the machine still assembles and still turns.

What the redundancy buys

A count that reports a structure where there is a mechanism is normally bad news. Here it is the price of something specific and it is worth restating what.

An unbraced compiled machine has 2k2^k assemblies for kk parallelograms, and on the hyperbola’s twenty-bar version four of the eight that close draw a different curve at the same closure residual. Bracing removes them: four spurious assemblies become none, and the machine still assembles and still turns.

So the hundred and fifty-four redundant equations are not waste. They are what makes the machine’s configuration space have one component where it had many, and there is no cheaper way to do it: a constraint that removes an assembly without removing a freedom has to be redundant, because a non-redundant one would remove a freedom by definition.

That sentence is the general form of the finding and it is the piece worth carrying to another field. Redundancy is not merely a hazard to be counted around. It is the only mechanism available for excluding a configuration while preserving a motion.

The count is right about how much was spent

There is a reading of the minus a hundred and fifty-three that makes it useful rather than merely wrong, and it is worth having.

The gap between the counted mobility and the measured one is exactly the number of redundant equations, and the number of redundant equations is exactly the number of braces. So the count, subtracted from the rank’s answer, measures the deliberate redundancy — which is a quantity somebody designing this machine would want and which neither instrument reports directly.

That is a small reframing and it generalises. On a mechanism whose overconstraint is accidental, the same difference measures how many of its constraints are coincidences — how much of its behaviour depends on an exact geometric condition being held. The spatial field reads it that way for Bennett’s linkage and for Sarrus, and the reading is the same arithmetic with a different story attached.

A count that disagrees with a rank is not a broken instrument; it is a second measurement whose difference from the first is the interesting number. That sentence is worth more than either of the two readings on its own, and this rung is the case that makes it unmissable, because the difference is a hundred and fifty-four and nobody can pretend it is a rounding error.

Every curve in the catalogue, compiled and counted. The bar count is not a formula: it is the number of bar constraints the compiled mechanism actually carries, asked of the object rather than predicted. A line costs five bars and a quintic four hundred and thirteen, which is the field's central number — exactness is paid for in size. The all-revolute column replaces the one prismatic pair that closes the chain with a Peaucellier cell, which costs seven bars whatever the curve, so it is more than half the machine on a line and a rounding error on a quintic. The last column is the polynomial evaluated at the traced point, worst over each machine's working arc.
Fig. 7 The machines the two instruments disagree about, and how large each of them is.

Reading the two instruments together

The site’s habit is to have two routes wherever two exist, and this is the clearest demonstration of why.

Alone, the count says the machine is a structure a hundred and fifty-three times over. Anybody acting on that would conclude the design was hopeless.

Alone, the rank says the mobility is one. True, and it says nothing about how much apparatus was spent getting there — a rank of one thousand and ninety-five on one thousand and ninety-six unknowns is the same reading whether the machine has zero redundant equations or a hundred and fifty-four.

Together they say what is actually true: one freedom, delivered by constraints of which a hundred and fifty-four are duplicates, deliberately. The difference between the two readings is the quantity of interest, and neither instrument reports it on its own.

The spatial field makes the same argument about mechanisms whose overconstraint is accidental, and this rung is that argument with the sign of the intention flipped.

What the ratio is asserted on, rather than the totals

The site’s gate for this field checks a ratio and not a total, and the choice is deliberate.

The totals depend on the curve: the hyperbola has four braces and four redundant equations, the ellipse seven and seven, the cubic twenty-nine and twenty-nine, the quintic a hundred and fifty-four and a hundred and fifty-four. Asserting any of those pins the gate to one catalogue and breaks it the moment a curve is added.

What is asserted is that the redundancy divided by the brace count is exactly one, and that the measured mobility is unchanged by bracing. Both are properties of the construction rather than of any particular input, and both would break if a brace were ever mis-sized — a brace bar of the wrong length is not redundant, it is a contradiction, and the machine would fail to close rather than gain a redundant equation.

That last point is worth having: the redundancy is evidence that the brace is right. A brace that asserted something false would show up as a machine that will not assemble; one that asserted something already true shows up as exactly one extra rank deficiency. The gate is reading the second and would notice the first.

Where the count goes wrong is not where it is usually blamed

A last note for a reader who has met Grübler’s formula in the usual way.

The textbook caution about the count is that it does not know about special geometry — parallel axes, concurrent axes, links of equal length — and that a mechanism satisfying such a condition may move when the count says otherwise. That is a real caution and it accounts for most of this site’s earlier examples.

It does not account for this one. There is nothing special about a compiled machine’s geometry: its link lengths come out of a polynomial’s coefficients and its angles from a scan, and no two of them satisfy any condition anybody imposed. The count is wrong here because equations were added that were already implied, which is a different route to the same arithmetic.

So the assumption Grübler’s formula actually needs is not no special geometry. It is no redundant constraints, and special geometry is one way to acquire them. Naming the assumption at the right level of generality is what makes the braced machine a comprehensible case rather than an anomaly, and it is worth doing because the usual phrasing would leave a reader looking for a coincidence that is not there.

A machine with one thing in it that must be solved at once. Each compiled machine's joints, walked in the order they can be placed: a joint goes down as soon as two things already placed decide where it is. Every one of these machines unwinds completely, and each contains exactly one pair that has to be solved together — the arm and the parallelogram that carries its second angle back to the pivot. Nothing larger than a dyad appears in a machine of two hundred and forty joints. The topology field's four-bar, at four, has no such decomposition at all; a compiled linkage is enormous and structurally trivial, which are not the same axis.
Fig. 8 The same machines walked joint by joint. Nothing in them has to be solved together beyond a single dyad, which is why an unbraced one has no redundancy to find.

The scale, and why it matters

Every earlier example of this comparison on the site is small enough to check by hand. A four-bar has eight unknowns. Sarrus has eighteen. The largest example before this one is a twelve-bar network.

Here it is one thousand and ninety-six unknowns and one thousand two hundred and forty-nine equations, and nobody is checking that by hand. Two things follow.

The rank computation has to be trustworthy at that size. It is a row reduction with a tolerance, and a tolerance is a decision about which numbers are zero — so the gap between the smallest value kept and the largest discarded is what makes the answer a measurement. On these machines that gap is enormous, because the redundant rows are exact duplicates in content rather than near ones.

The machine’s motion is not in doubt. Whatever the count says, the machine was driven through a hundred and sixty positions, each a converged solve, with its tracing point on its curve throughout. A disagreement between two instruments is much easier to adjudicate when a third observation settles it, and here the third observation is that the thing turns.

There is one more reason the size is worth reaching. The site’s own solver had to be able to handle a redundant system before any of this was possible, and it could — the Levenberg escalation in it was added three phases ago for a parallelogram with a third parallel bar, a mechanism with seven equations, six unknowns and rank five. That was a three-bar example and the machinery it needed is the machinery a machine with a hundred and fifty-four redundant equations uses unchanged.

A shared piece written for the smallest case carried the largest one without alteration, which is the ordinary way this fleet’s machinery earns its place and is worth recording when it happens.

That is the honest reason this rung is not alarming. A count of minus a hundred and fifty-three about a mechanism nobody has seen move would be a problem; about one that has been driven, position by position, with a residual published at each, it is a fact about the count.

Where this goes next

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. 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. 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.

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.

Compiled linkageConstraint rankDegrees of freedomGrübler's criterionMobilityOverconstraintRedundant constraint