Two ways to be overconstrained

The last essay left two mechanisms holding identical paperwork.

A planar four-bar built as a spatial loop: four links, four revolute joints, Kutzbach −2, rank 3, three redundant constraints, mobility 1. Bennett’s four-bar: four links, four revolute joints, Kutzbach −2, rank 3, three redundant constraints, mobility 1.

Every number matches. And the two mechanisms are not alike at all.

Any four bars whatever will make a planar four-bar. Pick four lengths, pin them end to end with parallel axes, and the thing turns. There is no condition to satisfy — the Grashof condition decides which link can rotate fully, not whether anything moves.

Almost no four bars will make a Bennett linkage. The lengths and the twists must satisfy $\sin\alpha/a = \sin\beta/b$ exactly. Miss it by a thousandth and the mechanism is a structure.

Both are called overconstrained. This essay is about the fact that they are overconstrained in genuinely different ways, and about the measurement that says so.

What a subgroup is, and why it settles the question #

The rigid displacements of space form a group: do one, then another, and the result is a third. Some of its subgroups are familiar.

Planar motion is a subgroup. Every displacement that keeps a body in a given plane composes with every other to give a third that does the same, and the collection is three-dimensional — two translations and a rotation.

Spherical motion, keeping one point fixed, is a subgroup. Three-dimensional: three rotations about that point.

Translation is a subgroup, and so is translation along one line, and rotation about one line, and the screw motions of one fixed pitch about one line. There are a handful more, and the list is short and classical.

Now the connection. A mechanism whose motion lies inside one of these subgroups has, at every position it can reach, a screw system that is the same subspace — the subgroup’s own tangent space. It cannot be otherwise: the mechanism never leaves the subgroup, so its instantaneous motions never leave the subgroup’s set of instantaneous motions.

A planar four-bar’s screw system is $\{(0,0,\omega_z\,;\,v_x,v_y,0)\}$ at every position, for every set of link lengths, forever. That is why any four bars will do. The overconstraint is not a coincidence of dimensions — it is a structural fact about the motion the mechanism is confined to, and no length appears in the argument anywhere.

Bennett’s linkage lies in no subgroup. Its screw system is three-dimensional at every position and it is a different three-dimensional subspace at each one.

The measurement #

That is a statement about subspaces, so the instrument is the one the reciprocity essay ended on: principal angles.

Sweep the mechanism. At each position, take the span of its joint screws. Compare that subspace with the one at the start of the sweep, by the largest principal angle between them. Report the largest such angle over the whole range.

For a subgroup mechanism the answer must be zero. For a paradoxical one there is no reason for it to be, and every reason for it not to be.

Measured over thirty positions each:

The first four are the arccosine’s noise floor and not a movement. The last two are the subject.

Getting this wrong, which took two attempts #

The first version of this measurement reported 89 degrees for the planar four-bar.

It was not a bug in the arithmetic. The principal-angle routine was right, the sweep was right, the linkage was right. It was measuring the wrong subspace: the union of the two legs’ constraint systems rather than the whole loop’s screw system.

Those are different objects and only one of them is what the claim is about. A leg’s constraint system depends on where that leg’s own joints have got to, and a leg’s joints move in every mechanism there is — subgroup, paradoxical, generic. So the union of two legs’ constraints moves for everything, and measuring it produced a number that was correct about a quantity nobody wanted.

The failure has no symptom. The number was plausible. The code was clean. An assertion written after seeing it, with a threshold chosen to accommodate it, would have passed. And this essay would have contained a measured falsehood with a figure to support it.

What caught it was that the answer had been derived on paper first. A planar four-bar carries a force out of the plane and two bending couples, always, and that derivation takes four lines and no computer. When the measurement said those three wrenches turn through 89 degrees, one of the two was wrong, and there was no way to leave it alone.

The lesson is narrow and worth keeping: a measurement that agrees with nothing in particular cannot be checked. What made this catchable is that the subgroup cases have known answers. The paradoxical cases do not, which is exactly why the instrument had to be calibrated on the cases that do.

What Bricard’s linkage is, and how it got built #

Bennett’s is the only mobile four-revolute spatial loop, and it has been on this site since the expansion. The six-joint paradoxical loops are a family rather than a single mechanism, found by Bricard, and this phase added one of them.

A generic six-revolute closed loop does not move: Kutzbach counts zero and the count is right. Bricard found six families that move anyway, and the one built here is the line-symmetric family. There is a line about which a half-turn carries the linkage onto itself, sending each axis to the one three places round the loop.

The construction is the proof. Three axes are chosen freely; the other three are the half-turn images of those three, in order; nothing else is imposed. If the symmetry is what supplies the mobility, then any three axes will do — and the site’s check runs three unrelated seeds and gets rank 5 and mobility 1 from all of them, against a control of six axes with nothing relating them, which gives rank 6 and mobility 0 every time.

The refusal half is the same instrument Bennett’s has. bricard6R({ detune }) rotates the third mirrored axis out of where the symmetry puts it, by 0.06 radians. Rank goes from 5 to 6, mobility from 1 to 0, and the joint graph, the link count and Kutzbach’s zero are unchanged throughout.

So the site now has two paradoxical mechanisms built from two different principles, and both drift.

The other reading, at one position #

The drift is a measurement over a range. There is a second signature, available at a single position, and it says the same thing in a different vocabulary.

The reciprocal of a subgroup’s tangent space is a set of forces and couples. That is not a coincidence: the subgroups of rigid motion are the ones whose reciprocals are spanned by zero-pitch and infinite-pitch screws, which is what forces and couples are. Planar motion is reciprocal to a force and two couples. Spherical motion is reciprocal to three forces. Translation along a line is reciprocal to a couple.

So a mechanism confined to a subgroup carries forces and couples, and nothing else. A paradoxical one carries screws — things with an axis and a finite pitch, which no subgroup has as its reciprocal, and which is therefore a signature at one position of a property defined over the whole motion.

Across the site’s mobile overconstrained loops:

assertConstraintKindsSeparateThem requires the subgroup mechanisms to carry no finite-pitch screws and the paradoxical ones to carry nothing but. Both halves, because a check on one half would pass on a routine that had started classifying everything as a screw.

One qualification, and it is the reason this signature is a companion to the drift rather than a replacement for it. The generic four-bar — a structure — also carries screws. Of course it does: it is not in a subgroup either, and being in no subgroup is not the same as being a mobile paradoxical mechanism. The signature reads this mechanism’s motion lies in no subgroup, which is only interesting once the mechanism is known to move.

Why this distinction has consequences #

Three, and the last is the one that decides whether anyone builds the thing.

Predictability. A subgroup mechanism can be designed from its topology. Decide the motion is planar, put the axes parallel, choose any lengths. A paradoxical mechanism has to be solved for: the condition is an equation in the dimensions, and the set of solutions is a surface of measure zero in the design space.

Robustness to manufacture. The redundancy essay makes the point that ν measures how much a mechanism depends on being made accurately. Both families are exposed to that, and the paradoxical one is exposed twice over: a planar four-bar built with slightly non-parallel axes binds, but the design was still valid; a Bennett linkage built to the wrong ratio was never a mechanism at all.

Scaling. Subgroup overconstraint is inherited by anything built out of subgroup pieces. Three planar four-bars sharing a frame are still planar. Paradoxical mobility is not inherited: two Bennett linkages joined at a link are generically immobile, and the constructions that do produce mobile fusions — Goldberg’s five-bar, Myard’s — are named after the people who found them precisely because finding them was the work. This site builds Bennett’s and Bricard’s line-symmetric family and does not build the fusions; naming what is not here matters more in this field than in most, because the family is not closed under the obvious operations.

The count of what this separates #

Nine loops, three categories, and the categories cut across every number computed before this essay.

Four are subgroup mechanisms: the planar four-bar, the universal joint, the spherical four-bar, the Sarrus linkage. All overconstrained, all with ν above zero, all with a Kutzbach count that is wrong, and all buildable from any dimensions at all.

Two are paradoxical: Bennett’s and Bricard’s. Same description, one line at a time — overconstrained, ν above zero, Kutzbach wrong — and buildable only at a condition.

Three are generic: the spatial four-bar with an ordinary twist, the six-joint loop, the seven-joint loop. ν of zero or two, and the first two do not move.

Reading down the ν column separates the third group from the first two. Reading down the drift column separates the second from the first. Nothing before this essay could do the second separation, and the numbers that could not do it — the count, the rank, the redundancy, the joint graph — are the numbers a mobility analysis produces.

The word that was doing the work #

One reason this distinction went unmeasured for as long as it did is that the vocabulary hides it.

“Overconstrained” is a statement about a count: more constraints than the freedoms they remove. It is true of both families and it is the same sentence in both cases. Nothing about the word suggests there is anything else to ask.

The literature’s own name for the second family is paradoxical, which is honest about the situation and unhelpful as a definition — it names the reader’s surprise rather than a property. Bennett’s linkage is not paradoxical to Bennett’s linkage. What is actually true of it, and is checkable, is that its motion lies in no subgroup of the rigid displacements; and the consequence of that is a screw system which does not stay put.

So the two words in circulation are a count that does not separate them and an expression of astonishment. The drift is a third thing: a number, computed from the mechanism, that comes out at the arccosine’s noise floor for one family and at tens of degrees for the other. It does not need anyone to be surprised.

Whether the drift is the right invariant is a separate question this site has not settled. It is sufficient to separate the six mechanisms here, and it is a consequence of the subgroup property rather than equivalent to it — a mechanism could in principle have a screw system that moves and returns, or that moves within a family of subspaces with some structure of its own. The claim made here is the narrow one: on the mechanisms this site builds, the drift separates the two families cleanly, with four orders of magnitude between the two groups and nothing in between.

Both conditions are dimensional; one is a condition manufacture hits #

The distinction is stated above as any four bars whatever against almost no four bars, and that is nearly right and worth sharpening, because the sharpened version is what decides which mechanisms get built.

A planar four-bar is overconstrained because its motion lies in a subgroup, and it lies in that subgroup because its four axes are parallel. Parallelism is a dimensional condition. Build the same four bars with the axes half a degree out of parallel and the mechanism is not in the planar subgroup, its screw system is not the planar one, and it binds — which is exactly what a badly made four-bar does, and is why every real one runs with clearance in its bearings or a ball joint somewhere.

So both families depend on a condition among their dimensions, and the honest difference is not whether the condition exists but whether ordinary manufacture is organised to hit it. Parallel axes are what a flat plate with drilled holes produces, what a jig boring machine is for, and what a designer gets for free by putting all four pins in one casting. Coplanar axes through a point are what a spherical joint’s construction produces. Those conditions are met not because they are easy in principle but because the whole apparatus of the workshop is built around producing them.

Bennett’s condition is $\sin(\alpha/2)/a = \sin(\beta/2)/b$, a numerical relation between two lengths and two twist angles. Nothing about a machine tool produces it, no fixture enforces it, and there is no process whose natural output satisfies it. It has to be hit by measurement, and any error in any of the four quantities takes the mechanism off it — at which point it is a generic four-revolute spatial loop, which does not move at all.

That is the whole practical content of the distinction, and it is sharper than robustness to manufacture. A subgroup mechanism’s condition is one a workshop already knows how to hold; a paradoxical mechanism’s is one nobody has a process for. Both bind when they are missed, and only one of them is routinely missed.

It also explains the asymmetry in how the two are used. Planar and spherical four-bars are everywhere and their overconstraint is invisible, because it is absorbed by clearances that were going to be there anyway. Bennett’s linkage appears in deployable structures and in papers, is made to tighter tolerances than its size suggests, and is usually built with a compliant joint somewhere — which is the same admission the four-bar’s bearing clearance makes, paid for deliberately rather than incidentally.

Which is why the drift measurement is worth having beyond its own elegance. It separates a mechanism whose condition a workshop can hold from one whose condition it cannot, and that is a question about what will be built rather than about what exists.

What is left unmeasured #

Two things this site cannot currently say, both stated so the gap is on the record.

The drift is measured over each mechanism’s own range, because the ranges differ by a factor of fifty — Bennett’s turns through 225 degrees and Bricard’s assembles over a few tens. The comparison is therefore between “how far each turned through its own motion”, which is the right comparison for the question at hand and is not a rate. Whether the drift is fast or slow, in any sense that would let two paradoxical mechanisms be ranked, is not answered here.

And Bricard’s range depends on the step the walk takes: 18 degrees at a step of 0.02 radians and 109 at 0.005, because each solve is seeded by the last and a smaller step keeps the seed nearer. That is a property of how the range was found, not of the linkage, and motionRange returns the step alongside the answer for exactly that reason. Any figure quoting a range without the step would be quoting the sampling.