Machine Kinematics
solved, then drawn
What a joint is

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.

Assumes Two planes meeting in a line and Two ways to be overconstrained.

16 min read 8 figures The singularities are the pointTwo routes to the same numberSolved, not drawn

A group is a set closed under composition. That is not a definition to be admired; it is an experiment.

Take a mechanism. Sample the displacements one of its links actually reaches, from converged configurations only. Take two of them, compose them, and ask whether the result is the same kind of thing as the ones that went in.

Compose two positions and see where you land. The group axiom run as an experiment. For each loop, take two displacements the moving link actually reaches, compose them, and measure how far the result lies outside the set the reached displacements span. The four trivial loops come back at the floor — and the floor here is the solve, not the arithmetic, because a configuration is a root found to 10⁻¹³ and its logarithm inherits that. Bennett's linkage and the Bricard six-bar come back at two to three tenths. There is no threshold between the two answers; there are twelve orders of magnitude.
Fig. 1 The experiment, on six mechanisms. Four answer at the solver’s own floor and two answer at two to three tenths.

What is measured

The measurement needs one careful choice and then it is mechanical.

Take the logarithm of each reached displacement — the twist whose exponential is it — and find the smallest subspace containing all of them. Call its dimension the span. Then compose pairs of reached displacements and measure how far each product’s logarithm lies outside that same subspace, as a fraction of its own length. Call the worst of those the defect.

Two numbers, and between them there is no ambiguity anywhere in this field:

  • Span below six, defect at the floor. The motion is inside a proper subgroup and composing stays inside it.
  • Span of six. There is no proper subgroup to be inside, and the first number has already answered.
  • Span below six, defect of order one. The reached displacements occupy a slice, and the slice is not closed. This is the case nobody would guess exists.

The third row is Bennett’s linkage.

The six loops

The mechanisms are the spatial field’s, unchanged: a planar four-bar built as a spatial loop, a spherical four-bar, a universal joint, Sarrus’s linkage, Bennett’s and a Bricard six-bar. Every one of them is overconstrained by every count this site has.

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.
Fig. 2 Four instruments on six mechanisms. Kutzbach and the rank give a planar four-bar and Bennett’s linkage identical readings; the last two columns do not.

The planar four-bar spans three and closes at three: planar motion, defect 4.4×10−164.4 \times 10^{-16}.

The spherical four-bar and the universal joint span three and close at three: spherical motion. That the two report the same group is the group-theoretic form of a fact the spatial field established mechanically — a universal joint is a spherical four-bar, with the two shaft yokes as two of its links.

Sarrus spans one and closes at one: a translation, defect 1.6×10−121.6 \times 10^{-12}. That is the straight line, reported as a dimension.

Bennett’s linkage spans four and closes at six, defect 0.310.31.

The Bricard six-bar spans four and closes at six, defect 0.200.20.

Four dimensions from one freedom

Bennett’s numbers are worth reading twice, because the first of them is strange before the second is alarming.

Bennett’s linkage has one degree of freedom. Its coupler’s reachable displacements are a one-parameter curve in the six-dimensional group of rigid displacements: a line’s worth of poses, no more. And the logarithms of those poses span four dimensions.

A one-parameter curve whose points need four coordinates to describe is not a contradiction — a curve in space needs three coordinates and is still a curve — but it is a strong statement about how the curve sits. A planar four-bar’s coupler is also a one-parameter curve, and its logarithms span three, because the curve lies in a three-dimensional subspace and cannot leave it. Bennett’s curve leaves every three-dimensional subspace, and every four-dimensional one that is a subalgebra, and the smallest thing containing it after brackets are taken is the whole six.

Bennett's linkage, and the group it moves inFour pins with skew axes and a condition on the lengths. Nothing about it is planar, spherical or translational. The dashed stubs are the joint axes, the solid marker is one point of link 1 and the faint curve is everywhere that point goes. Every frame is a **solve**: the joint angles are the unknowns, the closure of the loop is the equation, and a frame is drawn only where the residual comes below 10⁻⁹. What this field adds to the picture is one number. Take the displacements this link reaches, take their logarithms, and close them under the bracket: the answer is **6**, so no group smaller than all the rigid displacements contains it — the displacements occupy 4 dimensions and their products leave those 4 by 0.31 of their own length. positioned by solving, not by drawing.4 joints · Bennett's linkageinside nothing smaller than SE(3)
Fig. 3 Bennett’s linkage, drawn from solved configurations, with the marked point’s path. Four pins with skew axes, a condition on the lengths and twist angles, and a motion inside no proper group at all.

Then the defect. Take two of the poses Bennett’s coupler reaches, compose the displacements, and the result’s logarithm lies 0.31 of its own length outside the four dimensions the reached poses occupy. That is not a numerical artefact; it is nearly a third, and the same measurement on a planar four-bar returns 4×10−164 \times 10^{-16}.

The mechanism reaches a set of displacements, and composing two of them lands somewhere the set does not go.

That is the plainest statement of what makes Bennett’s linkage different from every other overconstrained mechanism the site has drawn, and it needed none of the classification machinery to say. Two poses, a multiplication, and a distance.

What composing two configurations means physically

It is worth pausing on what the experiment is doing, because “compose two displacements” is an algebraic phrase and the thing it describes is concrete.

A mechanism’s link is somewhere. Drive the mechanism from its reference configuration to configuration AA and the link has undergone a displacement — a rotation and a translation, together, the thing a four-by-four transform is. Drive it to configuration BB instead and it has undergone another.

Now apply the second displacement to the link after the first. Not by driving the mechanism: by picking the link up and moving it. The question is whether the pose the link ends up in is one the mechanism could have put it in.

For a planar four-bar the answer is not necessarily the same configuration, but certainly the same kind of pose: the link is still in the plane, still at some position and angle a planar displacement gives, and every such pose is in the group even if the particular linkage cannot reach all of them. For Bennett’s linkage the answer is that the pose is not even the same kind of thing — it is outside the four dimensions its own reachable poses live in.

A trivially overconstrained mechanism’s link stays in a world; a paradoxical one’s does not have a world to stay in. That is what the experiment shows, and it is why the composition is the right test rather than a formal one.

The floor is the solve

One detail keeps the small numbers honest, and it is the sort worth carrying.

The trivial mechanisms’ defects come back at 10−1210^{-12} to 10−1610^{-16}, and the spread between those is not noise about the mechanisms. A configuration of a closed loop is a root found by Newton, driven to 10−1310^{-13} and judged at 10−910^{-9}; its logarithm inherits that error; so a defect computed from solved configurations cannot be smaller than the solve.

Sarrus’s 1.6×10−121.6 \times 10^{-12} is the solve. The planar four-bar’s 4×10−164 \times 10^{-16} is the arithmetic, because its configurations happen to be found more accurately. Neither is a statement about the mechanism, and both are the floor.

The number that is a statement about a mechanism is three tenths. Twelve orders of magnitude separate the two populations, and the separation does not depend on knowing which floor applies.

Span, and what it closes to. Two bars per loop: how many dimensions the reached displacements occupy, and how many they occupy after the brackets are added. For the four trivial loops the two bars are equal — the motion is already inside a group and bracketing adds nothing. For Bennett's linkage and the Bricard six-bar the first bar is four and the second is six, and the gap between the two bars is the whole of what paradoxical means: a one-degree-of-freedom motion that occupies four dimensions of displacement and generates all six.
Fig. 4 Two bars per loop: the dimensions the reached displacements occupy, and the dimensions they generate. Equal bars mean a group; the gap between four and six is what “paradoxical” means.
A Bricard six-bar, and the group it moves inSix pins with a line symmetry. Kutzbach counts −3. The dashed stubs are the joint axes, the solid marker is one point of link 2 and the faint curve is everywhere that point goes. Every frame is a **solve**: the joint angles are the unknowns, the closure of the loop is the equation, and a frame is drawn only where the residual comes below 10⁻⁹. What this field adds to the picture is one number. Take the displacements this link reaches, take their logarithms, and close them under the bracket: the answer is **6**, so no group smaller than all the rigid displacements contains it — the displacements occupy 4 dimensions and their products leave those 4 by 0.20 of their own length. positioned by solving, not by drawing.6 joints · a Bricard six-barinside nothing smaller than SE(3)
Fig. 5 The Bricard six-bar, drawn from solved configurations. Kutzbach counts nought, the rank is five of six, and the closure of its reached displacements is six — the second of the two paradoxical rows, at a defect of 0.20.

The measurement is made on one link per mechanism, and running it on all of them is worth a paragraph because the pattern is informative.

The frame link never moves, so its displacement set is the identity and its closure is nought. The two links adjacent to the frame each turn about a fixed axis, so their displacement sets are one-dimensional rotation groups and their closures are one — every single-joint link is trivially a group, which is the census result about one-dimensional subspaces showing up in a mechanism.

The interesting link is the one furthest from the frame, and it is the one every row of the table reports. For Sarrus’s linkage that is the platform, two joints from the frame on either side; its closure is one, and the intermediate links on each arm report three, the planar group of their own arm. So the mechanism’s own structure is legible in the column: two arms each confining their links to a plane, and a platform confined to the intersection.

For Bennett’s linkage every moving link but the two next to the frame reports six.

A universal joint, and the group it moves inThe joint in every propeller shaft, and a spherical four-bar in disguise. The dashed stubs are the joint axes, the solid marker is one point of link 1 and the faint curve is everywhere that point goes. Every frame is a **solve**: the joint angles are the unknowns, the closure of the loop is the equation, and a frame is drawn only where the residual comes below 10⁻⁹. What this field adds to the picture is one number. Take the displacements this link reaches, take their logarithms, and close them under the bracket: the answer is **3**, so the motion lies inside rotations about a point and composing two of its displacements gives a third one it also reaches, to 3.5e-16. positioned by solving, not by drawing.4 joints · a universal jointinside S
Fig. 6 A universal joint, drawn from solved configurations. It reports the same group as a spherical four-bar because it is one — and the two shaft yokes are two of its four links.

What this adds to the drift

The spatial field already separated these two populations, and it did it well, so the honest question is what a second instrument is for.

Two ways to be overconstrained measures how far a mechanism’s screw system turns through its own motion. A planar four-bar’s stands still — 2×10−62 \times 10^{-6} degrees over a sweep. Bennett’s turns 89 degrees and a Bricard six-bar’s 22. That is a real separation and it was the site’s answer until now.

The two instruments agree on every row, which is the first thing to check and it is not automatic. They are also different in kind, and the difference is what the new one adds.

The drift is a derivative. It reads the twists available at a configuration and asks how that subspace changes as the configuration moves. It is evaluated pointwise and integrated over a sweep, and every quantity in it is a velocity.

The closure is finite. It reads displacements, composes them, and asks what set they are in. Nothing in it is a rate.

And the closure answers a question the drift cannot: which group. A screw system that stands still says the motion is inside some subgroup; a closure dimension of three with type GG says it is planar motion. Sarrus’s drift is small and its closure is one — a translation — and the drift has no way to say that.

The relation between them is exactly what the range measurement makes precise, and it is one rung further on.

Why span four and not span six

The middle number in Bennett’s row is the one that took the longest to understand, and it is the one that says the mechanism is not simply generic.

A random spatial loop’s coupler reaches a one-parameter curve too, and its logarithms span six. Bennett’s span four. So Bennett’s curve is not sitting in the displacement group like an arbitrary curve; it lies inside a four-dimensional subspace, and it is only when the brackets of that subspace are taken that the answer becomes six.

A four-dimensional subspace of the twists that is not a subalgebra is not a group, is not a screw system anybody would name, and has no standing in the subject. It is a real feature of this mechanism’s motion nonetheless, and the measurement reports it because reporting the span separately from the closure costs nothing.

What it means is that Bennett’s linkage is closer to being trivially overconstrained than a generic loop is, in a sense that has a number attached: its displacements occupy four of six dimensions rather than all six, and the two dimensions they miss are the trace of the exact condition its lengths satisfy. That is a finding this field did not set out to make and it is the kind of thing a second instrument produces — the drift measurement has no place to put it, because a screw system that turns has turned and there is no partial credit.

Whether the four dimensions mean anything further is not settled here, and the essay says so rather than speculating. What is settled is that they are not a subalgebra: closing them under the bracket gives six, and composing two displacements inside them lands outside.

A definition, as an integer

This collection has used the word paradoxical for mechanisms like Bennett’s since the spatial essays were written, and it has meant “overconstrained, and not for one of the reasons already understood”. It can now be a number.

A loop is trivially overconstrained when the smallest subalgebra containing the logarithms of its reached displacements has dimension below six, and the classification names it. It is paradoxical when that dimension is six.

That is a definition with a computation attached, and it does what a definition should: it sorts the mechanisms the same way the earlier instruments did, it applies to mechanisms nobody has classified, and it produces the group as a by-product rather than merely a verdict.

It also explains the word. A trivially overconstrained mechanism moves because its joints share a group and the count did not know; there is nothing paradoxical about it once the group is named. Bennett’s moves and shares nothing, so the redundancy in its constraints is not explained by any group — it is a coincidence in four lengths and four twist angles, and the fact that it exists at all is a genuinely surprising piece of geometry that the spatial field’s own essay is about.

Four loops, and where one point of each of them goes. The orbit of one point of the moving link in four overconstrained loops, drawn from solved configurations. Sarrus's platform runs along a straight line, because its displacements are a one-dimensional group of translations. A spherical four-bar's coupler point stays on a sphere. A planar four-bar's stays in a plane. Bennett's does none of those, and the reason is not that its curve is complicated: its displacements are inside no proper subgroup at all, so there is no surface for the point to be confined to. The first three paths are orbits of groups and the fourth is not an orbit of anything.
Fig. 7 One point of the moving link in four loops. Three of the paths are orbits of groups; the fourth is a curve that is not part of any surface, because there is no group whose orbit it could be.

What it does not say

Three limits, all worth stating in the essay that makes the field’s central claim.

It does not explain why Bennett moves. The closure says there is no group; it does not say what there is instead. Bennett’s mobility comes from a relation between its lengths and twist angles that has to be satisfied exactly, and detuning it by two per cent stops the loop closing at all — which the spatial field checks and this field’s own machinery re-checks, by requiring that a detuned Bennett loop produce almost no solved frames to measure.

It does not scale to a mechanism that has not been solved. Every displacement in the measurement comes from a converged configuration. A mechanism that will not assemble has nothing to compose, and the test correctly returns nothing rather than a number.

It is a statement about the ideal geometry. A Bennett linkage made to a tolerance satisfies its condition approximately, and what it reaches is approximately Bennett’s motion. The defect is the right quantity there too, and it does not distinguish a mechanism that is nearly Bennett from one that is nearly planar without the span beside it.

Why both instruments are worth running

The drift and the closure agreeing on every row is described above as not automatic, and it is worth saying what a disagreement would look like, because the possibility is what makes the pair worth the cost of computing both.

The drift is a derivative: it asks whether the screw system available at a configuration is the same subspace at the next configuration. The closure is finite: it asks whether the displacements actually reached compose into displacements of the same kind. Those are different questions, and there is a shape of mechanism that would separate them.

Suppose a mechanism whose screw system stands perfectly still along its motion — drift at the floor, a constant subspace — but whose constant subspace is not a subalgebra: a four-dimensional set of twists closed under nothing, held fixed as the mechanism runs. The drift would report zero and conclude the mechanism sits in a group; the closure would compose two reached displacements, land outside, and report a defect. The two instruments would disagree, and the closure would be right.

Nothing in this collection is that mechanism, and the reason to name the possibility anyway is that it is exactly the case the drift cannot detect. A constant subspace is necessary for a group and not sufficient, and the drift measures only the necessary half. Every row where the two agree is a row where the sufficient half has been checked separately rather than inferred.

Which makes the agreement a genuine result rather than a formality. Six mechanisms, two instruments computed from different data — twists at a configuration against displacements over a range — and no disagreement anywhere. That is evidence that on these mechanisms the constant subspaces really are subalgebras, and it is evidence obtained rather than assumed.

The relation between the two also explains their division of labour, which the field settles into naturally. The drift is cheap, local, and answers is anything moving. The closure is expensive, global, and answers which group, and by how much is it not one. A survey runs the drift; a mechanism that surprises the survey gets the closure — and the one that did, on this collection, is the one whose name the field has been arguing about since the spatial essays.

The row that is not in the table

One useful negative to end on.

Run the same measurement on a generic spatial loop — seven revolutes with axes at random, the mechanism the spatial field uses as its control — and the span is six and the closure is six, with a defect that has nothing to measure. That is the ordinary answer, it is what eighty thousand random subspaces give, and it is why the four trivial rows are findings.

How often a set of screws is a group. Take a subspace of the twists at random and ask whether it is closed under the Lie bracket — whether doing two of its motions in one order and undoing them in the other leaves you inside it. Every one-dimensional subspace is, trivially and importantly: a single screw always generates a one-parameter subgroup, which is the same statement as every screw is a joint somebody could build. Above one dimension, not one of eighty thousand is a group, and every one of them generates the whole of the rigid displacements at the first bracket. So a mechanism whose motion lies inside a proper subgroup is not merely unusual; it is a coincidence of measure zero — and it is the coincidence every planar mechanism, every spherical one and every Sarrus linkage on this site is built on.
Fig. 8 The background the six loops are read against. Above one dimension, no subspace of the twists drawn at random is a group, and every one generates all six — so a closure of six is what the world gives by default.

Closing at six is only remarkable when the mechanism was expected to be in a group, and Bennett’s linkage is the only mechanism on this site of which that is true and false at once: every count says it is overconstrained, and there is nothing for it to be overconstrained into.

Where this goes next

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. 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. 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.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

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

What links here

The 8 of 9 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Bennett's linkageBricard's linkageDisplacement subgroupLie bracketMobilityOverconstraintParadoxical mechanismRankScrew systemSubalgebra