Machine Kinematics
solved, then drawn
What a joint is

Almost nothing is a group

Eighty thousand subspaces of the twists were drawn at random and closed under the Lie bracket. Above one dimension, not one of them was already closed, and every single one generated the whole of the rigid displacements at the first bracket. Two pins with parallel axes close at three; move one axis a hair and they close at six.

Assumes Twelve kinds of freedom.

14 min read 8 figures The singularities are the pointTwo routes to the same number

A subspace of the twists is not a group. It is a group’s algebra only if it is closed under the Lie bracket — if doing two of its motions in one order and undoing them in the other lands inside it rather than somewhere new.

That extra condition looks like a technicality. This rung is about how far from a technicality it is.

The bracket, on the smallest case

Two revolute joints. Two twists, spanning a two-dimensional subspace, whichever way the axes are arranged: no count on this site can tell one arrangement from another, and the constraint rank cannot either, because both are measuring the same dimension.

Two pins, and what their bracket costs. The bracket, on the smallest case there is. Two revolute joints span a two-dimensional set of twists whichever way they are arranged, and no count on this site can tell the two arrangements apart. Their brackets can. Two parallel pins bracket to a translation, which was not in the span, and the span closes at three — planar motion, which is the group the pair of them lives in. Two skew pins bracket to something that closes at six: nothing smaller than the whole of the rigid displacements contains them. The defect column is how far the bracket lies outside the original span as a fraction of its own length, and in both cases it is of order one — which is the ordinary case, and is why a mechanism confined to a subgroup is the exception.
Fig. 1 Two pins, and what their bracket costs. The span is two either way; the closure is three or six, and the two cases are the difference between a mechanism confined to a plane and one that is confined to nothing.

Put the axes parallel and the bracket of the two twists is a translation, perpendicular to both. It was not in the span, so the span was not closed; add it and the set closes at three, which the classifier names as planar motion. That is exactly right and is the reason a chain of parallel pins stays in a plane.

Move one axis so that the two are skew and the bracket is something else entirely, and adding it does not settle the matter: the new set brackets again, and again, and the closure is six. Nothing smaller than the whole of the rigid displacements contains two skew revolutes.

Four joints that give a group, and four that do not. Two chains of four revolute-and-slide joints, each drawn at its home position with its joint axes dashed, and each with a cloud of the tool positions it reaches. The counts are identical: four joints, four freedoms, the same Jacobian rank everywhere off a singularity. On the left the three pins are parallel and the slide is along them, and the displacement set is the Schoenflies group — every translation and one rotation direction, four dimensions, closed. On the right the axes are at random and the set is four-dimensional too, and it is inside no group smaller than all the rigid displacements. The instrument is in the caption of each panel: take the logarithms of the displacements the chain reaches and count the dimensions they occupy. Four means a group. Six means there is nothing to be inside.
Fig. 2 The same distinction one size up. Four joints on the left whose displacements occupy four dimensions, four on the right whose occupy six, and every count anybody would apply to the two is identical.

In both cases the bracket lies a full hundred per cent outside the original span — the defect is of order one, not of order a tolerance. That is worth noticing because it is the shape of every closure measurement in this field: a subspace is closed to within 10−1610^{-16} or it is open by a number near one, and there is nothing in between.

Eighty thousand tries

The two-pin case is suggestive and it is two examples. So: draw subspaces of the twists at random, of each dimension, and count how many are already closed.

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. 3 Twenty thousand subspaces at each dimension. The one-dimensional row is a full bar and every row above it is empty; the closure column, printed in the library’s own output, is six for every single one of the eighty thousand.

Dimension one: all twenty thousand. Every one-dimensional subspace is closed, and the reason is worth saying out loud because it is the whole first rung in one line — a single screw always generates a one-parameter subgroup, which is the same statement as every screw is a joint somebody could build. Give a surface the right thread and it permits that screw and nothing else. There is no such thing as a one-freedom motion that fails to compose.

Dimensions two, three, four and five: none. Not one subspace in eighty thousand is closed, and every one of them closes at six. There is no middle: no random three-dimensional subspace closes at four or five on its way to being open.

The census is deterministic and repeatable — a fixed seed, a stated generator — so it is a measurement rather than an anecdote. And it says something the earlier rungs only implied.

How the closure is computed, and when it stops

The procedure is four lines and it is worth stating, because it is the instrument every later rung reads.

Span the twists. Bracket every pair of basis elements. Span again. Repeat until the dimension stops growing.

It terminates in at most five rounds, because the dimension cannot exceed six and must strictly increase for the loop to continue. In practice it takes two: a set that is going to close at six almost always gets there at the first bracket, which is what the census’s closure column says — eighty thousand subspaces, every one reporting six after one round.

The rank decisions inside it are the same ones the rest of this site makes, with every twist normalised first. A set containing one screw of magnitude a thousand and five of magnitude one has a rank decision about its scaling rather than about its geometry, and normalising is what makes the decision be about the mechanism.

The defect is the second number the test reports and it is the one that keeps the first honest: the largest fraction of its own length that any bracket lies outside the span. A genuine subalgebra returns 10−1610^{-16}. A subspace picked at random returns a number of order one. Between those two there is nothing, on any example this field has produced, which is why the tolerance can be as loose as 10−710^{-7} without anybody having to defend the choice.

Why this measurement is not a derivative. The honest failure mode of the field's instrument, drawn rather than hidden. Every set looks like its own tangent space near the identity — that is what a tangent space is — so a chain sampled over a thousandth of a radian reports the dimension of its velocities, which is the number the screw system already gives. Four pins at random has four joints, and sampled over 10⁻⁹ radians its displacements occupy four dimensions; sampled over two radians they occupy six. The step is at 10⁻⁶, which is where the departure from the tangent space falls below the rank tolerance — so the position of the step is a fact about arithmetic and the two plateaux are facts about the mechanism. A group is a statement about displacements you could compose, and no derivative can make it.
Fig. 4 The same instrument, misused on purpose, to show what it depends on. Sampled over a range small enough, a chain’s displacements cannot be told from its velocities and the closure reports the tangent space; the plateau on the left is the honest answer and the one on the right is what a derivative would have said.

What eighty thousand is evidence for

A census over random subspaces is evidence about a distribution, so it is worth being exact about which one and what it can support.

The subspaces are drawn by taking as many six-vectors as the dimension calls for, each with independent coordinates uniform on [−1,1][-1, 1], and orthonormalising. Draws that come out degenerate are discarded. That is a perfectly ordinary way to hit a subspace at random and it is not a canonical one — there is no rotation-invariant measure on subspaces of twists that everyone would agree on, because the twists have no natural inner product to be invariant under.

None of that weakens the finding, because the finding is not a probability. It is that the closed subspaces form a set of measure zero, and a set of measure zero is missed by any reasonable distribution, not just this one. Eighty thousand misses is what missing a measure-zero set looks like from the inside of a computation, and a single hit would have been a bug rather than a discovery.

What the census cannot say is anything quantitative about nearly closed subspaces, and that is a real question — a mechanism made to a tolerance is near a group rather than in one. The defect answers it: it is a continuous number, it is what the test actually measures, and the rest of this field reads it rather than the yes-or-no.

Being a group is a coincidence

Take the census seriously and the twelve-entry classification changes character.

The twelve groups are not twelve common cases with a long tail of others. They are twelve exceptions, sitting inside a space where the general answer is everything. A subspace of twists picked without care is not the algebra of anything; a mechanism whose motion happens to lie in a proper subgroup has satisfied a condition that nothing in its construction required.

This reframes a great deal of what the site has already measured. The spatial field’s list of overconstrained mechanisms — a planar four-bar built as a spatial loop, a spherical four-bar, a universal joint, Sarrus’s linkage — are all mechanisms whose motion lies inside one of the twelve. Each of them requires its axes to satisfy an exact condition: parallel, or concurrent, or in two perpendicular planes. Perturb any of those conditions and the group is gone, immediately and completely, exactly as the two-pin case above loses three dimensions when one axis moves a hair.

That is what overconstraint is, restated: not a redundancy in an equation count but a coincidence in a group. And it is why every overconstrained mechanism in engineering practice is a mechanism whose parts are made to a condition — a jig, a boring bar, an alignment procedure — rather than one that happens to satisfy it.

How many dimensions each chain's displacements occupy. Every chain in the field, with the dimension its reached displacements' logarithms occupy. A chain of n joints always has n freedoms; what varies is whether those freedoms compose. Where the bar equals the joint count the motion is inside a group and the group is named; where it reaches six there is no proper group containing the motion, and the two chains that do are the ones whose axes were chosen at random. Nothing about the joints themselves differs — three pins are three pins, and the two rows differ only in where the axes point.
Fig. 5 The same point on chains rather than subspaces. Three pins that are parallel give three dimensions; three pins that are nearly parallel — the axes tilted by six hundredths of a radian — give six. Nothing else about the two rows differs.

The pair of rows in the middle of that chart is the census made physical. Two chains of three revolutes, identical joint counts, identical mobility, identical rank, and the difference between them is that one set of axes is exactly parallel and the other is nearly exactly parallel. One motion is inside a three-dimensional group and the other is inside nothing.

The same word, a different object

The bracket has appeared on this site before, and the collision of names is worth handling directly rather than leaving as a trap.

The rolling field is built on a Lie bracket. A wheel forbids a sideways velocity and does not forbid a sideways position, and the reason is that the forbidden direction is the bracket of two permitted ones — so it is reachable at second order in the size of a shuffle, which is why a car can be parked sideways and why the exponent is two.

That bracket and this one are the same operation and not the same object.

The rolling field’s is a bracket of vector fields on a configuration manifold. It is computed numerically, by finite differences, and it varies from point to point: a distribution can be integrable here and not there. What it is used for is to show that a set of directions is not closed, because non-closure is what makes a wheeled vehicle able to go everywhere.

This field’s is a bracket on a six-dimensional algebra with constant structure. It is exact, it is the same everywhere, and it is used to show that a set of twists is closed, because closure is what makes a mechanism’s motion a group.

So the two fields want opposite answers from the same operation, and both answers are usually the same one: not closed. For a wheel that is a liberation and for a mechanism it is the ordinary case. The two are given different names in this site’s own machinery for exactly that reason, with the reason written beside each. A shared name is not a shared function, and this collection has already found three routines sharing one name that turned out to be three different functions; this is the same hazard, caught before it landed.

What the exceptions have in common

If being a group is a coincidence, it is fair to ask what kind of coincidence.

Every one of the twelve has a geometric condition behind it, and the condition is always about axes agreeing on something:

  • RR, TT, HH, CC: one axis. Everything is on it.
  • SS: one point. Every axis passes through it.
  • GG and YY: one direction. Every rotation axis is parallel to it and every translation is perpendicular.
  • T2T_2, T3T_3: no rotations at all.
  • XX: one rotation direction, with translations unrestricted.

Each is an exact statement about alignment, and each fails under an arbitrarily small perturbation. That is what makes them measure zero, and it is also what makes them buildable: a machinist can hold a condition on alignment to a tolerance, and the mechanism then approximates a group to that tolerance.

The twelve kinds of freedom, and which are joints. Every connected group of rigid displacements, up to where its axis points and where its origin sits. There are twelve, the height on the page is the dimension, and a line means the lower one is contained in the upper — computed by asking whether each generator of the smaller lies in the span of the larger, with all twelve built about a common axis. Filled discs are joints: six of the twelve are the symmetry group of a surface and can be a single pair, and six are not and have to be built out of a chain. There is nothing at dimension five, which is not obvious and is checked rather than assumed: twenty thousand random five-dimensional subspaces of the twists were closed under the bracket, and every one generated the whole of the six.
Fig. 6 The twelve, with the conditions above as the reason the lattice is the shape it is. Every line in it is one alignment condition being added to another.

Which raises the question this field will have to answer honestly, and does, three rungs from now: a mechanism made to a tolerance is not exactly in a group. The bracket closure is computed on ideal geometry, and a real four-bar’s axes are parallel to a few arcminutes. What the measurement then says is that the motion lies within a bounded distance of a group, and the distance is the alignment error — which is precisely how the practice field reads every other exact statement on this site.

The two ways a mechanism gets into a group

Given how rare closure is, it is worth asking how any mechanism ever manages it, and there turn out to be exactly two routes. Both are the subject of later rungs and both are visible from here.

By having joints whose groups are already inside one. Three revolutes with parallel axes are three one-dimensional groups, all inside the same planar group; their product is therefore inside it too, because a product of subsets of a group is a subset of the group. Nothing about the chain achieves anything — the alignment of the axes does all of it, and the chain merely fails to leave. That is how a chain multiplies, and it is why the interesting question about a chain is which group its joints share rather than how many joints it has.

By intersecting. Two legs each confine a platform to a group, and the platform gets both, so it gets the intersection — and an intersection of groups is a group with no coincidence required at all. This is the only construction in the whole field that produces closure for free, and it is why parallel machines can be designed for a motion type rather than discovered to have one.

What every pair of groups meets in. The intersection of two subgroups is always a subgroup — that needs no computation — and which one is the useful part. This is the design rule behind every parallel machine on this site: choose legs whose groups meet in the motion the platform is wanted to have, and it has that motion whatever the leg lengths are, with no synthesis and no tolerance. The row and column are built about different axes, at right angles, because an intersection is a statement about particular subgroups rather than about their kinds: two planar groups with the same normal meet in the whole of themselves, and two with different normals meet in a line.
Fig. 7 The second route, tabulated. Every cell is a group, because an intersection of groups always is — which is what makes this a design tool and a product of groups a hazard.

The census measures the space, not the practice

The rarity is a statement about randomly drawn subspaces, and it is worth being exact about what population that is, because the population of mechanisms anybody has built is a completely different one and the two conclusions run opposite ways.

Draw a subspace at random and it is not closed — eighty thousand times out of eighty thousand, above one dimension. Draw a mechanism from the ones that exist and it is very often inside a group: every four-bar, every slider-crank, every SCARA arm, every Cartesian machine, every wrist, every planar linkage on this site. The set of built mechanisms is concentrated almost entirely on the measure-zero set the census says is empty.

That is not a contradiction and it is the useful reading of both facts together. Engineers build the coincidences on purpose. Parallel axes, concurrent axes, perpendicular planes — every one is a condition that a random draw never satisfies and that a jig bore satisfies routinely, and the whole apparatus of manufacture exists to hit conditions of exactly that shape.

So the census is a measurement of the space and the repertoire is a measurement of the practice, and neither predicts the other. A statistical intuition formed on random subspaces is wrong about every mechanism in a catalogue; an intuition formed on the catalogue is wrong about what happens when a mechanism is perturbed. Both are correct about their own population and neither transfers.

This site has met the same pair of populations before from the other side. A rule tested on a repertoire is tested on inputs pre-selected to satisfy it, and the census is what exposes the selection. Here the roles are reversed: the census is the background against which the repertoire’s specialness is measured, and the repertoire is the interesting object rather than the biased sample.

Which gives the rarity its proper standing in this field. It is not a warning that mechanisms are usually not groups — they usually are, because they were made to be. It is the measurement of how much that costs: a condition of measure zero, held by manufacture, lost to any perturbation, and worth the whole apparatus of alignment that machines are built with. The census says what the alignment is buying.

The instrument this sets up

Two things are now available that were not.

The first is a test: given any set of twists, close it and read the dimension. Six means no proper group contains them; anything less names one.

The second is a prior. Because closure is so rare, a mechanism that turns out to be inside a proper subgroup says something strong about itself, and one that closes at six says nothing at all — it is doing what almost everything does. That asymmetry decides how the rest of the field reads its own numbers: a closure of three is a finding, and a closure of six is a finding only when the mechanism was expected to be in a group.

Bennett’s linkage is that case. Every count says it is overconstrained; every rank says it is overconstrained; it satisfies an exact condition on its lengths and twists, and if it were doing what the other overconstrained mechanisms do, its motion would be inside one of the twelve. Its closure is six.

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. 8 The instrument applied. Four mechanisms whose two bars are equal — the motion is inside a group and bracketing adds nothing — and two whose bars are four and six.

That is the shape of the field’s central result, and the reason it needed this rung first. Closing at six is only remarkable against the knowledge of how ordinary it is.

There is one more thing the rarity buys, and it is a piece of intellectual hygiene rather than a result. A test that almost always says no is a test whose yes is worth something. The instruments this site has been using for overconstraint — a count that comes out negative, a rank that comes out short — say yes to a great many mechanisms, including every one that is merely badly modelled. This one says yes to four mechanisms out of six, and to none of the eighty thousand subspaces that were not built to satisfy an alignment condition. When it does say yes, it names the group, and the name is checkable against the mechanism’s own geometry: parallel axes give the planar group, concurrent axes give the spherical one, and the two arms of Sarrus’s linkage give a translation along their common perpendicular. Every yes comes with a reason that can be read off the drawing, which is what distinguishes a measurement from a coincidence in the other direction.

Where this goes next

The twelve kinds of freedom, and which are joints. Every connected group of rigid displacements, up to where its axis points and where its origin sits. There are twelve, the height on the page is the dimension, and a line means the lower one is contained in the upper — computed by asking whether each generator of the smaller lies in the span of the larger, with all twelve built about a common axis. Filled discs are joints: six of the twelve are the symmetry group of a surface and can be a single pair, and six are not and have to be built out of a chain. There is nothing at dimension five, which is not obvious and is checked rather than assumed: twenty thousand random five-dimensional subspaces of the twists were closed under the bracket, and every one generated the whole of the six. Twelve kinds of freedom Every set of displacements that is closed under composition is one of twelve, up to where its axis points. Six of them are joints somebody sells. Four are motions a designer may perfectly well want and cannot buy at any price. And there is nothing at all of dimension five — checked here on twenty thousand random subspaces, every one of which generated the whole of the six. Four instruments, and only the last one names the group. Every instrument this site has for an overconstrained loop, on the same six mechanisms. Kutzbach's count gives −2 for a planar four-bar and −2 for Bennett's. The rank of the constraint Jacobian gives three and three. Both are right and neither separates them. The last two columns are this field's: the span is how many dimensions the logarithms of the displacements the moving link actually reaches occupy, and closes at is the dimension after those are closed under the bracket. A planar four-bar closes at three and the three are planar motion; Sarrus closes at one, a translation, which is the exact straight line the spatial field measured by solving the mechanism sixty times. Bennett closes at six: its displacements occupy four dimensions and no group smaller than all of them contains those four. That is what "paradoxical" has meant on this site for six phases, stated as an integer. Compose two positions and see where you land Take two configurations a mechanism actually reaches, compose the displacements that got it there, and ask what kind of thing the result is. A planar four-bar lands inside planar motion, to 4 × 10⁻¹⁶. Sarrus lands on its own line. Bennett's linkage lands three tenths of a radian outside the four dimensions its own displacements occupy — a one-freedom motion that generates all six. Two pins, and what their bracket costs. The bracket, on the smallest case there is. Two revolute joints span a two-dimensional set of twists whichever way they are arranged, and no count on this site can tell the two arrangements apart. Their brackets can. Two parallel pins bracket to a translation, which was not in the span, and the span closes at three — planar motion, which is the group the pair of them lives in. Two skew pins bracket to something that closes at six: nothing smaller than the whole of the rigid displacements contains them. The defect column is how far the bracket lies outside the original span as a fraction of its own length, and in both cases it is of order one — which is the ordinary case, and is why a mechanism confined to a subgroup is the exception. One bracket, two subjects A wheel can be parked sideways because the forbidden direction is the bracket of two permitted ones. A mechanism's motion is a group when the brackets of its permitted twists are already permitted. Same operation, same two plateaux, and the two fields want opposite answers — which is why the two are named for different things here, with each one saying beside itself that the other exists.

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

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.

ConstraintDegrees of freedomDisplacement subgroupLie bracketOverconstraintRankScrew systemSubalgebraTwist