Twelve kinds of freedom

The first rung turned a joint into a group of displacements, and the second found that surfaces produce six of them. That leaves an obvious question hanging: six out of how many?

Twelve. And the way the twelve are told apart is short enough to state completely.

Three integers #

A group of displacements is easier to work with through its algebra — the twists that generate it, which form a subspace of the six-dimensional space of twists. Composition in the group becomes the Lie bracket in the algebra, and a subspace is the algebra of a group exactly when it is closed under that bracket.

Deciding which of the twelve a subalgebra is takes no search. Three numbers separate every case and all three come off the basis:

The dimension $d$ — how many independent twists there are. The rotational rank $r$ — how many independent directions appear among the angular halves of those twists. The translation dimension $t$ — how many combinations of the basis have no angular part at all, which is a null space inside the null space and comes out of the same eigen-decomposition.

Then:

• $d = 1$. No rotation, and it is a translation $T$. With a rotation, its pitch decides: nought gives a revolute $R$, anything else a screw $H$ at that pitch.
• $d = 2$. Two translations gives $T_2$; one translation and one rotation direction gives the cylindrical group $C$.
• $d = 3$. Three translations gives $T_3$. Three rotation directions and no translations gives the spherical group $S$. One rotation direction and two translations gives planar motion $G$ if the pitch is nought and the pitched planar group $Y$ if it is not.
• $d = 4$. One rotation direction and three translations: Schoenflies motion $X$.
• $d = 6$. Everything.
• $d = 0$ is the identity, and $d = 5$ does not occur.

Twelve entries. Every combination of the three integers that is not in that list belongs to no subalgebra, which is what makes the classifier entitled to refuse rather than guess — and it does, rather than returning a nearest match. A naming routine that always names something is not evidence about anything.

The check is a round trip: build the canonical basis for each of the twelve, hand it back to the classifier, and require the same answer. Twelve for twelve, each with a bracket defect of exactly zero.

Six that are on sale #

The six that are lower pairs are the six that are the full symmetry group of a surface, and the previous rung is the census that finds them: $R$, $T$, $H$, $C$, $S$ and $G$.

The other six are where this rung earns its place, so take them one at a time.

The identity. Zero-dimensional, and it is a weld. It is in the classification because a classification with a hole in it is not one, and because almost every surface gives it — two bodies touching over a shape with no symmetry cannot move at all.

All the rigid displacements. Six-dimensional, and it is two bodies that are not joined. Also in the list for completeness, and also the answer a great many computations in this field return, because it is what a set of twists closes to when nothing is special about it.

Two translations, $T_2$. A drawing board’s parallel motion, an X–Y table, a scissor stage. A perfectly ordinary motion, a genuine group, and no pair gives it. The reason is the one the previous rung establishes: a surface invariant under two translations is a plane, and a plane’s own symmetry group also contains the turn about its normal. A lower pair permits everything its surface permits, so the surface that would give $T_2$ gives $G$ instead. Every X–Y table in the world is therefore two prismatic joints in series, and that is not a manufacturing convenience.

All three translations, $T_3$. A Cartesian machine’s spindle relative to its bed. Same story, worse: the orbit of a point under $T_3$ is a three-dimensional region and no surface is invariant under it at all. Three slides.

The pitched planar group, $Y$. Translations in a plane, together with a screw about the plane’s normal. Three-dimensional, closed, and it has no common name because almost nobody builds it deliberately: a thread and two slides at right angles to its axis.

Schoenflies motion, $X$. Four-dimensional: all three translations and a rotation about one fixed direction. This is the useful one. It is what a pick-and-place machine needs — put the part anywhere, at any angle about the vertical, with its face still level — and it is produced by three parallel revolutes and a slide along their axis, which is a SCARA arm.

A SCARA arm has four joints because $X$ is four-dimensional and is not the symmetry group of any surface. That is the practical content of this classification in one sentence, and it is a fact about groups rather than about robotics.

Why a group and not just a list of poses #

It is worth being blunt about what the closure condition buys, because a set of reachable poses is a perfectly good object and most of this site’s fields work with one.

A mechanism’s reachable set is generally a manifold: a smooth family of poses with a dimension, which is what mobility counts. Almost none of them is a group. What a group adds is that the set is closed under composition, and the consequence is an equivalence a manifold does not have: every part of the set looks like every other part.

Take a planar four-bar. Its coupler’s reachable displacements are a one-parameter curve, and that curve is not a group — but every one of its displacements lies inside planar motion, which is. So a statement about what the coupler may do near one configuration is a statement about what it may do near every configuration, transported by an element of the group. That is why a planar mechanism’s coupler stays in its plane for the whole of a sweep rather than for the part of it somebody checked, and it is a stronger guarantee than any number of sampled configurations.

The same argument is what makes an intersection of groups useful and an intersection of manifolds a mess. Two manifolds meet in something whose dimension depends on where they meet; two groups meet in a group, everywhere, with one dimension. Every parallel machine on this site is built on that difference.

And it is why the closure test is worth running on a mechanism at all. Asking whether a set of poses is a group is asking whether the mechanism’s behaviour is the same everywhere in a very strong sense, and the answer is usually no. The mechanisms for which it is yes are exactly the ones this site has long been calling overconstrained without a name for what they had in common.

Nothing of dimension five #

The gap in the table is the interesting part, and it is not a gap anybody would guess.

Every dimension from nought to four has at least one entry and six has one. Five has none. There is no five-dimensional subgroup of the rigid displacements, which means there is no joint, no chain and no mechanism whatever whose displacement set is a five-dimensional group. A body constrained by exactly one condition, in a way that composes, does not exist.

That is checked here rather than quoted. Twenty thousand five-dimensional subspaces of the twists were drawn at random and closed under the bracket, and every one of them generated the whole six-dimensional algebra. No subalgebra of dimension five appears anywhere else in the field either — not from a surface, not from a chain, not from a loop, not from an intersection of two groups.

The argument behind the number is short. The translations form a three-dimensional ideal, and a subalgebra of dimension five either contains it or meets it in a plane; containing it leaves two rotation directions, whose bracket is the third; meeting it in a plane fails for the same reason one step down. Either way the subalgebra is all six. The search does not prove it and is not meant to: it says that nothing in twenty thousand tries contradicts it, which is what a check is for.

What a designer takes from it is concrete. A joint that removes exactly one freedom in space and permits everything else does not exist as a lower pair, and does not exist as a chain either. The nearest thing is a higher pair — a contact along a line or at a point, which removes one freedom — and what it permits is not a group at all. The classification and the engineering agree: five is not available, and the thing that looks like it is not closed.

Up to where it points #

One piece of bookkeeping decides how the twelve are read, and getting it wrong turns the lattice into nonsense.

The twelve are types, not subgroups. A rotation about one axis and a rotation about another are the same type and different groups. A planar group with one normal and a planar group with another are the same type and different groups. The table is a list of conjugacy classes, and there are infinitely many actual subgroups.

Everywhere in this field where an actual containment or an actual intersection matters, the groups are built about a stated common axis and the answer is about those particular subgroups. The lattice above is drawn that way. And the moment it matters most is two rungs from now: two planar groups with the same normal are one group and intersect in the whole of themselves, while two with different normals intersect in a line — which is the difference between a flat parallelogram and Sarrus’s linkage, and is the whole reason that mechanism draws a straight line.

What contains what #

The lines in the lattice are computed rather than drawn from memory: each generator of the smaller group is required to lie in the span of the larger, to the same tolerance every rank decision in this field uses.

A few of the containments are worth reading off, because they are design facts.

A screw is inside a cylindrical group. A helical pair of any pitch permits motions a plain cylindrical pair also permits, which is why a nut on a thread and a shaft in a bore feel related and why replacing one with the other adds exactly one freedom.

A rotation is inside both the spherical and the planar group. So a revolute can be substituted for either wherever only the rotation is wanted — and this is the substitution behind a wrist being three joints and one point.

Everything of dimension three or four except the spherical group is inside Schoenflies motion. $T_3$, $G$ and $Y$ all sit under $X$; $S$ does not, because Schoenflies motion has only one rotation direction. That single line explains why a SCARA arm cannot orient a part arbitrarily however many joints of that kind are added to it — the group it lives in has no room for a second rotation direction, and a chain of joints all of whose groups lie in $X$ produces displacements that lie in $X$.

What the classification does not say #

Three limits, all of them boundaries this field keeps.

It is a classification of connected subgroups. A screw of irrational pitch generates a subgroup whose closure is larger than itself; a discrete group of rotations by a fixed angle is a subgroup and is not in the table. Nothing here is a mechanism: a joint moves continuously from where it is, and the set it reaches is connected by construction. The classification is the right one for the question and it is not the classification of all subgroups.

It says nothing about range. A revolute pair that turns thirty degrees and one that turns freely are the same type, because the type is a property of the surfaces and the range is a property of where they run out. Every mechanism on this site has ranges — a crank that does not turn fully is the four-bar field’s second essay — and none of that is visible here. A group is what the joint permits; how much of it a particular pair of parts can deliver is a separate and entirely real question.

It says nothing about force. Nothing in this field has a load, a friction cone or a stiffness. A contact that only pushes permits a cone of motions rather than a subspace, which is the holding field’s object and is emphatically not a group — a cone is not closed under inverses, since undoing a motion that separates two parts drives them together. The two fields are next to each other and their objects are different in exactly that way.

The four that must be built rather than bought #

Four of the twelve are motions a designer may want and cannot buy as a pair, and the reason is stated above — they are not the full symmetry group of any surface. What follows from that is worth setting out, because it is the practical content of the whole classification.

Each of the four is reached by a chain instead. Two translations need two prismatic pairs in series; three need three; Schoenflies motion needs three prismatics and a revolute, which is a SCARA arm; the pitched planar group needs a helical pair and two translations. In every case the group is realised by composing joints rather than by one interface, and the joint count is the group’s dimension because each pair contributes one freedom.

The cost of that substitution is not the extra links. It is that a chain’s group membership is a coincidence among its axes rather than a property of a surface. A cylindrical pair permits its two motions because a cylinder slides on itself, and no assembly error changes that — a badly made cylinder is a loose joint, still cylindrical. A pair of prismatics permits two translations because their two directions are what they are, and if one is out of square the chain still has two freedoms and they are no longer two translations of a plane.

So the six on sale and the four that are not differ in exactly the way an aligned chain differs from a misaligned one. A surface enforces its own symmetry; a chain has to be assembled into its symmetry and can be assembled out of it. That is why a lower pair is a component with a tolerance on its form, and a Schoenflies chain is an assembly with a tolerance on the relations between its parts.

It also explains the two ends of the table, which look like curiosities and are not. The identity is a weld, and a weld is the only zero-dimensional group because there is only one way to permit nothing. All six dimensions is two bodies not joined at all, and it is the only six-dimensional group for the same reason. Both are on the list because the classification is of subgroups and both are subgroups; neither is a joint, and neither needs to be built.

Which leaves the classification with a clean reading as a shopping list. Six can be bought, four must be assembled, and two are not mechanisms. A designer wanting one of the middle four knows in advance that the alignment is a functional requirement rather than a quality one, and knows it from the table rather than from experience — which is as much as a classification can be expected to do.

The count, restated #

The mobility count this site has used since its first field is now readable as a statement about this table.

A joint’s contribution is its group’s dimension. A mechanism’s mobility is a dimension too — of the set of configurations rather than of a group, since a mechanism’s configuration set is generally not a group at all. And every constraint the counting rule applies is an assertion that some dimension is what it is.

None of that changes. What changes is that a dimension is now known to be the dimension of something, that the something has twelve possible kinds, and that the kind decides everything the dimension does not.

The next rung asks how special it is to be one of the twelve at all, and the answer sets up everything that follows: among subspaces of the twists picked at random, above one dimension, the number that are groups is nought.