The arm that is a group

The serial field’s account of an arm is that its pose is a product of transforms with nothing to converge, that the difficulty moves to the other end where one tool pose has eight answers, and that a straight line can cost more than the machine has.

None of that distinguishes one arm from another. This rung is about a distinction between arms that no count can make, and it applies to the most common four-axis machine there is.

Why four joints #

A SCARA arm has three parallel revolutes and a prismatic joint along their axis direction. Four joints, four freedoms.

The usual explanation is that four is enough for the job — put a part anywhere on a table, at any angle about the vertical — and that a fifth and sixth would be wasted. That is true and it is a statement about requirements. The stronger statement is about what the job is.

The set of displacements that job requires is: every translation, together with a rotation about one fixed direction. That set is four-dimensional and it is closed under composition — two such displacements composed give another — so it is a group, and it is one of the twelve: Schoenflies motion.

And no surface has Schoenflies motion as its symmetry group, because a point’s orbit under it is a three-dimensional region and no surface is invariant under something that moves a point over a volume. So it cannot be a single joint. It has to be a chain, and the shortest chain that produces it is four joints.

Four is not a choice. It is the dimension of the group, and the group is not a pair.

The measurement #

The claim that a SCARA’s four joints give Schoenflies motion is checkable rather than definitional, and this site checks it.

Sample the arm over random joint values across ±2.2 radians. Take the logarithm of each pose it reaches. Find the dimension of the subspace those logarithms occupy: four. Close that subspace under the Lie bracket: still four. Hand it to the classifier: one rotation direction, three independent translations, dimension four — $X$.

Then compose: take pairs of reached displacements, multiply them, and measure how far the products’ logarithms lie outside the same four dimensions. The worst is $7 \times 10^{-16}$.

The measurement matters because the claim could have been false in an instructive way. Three parallel revolutes give planar motion, which is three-dimensional; a slide along their common direction gives a one-dimensional translation group; and the product of a three-dimensional group and a one-dimensional one is four-dimensional if the two intersect trivially and if the product is closed. Neither is automatic — a product of groups is almost never a group — and here both hold because the planar group and the perpendicular translation together generate exactly $X$.

The order of the joints does not matter to the answer, which is worth noting since a real SCARA puts its slide at the end and some put it in the middle. The group a chain lies in is decided by which joints it has and where their axes point, not by the sequence; everything else about the machine — reach, stiffness, where the cables go — is decided by the sequence and not by the group.

What the group guarantees #

Here is the payoff, and it is not a restriction but a guarantee.

If the arm’s reachable displacements are a group, then whatever is true of the arm near one pose is true near every other, carried over by the group element that connects them. The tool face is level at every configuration the arm can reach — not at the configurations somebody tested, and not to within an interpolation between them.

That is why nobody checks a SCARA arm’s levelness pose by pose. The geometry guarantees it, the guarantee is group-theoretic, and it holds for the machine’s whole workspace at once.

A four-axis arm with unrelated axes has no such property. Its tool attitude varies over its workspace, has to be commanded, and any statement about it holds where it was measured. That is the ordinary situation for a serial arm and the SCARA’s exemption from it is precisely what its geometry buys.

It is worth being precise about the form of the guarantee, because “the same everywhere” can be read too strongly. The arm does not reach every element of Schoenflies motion: its links are of finite length, so it covers a bounded piece of the group and no more. What is guaranteed is that every pose it does reach is an element of $X$, and that the relationships between reached poses are the group’s relationships.

So the arm’s workspace is a bounded set and its motion type is a group, and the two are separate facts about it. Where the hand can go is the first question and the serial field answers it; which group the poses live in is the second, and it is the one that carries the levelness guarantee.

The alignment is the specification #

Which turns the arm’s manufacturing tolerance into a functional requirement rather than a quality one.

The group exists because three axes are parallel. Tilt two of them and the span jumps from three to six — for the three-pin chain the numbers are exactly that, and the same happens to the SCARA’s four. There is no intermediate value: the dimension is a threshold on a continuous quantity, and being in a group is a measure-zero condition.

What varies continuously is the defect, and it is the number to specify. A perfectly aligned chain reports a defect at the floor of double precision; a chain tilted by $0.06$ radians reports one of order one; in between it scales with the tilt.

So the parallelism of three bores is what keeps the arm’s tool level, and no mobility count on the drawing identifies it as the dimension that matters. That is a real and slightly alarming consequence: the specification the machine’s guarantee depends on is not visible in any kinematic quantity the field has used until now.

What it costs at the wrist #

The other side of the guarantee is an impossibility, and it is the reason SCARA arms have the shape they do.

Schoenflies motion contains one rotation direction. The lattice shows that the spherical group is not inside it. And a chain all of whose joints’ groups lie inside $X$ produces displacements inside $X$.

So no number of additional parallel revolutes or vertical slides will let a SCARA arm tilt a part. Adding a fifth joint of the same kind adds a freedom that does nothing new — a redundancy of the sort the serial field has already met — and the machine still cannot orient. Getting orientation means adding a joint whose group is not inside $X$, which in practice means a wrist, and a wrist is three revolutes through a point whose group is spherical.

That is an impossibility proof from a lattice diagram, and it costs nothing to run. The alternative is trying configurations until the point is conceded.

The pair that would have replaced it #

A thought experiment that makes the joint count vivid.

If Schoenflies motion were the symmetry group of some surface, a SCARA arm would be one joint: two parts touching over that surface, four freedoms, and a pick-and-place machine with a single moving element. Nothing about the requirements would change and the machine would be unrecognisable.

It is not, and the reason is a dimension count that takes one line: a point’s orbit under $X$ is three-dimensional, no surface contains a three-dimensional set, so no surface is invariant. The same line disqualifies all the translations and the whole displacement group, and a slightly longer version disqualifies the two-translation group and the pitched planar group — those have two-dimensional orbits, but the surfaces concerned have larger symmetry groups, so a pair whose surface is a plane is a planar pair rather than a two-translation pair.

Six of the twelve survive, and every joint in every catalogue is one of the six. The four-joint SCARA, the three-slide Cartesian machine and the two-slide cross table are all machines built to synthesise a group that could not be bought.

Other arms, other groups #

The same reading names several standard architectures, and the naming is the useful output.

A Cartesian machine — three orthogonal slides — is $T_3$: all the translations, three-dimensional, no rotation, no pair. Three joints for a three-dimensional group.

A wrist — three revolutes through a point — is $S$: the spherical group. Three joints, and there is a pair that gives it, which is why a ball joint and a wrist are interchangeable in a kinematic model and not on a drawing.

A cylindrical arm — a revolute and a slide on one axis — is $C$, and that one is a single lower pair, so the two-joint chain is a way of building a joint that already exists.

And a six-axis industrial robot is nothing at all: its joints share no proper group, its reached displacements span six, and it is built that way on purpose. The point of a general-purpose arm is that it is inside nothing.

Where a group meets a singularity #

One question the group view does not answer, and it is worth heading off because it looks as though it should.

Being in a group says nothing about singularities. A SCARA arm has them: at full extension the elbow locks and the arm loses a direction, exactly as any two-link planar arm does. The Jacobian drops rank there, the inverse solution’s two branches merge, and none of that is affected by the reached displacements forming a group.

The reason is that a singularity is a property of the map from joint values to poses, and a group is a property of the map’s image. The image can be a perfectly well-behaved four-dimensional group while the parameterisation of it folds. A SCARA arm at full extension is still producing a Schoenflies displacement; it is just producing it in a way that has lost a degree of controllability.

So the two instruments answer questions that do not overlap: the Jacobian’s rank says where the machine is in trouble, and the closure says what kind of motion it is making everywhere including there.

What the field already had #

Two of the serial field’s existing results read differently in this light, and neither is contradicted.

Eight ways to hold the same tool counts the inverse solutions of a six-axis arm — eight poses of the joints for one pose of the tool. That is a statement about a fibre of the forward map, and it is unaffected by whether the image of that map is a group. A SCARA arm has two inverse solutions for a reachable pose, for the ordinary elbow-up and elbow-down reason, and being in a group does not reduce it to one.

What a calibration cannot see is about parameters with no effect on the tool pose, and it acquires a sharper example here: a calibration model written for a SCARA arm inside the Schoenflies assumption has no parameter for axis tilt at all, because within the model the tilt does not exist. The tilt then shows up as unexplained residual and is attributed to whatever parameter is nearest. That is a real failure of a real procedure and its cause is that the model came from a group the machine is not exactly in.

A parallel machine gets it for nothing #

The last comparison is with the other way of building the same motion, because it is where the asymmetry between serial and parallel architectures is sharpest.

A serial chain’s displacement set is the product of its joints’ groups, and a product of groups is a group only under an alignment condition that fails under any perturbation. A parallel machine’s platform gets the intersection of what its legs permit, and an intersection of groups is a group with no condition at all.

So a SCARA arm is in Schoenflies motion because three axes are exactly parallel, and a Delta machine is in the translation group because its legs’ groups happen to meet there — and the second is robust across an open set of geometries where the first sits on a knife edge.

That is a genuine architectural advantage of parallel machines and it is usually stated in terms of stiffness and inertia. This is a different reason for the same preference, it is purely kinematic, and it is the one the group view makes visible.

What a group saves in testing #

The guarantee is stated above as whatever is true near one pose is true near every pose, and its most concrete cash value is in acceptance testing — which is worth spelling out, because it is where a designer meets the group without ever hearing the word.

A machine that must keep its tool face level has to be shown to do so. Without a group, that means sampling: measure the levelness at a grid of poses across the workspace, across orientations, at several heights, and hope the grid is fine enough to catch the worst place. The number of measurements grows with the workspace and there is no argument that says when to stop.

With a group, one measurement stands for all of them. The reachable displacements are closed under composition, so the arm at any pose is the arm at any other pose composed with something the arm can do — and level is preserved by everything the arm can do. Measure the levelness once, anywhere, and the answer holds everywhere the arm reaches.

That is the practical reason a SCARA arm’s data sheet quotes a single flatness figure and a six-axis arm’s quotes accuracy at a stated pose with a repeatability caveat. A group turns a sampling problem into a single measurement, and the saving is the difference between a grid of poses and one.

The caveat is that it transports the type and not the accuracy, so the single measurement has to be the right one. What is exactly preserved is that the arm has no rotation available except about the vertical; what is preserved to within the arm’s own defect is that the tool face is actually level. So the measurement to make once is the defect — how far the axes are from parallel, expressed as a tilt — and it is that number, not a pass, that transports.

Which sharpens the manufacturing argument one last time. A grid of levelness measurements on a well-built SCARA would find nothing anywhere, because the group is exact; on a badly built one it would find the same small tilt everywhere, because the defect is also uniform. Neither result distinguishes a good arm from a bad one by sampling, since the failure is not a place in the workspace — it is a property of the whole machine, and the only measurement that sees it is one aimed at the alignment.

The shortest version #

A SCARA arm is a machine built to be inside a group.

The group is Schoenflies motion, its dimension is four and it is not a pair, so the arm has four joints. Being inside it makes the tool face level everywhere at once rather than pose by pose. Being inside it depends on three axes being parallel, which is a tolerance rather than a topology. And getting out of it — orienting a part — takes a joint of a kind the arm does not have, which is a fact readable off a lattice rather than discovered by trying.

None of those five sentences is available from a joint count, and all five are about the most ordinary four-axis machine in industry.

The general form of the argument is worth keeping, because it applies to any machine built for a motion rather than a reach. Name the set of displacements the job requires. Ask whether it is a group. If it is, its dimension is the joint count, its membership of the six says whether it can be one pair, and the alignment condition that produces it is the specification the machine has to hold. If it is not a group, none of that applies and the machine is designed the ordinary way, by reach and by inverse kinematics.

The SCARA arm is the case where every step of that has a clean answer, which is why it is the example. Most machines are not, and knowing which kind a machine is is the first thing this field is for.