A chain multiplies

An open chain has no loop, which is why a serial arm’s pose is a product rather than a solve: multiply the joints’ transforms in order and the answer is where the tool is, with nothing to converge and nothing to refuse.

Read that product one level up and it says something about groups. Joint $i$ permits the displacements of its own group $G_i$ — a rotation, a slide, a screw — and the chain permits any element of $G_1$ followed by any element of $G_2$ and so on, so the set of displacements the chain reaches is the product

$$G_1 G_2 \cdots G_n = \{\,g_1 g_2 \cdots g_n \;:\; g_i \in G_i \,\}.$$

The question of this rung is whether that product is a group.

The product usually is not #

A product of two subgroups is a subgroup only under conditions, and the conditions are restrictive. In general $G_1G_2$ is not even closed: take $g_1h_1$ and $g_2h_2$ with $g$’s in the first and $h$’s in the second, and the middle two factors do not commute past each other.

The previous rung says how restrictive. Closure is a measure-zero coincidence among sets of twists, and a chain’s joints do not conspire to produce one unless somebody arranged them to. Two revolutes with parallel axes generate a three-dimensional group; move one axis a hair and they generate all six.

So the interesting chains are the exceptions, and the exception has a clean description: the product is a group when every joint’s group lies inside one common subgroup, and the product fills it. The joints do the arranging; the chain merely fails to leave.

The span is the test #

Measuring it needs care, and the measurement is the instrument the rest of the field uses.

Sample the chain over random joint values. Take the logarithm of each pose it reaches — the twist whose exponential is that displacement. Find the dimension of the subspace those logarithms occupy.

If the reached set is inside a group of dimension $k$, every logarithm lies in that group’s $k$-dimensional algebra, and the span is $k$. If it is not inside any proper group, the logarithms leave every proper subspace and the span is six.

There is a nicety in that argument worth stating, because it is where a reader could reasonably object. The span is a statement about the smallest subspace containing the logarithms, and being inside a subgroup is a statement about a subgroup. The two agree because every group in the classification is one whose exponential map is onto: every element of it is the exponential of a twist in its algebra, so its logarithms lie in the algebra and nowhere else. That is true of all twelve and is not true of every Lie group, which is why the argument is stated rather than assumed.

The chart reads as a list of joint counts against spans, and the rows where the two agree are the chains whose product is a group:

• A shaft in a plain bore — a revolute and a slide on one axis — spans two, and the group is cylindrical. That one is also a single lower pair, so the chain is a way of building a joint that already exists.
• Two slides at right angles span two, and the group is the two-translation group, which is not a pair: an X–Y table needs the chain.
• Three orthogonal slides span three: all the translations, and again no pair gives it.
• A wrist — three revolutes whose axes meet at a point — spans three, and the group is spherical. A ball joint gives the same group in one pair, which is why a wrist is three joints and one point and why the two are interchangeable in a kinematic model and not on a drawing.
• Three parallel pins span three: planar motion. Every planar mechanism this site has ever drawn lives here.
• A thread and two slides span three, and the group is the pitched planar group, which has no common name and no pair.
• A SCARA arm spans four: Schoenflies motion.

And two rows where the span exceeds the joint count. Those are the next rung.

The SCARA is the case worth knowing #

Of all of them the four-joint one is the one with an industry attached.

Schoenflies motion is every translation together with a rotation about one fixed direction. It is exactly what a pick-and-place machine needs: put the part anywhere in the volume, at any angle about the vertical, with its face still level. It is four-dimensional, it is a genuine group, and — by the argument in the classification rung — no surface has it as a symmetry group, so no single joint gives it.

Hence four joints. Not four because somebody chose four; four because the motion wanted is four-dimensional and is not a pair. The same reasoning fixes the joint count of every row above, and it is the practical content of the whole classification: what a motion costs in joints is decided by whether it is the symmetry group of a surface.

The check that the SCARA’s four really do give Schoenflies motion rather than merely four dimensions of something is the span: four hundred sampled poses, logarithms spanning four dimensions, closed under the bracket, and classified by the same three integers that name every other group in this field. It comes back as $X$.

Why the sampling has to be wide #

The span is computed from sampled poses, and how widely they are sampled decides the answer — which sounds like a defect and is the measurement’s most important property.

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 its joint count, whatever the axes are doing. Four random pins sampled over $10^{-9}$ radians report a span of four and look exactly like a SCARA arm.

Sampled over two radians they report six.

The joint values in every measurement here run over ±2.2 radians for that reason, and the number is stated rather than tuned. It is a rung of its own, because the same fact is what separates this field’s instrument from the screw system: a screw system is the tangent space, and no amount of care with it can answer a question about composing finite displacements.

A group is not a workspace #

There is a confusion available here that is worth removing before it does damage, because it is the one that broke the first version of the measurement.

A chain’s displacement set is bounded. Three parallel pins on links of finite length reach a bounded piece of the planar group and not the whole of it; a SCARA arm cannot put its tool a kilometre away. So the reached set is not literally a subgroup — it is a bounded subset of one — and asking whether the chain can reach the composite of two of its own poses is asking a question about reach.

The first version of this test did exactly that, and it reported a SCARA arm as not closed, because one composite landed outside the arm’s own workspace. The mechanism was fine and the question was wrong.

Being inside a group and being inside a workspace are different questions, and only the first one belongs to this field. The span answers the first without touching the second: the logarithms of a bounded piece of a three-dimensional group still span three dimensions, however small the piece is, as long as it is not degenerate.

The second question belongs to the serial field, which has asked it directly, and to the parallel field, where the workspace is not a shape anybody chooses.

The chains that build a group nobody sells #

Three rows of the chart are worth pulling out, because each is a group with no pair and each is a machine somebody has built.

Two slides at right angles give the two-translation group. A cross-slide table, a drawing board’s parallel motion, a plotter. It is two-dimensional and closed, and the reason it takes two joints rather than one is the census argument: the surface that would give it is a plane, and a plane’s own symmetry group is larger.

Three orthogonal slides give all the translations. Every Cartesian machine tool and every three-axis printer. Three joints for a three-dimensional group, again because no surface has it.

A thread and two slides give the pitched planar group, which has no name in any catalogue because almost nobody builds it deliberately. It is the group whose elements translate in a plane and screw about the normal, and it is the one entry of the twelve that is genuinely obscure. The chain that produces it is one helical pair and two prismatic pairs at right angles to its axis, and the span comes back as three with the pitch of the thread in it.

That last one earns its place in the chart precisely because nobody would look for it. The classification says the group exists; the chain says it is realisable; and the span says the realisation is the group rather than three dimensions of something else. A classification that only ever named things somebody had already built would not be evidence about anything.

The order of the factors #

One more feature of a product that a sum does not have: it is ordered, and the order matters.

The set $G_1G_2$ and the set $G_2G_1$ are different sets in general, which is a real statement about mechanisms: putting the slide before the three pins in a SCARA arm gives a different machine from putting it after, with a different structure and a different set of link geometries. What is not different is the group they end up inside, because both are inside the smallest group containing all the factors, and that group does not care about order.

So the field’s measurement is order-blind and the mechanism is not. The span of a chain’s logarithms is a property of which joints it has and where their axes are, and two chains with the same joints in a different order have the same span. Everything else about them — reach, singularities, stiffness, whether a cable can be got through it — is not.

That is a useful division of labour rather than a defect. It means the span answers the question it answers completely, and hands the rest back.

What a chain cannot escape #

The last thing the product view gives is a set of impossibility statements that are free, and they are the ones a designer notices first.

If every joint’s group is inside a common group $H$, then the chain’s displacements are inside $H$. No amount of adding joints of the same kind gets out. Add a fourth parallel pin to three parallel pins and the arm has four joints, three of which are redundant, and the motion is still planar.

A chain of joints all inside Schoenflies motion cannot orient a part arbitrarily. $X$ contains only one rotation direction, and the lattice shows that $S$ is not inside it. So a pick-and-place machine cannot be persuaded into a general orientation by adding more of the same joints; it needs a joint whose group is not inside $X$, which in practice means a wrist.

And a chain whose joints share no common proper group is not confined at all — six dimensions, and everything the arm can reach it can reach as a general displacement. Most arms are this, deliberately: a six-axis industrial robot is built so that its joints share nothing, because the point of it is to reach general poses.

The order matters and the answer does not #

There is a tension between two statements above worth resolving, because the resolution is a small piece of algebra that justifies the measurement’s order-blindness rather than excusing it.

The product $G_1G_2$ and the product $G_2G_1$ are different sets in general — that is a real fact about composing displacements, and a chain built in one order is not the chain built in the other. Yet the span measurement never looks at the order, and reports the same answer either way.

The resolution is that being a group is order-independent even though the set is not. A subgroup is closed under inverses, so if $S = G_1G_2\cdots G_n$ is a group then $S = S^{-1}$; and the inverse of a product is the product of the inverses in reverse order, which for subgroups is $G_n\cdots G_1$. So a chain whose product is a group has the same displacement set as its own reversal, and the two orders coincide exactly in the case the measurement is asking about.

Which is a pleasing arrangement rather than a coincidence. The measurement is blind to the order, the property being measured is invariant under it, and the cases where the order genuinely matters are precisely the cases where the answer is not a group — where the reversed chain reaches a different set, both of them six-dimensional, and neither of them closed.

It also gives the reversal a physical reading. A chain read from the tool back to the base is the same hardware with the roles exchanged, and its displacement set is the set of inverses of the original’s. For a group-valued chain the two sets are identical, so a SCARA arm bolted down at its wrist and holding the world at its base would reach the same Schoenflies motion — which is a strange sentence and is exactly what the algebra says.

For a generic chain the two sets differ, and the difference is not a nuisance so much as a fact nobody usually has to face, since an arm is bolted down at one end by construction. The order matters to the mechanism, the order does not matter to the classification, and the reason those two statements sit together is that the classification is asking about a property no ordering can change.

What this sets up for a closed loop #

A serial chain is the easy half, and the reason is worth naming before the field moves to loops.

An open chain reaches everything its product allows: pick any joint values, and the pose is reachable. So the reached set is the product, and asking about the product is asking about the mechanism. A closed loop is not like that. Its joint values satisfy a closure condition, so the set of poses a link reaches is a subset of the product, cut out by an equation, and generally a much smaller one — a four-bar’s coupler reaches a one-parameter curve inside a three-dimensional group.

That makes the loop question strictly harder and strictly more interesting. The product of a planar four-bar’s four revolute groups is the whole planar group; the coupler reaches a curve in it; and the curve is inside the same group, which is the trivial part. What the loop test has to find out is whether a curve that is not obviously inside anything is inside something — and Bennett’s linkage is the case where the answer is no despite every count saying it should be yes.

Which is the right note to end on. A chain being in a group is a constraint, not a capability. The SCARA’s four dimensions are four dimensions the machine has and two it does not, and that is exactly why it is cheap, stiff and fast. The general arm is in nothing and can do everything, and pays for it in joints.

The constraint is worth more than it looks, though, and this is the half a designer cares about. A machine inside a group has the same behaviour everywhere in its motion, in a strong sense: any statement about what it may do near one pose transports to every other pose by an element of the group. A SCARA arm’s tool face is level at every configuration it can reach, not at the configurations somebody checked. A planar mechanism’s coupler stays in its plane through the whole of a sweep and not merely through the sampled part of one. Guarantees of that kind are what a group buys, and they are why the next rung’s comparison — two chains that agree on every count and differ in this — is worth a whole essay.