Where the arm loses a direction

An arm has three singularities. They are usually presented as a list, which makes them sound like three cases of one thing, and they are not: they sit in three different places, they take away three different quantities, and only one of them is a problem worth engineering around.

The three #

The elbow singularity is the arm straight. The wrist centre is as far from the shoulder as it can get, and the tool cannot be moved further out by any combination of joint rates. The constraint screw has zero pitch and points along the arm itself: a pure force, radially outward, that the joints cannot resist.

Nothing useful is lost. The direction the arm cannot move in is the direction there is nothing to move into, and the same is true at the fully folded posture on the inside. A machine meets this singularity when it has been asked to reach further than it can, and the right response is to have asked for less.

The shoulder singularity is where the two ways of facing the target merge. On an arm with a shoulder offset — 0.18 m here — the wrist centre can never come nearer the base axis than that offset, and on the cylinder of exactly that radius the front and back solutions of the inverse problem become one. The constraint screw is again a pure force, this time across the arm’s plane rather than along it: the tool cannot be moved sideways, tangentially about the base.

This one is interior in one sense and boundary in another, and both are true. It is deep inside the region a datasheet calls the working envelope, and it is on the surface of a cylindrical hole the arm cannot reach into at all. Which description is right depends on whether the hole counts as part of the workspace.

The wrist singularity is the fourth and sixth axes in line, which happens when the fifth joint reads zero. It is the one that stops real machines, and it earns the attention on two counts. The tool is somewhere entirely ordinary — sampling 1,680 wrist-singular postures puts the tool anywhere from 0.495 m to 2.580 m from the base, which is essentially the whole workspace — and what is lost is not a pure force.

What is lost is a screw #

The natural way to say what a singularity takes away is “the direction the arm cannot move in”, and the natural way to compute it is the singular vector belonging to the smallest singular value: the collapsed axis of the manipulability ellipsoid.

That vector depends on the units. It is a twist — three components in radians per second and three in metres per second — and “orthogonal to every column of the Jacobian” is a statement about an inner product that adds a radian to a metre. Rescale the arm and the answer rotates.

Measured, at the wrist singularity of this arm. Written in metres, the collapsed direction is

$$(-0.346,\ -0.470,\ -0.145\ \mid\ 0.045,\ 0.708,\ 0.368)$$

and written in millimetres — the same arm, the same posture, the same machine — it is

$$(0.576,\ 0.781,\ 0.242\ \mid\ -0.000,\ -0.001,\ -0.001)$$

One says the arm has lost a mixture of a rotation and a translation. The other says it has lost a rotation and nothing else. They agree to 0.60, which is to say they do not agree.

The invariant object is the reciprocal screw: the wrench that does no work against any joint screw, under the same reciprocity form $\boldsymbol\omega\cdot\mathbf v' + \boldsymbol\omega'\cdot\mathbf v$ that this site’s screw field uses on closed loops. Its direction comes out identical in both unit systems — they align to fourteen decimal places — and its pitch scales as a length must: −0.6285 m and −628.5 mm.

So the singularity essays here quote reciprocal screws. Three quantities survive a change of units and they are the ones worth putting in a caption: the rank, which is a count; the fact of a singularity, which is a rank statement; and the constraint screw, which is a line with a pitch on it. A condition number is not among them, which the Jacobian essay measures — 6.55 in metres and 3234 in millimetres, for one arm at one posture.

Which is the same machinery the loops use #

That the constraint at a serial arm’s singularity is a reciprocal screw is not an analogy. It is the same computation, on the same objects, as the one the spatial field runs on closed loops.

A loop’s joint screws span a system; the wrenches reciprocal to that system are the constraints the loop imposes; and the whole overconstraint argument is about measuring that system rather than counting it. An open chain away from a singularity has six independent joint screws, so its reciprocal system is empty — there is no wrench it cannot act against, which is another way of saying it can move any way it likes. At a singularity the span drops to five and one reciprocal screw appears.

That is the cleanest statement this field has of what a singularity is: a posture at which an open chain acquires a constraint.

And it gives the pitch a meaning. A zero-pitch reciprocal screw is a pure force: there is a line along which the tool cannot be pushed. A finite-pitch one is a force and a couple locked together in a fixed ratio: the tool cannot be pushed along that line without also being twisted about it, and no combination of joint rates produces one without the other. The wrist singularity is of that kind, and it is the same distinction the spatial field drew when it separated subgroup mechanisms, whose constraints are forces and couples, from paradoxical ones, whose constraints are screws of finite pitch.

Losing a freedom, against gaining one #

The parallel field made a distinction that this essay is the other half of, and the pair is worth setting out together because the vocabulary is identical and the machines fail in opposite ways.

A parallel mechanism has two kinds of singularity. At the first, the inverse one, a leg runs out of reach and the platform reaches a boundary. At the second — the direct one, in the middle of the workspace — the platform gains a freedom: it can move with every actuator locked, and no amount of motor torque will hold it.

A serial arm’s singularity is the mirror image. The arm loses a freedom: there is a direction the tool cannot be moved in, whatever the motors do. The machine does not run away; it refuses.

Both are rank deficiencies of a Jacobian, both sit inside the workspace, and the consequences could hardly be more different. A parallel platform at a direct singularity is dangerous. A serial arm at a wrist singularity is merely unable, and what makes it dangerous is the controller’s attempt to be able anyway — which is the next essay, and the demand goes as one over the distance.

The ellipsoid does not shrink, it flattens #

The manipulability ellipsoid is the set of tool velocities the arm can produce for a joint rate of one, and its axes are the Jacobian’s singular values. It is the most useful picture in this field and the most easily misread.

Approaching a singularity, the ellipsoid does not get smaller. One axis gets shorter and the others do not: at an ordinary posture of this arm the six singular values run 2.614, 1.700, 0.799, 0.512, 0.267, 0.254, and at the wrist singularity the last is zero and the rest are much as they were. The arm can still move quickly in five directions. It cannot move at all in the sixth.

That distinction matters for what a machine should do about it. A uniformly slow region can be traversed slowly. A collapsed direction cannot be traversed at all, and a controller that responds by slowing down is solving a problem it does not have while ignoring the one it does.

Two things called a singularity #

The word has been used for two different objects in this field already, and the wrong field’s habit of separating homonyms applies here as much as it did to jamming.

A singularity of the mechanism is a configuration at which the Jacobian loses rank. It is where the arm cannot move in some direction, it is a property of the machine, and no change of description removes it. All three above are of this kind.

A singularity of a parameterisation is a place where a chosen set of numbers stops being able to describe a rate of change. Euler angles at their pole are the standard case, and the Denavit–Hartenberg table at parallel axes is another: the parameters are undefined there, and the arm is perfectly ordinary.

The two are separated by exhibiting a posture where one holds and the other does not, and this arm has one: at joint values (0.30, 0.04, 1.56, 0, −1.60, 0.20) the Jacobian is rank 6 with a smallest singular value of 0.502 — as far from singular as this arm gets — while a z-y-z description of the tool’s orientation is exactly at its pole. A controller working in those angles reports an infinite demand from a machine that is standing there entirely capable.

It is worth noticing that the mistake runs in both directions. Taking a coordinate singularity for a mechanism one makes a machine avoid a region for no reason. Taking a mechanism singularity for a coordinate one leads to the belief that a better parameterisation would fix it, and no parameterisation will: the wrist genuinely has five independent axes there instead of six.

Near, rather than at #

Exactly singular configurations are a measure-zero set and no machine ever visits one. What machines visit is their neighbourhood, and there the relevant quantity is not the rank — which is 6, right up to the instant it is 5 — but how small the smallest singular value has become.

That is why σ_min is what these essays plot and why the rank is what they assert. The rank is a discontinuous function of a matrix and there is no tolerance-free way to compute it in floating point; the site’s own rank routine reports how close the decision was for exactly that reason. A σ_min of $10^{-9}$ and a σ_min of 0 have the same practical meaning and different ranks, and any statement that turns on which side of a threshold a number fell should be treated as a statement about the threshold.

The shoulder singularity in the table above is the case in point: its σ_min is $4.2\times10^{-9}$ rather than zero, because the posture that produces it is computed from a cosine rule and lands on the cylinder to within floating-point noise rather than exactly on it. The rank routine calls it five. A tolerance ten times tighter would call it six, and the arm would be no different.

Eight becomes four #

There is a third way to see each of these, and it needs no Jacobian at all: count the solutions of the inverse problem.

At a general pose there are eight postures. Ask for a pose that sits on a singularity and the construction still returns eight — it is arithmetic, and it does not know — but they are no longer eight different postures. At the elbow singularity, elbow up and elbow down have become the same arm, and the eight collapse to four distinct sets of joint values. At the shoulder singularity, front and back have merged, and again four.

The wrist is different and more interesting. There the fourth and sixth joints have become the same axis, so only their sum is determined: the true solution set is a one-parameter family rather than a list, and the construction — which has to return something — resolves the ambiguity by a stated convention, putting the whole sum into one joint. Counting distinct answers gives seven, which is not a meaningful number; it is an artefact of that convention meeting floating-point arithmetic, and it is exactly the sort of number that would be reported as a finding by anybody who did not know where it came from.

That the branch count drops is the same event as the rank dropping, seen from the inverse side. Two solutions merging is two branches touching, and the configuration where they touch is the only place a mechanism can pass from one to the other — which is the statement the space of configurations makes precisely.

The one measure that transfers #

If a condition number cannot be quoted across arms and a singular vector cannot be quoted across units, the obvious question is what can be compared, and there is an answer.

A fitted exponent. How the demand grows as a singularity is approached — measured over a spread of distances and fitted by least squares through the logs — is dimensionless, because it is a ratio of ratios. The next essay measures it at −1.010 and checks it against σ_min’s own exponent of +1.008, which must be equal and opposite. Neither number changes if the arm is rewritten in millimetres, in inches or in furlongs.

That is the general escape from a units problem and this site has taken it before: the Panhard rod’s error is second order and a Watt linkage’s is fifth, fitted at 2.001 and 5.011, and those two numbers are comparable between two mechanisms of different sizes precisely because they are exponents. A millimetre is not comparable with anything until it is divided by another millimetre.

The other thing that transfers is a ratio to the arm’s own scale: σ_min divided by the largest singular value, or a joint rate divided by the rate available at an unremarkable posture. Both are dimensionless and both are honest. What is not honest is a bare σ_min with no arm and no unit attached, and that number appears in a good deal of published work.

What a machine does about each #

Three singularities, three different responses, and only one of them is interesting.

The elbow. Do not ask for poses outside the workspace. This is a planning question and it is answered before the machine moves.

The shoulder. Do not put the fixture directly over the base. Every installation guide says this and now there is a reason: it is not that the arm is weak there, it is that the two ways of reaching there have merged and the arm cannot move tangentially at all.

The wrist. This one cannot be planned away, because the singular set covers the workspace and a perfectly ordinary task — point the tool along the forearm — sits on it. The responses in service are all compromises: damp the inverse solve and accept that the tool does not go quite where it was told; project the demand onto what the arm can do and accept that the orientation drifts; detect the approach and stop. Each of those is a decision about which error to prefer, and the kinematics can say what each costs but not which to choose.

The three singularities differ in where they are and they agree in something worth naming: each drops the rank by exactly one. Not two, not a range — one, in every case, on this arm. That is what makes the smallest singular value a usable measure across all three: a single scalar approaching zero, with the same meaning at the elbow, the shoulder and the wrist, so one number in a controller covers three geometrically unrelated events. It is also why the ellipsoid flattens rather than shrinking. A rank dropping by one collapses exactly one axis of the velocity ellipsoid while leaving the other five where they were, so the arm loses one direction and keeps its speed in the rest — which is the geometric content of losing a direction and is a much more specific statement than the arm slows down. A mechanism whose rank fell by two would flatten two axes at once and would need two numbers to describe; that this arm never does is a property of its architecture rather than of singularities in general, and it is worth knowing that the convenient single measure is not guaranteed elsewhere.