The arm that hits itself

The serial field’s founding observation is that an open chain has nothing to solve. Give it joint angles and the links go where they go; there is no loop, no closure equation, no Newton, and no configuration it cannot have. The configuration space of a three-joint planar arm is a three-torus and every point of it is a pose.

Give the links bodies and about a fifth of that torus is not available.

What can meet what #

An arm of three links has three pairs of links, and two of them are excluded for the reason the whole field uses: consecutive links share a joint, both surround the pin, and testing them reports every arm as colliding everywhere.

So only links 1 and 3 can meet, and the question is two-dimensional rather than three. That is not an approximation; it is the whole of it, for an arm of three links. A four-link arm has three testable pairs and a genuinely three-dimensional forbidden set.

Why the base angle drops out #

The first joint does not enter the answer, and the reason is worth stating because it is the one structural fact in this essay.

Turning the whole arm about its base is a rigid motion of every link at once. Distances between rigid bodies are unchanged by rigid motions, so whether links 1 and 3 are inside each other cannot depend on θ₁ — it depends only on θ₂ and θ₃, the angles between consecutive links.

That makes the forbidden set a two-dimensional region in a three-dimensional space: a cylinder over a picture, invariant along the θ₁ direction. It is why this essay has a picture rather than a description of a volume, and it is a genuine structural saving rather than a convenience — the whole set is determined by a plane section of it.

It is also checked rather than assumed: forty pairs of relative angles, each tested at two different base angles, with zero disagreements. That is the cheapest possible test of an argument that is obviously right, and it is exactly the kind of argument that turns out to be wrong when a placement has a bug in it.

The number #

At 0.13 half-width on links of 1.6, 1.2 and 0.9, over a 101 × 101 grid of the two relative angles: 21.4% of the arm’s joint space is forbidden by its own material.

That is a fifth of everything the arm can be commanded to do, on an arm whose links are around a tenth of their length wide. It is not a corner case, it is not near a limit, and it is not something a joint-limit table would catch — the forbidden region is in the middle of the range of both joints, shaped like a broad band across the square, and every angle in it is a perfectly ordinary number that a controller would accept.

What it depends on #

Three parameters, and they behave differently enough to be worth separating.

The width. Doubling every link’s half-width from 0.13 to 0.26 grows the band in the picture and raises the share, roughly linearly at first — the band’s area goes as the width, since it is a strip whose length is set by the geometry and whose thickness by the material. That makes the forbidden share a nearly-linear cost of building the arm out of anything.

The link lengths. A long third link reaches further round and meets the first over a larger range of angles; a short one may not reach at all. There is a proportion below which the share is zero — a third link shorter than the second cannot reach back past it — and the arm here is above it.

And the joint offsets, which are not modelled. A real arm’s consecutive links are offset sideways from each other so that they fold past rather than into each other, which is precisely a designer paying material to buy back joint space. This arm is planar and coplanar, so it has none of that, and its 21.4% is the price of building the whole arm in one plane.

That last one is the layers question arriving in an open chain, and the answer is the same: put links 1 and 3 in different planes and the self-collision disappears entirely. An arm built that way trades a fifth of its joint space for an offset — which is exactly what industrial arms do, and it is why they are shaped the way they are rather than being flat.

What the region looks like #

A band, and it is where a reader would expect once it is drawn: the arm folded back on itself, with the third link swung round into the space the first one occupies.

It is not a neighbourhood of the fully folded pose, which is the shape people expect. Fully folded — both relative angles at π — is one point, and the region round it is only part of the band; the band continues to configurations where the arm is bent but not folded, because a long third link swung far enough reaches the first link without the second doing anything unusual.

And it has a hole in it, at the origin of the plane: the arm straight out has links 1 and 3 collinear and a long way apart, which is as far from a self-collision as it gets.

Reachable, and attainable #

The second half of this essay is a different set, and the difference between them is what the field is for.

The workspace of this arm is the disc of radius 3.7 — the sum of the lengths — which is a statement about link lengths and nothing else. Put a square obstacle half a unit across in the way, at (0.55, 1.15), and ask instead where the tip can be put with every link clear of the obstacle and of itself.

The answer is 2.7% smaller. And the missing part is not a bite out of the disc round the obstacle: it is wherever the arm would have to reach through it, which is a set with a shadow-like shape running out from the obstacle in the directions no admissible arm posture can serve.

That is a small number and it is the interesting one, because of what it takes to be small.

Redundancy is what pays for the obstacle #

A three-link planar arm placing a tip has three joints and two things to satisfy, so it has a spare freedom: at each reachable point there is a whole circle of postures, and the arm that does nothing useful with its extra joint is the field’s essay on what that freedom is for.

Here is what it is for. Run the same measurement on a two-link arm — same obstacle, same widths — and the loss is 5.8%, more than twice as much. A two-link arm has exactly two postures at each point, elbow up and elbow down, and if the obstacle blocks both then the point is gone. A three-link arm has a continuum to choose from and needs only one of them to be clear.

So redundancy buys obstacle tolerance, and the price is measured here rather than asserted: a spare freedom halves what one obstacle costs the workspace.

That is the same freedom the serial field has already priced twice — once as the self-motion that moves the elbow without moving the tool, and once as the extra solutions it adds to the inverse problem. This is a third reading of it, and it is the one that says why arms are built with more joints than their tasks need.

Where the loss is, and where it is not #

The 2.7% is worth taking apart, because the shape of what is lost says more than the fraction.

Near the obstacle itself almost nothing is lost. A point a little to one side of a small obstacle is reachable by an arm coming in from the other side, and the redundant freedom finds that posture. The obstacle is a hole in the material, not a hole in the workspace.

What is lost is behind it, in the shadow: points that can only be reached by a wrist route that passes the obstacle on a particular side, where all the alternatives run out. And a second, thinner set is lost at the edge of the reachable disc, where the arm is nearly straight and has no spare posture to choose from — the redundancy is only useful where there is room to use it, and at full stretch there is none.

So the loss concentrates in exactly the two places a designer would least like: at the far edge of the reach, and in the region the obstacle occludes. A fraction of 2.7% distributed evenly would be a nuisance; the same fraction in a shadow behind an obstacle and a rim at full reach is a workspace with two specific holes in it.

How the attainable set is computed #

By inverse kinematics with a body test on each solution, and the method is worth stating because it is where the honesty of the number lives.

For each grid point inside the reachable disc, sweep the last link’s direction ψ through 48 values. For each ψ, the wrist is at a known place, and the remaining two-link problem has the closed-form solution the serial field derives — two of them, elbow up and elbow down. For each of those, place all three links as bodies and test them against the obstacle and against each other. The point is attainable if any posture is clear.

So the answer is a lower bound on attainability: a point is called blocked if none of 96 sampled postures works, and a finer sweep of ψ can only find more. At 48 directions the number has stopped moving, which is the evidence that the sampling is adequate — and it is a different kind of adequacy from the swept certificate’s, because here the samples are over a choice rather than over a motion, and missing one loses a point rather than a collision.

A picture of a constraint, rather than a list of limits #

It is worth comparing what has been produced with how this constraint is usually carried.

A robot controller normally holds joint limits — a lower and an upper bound per joint — and possibly a handful of hand-written exclusion rules for postures known to be bad. Limits are cheap, they compose trivially, and they are a box in joint space.

The self-collision set is not a box. It is a band across the θ₂–θ₃ square, curved, with a hole in it, and no product of intervals contains it without also excluding a great deal that is perfectly fine. Approximating it by limits either forbids good configurations or admits collisions, and which of the two depends on how the limits were chosen.

Having the set as a picture is therefore not a presentational nicety. It is the difference between a constraint that can be reasoned about — this band is what forbids folding back; this hole is the straight-arm pose — and a table of four numbers that stands in for it.

And the picture is small: two angles, one region, invariant along the third. That is the payoff of the base-angle argument above, and it is the reason this constraint is drawable at all when the general version — a four-link arm’s forbidden set, three-dimensional and not invariant along anything — is not.

What is not being computed #

Not a path. The attainable set says where the tip can be put; it says nothing about whether the arm can get from one attainable point to another, which is a question about the connectivity of the free set in joint space — the same question free arcs answer for a one-freedom mechanism, one dimension up. It is a real question and it is the beginning of a subject about algorithms, which is not this site’s.

Not joint limits. A real arm’s joints do not turn all the way round, and the limits interact with the self-collision region in a way that can be helpful — a limit that excludes the folded band is a limit that costs nothing.

Not the tool. The arm here ends at a point. A real one carries something on the end — a gripper, a torch, a probe — which is a body of its own, usually the largest one on the arm, and which collides with everything the third link does and more.

And not the base. The arm here is pinned to a point with no housing round it. A real base is a body, usually a large one, and the first link folding back onto it is a commoner collision than links 1 and 3 meeting.

What a fifth means in practice #

The share is a measure of joint space, and joint space is not what an operator experiences, so it is worth converting it once.

A fifth of the joint space unavailable does not mean a fifth of the tasks are impossible. The arm has a spare freedom, so a task specifying only the tip’s position can usually be done in a posture outside the forbidden band — which is the same argument that makes the obstacle cost only 2.7%. What the forbidden band costs is choice: the circle of postures at a given tip position has a piece missing, and the fraction missing varies from nothing to nearly all of it depending on where the tip is.

Where it bites hardest is a task that specifies the posture as well as the tip — a tool at a particular angle, an approach direction, a cable route that must not twist. Those pick a point in joint space rather than a set, and a point in the band is simply unavailable.

So the honest reading is that self-collision is a constraint on how a task is done rather than on which tasks are possible, and it becomes a constraint on the tasks themselves exactly when the task starts specifying the how.

An open chain has a constraint after all #

There is a tidy reversal here worth stating, because it is the serial field’s founding observation being qualified for the first time.

The whole point of an open chain is that it has no constraint equations: nothing to solve, nothing to satisfy, every joint value admissible. That is true of the equations and it stops being true of the machine. A fifth of a three-link arm’s joint space is unavailable, and the condition that excludes it is a genuine constraint — it is just an inequality between bodies rather than an equality between coordinates, so it enters nowhere in the mathematics that made the chain easy.

The closed-chain fields have the reverse asymmetry: their configurations are hard to find and, once found, mostly legal. An arm’s configurations are trivial to find and a fifth of them are not there.

That is the same trade in two directions, and it is the reason this field belongs to the whole site rather than to the loop fields: a mechanism’s difficulty moves between the equations and the material, and a description that only carries one of them will always report the wrong one as easy.

The forbidden set not depending on the base angle is the property that makes the whole rung drawable, and it is worth naming what kind of property it is. Self-collision is a question about the arm’s shape, and the base joint changes where the arm points without changing its shape at all — so the base angle cannot appear in the answer. That is a symmetry argument, it takes one sentence, and it reduces a three-dimensional question to a two-dimensional picture. The same argument works wherever a joint’s motion is a symmetry of the thing being asked about: a wrist roll cannot affect whether a tool clears an obstacle that is round about the roll axis, a turntable’s angle cannot affect a mechanism’s own internal clearances, and in each case one coordinate drops out for free. Look for the joints the question is invariant under before computing anything, because each one removes a dimension from whatever has to be swept — and on a problem that is otherwise a volume, one such joint is the difference between a picture and a data set.