One placement of every placement

Two orientations no position can rescue found a planar platform that is singular at every position it can be put in, at exactly two orientations. Three extending legs run from base pivots to platform points; three lines through corresponding vertices of two triangles meet at one point whenever one triangle is a scaled copy of the other with no turn between them; and if the platform’s triangle is similar to the base’s there are two orientations at which it is such a copy. At either of them the three leg lines meet wherever the platform is put, and three concurrent lines cannot resist a turn about the point where they meet.

What a fourth leg buys had already measured what a redundant leg does to a different platform. On a 3-RRR, whose legs are jointed arms, the singular curves of each orientation’s slice shrink to isolated points. The obvious question for the 3-RPR is whether a fourth leg helps there too, and the obvious answer — that a fourth row in a Jacobian can only raise its rank — turns out to be wrong in one case and right in every other.

The one leg that changes nothing #

Three legs are singular when their lines meet, and a fourth line through the same point leaves them singular: the fourth row of the Jacobian lies in the same two-dimensional space the first three did, so the rank does not move.

That fixes the placement exactly. At the singular orientation the three lines meet at the centre of similarity $c$, the point for which $p_i = c + \kappa(b_i - c)$ with $\kappa$ the similarity’s scale factor. A fourth leg joins its base pivot $b_4$ to its platform point $p_4$, and its line passes through $c$ exactly when

$$p_4 = c + \kappa(b_4 - c).$$

For each choice of base pivot that is one point, and every other choice of platform point takes the rank back to three.

It is not an approximate statement. With the fourth base pivot at $(0, -2)$ and the attachment placed where that formula puts it, the fourth leg’s line passes $2.8 \times 10^{-17}$ from the meeting point, the smallest singular value of all four rows is $2.6 \times 10^{-9}$ — the eigen solver’s own floor on a matrix whose entries are order one — and the platform is exactly as singular as it was with three legs. Move the attachment 0.28 round the platform and the fourth line misses the meeting point by 0.399, the four legs hold at 0.291, and the singularity has gone.

That a redundant leg can be useless is not itself a surprise — a fourth bar on a parallelogram is the site’s standing example of a constraint that repeats one already there. What is unusual here is how narrow the uselessness is. The parallelogram’s third crank is redundant for a whole family of lengths; this fourth leg is redundant at one point of a continuum, and a millimetre either side of it the leg does its job.

Swept round the platform #

The formula is a prediction, so the way to test it is to look for the bad placement without using it.

The sweep moves the fourth attachment through three hundred and sixty degrees in three-degree steps, and at each placement measures the worst holding over a square of positions 1.8 across at the singular orientation. The curve touches nothing once, at −90°, and the similarity predicts −90.0° without being consulted.

Everywhere else the platform is held. The best placement, 27° round from the bad one, gives 0.058 — and the reason the best is near the worst rather than opposite it is worth a sentence, because it is the geometry rather than an accident. What a fourth leg contributes is the perpendicular distance from the meeting point to its own line, times the leg’s own leverage; moving the attachment away from the killing point increases the first quickly at first and then starts trading it against the second, and the trade turns over well before the far side of the platform.

So the answer to “which placements of a fourth attachment remove the singular orientation” is: all of them but one, and the one is named in advance by the same similarity that created the singularity.

It is worth saying what the measure is, because the whole argument is carried by one number. The holding is the smallest singular value of the four rows $[\hat u_i, \; \hat u_i \cdot E r_i]$, with $\hat u_i$ the unit direction of leg $i$ and $r_i$ its platform point measured from the platform’s reference — the same quantity the map of where a platform fails is drawn from, and the same one the hexapod’s singular yaw uses in six dimensions. It is nought exactly when the rows fail to span, which for a planar platform means the leg lines are concurrent or parallel, and it is otherwise a measure of how far from that the design is.

Proportional, with no threshold #

Between “one placement fails” and “the rest work” there is a question about how sharply the failure is confined, and it has a clean answer.

The holding is proportional to the offset. Over the first four steps the ratio is 0.4790, 0.4777, 0.4755 and 0.4711, drifting only as the offset becomes comparable to the platform’s own size. There is no threshold, no neighbourhood of bad placements, and no sudden recovery: a fourth leg misplaced by a thousandth holds the platform at a thousandth of what a well-placed one gives, which is a small number and not nothing.

That is the practically important half. A designer cannot accidentally build the bad platform — it requires hitting one point exactly — but a designer can build a platform whose fourth attachment is nearly at that point, and such a platform has an orientation at which it is nearly singular everywhere. The linear law is what turns “avoid one point” into a tolerance: keeping the holding above some floor means keeping the attachment more than a stated distance from the similarity’s own placement, and the stated distance is the floor divided by 0.479.

What the sweep was not told #

Two routes meet at the placement, and it is worth separating them cleanly because the agreement is the evidence.

The similarity route never builds a platform. It reads the base and platform triangles, fits the direct similarity carrying one onto the other, takes its scale factor and its angle, and from those computes where the centre of similarity sits at a stated pose and where the fourth attachment would have to be for its line to pass through that centre. Nothing in it is a singular value, a position or a sweep.

The sweep route never mentions a similarity. It places the fourth attachment at a hundred and twenty angles round the platform, builds the four-legged Jacobian at each of several hundred positions, takes the smallest singular value of each, keeps the worst, and reports where the worst is worst.

They agree to the resolution of the sweep. The prediction is −90.0° and the sweep’s minimum is at −90°, which is as close as a three-degree step can come, and the minimum’s own value is at the solver’s floor rather than merely small.

Both orientations, and the whole turn #

The similarity has two orientations, at $\kappa = +k$ and $\kappa = -k$, and everything above has been measured at one of them. The sweep over the whole turn says what happens at both.

With three legs the curve falls to the floor at two orientations, 0° and −180°, and those are the two the attachment points predict. A fourth leg placed where the similarity puts it leaves both of them exactly as they were — which is not obvious, since the killing condition was derived at one orientation, and it holds at the other because the fourth pair of attachment points sits at the same two radii as the other three and is therefore carried by the similarity at either sign of $\kappa$.

A fourth leg 0.28 round the platform from there removes both. The worst holding anywhere in the workspace at any orientation becomes 0.074, and the curve has no minimum touching the floor at all.

The position map is the same statement one level down, and the left panel is the picture the whole argument turns on: it is blank, because the smallest singular value is nought at every position and the shade has nothing to vary. With the fourth leg the same square is held everywhere, nowhere worse than 0.106, and the shading is a holding that varies from place to place rather than a singularity anywhere in it.

Both orientations being lost or kept together is also the reason this platform’s trouble reads as an architecture problem rather than a pose problem. A platform’s singular curves move with orientation and a designer can plan a path around them; a cusp can even be gone round to change assembly without passing through a singularity. Neither move is available here, because there is nowhere in the orientation to go: the whole slice is singular, and the only escape is a change to the machine.

What a redundant leg does, and what it does not #

Set beside the 3-RRR, the two results say the same thing in different vocabularies.

On the 3-RRR a fourth leg reduced the singular curves of each orientation’s slice to isolated points: the singularity was a condition on the pose, the fourth leg added a condition, and two conditions on two coordinates leave points. On the 3-RPR the singularity is not a condition on the pose at all — it is a condition on the architecture, and every pose at that orientation satisfies it. A fourth leg adds a row rather than a condition on where the platform is, so it either fixes the whole orientation or none of it.

That is why the answer here is binary where the answer there was dimensional. A singularity a mechanism has by design cannot be reduced by geometry to a smaller set of poses; it can only be removed or left. And the one way to leave it is to build the fourth leg out of the same design.

The general shape of it is worth stating without the platform. A redundant actuator removes a rank deficiency unless it is itself in the deficient subspace, and for a planar platform that subspace has a picture: the pencil of lines through one point. Anything whose line passes through the meeting point is already saying what the other three said.

Why anybody would build the bad one #

A result of the form “avoid one point in a continuum” usually needs no further attention, because nobody hits a point by accident. This one does need some, because there is a reason a designer would aim at it.

Similar triangles are what a symmetric platform has. Base pivots at 120° on one circle and platform points at 120° on another is the natural design — it is what the smallest parallel robot and most textbook figures draw — and it is exactly the condition that produces the two everywhere-singular orientations. Having found that, the natural repair is to add a fourth leg, and the natural way to add it is to keep the symmetry: a fourth base pivot on the same circle and a fourth platform point on the same circle at the same angle. That is a four-fold symmetric platform, and its base and platform quadrilaterals are similar, and similar quadrilaterals means the fourth leg’s line passes through the same centre.

So the bad placement is not an obscure point somebody might stumble onto. It is what taste produces. A four-legged platform built with all its attachment points on two concentric circles at matching angles has the same two singular orientations as the three-legged one it was meant to cure, and every further leg added on that pattern adds nothing.

They are the same point at every angle, to nought. With the fourth pivot at −90° on the base circle both read $(0.0000, -0.6000)$; at 137° both read $(-0.4388, 0.4092)$. The similarity does not have to be consulted for the symmetric design to land on it, which is exactly why the trap is a trap.

The repair is to break the pattern: put the fourth pivot on a different radius, or at an angle that is not the platform point’s, or both. The sweep says the cost of doing so is nothing — almost every placement works and the good ones work well — but it has to be done deliberately, because the symmetric choice is the one a drawing falls into.

What this does not settle #

Only the attachment moves. The fourth base pivot is fixed at $(0, -2)$ throughout, and the sweep is over where the platform point goes. The other design variable — where to put the fourth pivot on the ground — is not swept, and the killing attachment moves with it, so the full answer is a surface of bad placements in the four-dimensional space of fourth-leg attachment points rather than the one point measured here.

Holding is not force. The smallest singular value says how nearly the four leg lines fail to span; it is not a torque, a stiffness, or a load a platform can carry. Redundant actuation also brings its own difficulty — four motors commanding three freedoms must agree, and what happens when they do not is a question about misfit rather than about rank.

The triangles are exactly similar. A nearly similar platform has no exactly singular orientation and a worst one instead, held in proportion to the misfit. A fourth leg on such a platform has no exact placement to avoid, only a worst one, and whether the linear law above survives that softening is not measured.

Legs are lines, not bodies. Each leg is treated as a line of action from pivot to platform point, which is what an extending leg with a revolute at each end contributes. It has no thickness, so nothing here asks whether the fourth leg at its best placement would foul one of the other three — and on a platform 1.2 across with four legs converging on it, that is a question somebody has to ask. Three legs and one plane is where the same site meets it from the other side.

One workspace. All the holdings are worsts over a square 1.8 across centred on the base. A larger square includes poses where a leg is nearly at nought length and the measure means less; a smaller one is easier everywhere.

Still open: the fourth leg that helps most #

The sweep answers where a fourth leg may not go and shows, in passing, that the best placement is 27° round from the worst — but it answers that at one radius, for one base pivot, and at one orientation.

Its distinct argument would be the design chart: over both anchors of the fourth leg, the worst holding it leaves across the whole turn of orientation and the whole workspace, so that the bad set appears as a curve in that space rather than as a point in a slice of it, and the best placement appears as a maximum. Two things would come out of it. Whether the best fourth leg is always near the worst one, which the trade described above suggests and one sweep cannot establish; and whether a fourth leg can be placed so that the platform’s worst holding over everything is better than the three-legged platform’s best — which would be the statement that redundancy buys conditioning rather than merely removing a singularity, and is the question the hexapod’s singular yaw will ask in six dimensions.