A seventh leg must lean its own way

A seventh leg opens the dead yaw gave the paired Gough platform a way across the yaw at which its six legs fail together. Six legs are singular where one twist does no work on any of them, which is one condition on the pose. Seven legs are singular only where that twist also does no work on the seventh, which is a second condition. A path that has to cross the six legs’ singular set generically misses the seven legs’ set, as a curve in a plane misses a point. The best placement found kept the seven legs’ smallest singular value near 0.029 through the dead yaw on every random path tried.

That essay left the seven legs’ own singular set unmapped. The count says it exists: two conditions in a six-dimensional space of poses leave a four-dimensional set, and a set that large has to be somewhere. If it runs through the neighbourhood of the dead yaw, a task that crosses there might be forced near it. If it does not, the seventh leg removes the dead yaw from the machine, and not merely from the paths that were tried.

The answer turns on a single angle of the seventh leg, and the angle is one the six legs already have. Seen from above, every leg runs from the base circle to the platform circle with its platform attachment some way round the axis from its base attachment. Call that angle the leg’s lean. At the dead yaw, the six legs lean 82.5° or 97.5°, and that equality is the whole cause of the dead yaw. A seventh leg that leans the same way is just one more leg of the same kind and fails with the six everywhere. A seventh that leans otherwise does not fail anywhere on the level dead yaw, and tilting the platform brings its singular set in only for leans near the six’s.

The six legs lean alike #

The paired platform has its base attachments in three pairs, 12.5° either side of 0°, 120° and 240°, on a circle of radius 2.2. Its platform attachments are in three pairs 20° either side of their own centres on a circle of radius 1.1. At the dead yaw the platform is turned so that each leg’s platform attachment sits 82.5° or 97.5° round the axis from its base attachment. One leg of each pair has the first lean and the other the second.

Those two angles have the same sine. A leg running from a point on the base circle to a point on the platform circle has a moment about the vertical axis of $R_b R_p \sin(\text{lean})$, the two radii times the sine of its lean, and here that is 2.2 × 1.1 × sin 82.5° = 2.399 for every one of the six. Tilted near the dead yaw gave that as the reason the dead yaw exists: the six legs’ moments about the vertical are equal, so together they cannot resist a turn.

The twist they cannot resist has a closed form at a centred position. It is a screw about the vertical axis, turning and sliding at once, with a pitch of −2.399 divided by the platform’s height. The measured pitch is −0.9997 at a height of 2.4 and −0.7498 at 3.2, both matching. A line does work on such a screw equal to its moment about the axis plus the pitch times its rise. Every leg rises the platform’s height, so the pitch term contributes −2.399 to every leg, and each of the six legs’ moment cancels it exactly. That is what makes the six singular.

A seventh leg joins them or does not #

The same arithmetic applies to a seventh leg between the same two circles. Its work on the free twist is its own moment, $R_b R_p \sin(\text{lean})$, less 2.399. That is nought when the sine of its lean equals the sine of 82.5°, at the two leans 82.5° and 97.5°, and at no others.

The measurement finds those two roots by bisection on the computed work, without the closed form, and they agree with 82.5° and 97.5° to a billionth of a degree. The work is positive between the two roots, where a leg’s moment exceeds the six’s, and negative everywhere else. It is most negative near 278°, a leg leaning almost a quarter turn the other way. The best leg of the earlier placement map leans 285° and does work −0.506 against an extreme of −0.507. That map was searched over the least singular value across the workspace, a different measure, and it found a leg within seven degrees of the one this argument would pick.

The closed form holds only at a centred position. Off the centre the free twist’s axis tilts, and the argument from moments about a vertical axis no longer applies directly. The measurement takes over there. A seventh leg leaning 82.5° or 97.5° does no work on the free twist at every sampled level position across the workspace, to 10⁻¹², from base attachments at 0°, 40° and 200°. The seven legs’ smallest singular value at those poses is nought. Every other lean on the scan, in half degrees within sixty of square and two degrees beyond, keeps one sign of work across the whole level slice, so it fails nowhere.

So on the level dead yaw the seven legs’ singular set is all or nothing. A seventh leg that leans like the six is singular wherever the six are, which on the level dead yaw is everywhere. Any other seventh leg is singular nowhere on it. The earlier essay’s warning, that some seven-leg singularity must exist, is true, but it does not have to be at the dead yaw, and for almost every leg it is not.

Tilt brings in the legs that lean nearly alike #

A level dead yaw is a slice of the six legs’ singular set, not the whole of it. The dead yaw opens into a wedge found that once the platform tilts, each position is singular at its own yaw, displaced from the dead yaw in proportion to the tilt and to the position’s offset. The six legs’ singular poses near the dead yaw form a five-dimensional surface, parameterised by three coordinates of position and two of tilt, with the yaw solved at each.

On that surface the six legs’ moments are no longer exactly equal, and the free twist is no longer the level one. A seventh leg’s work on it can then change sign across the workspace, for a leg leaning close to the six. Where it changes sign, the seven legs are singular too.

The measurement samples that surface: 122 positions of the workspace, x and y within ±0.8 at heights 1.8, 2.4 and 3.2, each tilted 1°, 2°, 4° or 8° about eight axes and turned to its own singular yaw. That makes 4,026 poses, each with its free twist computed once. For a seventh leg at every lean round the axis, the question is whether its work takes both signs over the poses up to a given tilt.

Level, the failing leans are the six legs’ own two, 82.5° and 97.5°. Up to 1° of tilt they spread to 81.5°–83.5° and 96.5°–98.5°. Up to 2° they reach 79.5°–85° and 94.5°–100.5°. By 4° the two bands have merged, 74.5° to 107°, and by 8° they cover 44.5° to 137.5°. No lean beyond 180°, on the far side of the axis from the six, is reached at any tilt sampled.

A leg 1.5° from the six’s lean, at 84°, is clear of the level dead yaw. Tilt the platform 2° and its singular set enters the workspace. A slice through the workspace at a 4° tilt shows the set as a curve across it.

That curve is the four-dimensional set the count promised, cut by a two-dimensional slice. It divides the slice into 417 positions where the 84° leg’s work is positive and 208 where it is negative. A platform with this seventh leg, tilted four degrees and turned to the six legs’ singular yaw, fails along that curve. A path across the dead yaw at that tilt can still miss it, but the set is in the workspace, and the margin the seventh leg was added to buy is gone near it.

How fast the band widens #

The bands do not widen in proportion to the tilt. The lower band’s edge sits 1° from the six legs’ lean at a tilt of 1°, 3° at a tilt of 2°, 8° at a tilt of 4° and 38° at a tilt of 8°. The first two doublings of the tilt allowed roughly triple the range of leans that fail, and the third nearly quintuples it. A designer who expects the platform to tilt a little can lean the seventh leg a few degrees away from the six and be safe. One who expects eight degrees has to keep it outside 44.5° to 137.5°, which in practice means the far side of the axis.

The work suggests why. Level, every leg’s work on the free twist is its moment less a common term, and the six legs’ moments are exactly equal. Tilted, the six legs no longer rise equally: the platform attachments on the high side are lifted and those on the low side lowered, so each leg’s pitch term is different. The free twist moves to balance six legs whose moments and rises have spread apart, and a seventh leg fails where its own moment and rise happen to match the new balance somewhere in the workspace. The spread grows with the tilt, and it grows with the workspace’s offset from the centre, which is why the bands in the figure are wider than any single position would give.

None of this touches the far side of the axis. A leg leaning between 180° and 360° has a moment about the vertical of the opposite sign to the six legs’ 2.399, and no tilt sampled moves the free twist far enough for that to be matched. A yaw that is singular everywhere found the dead yaw from one line of trigonometry about the six legs, and no path across the dead yaw misses it found that six legs cannot be steered round it. A seventh leg on the far side changes both findings, at every tilt sampled.

The best leg leans the other way #

The best leg of the placement map leans 285°, on the far side of the axis, and none of the bands reaches it. Its work on the free twist stays between −0.64 and −0.18 at every one of the 4,026 sampled poses, level and tilted up to 8°. The seven legs’ smallest singular value on the six legs’ singular poses stays at or above 0.0089, which is the eight-degree tilt’s worst. Level it is about 0.022.

The other legs fall in order of how close they lean to the six. A seventh leg leaning as the six do is singular level, everywhere. One 1.5° away enters at a 2° tilt, one 7.5° away at 4°, and one 22.5° away at 8°. The best leg does not enter at any tilt tried. The order is the one the moments predict: the further a leg’s moment about the vertical is from the six legs’ common moment, the more tilt it takes to bring the six legs’ moments far enough apart for the seventh to match the free twist.

That answers the question the earlier essay left. For the best leg, within the workspace and within eight degrees of tilt, the seven legs’ singular set does not come near the dead yaw. The seventh leg removes the dead yaw from that neighbourhood of the machine, not just from the paths tried. Where the four-dimensional set lies for this leg is not measured here. It has to be somewhere among the six legs’ singular poses, and those extend far beyond the dead yaw’s wedge, to other orientations and to positions outside the box.

A rule for placing the seventh leg #

The result gives a designer a rule that needs no search. Lean the seventh leg away from the six. The six legs’ common lean is what makes the dead yaw singular; a seventh leg that shares it inherits the defect, and one that leans nearly as they do inherits it as soon as the platform tilts. The safe leans are on the far side of the axis, where the sine of the lean is negative and the leg’s moment about the vertical opposes the six legs’.

This is the same lesson one placement of every placement found for a planar platform, in a different form. There, a fourth leg removed a similar platform’s two singular orientations unless its own attachment points were related by the same similarity as the other three, and then it changed nothing. A redundant leg helps only where it differs from the legs whose failure it is meant to cure. Here the difference is one angle, and it can be read off the attachment points before anything is built.

It also says what the dead yaw is a design choice changes. Rotating the platform’s pairs by an angle against the base’s moves the dead yaw by the same angle the other way, so at the new dead yaw every platform attachment comes back to where it was round the axis and the six legs lean 82.5° and 97.5° again. A seventh leg fixed to the platform does not come back: its lean at the new dead yaw is shifted by the rotation. A leg placed for one rotation of the pairs has to be checked again for another, since the safe side of the axis is defined by the six legs, not by the platform.

It also says why the earlier placement map found most placements adequate. A leg’s lean ranges over the whole circle, and on the level dead yaw only two leans fail. The 15° grid of that map contained no leg at exactly 82.5° or 97.5°, so every placement it tried was clear at the level dead yaw. That is why none of its 576 entries reached nought. Its weakest entries were the legs near those leans, whose work on the free twist is small though not zero.

What the slices leave out #

Legs between other circles. Every seventh leg here runs between the same two circles as the six. A leg with its own radii has a moment $R_b' R_p' \sin(\text{lean})$, and at a centred position the closed form says it fails where that equals 2.399, whatever its lean. Off the centre the measurement has not been made.

Tilts beyond eight degrees, and other yaws. The sampled surface is the six legs’ singular poses within eight degrees of tilt of the dead yaw. The six legs are singular elsewhere too, and the four-dimensional set has to lie somewhere among those poses. A task that tilts the platform further, or turns it far from the dead yaw, would need its own map.

Stiffness, not only singularity. The least singular value of 0.0089 at an eight-degree tilt keeps the best leg’s platform nonsingular, but the earlier essay found it lifted no further than the six legs’ next-weakest direction allows. A singular set kept out of the workspace is not the same as a platform held firmly, and how stiff it is near the wedge’s edge is a separate number.

The cost in internal force. A seventh leg makes the platform redundantly constrained, so a length error in one leg is fought by the others. The earlier essay found that at the dead yaw the seventh carries none of that internal force. Whether that stays true for the best leg away from the level dead yaw is not measured here.

Still open: a seventh leg’s own dead yaw #

The six legs’ dead yaw is the yaw at which their moments about the vertical are equal. A seven-legged platform turned to a different yaw puts the seven legs at different leans, and at some yaw the seventh leg’s lean may come round to match the six legs’ common lean. The seven would then be singular level at every position, a dead yaw of their own.

The distinct argument would be that yaw found. Every leg’s lean changes by the same angle as the platform turns, so the six legs’ common sine and the seventh leg’s sine change together, and the condition for seven legs to fail level at every position is one equation in the yaw. The measurement would find the yaws at which the best leg fails with the six, if any, and ask how far from the six legs’ own dead yaw they are. If they fall outside the platform’s working range of yaw, the seventh leg has no dead yaw of its own in use. If one falls inside, the rule for placing it would have to say which yaws the task needs before it says where the leg goes.