A seventh leg opens the dead yaw

A yaw that is singular everywhere found a Gough platform with its legs in pairs that loses control at one yaw, wherever it is in its workspace, because there the six legs’ lines all lie in one linear complex. No path across the dead yaw misses it showed that this cannot be planned around. The determinant of the six leg screws has one sign a few degrees below the dead yaw and the other a few degrees above, at every tilt and position tried, and a continuous path from one side to the other must pass a pose where it is nought. Tilting only moves the crossing, and the long way round meets the second dead yaw.

That essay’s argument rested on the determinant, and it ended by noticing that a seventh leg would take the determinant away. What a fourth leg buys found the planar version. Three leg lines fail to hold a planar platform when they meet at a point, which is one condition, so failures form curves. Four fail only when all four meet, which is two conditions, so the curves become isolated points. The spatial counterpart is a seventh leg on the paired platform, and the question is whether it opens a way across.

It does more than that. Placed well, it removes the dead yaw, and every placement tried removes the singularity itself.

Two conditions instead of one #

The six legs are singular where some twist of the platform does no work on any of their lines. At the dead yaw that twist exists at every position, and near it the singular poses form a hypersurface, one condition in the six coordinates of pose, which every path from one side to the other must cross.

A seventh leg adds one line. For the seven to be singular, a twist must do no work on all seven. Near the six’s singular set the only candidate is the twist the six leave free, so the seven are singular only where that twist also does no work on the seventh line. That is a second condition. The seven’s singular poses, if there are any near the dead yaw, form a set of codimension two, and a path, which is a curve, misses such a set the way a curve drawn in a plane misses a given point. No sign argument survives it. Seven screws make a seven-by-six matrix, which has no determinant to change sign.

In any two-dimensional slice of pose space, yaw against one coordinate of position say, the six legs’ singular set is a curve that cuts the slice in two, as the earlier essay drew it. The seven legs’ singular set in the same slice is at most a scatter of isolated points, where the curve happens to pass through a pose at which the seventh line does no work on the free twist. That is the planar fourth leg’s picture one dimension up, and it is also why the leg lines are best thought of as lines a leg is pushed along. What matters about each leg is the line its force acts on, and a seventh line in a different place is a seventh independent constraint.

The hero figure shows the result at the simplest pose, level at home. The six legs’ smallest singular value falls to $9 \times 10^{-9}$ at the dead yaw, a sharp V. With a seventh leg from the base circle at 60° to the platform circle at 255°, it is 0.0288 at the dead yaw and no less than 0.0287 anywhere in sixty degrees of yaw. That is flat. The six legs, without the seventh, reach 0.0141 ten degrees off the dead yaw and 0.0320 thirty degrees off. With the seventh leg the dead yaw is no longer the platform’s worst orientation. It is not distinguishable from its average one.

Why six fail together #

The dead yaw is unusual even among singularities, and its unusualness is what a seventh leg exploits. Tilted near the dead yaw found that a level paired platform is singular at one yaw at every position at once. The six legs’ moments about the vertical are then equal, and together they cannot resist a turn. An ordinary singularity happens at particular poses because the legs’ lines happen to fall into a dependent arrangement there. This one happens because the platform’s symmetry forces the arrangement, at every position, for one orientation.

So the six legs’ failure at the dead yaw is not a failure of any one leg. It is a property of the six as a set. Legs intersect put the general rule for a parallel machine: the platform gets the intersection of what its legs permit, and here the six legs’ constraints intersect in one direction too few. A seventh leg is one more constraint in that intersection, and the only question is whether it constrains the missing direction. The map says that almost every placement does.

Where the seventh leg can go #

A seventh leg that does no work on the free twist would change nothing, and the question is how many placements are that bad.

Of the 576 placements on a 15° grid, none leaves the platform singular at any of the 49 workspace poses sampled at the dead yaw. 456 keep the smallest singular value above 0.01, and 72 fall below 0.005. The best, from 60° on the base to 255° on the platform, keeps it above 0.0226 over the sample. Checked over the whole workspace of 243 positions, at five tilts up to four degrees and at nine yaws from eight degrees below the dead yaw to eight above, its worst is 0.016, at a low corner tilted and eight degrees off. The seventh leg does not need careful placement to remove the singularity, only to make the removal worth having.

The pattern in the map is the free twist. At the dead yaw, at home, the six legs’ lines leave free a screw about the vertical axis, a turn coupled with a rise, and a seventh leg resists it in proportion to the work the screw does on its line. A leg lying in a plane through the vertical axis, like the one from 0° on the base to 0° on the platform, has no moment about the axis and does almost no work on the screw. A leg that runs a long way round, from 60° to 255°, has a large moment and does a great deal.

How high one leg can lift the platform #

The flat curve in the hero figure invites a larger claim than is true: that a seventh leg makes the platform as stiff at the dead yaw as anywhere. It cannot, and the reason is a theorem about matrices rather than about mechanisms.

Where the six are singular, adding a seventh row to their matrix can raise the smallest singular value only so far, and there are two ceilings. One is the seventh line’s own work on the twist the six leave free. Take the free twist as a trial motion and the seven lines resist it only through the seventh. The other is interlacing. Adding a row to a matrix cannot raise its smallest singular value above the next-smallest of the matrix without it. So the seventh leg fills the one direction the six had lost, and no more. Where the six were already weak in a second direction, the seven are no stronger there.

Measured over a sample of the workspace for five placements, every point lies under both ceilings. The weak legs sit at about eight-tenths of the first, their work on the free twist. The leg from 0° to 0° does a fiftieth of the best leg’s work and lifts the dead yaw at home only to 0.0018. The strong legs are held down by the second ceiling, sitting at seven-tenths to all of it. For the best leg that ceiling is the paired platform’s next-weakest direction at the dead yaw, and no seventh leg, however placed, can do better than the platform’s second-worst direction allows.

What the number means as stiffness #

A smallest singular value is an abstraction, and it has a physical reading. With every leg equally stiff, the platform’s stiffness in its weakest direction is the leg stiffness times the square of the smallest singular value. At home, ten degrees off the dead yaw, the six legs give 0.0141, so a weakest-direction stiffness of $2.0 \times 10^{-4}$ of a leg’s. The seventh leg at the dead yaw gives 0.0288 and so $8.3 \times 10^{-4}$, about four times as much. The comparison that matters to a builder is with the six legs at an orientation they handle well, and thirty degrees off the dead yaw the six give 0.0320, a stiffness of $1.0 \times 10^{-3}$. The seventh leg brings the dead yaw to four-fifths of that.

The square matters near the dead yaw itself. The six legs’ smallest singular value falls linearly as the yaw approaches the dead one, so their weakest stiffness falls as its square. A degree off the dead yaw the six give about a tenth of their ten-degree value, a hundredth of the stiffness. Three degrees off, where the earlier essay’s random paths began, the six give 0.0042, a weakest stiffness of $1.8 \times 10^{-5}$ of a leg’s, forty-six times less than the seventh leg gives at the dead yaw itself. Where the six-legged platform is soft enough to be useless for a band of yaws, the seven-legged one is uniform, and the band disappears rather than narrowing.

Every path across #

The earlier essay’s hundred random paths are the direct test. Each runs from three degrees below the dead yaw to three above, wandering over the workspace in position and in tilt up to four degrees, and each was followed with six legs and with seven.

With six legs every path crosses the singular set an odd number of times, and its least singular value is below $2 \times 10^{-8}$. That is nought to the precision of the computation, as the sign argument requires. With the seventh leg no path falls below 0.0234. At the very poses where the six are singular the seven are between 0.0234 and 0.0321. The seven’s singular set, if it reaches into this region at all, is missed by all hundred paths by a margin comparable to the platform’s ordinary stiffness. Nothing in the census comes near it.

The two routes to that result share almost nothing. The six-leg crossings are found by the sign of a determinant and refined by bisection. The seven-leg values come from a singular value decomposition of seven screws. They meet only in agreeing that where one says nought the other says a number of ordinary size.

What the extra leg costs #

Seven legs on a platform with six freedoms leave one combination of leg forces that loads the platform not at all, its internal force. In a redundantly driven platform that combination is the price. If the legs’ lengths disagree by a manufacturing error, the platform cannot satisfy all seven, and the legs fight each other along that combination.

At the dead yaw the internal force is a surprise. It is the six legs’ own self-stress, 0.405 and 0.411 alternating round the six, and the seventh leg carries none of it. That is exactly what dependence means. At the dead yaw the six lines are linearly dependent by themselves, so a combination of their forces already balances with nothing from the seventh. Ten degrees off, the six are independent, and the internal force spreads over all seven, the seventh taking 0.027.

The consequence is practical and one-sided. At the dead yaw an error in the seventh leg’s length is not fought by the others at all. It simply moves the platform along the twist the six leave free. The seventh leg alone decides where the platform sits in that direction, which is what one would want from a leg added to control it. Errors among the six, by contrast, are fought among themselves through their self-stress, and how hard depends on the legs’ stiffness and on how far the error is from one the platform could absorb by moving. That fight is already present in the six-leg platform at its dead yaw, where the self-stress is what the singularity is.

What a designer can take from it #

A seventh leg removes a singularity that no path planning could. The sign argument that made the dead yaw unavoidable needs a square matrix, and a redundant leg takes it away. Of 576 placements tried, none failed to remove it, and three in four did so decisively.

Place it for work on the free twist. At the dead yaw the free twist is a screw about the vertical, so the leg that helps most runs a long way round between base and platform. Its benefit is capped by the platform’s next-weakest direction, so beyond a certain point placement buys nothing more.

The extra leg is a positioner in the direction it rescues. At the dead yaw the other six do not fight it, which makes its length a direct control of the platform along the twist they cannot hold. The dead yaw is a design choice found that rotating the pairs moves the dead yaw. A seventh leg is the alternative to moving it: leave it where it is and stop it mattering.

What the seventh leg does not settle #

Actuation. A seventh actuated leg must be driven in agreement with the other six, or it and they fight. The internal force says how they fight, not how a controller should keep them from it. Commanding it costs nothing new: six legs and a square root found each leg’s length to be one subtraction and one square root, computed without consulting another, and a seventh leg is a seventh square root. What changes is the forward problem, which with seven lengths for six freedoms has a solution only if the lengths agree.

The seven’s own singular set. It is a set of codimension two and none of the poses or paths tried came near it, but it has not been located. Somewhere in pose space the free twist of the six does no work on the seventh line, and a path that happened to pass through such a pose would meet a singularity. None of the hundred did.

Leg collisions and reach. The best leg runs a long way round, from 60° on the base to 255° on the platform, and across the platform’s interior. Whether it clears the other six legs through the workspace is a question of their thickness and is not checked.

Still open: the seven legs’ own singular set #

The census found no seven-leg singularity anywhere it looked, and the count says one exists somewhere. Two conditions in six dimensions leave a four-dimensional set, which is large even though a path misses it.

The distinct argument would be that set found. Restrict attention to the six legs’ singular hypersurface near the dead yaw, where the free twist is known at every pose, and trace where the seventh line’s work on it passes through nought. The measurement would map that zero set over the dead yaw’s neighbourhood for the best leg and for a weak one. It would ask whether it enters the workspace at all, and how close to it the best path across can be forced. If the set lies outside the workspace for the best leg, a seventh leg removes the dead yaw from the machine altogether, and not just from the paths tried.