No path across the dead yaw misses it

A paired Gough platform has a yaw at which it is singular everywhere. A yaw that is singular everywhere found it, and it is where the six legs’ moments about the vertical all become equal and the legs stop being able to resist a turn. The dead yaw is a design choice placed it by rotating the platform’s pairs against the base’s, at 90° less that rotation. The dead yaw opens into a wedge found what tilting does to it. Level, the platform is singular at every position at one yaw. Tilted, each position is singular at its own yaw, displaced from the dead one in proportion to the tilt and to the position’s offset over its height, so the bad orientations fill a wedge standing on the dead yaw.

That last essay ended on a task that has to turn through the dead yaw — a platform that must present a tool at yaws on both sides of it. Level, such a task crosses at one instant and at that instant is singular everywhere. Tilted, a singular surface sweeps across the workspace, meeting each position at that position’s own yaw. The hope it put forward was that a path combining position and orientation could be arranged so that the surface was always somewhere else: the platform turning through the dead yaw while moving so that the singular surface never reached it.

The hope is false, and the reason is an argument that needs no search. The measurements below confirm it and then turn to the question that is left, which is not whether a crossing happens but how exposed it is.

The determinant has a sign #

The instrument the wedge essay used is the one that settles this. The six leg screws of a Gough platform, stacked, make a square matrix, and the platform is singular exactly when its determinant is nought. Away from that set the determinant has a sign, and the sign cannot change without passing nought, because the determinant is a polynomial in the legs’ coordinates and those are smooth functions of the pose. Six legs and a square root is why: given a pose, every leg’s line is known outright, with no branch to jump between.

So the question is what sign the determinant has on the two sides of the dead yaw. The platform is the one the earlier essays used, with its pairs unrotated so that the dead yaw is at 90°, and the workspace is theirs too. It holds horizontal offsets within ±0.8 of home, heights 1.8, 2.4 and 3.2, and 243 positions.

Three degrees below the dead yaw it is negative, and three degrees above it is positive. The same holds at every one of the 243 positions, tilted by nought, one, two or four degrees about any of five axes: 1,215 poses on each side, and not one exception. Tilted, each position’s curve crosses nought at its own yaw, which is the wedge. But every curve crosses from the same sign to the same sign.

That is the whole argument. A task begins three degrees below the dead yaw and ends three degrees above it, at any positions and tilts inside this range. Its determinant begins negative and ends positive. Whatever route it takes between them, through any positions, tilts and yaws, the determinant varies continuously along it and so passes nought. At that pose the platform is singular. Tilting moves the singular set, but it does not open a gap in it.

A line that leans and never parts #

The figure below makes the argument visible on one slice of the poses. Across is the yaw; up is the position along the workspace’s low diagonal, from one corner at height 1.8 to the other; each cell is shaded by the determinant’s sign. The line between the shades is the singular set.

Level, that line runs straight up the slice at the dead yaw, since every position is singular there. Tilted two degrees, it leans: the low corners are singular about half a degree either side. Tilted four, it leans further. Three paths are drawn from the left edge to the right. One holds its position, one moves from corner to corner, and one wanders back and forth across the line. The first two meet the line once, and the third meets it three times.

A leaning line in a plane can be crossed at many places, and the choice of place is what tilting offers. But it runs from the bottom of the slice to the top, so a path from the left edge to the right cannot go round it. In the full space of poses, six dimensions, the same is true of a surface. The sign argument is the reason it runs from one side of the workspace to the other.

This is exactly what did not happen in round a cusp into another assembly. There a planar platform, driven round a loop of motor angles, left one assembly and arrived in another without ever being singular. It could, because the two assemblies lay on the same side of the singular set and a cusp let the path go round the end of it. A cusp is a place where the singular set stops. Here it does not stop anywhere between the two sides, and the determinant’s sign is the certificate that it does not.

Two hundred paths, and every one crosses #

The argument is short enough to be worth testing against paths nobody designed. Each of two hundred seeded random paths starts three degrees below the dead yaw and ends three above. On the way it wanders through the workspace in all three coordinates, tilts about both horizontal axes by up to 3.8°, and lets the yaw wander back and forth as well. Along each path the determinant is sampled at six hundred points, and every change of sign is refined by bisection.

All two hundred cross. 184 cross once and 16 cross three times, and none crosses twice or not at all. The count is always odd because a path that starts on the negative side and ends on the positive must change sign an odd number of times. At every crossing the smallest singular value of the leg screws is below $10^{-7}$, the arithmetic’s floor, so the determinant’s zero is a real singular pose and not an accident of its scaling.

Every crossing also lies inside the wedge. The random paths cross near the dead yaw at small tilts and further from it at larger ones, never more than 0.2° away, because they pass through positions the plane law puts near the middle of the band. That is what the wedge predicted: tilting spreads the places where a path can cross over an interval of yaw, and it does not remove any of them.

Turning the other way meets the other dead yaw #

One route has not been tried: going round. A platform that must get from 87° to 93° of yaw could instead turn the other way, from 87° down through nought to −87°, which is 273°. Nothing about a Gough platform prevents a large turn in principle, and the argument above only compares the two sides of one crossing.

Half a turn from the dead yaw there is another. The condition that the six moments be equal is met twice in a turn, at 90° less the rotation and 180° from it, and the dead yaw is a design choice found both. The determinant changes sign at the second as well: three degrees before it all 243 positions are positive, and three degrees after it none is, level and tilted four degrees alike. So the long way round passes a singular pose too. It meets the second dead yaw, where the platform is singular everywhere at once, and the determinant’s sign after it matches the short way’s.

More than about twenty degrees from either, the smallest singular value sits between 0.032 and 0.041, and the platform is sound. A task confined to one half-turn never meets either, and one that must cover more than a half-turn meets one of them whichever way it turns.

What a crossing costs #

A path that must cross meets a singular pose for an instant. The instant itself is a single point on the path and costs nothing measurable. What costs is the time the platform spends near it, where its legs must push hard to resist a small load, since the forces grow as one over the smallest singular value. So the question for a task that has to cross is how long it spends with the smallest singular value below some threshold ε.

Near a crossing the smallest singular value grows in proportion to the distance, in yaw, between the platform’s yaw and the singular yaw of the position it is at. The slope is about $1.4 \times 10^{-3}$ per degree at home and between 1.2 and 1.5 thousandths everywhere in the workspace. A path spends a share of its length under ε equal to $2\varepsilon/(k\,|r|)$. Here $k$ is that slope, and $r$ is the rate at which the path’s yaw leaves the singular yaw of the position it is passing through. Level, the singular yaw is the same everywhere, so $r$ is the path’s own rate of turn. Tilted, the singular yaw moves under the platform as it moves, and $r$ is the rate of turn less that drift.

Each crossing is a V, reaching nought at its bottom, and the only difference between them is the steepness of the sides. A level crossing held at home turns six degrees in the path. Tilted four degrees and moving along the workspace’s low diagonal from one corner to the other, the singular yaw drifts by 1.85° over the path. Moving so that the drift runs against the turn makes $r$ = 7.85, and the V steepens. Moving so that it runs with the turn makes $r$ = 4.15, and the V flattens. Holding the yaw still and crossing by moving alone makes $r$ = 1.85, the drift and nothing else, and the V is flattest of all.

Seven ways to cross, measured #

The rule makes a prediction for every way of crossing, and it can be checked against a direct count of how much of each path lies under the threshold.

The prediction holds to within 1% on all seven, and to within 2% at the lower threshold of $3 \times 10^{-5}$, where every dwell is smaller in the same proportion. So the comparison between strategies does not depend on the threshold.

The level crossing at home spends 2.35% of its path under $10^{-4}$. Held at a low corner instead, level or tilted, it spends 7% to 11% longer, because the slope there is a little smaller. Tilted and held at home, it crosses 0.075° early, the quadratic shift of home’s singular yaw that tilted, near the dead yaw measured. Its dwell is the level one’s within 1%. Tilted and held at the corner, it crosses 1.03° early and dwells about as long as level. Neither alters anything but where the crossing happens.

Moving against the drift is the one strategy that shortens the crossing: to 0.74 of the level dwell at four degrees of tilt and a full traverse of the workspace. That is the most a tilt of four degrees can buy on this workspace. The drift over the path is the plane law’s slope times the distance travelled over the height, and the band is only a degree or so across. Moving with the drift lengthens the crossing to 1.40. And crossing by translation alone, the yaw held and the platform carried through the leaning surface, is the slowest of the seven at 3.16 times the level dwell. The surface moves under a moving platform at under two degrees of yaw per traverse, which is far slower than a platform can simply be turned.

Tilt adds a way to be caught #

That last row says something about tilt that the wedge alone did not. A level platform held at a fixed yaw, even a fifth of a degree from the dead one, can be carried corner to corner across the whole workspace and never comes near a singularity. Its smallest singular value stays above $2.6 \times 10^{-4}$ throughout. The singular set is at one yaw, and a platform that does not turn does not reach it.

Tilted, that is no longer true. Inside the wedge’s band of yaws the singular set is a surface through the workspace, and a platform that is only moving, not turning at all, can pass through it. The map of where it fails drew such surfaces for an ordinary platform. A tilted paired platform acquires one in the band of yaws the wedge covers. So tilting adds a way of crossing that a level platform does not have: by translation, at a held yaw. It is the slowest crossing there is, and it can happen to a task that was never meant to cross at all.

What a task specification can now say #

The wedge essay gave a rule for avoiding the dead yaw: keep the yaw more than about 0.28 times the tilt from it. This essay adds the rule for the tasks that cannot avoid it.

• A task whose yaw must pass from one side of a dead yaw to the other passes a singular pose. No tilt, motion or route prevents it, and turning the other way meets the second dead yaw half a turn off. The only way to have no crossing is a task that stays within one half-turn between the two dead yaws.
• The crossing costs its dwell, $2\varepsilon/(k\,|r|)$. Turn briskly through it, since the rate of turn is almost the whole of $r$.
• If the platform is tilted while it turns, move against the singular yaw’s drift, which shortens the dwell by up to a quarter at four degrees, and do not let it drift with the turn.
• A tilted platform held near the dead yaw can cross by moving alone. A controller that watches only the yaw will not see it coming; the plane law gives each position’s singular yaw, and that is what must be watched.

What the sign does not settle #

One platform and one workspace. The endpoints were checked at tilts up to four degrees. At six degrees one position of the 1,215 has already joined the band three degrees below the dead yaw. For a larger tilt the endpoints of a task would have to be taken further from the dead yaw before the argument applies. The argument itself does not change, only the distance at which the endpoints must sit.

The forces are not computed. The dwell is a time with the smallest singular value under a threshold. How large the legs’ forces become depends on the load the platform carries and on the direction the singular value’s vector points, and a load aligned away from it costs little even at a small singular value.

Six legs. The whole argument rests on the matrix being square. A platform with a seventh leg has no determinant to change sign, and the argument does not apply to it.

Still open: a seventh leg #

What a fourth leg buys found, for a planar platform, that a redundant leg turns a singular curve into isolated points, because four lines failing together is two conditions rather than one. The spatial counterpart is a seventh leg on a Gough platform. It is singular only when all seven six-leg subsets are singular at once. The sign argument then has nothing to hold on to, because seven screws make no square matrix and no single determinant.

The distinct argument would be whether a seventh leg, placed well, opens a way across the dead yaw. At the dead yaw the six paired legs’ moments are equal and the six together cannot resist a turn. A seventh leg with a moment of its own about the vertical would resist it, so the dead yaw would stop being singular at all. The measurement would take the paired platform with a seventh leg at several placements and ask where the seven-leg platform is singular near the dead yaw, whether anywhere. It would measure the smallest singular value along the crossings above, and what the seventh leg costs in the forces the legs fight each other with once there are more legs than freedoms.