The dead yaw is a design choice

A yaw that is singular everywhere found that the paired hexapod used throughout the parallel field, held level and turned 30° about the vertical, is singular at every position it can be put in. It explained the angle with one condition. Each leg spans an angle from its base anchor round to its platform anchor; the legs come in two kinds, spanning $d_1$ and $d_2$; their vertical moments are proportional to $\sin(d_1 + \psi)$ and $\sin(d_2 + \psi)$ at a yaw $\psi$; and the moments are all equal — which frees a screw motion no locked leg resists — when $d_1 + \psi$ and $d_2 + \psi$ add to 180°.

The spreads within each pair of anchor points change $d_1$ and $d_2$ in opposite directions and leave their sum alone. What fixes the sum is the 60° by which the platform’s pairs are rotated from the base’s, and the essay ended by naming that rotation as the other free choice in the design, the one the derivation points at.

This essay makes that choice. It sets the rotation to every value from 0° to 120° and asks where the dead yaw goes, what the platform is like at home, and how far it can turn.

Where the dead yaw goes #

With the base’s pairs centred on 0°, 120° and 240° and the platform’s on ρ, ρ + 120° and ρ + 240°, a leg from the near anchor of one base pair to the near anchor of its platform pair spans $\rho - \tfrac12(s_p - s_b)$ and the other kind spans $\rho + \tfrac12(s_p - s_b)$. Their sum is $2\rho$. Equal moments need $2\rho + 2\psi = 180°$, so

$$\psi^* = 90° - \rho,$$

and, because the moments are sines, a second dead yaw half a turn away. At the usual ρ = 60° that is 30° and −150°, the pair the earlier essay found by sweeping.

The dragged figure shows it happening: as the rotation grows, the two dips to nought slide leftward across the revolution together, 180° apart, and the curves between them keep much the same shape. Every one of the four positions reaches nought at the same yaw, at every rotation — which is what “singular everywhere” means, and it survives the change of design intact.

Measured rather than derived, the dead yaw at thirteen rotations in steps of 10° lies on $90° - \rho$ to $7 \times 10^{-6}$ of a degree. It is found by golden-section search on the largest of the four positions’ smallest singular values, a quantity that is nought only where all four positions are singular together, and at the yaw found that largest value is $2 \times 10^{-8}$ or less at every rotation. Five degrees from the dead yaw every position is held at $6 \times 10^{-3}$ or more, and the 30° that is dead for the usual design is held at every other rotation tried.

The same free screw, turned #

The derivation also predicts what does not change. When the six moments are equal their common value is $|b||p|\sin\!\big(90° + \tfrac12(d_2 - d_1)\big)$, which has the spreads in it and not the rotation. So the free screw — a turn about the line from the base’s centre to the platform’s, with a slide along it set by the common moment — should be the same screw at every rotation, merely met at a different yaw. That is a prediction about a motion, not only about an angle, and it is the one most likely to be wrong if the moment argument were merely a coincidence of the 60° design.

With the pairs rotated 0°, the six moments cross at −90° and 90°, and they cross at 2.3993 — exactly the value the 60° design’s moments cross at, 60° further round.

At (0.4, −0.3, 2.4) and a yaw of 90°, the free screw’s axis is parallel to the line of centres and its pitch is −0.9581, the value measured on the 60° design at 30°. The rotation between the pairs relocates the dead orientation without changing anything about the motion the platform has there.

What rotating the pairs looks like #

In every design each base pair sends its two legs to one platform pair, and the rotation decides where that pair is. At 0° the platform pair sits inside its own base pair and the legs run almost radially, splaying by 7.5° either way. At 60° the platform pair has moved a sixth of a turn round, the legs lean round the platform in the same sense, and the platform pairs sit midway between the base pairs — the arrangement usually drawn for a Stewart platform, because from above it looks triangulated. At 120° the lean is a third of a turn and each pair’s legs cross its neighbour’s.

The appeal of 60° is the look of triangulation: legs meeting the platform from two directions at each pair suggest a stiff platform. What the singular value measures is how well the six leg lines together resist the worst-resisted motion, and that is a different question from how the legs look.

At home, and how far from it #

A designer cares about the dead yaws only through what they do to the orientations actually used, and a platform is usually used near home, turned a moderate amount either way.

Two numbers are charted against the rotation: how well the platform is held at home, and how far it can turn either way before that holding halves.

At a rotation of 0° the platform is held at 0.0411 and can turn ±75.5°. At 30°, 0.0396 and ±46°. At the usual 60°, 0.0363 and ±17°. At 90°, nought and nothing, because the home orientation is then itself a dead yaw. Past 90° both recover symmetrically: at 120°, 0.0320 and ±18.5°.

The usable range is the distance from home to the nearer dead yaw less a nearly constant margin — between 13° and 14.5° for every rotation up to 60° — because the smallest singular value falls into each dead yaw along nearly the same curve whatever the rotation. That makes the chart’s lesson short. The dead yaws are always half a turn apart, so the furthest either can be from home is a quarter-turn; that is ρ = 0; and ρ = 0 also holds the home orientation best. The usual 60° spends two thirds of the available distance.

What the holding numbers mean #

A smallest singular value is abstract until it is read as a force, and a parallel platform loses a freedom differently from a serial arm: its motors can be locked and it still moves. When the platform carries a load, the six leg forces that balance it are found by inverting the matrix of leg lines, and in the worst direction the leg forces are the load divided by the smallest singular value. So a platform held at 0.0411 carries its worst-case load with leg forces 13% smaller than one held at 0.0363, and a platform whose holding has halved needs twice the leg force it needed at home.

That is why the chart uses half the home value as the edge of the usable range. At that edge the motors see twice the force they see at home for the same load in the worst direction, and past it the force grows without bound as the dead yaw approaches, because a leg force that balances a load along the free screw does not exist there at all. The ±17° of the usual design is the range within which a load in the worst direction costs at most twice what it costs at home.

The four positions in the dragged figure show that the chart’s centred position is representative. Their curves differ in height — the platform is held better at the lower positions than the higher — but every one of them reaches nought at the same two yaws and falls into them along the same shape, which is the moment argument again: the dead yaws belong to the orientation, and position only scales the numbers around them.

Why the margin is nearly the same at every rotation #

The near-constant margin has a reason, and it is worth having because it is what turns the chart into a rule.

The singularity is the six moments agreeing. Away from the dead yaw they disagree by the difference between the two kinds of leg,

$$\sin(d_2 + \psi) - \sin(d_1 + \psi) = 2\cos\!\Big(\tfrac{d_1 + d_2}{2} + \psi\Big)\sin\!\Big(\tfrac{d_2 - d_1}{2}\Big),$$

and with $d_1 + d_2 = 2\rho$ the first factor is $\cos(\rho + \psi)$, which vanishes at $\psi = 90° - \rho$ and depends on the yaw only through its distance from that point. The second factor is fixed by the spreads. So the disagreement that holds the platform away from its free motion is the same function of distance from the dead yaw at every rotation; changing $\rho$ slides the whole picture along the yaw axis and changes nothing else about it.

The smallest singular value is not the moment disagreement itself — it depends on the whole geometry of six leg lines at each pose — but near the dead yaw it is governed by it, and the measured margins follow: 14.5° at a rotation of 0°, 14° at 20° and 30°, 13° at 50° and 60°. The margin shrinks close to ρ = 90° for a different reason. There home is itself near a dead yaw, so “half the holding at home” is half of a small number and is reached quickly: at 80° the platform is held at 0.0144 at home and can turn only ±4.5°.

Reading the formula as a specification #

Written as a design rule, the chart is two lines. A task that needs the platform to turn through ±Ψ of yaw from home, holding at least half its home stiffness, needs the nearer dead yaw at least Ψ plus the margin away:

$$|90° - \rho| \;\ge\; \Psi + 14°.$$

A task needing ±45° needs $\rho \le 31°$ or $\rho \ge 149°$. A task needing ±20° admits anything up to $\rho = 56°$, which the usual 60° narrowly fails. A task needing ±75° admits only a rotation within a degree or so of nought.

The rule has a hard ceiling that no rotation escapes: the dead yaws are half a turn apart, so no paired hexapod of this family can turn more than ±90° from home without meeting one, and with the margin the practical ceiling is about ±76°. Beyond that, the choice is no longer the rotation but the architecture — a different arrangement of legs, or an extra leg whose moment does not agree with the others’, which is what a fourth leg did for the planar platform.

The octahedral platform is the same rule #

The earlier essay calibrated its computation on the octahedral 3-3 platform — the spatial cousin of the planar platform with similar triangles, singular everywhere at two orientations — whose level poses Fichter showed in 1986 to be singular everywhere at a yaw of ±90°. Its anchor points are not paired in the same way — three base points and three platform points, each base point joined to the two nearest platform points — but its legs still come in two kinds, and their spans add to nought rather than to 120°: each base point’s two legs reach 60° either way round to its neighbouring platform points.

By the same arithmetic, the moments agree where $0 + 2\psi = 180°$, at a yaw of 90°: the octahedral platform behaves as a paired platform with a rotation of nought, and its classical dead yaw is the $\rho = 0$ entry of the rule. That is a second route to the formula that shares nothing with the sweep: a result from the literature on a different architecture, landing on the same line.

Why nobody builds the 0° platform anyway #

A chart that says the standard design is the worst but one of the thirteen tried deserves suspicion, and there are reasons it does not settle the choice.

The legs are nearly parallel in pairs. At ρ = 0 the two legs from one base pair run almost side by side to one platform pair. Physically they collide unless the pairs are spread wider, and a spread wide enough to clear them changes $d_2 - d_1$ — which, by the moment argument, leaves the dead yaws where the rotation put them but changes how steeply the singular value falls into them.

The smallest singular value is one number. It reports the worst-resisted motion and nothing about the others. A platform whose legs run radially resists rotation about the vertical through the leg’s small splay, and the torsional stiffness a milling or pointing task needs may be poor even while the singular value is good, because that value is dominated by a different direction.

Only level poses are examined. Everything here, like everything in the essay it continues, holds the platform level. Tilting changes the moments and may bring new singular sets closer to home on a design that looks best level. A map of where a platform fails over the whole workspace, rather than one slice of it, is what would find them.

Only one height is charted. The usable range is read at the centred position and a height of 2.4, and the singular value’s shape near a dead yaw changes with height. The dead yaws themselves do not, since they are singular at every position; the margin around them does.

What the choice actually is #

The rotation between a hexapod’s pairs is usually treated as settled by symmetry: 60° puts each platform pair midway between two base pairs, and the figure looks right. The measurement says it is a design variable with a clean effect on one thing and a smooth effect on others.

The clean effect is where the orientation singular at every position sits: exactly $90° - \rho$ and half a turn from it, with the free motion there unchanged. No other choice of anchor geometry within this family moves it, and a task that needs the platform to turn through a stated range of yaw can read the rotation it requires off that one formula, before anything else is designed.

Nothing that counts the machine sees the choice. Every rotation gives six legs, six spherical-prismatic-spherical chains, a mobility of six by any count and by the rank of the leg lines at a generic pose, and twenty-eight poses or fewer for a set of leg lengths. Two platforms that differ only in their rotation are the same mechanism to every instrument that ignores where the anchor points are, and one of them can turn four times as far as the other before its motors must double their effort. That is the pattern the parallel field keeps finding: the counts are the same and the geometry decides, and here the geometry is one angle chosen on a drawing before any leg is sized.

The smooth effects are the conditioning at home and the usable range, and on this platform both favour small rotations. How much of that survives collision constraints, a tilt, and a stiffness requirement in a particular direction is a question for the task, and what a fourth leg buys is the other lever when no rotation of six legs is enough.

Still open: tilted, near the dead yaw #

Every result here is on the level slice, where the moment condition is exact. A real task tilts the platform, and the question the earlier essay left open applies at every rotation: at the dead yaw, is a slightly tilted platform still singular everywhere, or does tilt break the equality of the moments and turn the everywhere-singular orientation into an ordinary singular surface through the workspace?

Its distinct argument would be that sweep — the smallest singular value over two tilt angles at the dead yaw, at several positions — made at two or three rotations. Two things would come out of it. Whether the set of dead orientations is a single level yaw or a curve through tilt space, which decides whether a tilting task can pass near the dead yaw by tilting; and whether that curve’s shape depends on the rotation, which would make the choice of ρ a trade between how far the dead yaw is from home and how much tilt it tolerates there. Two orientations no position can rescue found the planar version of a singularity that ignores position; the tilt sweep is what separates the spatial platform’s version from it.