A yaw that is singular everywhere

Every singularity the parallel field has drawn so far has been somewhere. The planar map found a curve of bad poses through the workspace at each orientation; a fourth leg shrank that curve to points. In both, a singular pose is a place, and a machine in trouble there can in principle be moved somewhere else.

The spatial platform has singularities of that kind too. Its forward problem is harder than the planar one, twenty-eight poses rather than six for its anchor points, and its direct singularities are where two of those poses merge, exactly as on the planar platform. It also has one of a different kind, and on the paired hexapod used throughout these essays it is at a yaw of 30°: turned by that much about the vertical and held level, the platform is singular wherever it is put. Moving it does nothing. This essay finds that yaw by sweeping, explains it with one condition on the anchor points, and identifies the motion the six legs cannot resist.

The platform, and the sweep #

The platform is the one six legs and a square root built. Its six base anchors sit on a circle of radius 2.2 in three pairs, the two anchors of each pair 25° apart. Its six platform anchors sit on a circle of radius 1.1, also in three pairs, 40° apart, with the platform’s pairs rotated 60° from the base’s so that the legs spread rather than stack. Each leg joins one base anchor to one platform anchor, and at the home pose the platform is 2.4 above the base.

The measure of how well it is held is the one the field uses throughout: the smallest singular value of the six-by-six matrix whose rows are the six legs’ lines, written as screws. It is zero exactly when some motion of the platform stretches no leg, which is a direct singularity.

The platform is held level, with no tilt, and turned about the vertical through a full revolution in steps of one degree, at four positions: centred at a height of 2.4, displaced to (0.4, −0.3) at the same height, centred but raised to 3.2, and displaced to (−0.5, 0.2) and lowered to 1.8. At each yaw the smallest singular value is computed.

At −150° and at 30° all four curves reach zero together, the largest of the four values being 9.7 × 10⁻⁹, which is rounding. One degree either side, the least of them is already 1.28 × 10⁻³. Nowhere else in the revolution does any of the four touch zero.

Every position at that yaw #

Four positions are four samples, and the claim is about all of them, so the yaw is held fixed and the position varied over a whole region.

Over a square 1.6 across at height 2.4, sampled on a 41 by 41 grid, with the platform turned 30°, the largest smallest singular value anywhere is 2.1 × 10⁻⁸. Every position in the square is singular. With the platform turned 0° instead, the values run from 0.0270 to 0.0363 across the same square, and no position is.

The two panels are on one colour scale, and the contrast is the finding. At a yaw of 0° the platform’s holding varies gently with position, as expected. At 30° it does not vary at all, because it is zero everywhere. This is a singularity of an orientation, not of a pose.

The condition, in one line #

A singularity that ignores position must come from something that does not involve position, and the quantity that fits is a moment.

For each leg, take its base anchor and its platform anchor, each measured from its own centre, turn the platform anchor by the yaw, and form the vertical component of their cross product. That number is |b|·|p|·sin(θ), where θ is the angle from the base anchor round to the turned platform anchor. It does not depend on where the platform is, only on how it is turned.

The six moments fall on two curves, because the legs come in two symmetric sets of three. At −150° and 30° the two curves cross, and all six moments are equal: 2.3993 at 30°, and −2.3993 at −150°.

When the six are equal, a particular screw motion does no work against any leg, at any position. It is a turn about the axis parallel to the line from the base’s centre to the platform’s centre, combined with a slide along that axis at a rate set by the common moment. Because the condition that makes it possible, six equal moments, involves no position, the motion exists at every position, and the platform is singular at every position.

Why 30°, and why the dimensions do not matter, is now arithmetic. Label each leg by the angle it spans at the home pose. With base pairs centred on 0°, 120° and 240° and spread by $s_b$, and platform pairs centred on 60°, 180° and 300° and spread by $s_p$, the legs span one of two angles:

$$d_1 = 60^\circ - \tfrac{1}{2}(s_p - s_b), \qquad d_2 = 60^\circ + \tfrac{1}{2}(s_p - s_b).$$

For this platform that is 52.5° and 67.5°. Turned by a yaw ψ, the two kinds of leg have moments proportional to sin(d₁ + ψ) and sin(d₂ + ψ), and these are equal when d₁ + ψ and d₂ + ψ add to 180°. But d₁ + d₂ is 120° whatever the spreads are, so the moments are equal when 120° + 2ψ = 180°, which is ψ = 30°, or 180° further round, ψ = −150°. The common value is 2.2 × 1.1 × sin 82.5° = 2.3993, which is the number measured.

The screw the legs cannot resist #

The condition predicts not only that a free motion exists but exactly what it is, and that prediction can be tested pose by pose.

At (0.4, −0.3, 2.4), turned 30°, the smallest singular value is 6.4 × 10⁻⁹, and the null direction of the six leg lines is the free motion itself. Read as a screw, its axis is parallel to the line from the base’s centre to the platform’s centre, and its pitch is −0.9581: each radian of turn about the axis carries a slide of 0.9581 along it.

The prediction for the pitch is minus the common moment times the platform’s height over the squared distance between the centres. Here that is −2.3993 × 2.4 / 6.01, or −0.9581, agreeing with the measured screw.

The check is repeated at eleven heights from 1.2 to 6 and three horizontal offsets of the platform’s centre. The largest disagreement between measured and predicted pitch is 4.4 × 10⁻¹⁴. The free motion is steepest, most slide per turn, when the platform is low, and it flattens towards a pure turn as the platform rises, because the height divided by the squared distance falls.

Why equal moments leave the legs unstretched #

At the centred positions the free motion can be seen without any screw algebra at all, and seeing it there explains why the moment is the right quantity.

Put the platform’s centre directly above the base’s, at height h. A leg’s squared length is then its horizontal span squared plus h squared. Turn the platform by a small angle about the vertical: the horizontal span of leg i changes, and the rate at which its square changes is exactly twice that leg’s vertical moment. Lower or raise the platform by a small amount: the vertical part of every leg’s squared length changes at twice h times the rate.

So every leg keeps its length, to first order, if the platform sinks by the moment divided by h for each radian it turns. For one leg that is always possible; it is just the motion that leg permits. For all six at once it needs the six moments to be the same number, since there is only one rate of sinking to choose. When the six moments are equal, one turn-and-sink motion leaves all six legs unstretched, and locked motors cannot resist it.

At a height of 2.4 that motion sinks the platform by 2.3993 / 2.4 = 0.9997 for each radian of turn, a pitch of almost exactly −1, and the pitch figure’s centred line is that value at every height: −2.3993 divided by the height. Off the centre the same argument picks up a horizontal term, the axis tilts to stay parallel to the line of centres, and the pitch becomes the moment times the height over the squared distance between the centres. The measured screw follows that more general form at every offset tried.

So the singularity has two routes to it that share nothing but the platform: a singular value computed from six leg lines at each pose, and a moment computed from anchor positions with no pose in it. They agree on where it is and on what the free motion looks like, to rounding.

The same argument, the classical yaw #

The best-known version of this phenomenon is on a different platform, and running the same computation on it is the calibration.

The octahedral platform has three base points and three platform points, each base point joined to the two nearest platform points, and it was shown by Fichter in 1986 to be singular at a yaw of ±90° when level, wherever it is. The sweep at the same four positions finds all four curves reaching zero together at −90° and 90°, the largest value 8.6 × 10⁻⁹, and one degree either side the least is 1.02 × 10⁻². At (0.4, −0.3, 2.4) turned 90°, the free screw is again parallel to the line of centres, with a pitch of −0.4832, and the six moments are all equal, at 1.2100.

So the same computation finds the classical singular yaw where the literature puts it, and a different singular yaw, 30°, on the paired platform, for the same reason.

The spread at which every yaw is singular #

The derivation says the singular yaw does not depend on the spreads, and it says something stronger about one particular spread.

The platform is rebuilt with its platform anchor pairs spread anywhere from 0° to 90°, keeping the base pairs at 25°, and each version is swept through a revolution of yaw at (0.3, −0.2, 2.6). The singular yaws at 30° and −150° are there for every spread: the largest smallest singular value along those two columns, over every spread, is 1.6 × 10⁻⁸.

And there is a pale row. When the platform’s pairs are spread by exactly the base’s 25°, d₁ and d₂ are equal, so the six moments are equal at every yaw, and the platform is singular at every yaw: the largest value along that row, over the whole revolution, is 7.1 × 10⁻⁹. That is not a singular orientation of a working machine. It is a machine that cannot be built to work, because its base and platform are similar hexagons, and it is the extreme case of the same one-line condition.

How close to thirty degrees is too close #

A singular orientation is only half the practical story. The other half is how quickly the platform recovers as the yaw moves away from it, since a machine is in trouble well before its smallest singular value reaches zero, for the reason the singularity essay gave: the leg forces needed to hold a load grow as that value shrinks.

At the centred position, height 2.4, the smallest singular value at a yaw of 0° is 0.0363. Turned towards the singular yaw it falls: 0.0294 at 10°, 0.0144 at 20°, 0.0071 at 25°, 0.0028 at 28° and 0.0014 at 29°, and zero at 30°. On the far side it rises almost symmetrically, 0.0014 at 31°, 0.0071 at 35°, 0.0141 at 40° and 0.0283 at 50°, and between 60° and 150° it sits nearly flat at about 0.032.

Close to 30° the fall is linear, 1.42 × 10⁻³ for each degree, which is what a simple zero looks like: the free screw appears at exactly one yaw and is resisted in proportion to how far the platform is turned from it. Further out the recovery is slow. Ten degrees from the singular yaw the platform is held only 40% as well as at 0°, and it takes twenty degrees to recover 81%. So the usable yaw range of this platform, for any task that needs it held at least half as well as at home, stops roughly thirteen degrees short of 30° and short of −150°, at every position, because the whole curve is a property of the orientation.

That is the design consequence of a singularity that ignores position. A planar platform’s singular curve can be steered round by choosing where to go; this one can be avoided only by limiting how far the platform turns, and the limit applies to the entire workspace at once.

What a designer does about it #

A singularity of an orientation cannot be avoided by path planning in position. What can be done follows from the condition.

Keep the yaw range away from 30° and −150°. The site’s platform is well held within a degree or two of those angles only in the sense that its smallest singular value is 1.28 × 10⁻³ one degree away, far below its 0.027 at 0°. A usable range of yaw has to stop well short of them, and how far short is set by the conditioning a task needs.

Do not tilt into it. Everything here holds the platform level. With tilt the moments change, and whether a tilted platform at 30° is held is a question these figures do not answer.

Do not make the spreads equal. The pale row is a design that would pass every check of its leg lengths and fail as a machine. It is the spatial relative of the delta robot’s parallelograms turned against the designer: a symmetry that constrains a motion there, and frees one here.

Nothing can change the singular yaws themselves by changing the spreads, because d₁ + d₂ is fixed at 120° by the pairs being rotated 60° from each other. Moving them would mean changing that rotation, which is the other free choice in this family of designs and the one the derivation points at.

What this essay does not establish #

Only level poses are examined. The singular set of the full six-dimensional workspace is not mapped, and the yaw singularity is one slice of it rather than all of it.

The derivation is for this family of paired platforms, with pairs centred 120° apart and rotated 60° between base and platform. It is confirmed by measurement on that family and on the octahedral platform, and is not claimed for other architectures.

Leg strokes, joint cone angles and collisions are not considered, so it is not established that the site’s platform can physically reach every position in the square at a yaw of 30°. The singularity is of the leg geometry whether or not a real machine could get there.

What comes next #

Tilted, and near the yaw. The level slice is where the condition is exact. How the smallest singular value behaves as the platform tilts at a yaw of 30°, whether the singular set there is a surface through the tilts or collapses to the level poses, is a sweep of two more angles with the same computation.

The rotation between the pairs as a design variable. The singular yaw is half of 180° minus the sum of the two leg angles, and that sum is fixed by the 60° rotation between base and platform pairs. Varying that rotation moves the singular yaw, and tracing where it goes, and how the platform’s best conditioning at 0° changes with it, would turn a derivation into a design chart.

The same question for a planar platform. The planar platform’s singular curves move with orientation. Whether any orientation of it is singular at every position, the planar analogue of this yaw, is decided by whether its three leg lines can be made concurrent for every position at once, and the answer can be read from its anchor points.