Tilted, near the dead yaw

The dead yaw is a design choice found that rotating a paired hexapod’s platform pairs by $\rho$ moves its singular yaw to $90° - \rho$, exactly, from $\rho = 0$ to 120°. The singularity itself came from a yaw that is singular everywhere: at that orientation the six legs’ vertical moments are all equal, the leg screws become dependent, and the platform is singular at every position it can reach rather than on a surface through its workspace. That is unusual: a platform’s singularities are normally places, and this one is a whole orientation.

Every one of those results is on the level slice. A real task tilts, and the exactness that produced the dead yaw — six moments equal, not nearly equal — is the kind of condition a tilt would be expected to break. So the question the earlier essay left is whether a tilting task can pass near the dead yaw by tilting, and it has two halves: whether the platform is held at a tilt, and if so, how much tilt it takes.

It is held, and it takes more tilt than it looks like.

The rise is second order #

At the home position, level, the smallest singular value of the six leg screws is $8.9\times10^{-9}$ — the solver’s own floor, which is what singular means here rather than a small number someone chose.

Tilt by a quarter of a degree and it is $4.4\times10^{-7}$. By half a degree, $1.7\times10^{-6}$. By one degree, $6.9\times10^{-6}$; by two, $2.7\times10^{-5}$; by four, $1.1\times10^{-4}$; by eight, $4.2\times10^{-4}$.

Every doubling of the tilt multiplies the value by four, and a fit over the small tilts gives an exponent of 1.987. The rise is quadratic, which is what an expansion about an exact coincidence gives when the first-order term vanishes — the mirror of a parallelogram a micron wrong, where a square-root law made a small error large — and the first-order term vanishing is the content: a small tilt does not break the moment equality at first order, it breaks it at second.

That is a worse rescue than a linear rise would be, and by a large factor. A designer who reads “the platform is held at a tilt” and allows a degree of it has bought one sixty-fourth of what eight degrees would buy. Near the dead yaw the useful statement is not tilt helps but tilt helps as the square, and a task that stays within a couple of degrees of level is a task that is still working within an order of magnitude of a singularity.

The exponent is not quite two and the departure is worth recording. Fitted over every stop including eight degrees it comes out 1.978 for $\rho = 0$ and 1.895 for $\rho = 30$ — the next term in the expansion contributing, differently for different designs. Two to three figures is the honest statement about the small tilts, and how far the large ones stray is a property of the platform rather than of the phenomenon.

Why the first-order term is missing #

The exponent is the surprising part and it is worth deriving rather than fitting, because it explains why the rescue is so weak.

The dead yaw exists because six numbers are equal. Each leg spans an angle from its base anchor to its platform anchor, the legs come in two kinds, and at the dead yaw the two kinds’ vertical moments coincide — which makes the six leg screws satisfy one linear relation and the platform singular. A quantity that measures the singularity therefore measures a difference between two moments, and that difference is nought at the dead yaw and level.

Now tilt. The two kinds of leg are related by a reflection of the platform’s own geometry: the paired arrangement puts them symmetrically about the plane the tilt is taken in. So a tilt that raises one kind’s moment raises the other kind’s by the same amount to first order, and the difference — the thing the singular value is built from — does not move until second order. The symmetry that creates the dead yaw is the same symmetry that protects it from a tilt. Why the platform stays flat is the delta’s version of the same bargain: a symmetry that buys a property and charges for it somewhere else.

That is a more useful sentence than the exponent, because it says where the exponent would change. A platform whose pairs are not symmetric about the tilt axis has no such protection and should come off its dead yaw at first order; the measurement here tilts about $x$ on a design that is symmetric about it, which is the worst case rather than a typical one. Tilting about an axis that breaks the pairing is the obvious thing to try and is not tried here.

What a tilt does to the workspace #

The second half of the question is the one that changes the answer, and it needs the workspace rather than one position in it.

Scan a grid of positions at the dead yaw and sort them by how well the platform is held. Level, the sorted curve is flat at the floor: the worst-held position in the whole workspace reads $2.5\times10^{-8}$, and the best reads the same. That is the sense in which the orientation is singular everywhere, and it is an unusual thing for a parallel platform — an ordinary direct singularity is a surface, and a position off that surface is held.

At three degrees of tilt the same grid runs from $3.4\times10^{-7}$ to $1.0\times10^{-3}$: a factor of three thousand. Most of the workspace is comfortably held. Some of it is not.

So the tilt has not removed the singularity. It has made it a property of position, which is exactly what an ordinary singularity is. The platform has gone from an architecture singularity — a defect of the design, present wherever the machine is — to a direct singularity, a surface the machine can be on or off.

The surface reaches nought #

A spread of three decades is suggestive and is not proof: a workspace with a low corner is not the same thing as a workspace containing a singularity. The test is whether the low places reach zero.

Driving a search downhill in position from the grid’s own minimum, at one, three and six degrees of tilt, the smallest singular value goes to exactly nought — not to a small number, to the zero the arithmetic returns when the six screws are genuinely dependent. The singular set is inside the reachable workspace at every tilt tried, and a slice through it at a fixed height shows what an ordinary direct singularity looks like: a curve across the square, with the machine held on both sides of it.

That result took two attempts and the first one was a design number that did not exist. A coordinate descent from one seed — the grid’s own minimum — reaches nought at three degrees and above and halts at $3.7\times10^{-7}$ at one degree. Read straight, that says the singular surface leaves the reachable workspace somewhere below about 1.25° of tilt, which would be a threshold worth knowing and worth designing to. Run from forty-eight seeds instead, one degree reaches nought at $(-0.70, 0.60, 1.05)$. The surface was there the whole time and the search was not, and a threshold fitted to where a descent happened to stall is a property of the descent.

An architecture singularity becoming an ordinary one #

The change the tilt produces has a name in this field and it is worth using it, because the two kinds of defect are handled completely differently.

An architecture singularity is a property of the design: the mechanism is degenerate at that configuration wherever it stands, and no amount of moving it helps. A yaw that is singular everywhere is one, and the only cure is to change the machine — which is what choosing $\rho$ does.

A direct singularity is a surface in the configuration space: the legs become dependent at particular poses and the machine is held at others. Every Gough platform has them, including ones where legs meet, and they are handled by path planning rather than by redesign.

What the measurement shows is that the tilt converts the first into the second. At nought degrees the platform has an architecture singularity at one yaw; at any tilt it has a direct singularity at a surface near that yaw. The defect does not go away and it changes category, and the category is what decides who has to deal with it — the designer or the planner.

There is a reading of that which is mildly encouraging and worth stating. A direct singularity is a thing every parallel platform already has and every parallel platform’s software already avoids, so a machine that tilts through its dead yaw is not asking its planner to do something new. What it is asking is that the planner know the surface is there, and a surface that appears only within a couple of degrees of one orientation is exactly the kind that a workspace map sampled at a few orientations misses. A platform that measures itself would not find it either, since a calibration is run where the machine is held.

What the two results say together #

Put side by side, the two halves answer the question in a way neither does alone, and the answer is not the one the rise suggests.

A tilt buys the home position a great deal — three orders of magnitude at six degrees — and buys the workspace nothing at all. The singularity is still reachable; it has moved from being unavoidable to being avoidable, and avoiding it now requires knowing where it is.

For a task that is the harder situation of the two. A dead orientation is a thing a planner can be told about once: do not yaw to 90°. A singular surface at a yaw of 90° and a tilt of three degrees is a thing a planner has to check at every pose, because the orientation alone no longer decides. The design has traded a defect that is visible in the specification for one that is visible only in a workspace map.

That is also what separates this from the planar case. Two orientations no position can rescue found a planar platform singular at an orientation whatever its position, and there the orientation is the whole story because a planar platform has one. Here the orientation has three components and only one slice of it carries the architecture singularity, so the defect survives as a surface in the rest.

The design reading for the rotation $\rho$ follows and it is narrower than it was. Choosing $\rho$ still chooses where the dead yaw is, and that choice is still worth making to put it away from a task’s working orientations. What it cannot do is make the neighbourhood of the dead yaw safe by tilting through it, because the neighbourhood of the dead yaw is where the singular surface lives, and it lives there at every tilt.

What a planner would have to be told #

It is worth writing down what the result asks of the software, because that is where the cost of the design choice actually lands.

Avoiding an architecture singularity is a constraint on one coordinate: stay away from $90° - \rho$ of yaw, by some margin, whatever else the machine is doing. It is checkable in a line and it is checkable at planning time from the commanded orientation alone.

Avoiding the surface is a constraint on the pose. Near the dead yaw, the platform’s condition depends on where it is standing as well as which way it is facing, so the check is a function of all six coordinates and has to be evaluated along a path rather than at its ends. That is the ordinary business of a parallel-robot planner and it is not free: the singular value has to be computed, which means the leg screws have to be formed, which is most of the inverse kinematics.

The margin is the part that is genuinely awkward. Three degrees of tilt buys the home position a factor of about three thousand and buys the worst position in the workspace nothing, so a margin expressed as “keep the singular value above some floor” excludes a region whose size depends on position in a way the orientation does not reveal. A planner given only the orientation margin would pass through poses whose singular value is four decades below the one it checked.

So the honest summary of what tilting buys is: it moves the problem from the specification into the trajectory. Whether that is a good trade depends on which of the two a particular machine’s software is better at, and this subject has no view on that.

What is not modelled #

One tilt axis. Everything is tilted about $x$, with $r_y$ held at nought. A tilt is two-dimensional and the set of dead orientations is a question about that plane; a sweep over one line through it cannot say whether the set is a curve, a point or something with structure. The measurement above shows that the level yaw is not isolated in position, not that it is isolated in tilt.

One platform family. The paired hexapod at $\rho = 0$ and $\rho = 30$, with the base and platform radii, spreads and height this field has used throughout. The exponent of two is an expansion about an exact coincidence and should survive any design that has one; the numbers attached to it will not.

A singular value is not a distance. The smallest singular value of the leg screws is the instrument this field uses and it has units and a scale. Saying the workspace spans a factor of three thousand is a statement about that quantity, not about how close in millimetres the platform is to a singular pose.

The workspace is a box of positions. A grid over $x$, $y$ and three heights, with reachability taken from whether the legs solve. A real platform’s workspace is bounded by leg lengths and joint limits, and a singular surface outside those bounds is not a problem the machine has.

The descent finds a singularity, not the surface. Forty-eight starts reaching nought establishes that singular poses are inside the workspace. Nothing here maps the whole locus, and the slice drawn is a shading of a grid rather than a traced curve.

Nothing is computed about behaviour near the surface. How large a force the platform cannot resist, how fast its stiffness falls as the surface is approached, and whether a controller can be told to avoid it are all outside a collection that computes no forces.

Still open: the dead set in the whole of tilt space #

The measurement above tilts about one axis and asks what happens to the workspace. The complementary question is what happens to the orientation: the dead yaw is one point in a three-dimensional space of rotations, and this looked along one line through it.

Its distinct argument would be the smallest singular value over a patch of tilt space — both tilt angles, at a few yaws either side of the dead one — reduced to the quantity that matters, which is the smallest value anywhere in the workspace rather than at home. Two things would come out of it. Whether the orientations at which the workspace contains a singularity form a surface in rotation space or fill a region, which decides whether “avoid the dead yaw” can be written as a condition on orientation at all; and whether the set’s shape depends on $\rho$ — which would make the choice of rotation a trade between where the dead yaw sits and how large a neighbourhood of it a task has to stay out of, rather than the single-quantity choice it is now.