Machine Kinematics
solved, then drawn
Concept

Architecture singularity — where it appears

A singularity that belongs to a parallel mechanism's design rather than to one pose: whole families of poses lose control together. It follows from how the anchor points are placed, so no path planning avoids it and only a different design removes it.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

Every position loses its hold at -150° and 30°. The smallest singular value of the six leg lines of the Gough–Stewart platform, held level, as it is turned through a whole revolution at 4 positions: (0, 0, 2.4), (0.4, -0.3, 2.4), (0, 0, 3.2), (-0.5, 0.2, 1.8). At -150° and 30° all 4 curves reach zero together — the largest of them is 9.7e-9 — and 1° either side the least is 1.28e-3. Nowhere else does any of them touch zero.

A yaw that is singular everywhere

Turn the standard hexapod platform 30° about the vertical, hold it level, and it is singular: not at one pose, but at every position it can be put in, with one screw motion that none of its six locked legs can resist. The angle is not a property of the dimensions. It comes out of one line of trigonometry that no spread of the anchor points can change.

parallel · Parallel A 3-RPR with similar triangles at 0°, its leg lines meeting wherever it is put. A planar platform with three extending legs, its base pivots on a circle of radius 2 and its platform points on one of radius 0.6 at the same angles, so the two triangles are similar, turned to 0° and drawn at two positions. Each leg's line is continued past its ends. With the platform's centre at (0.4, −0.3) the three lines meet at (0.5714, −0.4286), within 1 × 10⁻¹⁶ of every line, which is where a scale factor of 0.3 about that point carries the base triangle onto the platform; the smallest singular value of the three legs is 3 × 10⁻⁹. With the platform's centre at (−0.55, 0.3) the three lines meet at (−0.7857, 0.4286), within 2 × 10⁻¹⁶ of every line, which is where a scale factor of 0.3 about that point carries the base triangle onto the platform; the smallest singular value of the three legs is 0. The meeting point moves with the platform, and the three lines meet wherever it is.

Two orientations no position can rescue

A planar platform on three extending legs whose platform triangle is a scaled copy of its base is singular at every position it can be put in, at exactly two orientations: its three leg lines meet at one point wherever it is. The two orientations are read off the attachment points, a platform a few per cent from similar is held at its worst orientation only in proportion to how far from similar it is, and the revolute-legged 3-RRR, built on the same similar triangles, does not inherit any of it.

parallel · Parallel A fourth leg through the meeting point, and one beside it. The platform with similar triangles at its singular orientation, at one position, with a fourth base pivot at (0, -2). Left: the fourth platform point placed where the similarity puts it, so the fourth leg's line passes 2.8e-17 from the point the other three meet at, and the smallest singular value of all four is 2.59e-9 — nothing has changed. Right: the same base pivot with the platform point moved 0.28 round the platform, so the fourth line misses the meeting point by 0.3993 and the four legs hold at 2.914e-1. The ringed point is where the three original lines meet, and it moves with the platform.

One placement of every placement

A planar platform whose two triangles are similar is singular at every position it can be put in, at two orientations. A fourth leg removes both — unless its own pair of attachment points is related by the same similarity, and then it changes nothing at all. Swept right round the platform, exactly one placement of the fourth attachment fails, the similarity names it in advance, and the holding a placement buys is proportional to how far it sits from that one point.

parallel · Parallel A paired platform with its pairs rotated 60°, turned through a revolution. The paired Gough platform — base anchors in pairs 25° apart on a radius of 2.2, platform anchors in pairs 40° apart on 1.1 — with the platform's pairs centred 60° round from the base's, held level and turned through a revolution at four positions. All four curves reach nought together at 30° and −150°, which is 90° − ρ and 180° from it. At the centred position the platform is held at 0.0363 at a yaw of 0°, and it can turn ±17° before that falls to half. Dragging the rotation carries the two dead yaws across the revolution together.

The dead yaw is a design choice

A paired hexapod held level is singular everywhere at a yaw of 30° because its platform pairs sit 60° round from its base pairs. Rotate them by ρ instead and the dead yaws move to 90° − ρ and 180° from it, exactly, at every rotation from 0° to 120°. The furthest they can be from home is a quarter-turn each way, at ρ = 0 — where the platform is also best held at home and can turn ±75.5° before its holding halves, against ±17° for the usual 60°.

parallel · Parallel Where a tilted platform is still singular. A slice of the workspace at height 2.4, at the dead yaw and 3° of tilt, with each position shaded by how well the platform is held there — dark where the six legs are nearly dependent and pale where they are not. Level, this whole square would be uniformly dark. Tilted, the dark places are a curve through it, which is what an ordinary direct singularity looks like on a slice. Driving a search downhill from forty-eight starts reaches a smallest singular value of 0.00e+0, so the locus is a singularity rather than a shallow valley.

Tilted, near the dead yaw

A paired Gough platform held level is singular at one yaw wherever it stands. Tilt it and that stops being true — the six moments are no longer equal and the home position is held. It is a poor rescue: the rise is quadratic in the tilt, so a degree buys a sixty-fourth of what eight degrees buys, and what the tilt actually does is not remove the singularity but turn it into a surface through the workspace.

parallel · Parallel Where in the workspace each yaw is singular, once the platform tilts. A horizontal slice of the workspace at height 2.4, for the paired platform with its pairs rotated 0° and tilted 1° about an axis 0° from x (the arrow). Each square is shaded by the yaw, measured from the dead yaw 90°, at which that position is singular — found as the root of the six leg screws' determinant — blue where it is below the dead yaw and orange above, deepest at ±0.208°. Level, every position would be singular at the dead yaw exactly. Tilted, the singular yaw is a plane over the slice: over the whole workspace, the yaw times the height fits a·x + b·y to 0.7% of its spread, and the dashed line is where it is nought. The plane's slope turns with the tilt axis. Dragging turns the axis.

The dead yaw opens into a wedge

A paired Gough platform held level is singular everywhere at one yaw and nowhere a quarter of a degree either side of it. Tilt it and each position of the workspace becomes singular at its own yaw, displaced from the dead one in proportion to the tilt and to the position's horizontal offset over its height. So the orientations that put a singularity somewhere in the workspace are not a surface through the dead yaw but a wedge standing on it, about a quarter of a degree of yaw wide on each side for every degree of tilt — and the rotation between the pairs, which chooses where the dead yaw is, barely changes how much room it needs.

parallel · Parallel The singular set cuts the workspace's poses in two, and every path across it meets it. A slice of the paired Gough platform's poses, tilted 2° about x: the yaw's offset from the dead yaw across, and the position along the workspace's low diagonal, from one corner at height 1.8 to the other, up. Each cell is shaded by the sign of the six leg screws' determinant; the line between the two shades is where it is nought, the singular set. Tilted, it leans: each position is singular at its own yaw, the low corners about 0.48° either side. Three paths from −2.5° to +2.5° are drawn, one held still, one moving, one wandering, and each meets the line: held at the middle, 1 crossing; moving corner to corner, 1 crossing; wandering, 1 crossing. No path from one shade to the other can avoid it.

No path across the dead yaw misses it

A paired Gough platform that has to turn from one side of its dead yaw to the other cannot be steered round the singularity by tilting, by moving, or by turning the long way. The six legs' determinant has one sign everywhere on one side and the other everywhere on the other, so every path passes a singular pose — two hundred random ones do, an odd number of times each. Tilt chooses only where the crossing happens and how briskly: a quarter quicker at best, and threefold slower if it is done by moving instead of turning.

parallel · Parallel

Named alongside it

The objects these essays reach for when they reach for this one.

Singular valueSingularityDesign rulethe Gough–Stewart platformWorkspaceReciprocal screwPoseScrewDirect singularityPitchAssembly-modeConcurrency

All concepts