Field

Several legs, one platform

Everything else here has one path from the ground to the moving part. Give it three, and the easy problem and the hard one change places — and a new kind of singularity appears in the middle of the workspace, where the machine can move with every motor locked.

Three legs, one platform. A 3-RRR planar parallel mechanism at (0.20, -0.15) turned 11.5°, elbows up/up/up. The three actuator angles were computed one leg at a time and independently, which is what makes this direction cheap. The dashed lines extend each leg's second link: those are the three forces the legs can transmit to the platform, and the mechanism is controllable exactly while they stay independent. Here they miss one another by 0.651, and the smallest singular value of the three is 0.9550. At this position the platform can be turned through 206° in all before a leg runs out of reach.

The easy problem and the hard one change places

For a robot arm, working out where the hand is takes a walk down a chain and working out what the joints must be to put it somewhere is the hard part. Connect the platform by three legs instead of one and both statements reverse — and the reversal is not a matter of degree.

Losing a freedom and gaining one. Left, the nearest inverse-kinematic singularity: a leg is straight to 1.1e-5 and cannot reach further, so the platform has lost a freedom. That is the workspace boundary and it is a serial arm's singularity. Right, the nearest direct-kinematic singularity: the three leg lines pass through one point to 0.0001, so the three forces the legs transmit are no longer independent and the platform can turn about that point with every actuator locked. It has gained a freedom, and it is 1.97 from the other configuration and nowhere near the edge of the reach.

Locked, and still moving

A serial arm goes singular at the edge of its reach, where it loses a freedom, and the failure is visible as an arm gone straight. A parallel mechanism has a second kind with no serial counterpart — it gains a freedom, in the middle of the workspace, at poses nothing about the legs' reach marks out.

6 ways to assemble the same three actuator angles. The actuators are at 216°, 48°, 144° in every panel, so the three elbows are at the same three points throughout and only the platform differs. Each pose satisfies all three legs to 6.7e-16. Found by reducing the problem to one equation in the platform angle and scanning it at 0.100° — exhaustive to that resolution and no further, which is the honest thing to say about a root count. Over a survey of 6750 actuator triples this mechanism ranges from none to 6.

One command, six answers

Lock the three motors of a planar platform and the platform can be in as many as six different poses, every one of them satisfying every leg exactly. Which one it is in was decided by how it was assembled and where it has been since — and the number of answers is not a property of the mechanism but of where the motors happen to be.

Six legs, six numbers. A Gough–Stewart platform at (0.00, 0.00, 2.40) with a rotation vector of (0.00, 0.00, 0.15). The six leg lengths run from 3.077 to 3.263, and every one of them was computed from the pose by a single subtraction and a square root, with no leg consulting another. Going the other way — six lengths in, one pose out — takes an iterative solve and does not have one answer. The six leg lines span a screw system of order 6; the smallest singular value of the six is 0.0316, which is how far this pose is from a configuration where they become dependent and the platform stops being controllable.

Six legs and a square root

A Gough–Stewart platform's inverse problem is one subtraction and one square root per leg, computed six times without any leg consulting another. Its forward problem has forty solutions. That gap is the whole design, and it is why flight simulators are built this way and robot arms are not.

Three bars, and no rotation left. Each of a delta robot's legs ends in a parallelogram, which keeps the bar on the platform parallel to the bar on the arm. That leg therefore permits the platform no turn about either direction perpendicular to its bar: it imposes two couple constraints, drawn here as rings about the directions they act on. Three legs impose 6; together they span only 3, so 3 are redundant. What is reciprocal to them is three couples — three pure translations, and the platform cannot turn at all.

Why the platform stays flat

A delta robot has three legs and three freedoms, and there is no obvious reason those freedoms should be the three translations rather than some mixture. The reason is a parallelogram in each leg, and the argument from there to "the platform cannot turn at all" is a constraint computation that takes six wrenches and a rank.

Where the platform stops being controllable, at 0°. Every point is a position of the platform's centre at a fixed orientation of 0°, shaded by how far it is from a direct singularity — pale is near. Unshaded means unreachable, which is the workspace boundary and the ordinary kind of singularity. The line is det A = 0, traced through the field rather than tested for: it runs through the middle of the reachable region in 102 segments, and on it the platform can move with all three actuators locked. 3312 of 6561 sampled positions are reachable.

The workspace is not a shape you choose

A serial arm's reach is roughly a sphere and can be quoted as a number. A parallel mechanism's is the intersection of three reachability conditions, changes with every degree of orientation, and has a surface of uncontrollable poses cutting through the middle of it. There is no formula. There is a map, and it has to be computed.

What every pair of groups meets in. The intersection of two subgroups is always a subgroup — that needs no computation — and which one is the useful part. This is the design rule behind every parallel machine on this site: choose legs whose groups meet in the motion the platform is wanted to have, and it has that motion whatever the leg lengths are, with no synthesis and no tolerance. The row and column are built about different axes, at right angles, because an intersection is a statement about particular subgroups rather than about their kinds: two planar groups with the same normal meet in the whole of themselves, and two with different normals meet in a line.

Three legs and one plane

The parallel field argued that a platform stays flat from the structure of its legs, one leg at a time. The same fact is one line of linear algebra: each leg confines the platform to a group, the platform gets the intersection, and an intersection of groups is a group whatever the leg lengths are. The motion type is decided before a single dimension is chosen.

What a platform's own sensors settle. One three-legged platform, measured at 75 poses, with three different sets of sensors on it. Reading every joint angle determines 12 of the 15 numbers that describe the machine and leaves three undetermined, and one of the three is the scaling of the whole machine — every reading is dimensionless, so multiplying every length by a constant changes nothing any sensor sees. Laying a rule across two base pivots once recovers exactly one more, and the one it recovers is the size. Legs that report their own extension need no rule: a reading with a length in it breaks the scaling direction outright, and 7 of 9 come back. What none of the three determines is where the frame's origin sits, which is a convention rather than a defect.

A platform that measures itself

A three-legged platform with every joint read can be calibrated from its own sensors with no instrument in the room. Left to itself it shrinks the machine to a fiftieth of a per cent of its size — and reports a residual five orders smaller than the right answer's for doing it.

Where the assembly count changes, and a loop round a cusp. With the first motor held at 216°, each point of the square is a setting of the second and third motors, shaded by how many assemblies the platform has there, sampled on a 41 by 41 grid: 2 assemblies at 1,261, 4 assemblies at 347, 6 assemblies at 73. Each edge between two shades is a curve of direct singularities, where two assemblies merge and vanish, and two such edges meet in a sharp point. The 2 marked points are the cusps in this window, each found as a triple root of the closure equation. The dashed circle, of radius 8°, is the loop the motors are driven round; it encloses 1 cusp, the one at (101.35°, 185.32°), and starts at the open marker.

Round a cusp into another assembly

A parallel platform's assembly mode was supposed to change only through a direct singularity. Driven round a small loop of motor angles that encloses a cusp of the singular curve, the standard three-legged platform leaves one assembly and arrives in another, turned 52° from where it started, and at no point on the way is it nearer than 0.0716 to singular.

Two motors, two circles, two places for the hand. A planar five-bar with its motors 1.0 apart, arms 1 and distal links 1.25, at motor angles 100° and 60°. Each distal link holds the hand on a circle of radius 1.25 about its elbow, and the elbows are 1.678 apart, less than the 2.50 at which the circles would only touch, so they meet twice. The hand drawn solid is at (0.229, 1.850) with det A 0.995; the other assembly, dashed, is at (0.098, 0.001) with det A -0.995 — the same size and the opposite sign.

The smallest parallel robot

Two motors, two arms, and two links meeting at a hand: a planar five-bar is the smallest parallel robot there is. Its forward problem is two circles, so the hand has two places to be, and each is named by the sign of one determinant. That is the whole reason it cannot do what the three-legged platform does, and cannot change assembly without passing through the one configuration where the two meet.

Three legs and four, at 0°. The same slice of positions at a platform angle of 0°, shaded by how well the platform is held — pale is near singular. Left, three legs: 3,312 reachable samples and a singular curve through them in 102 segments. Right, the same three legs and a fourth: 3,198 reachable, because the fourth leg must reach too, and no curve. What is left of the singular set in this slice is 1 isolated point, at (-1.449, -0.811), where all four lines meet. Positions held above 0.1 go from 2,989 to 3,178, and at no sampled position is the four-legged platform held less well than the three-legged one.

What a fourth leg buys

Three leg lines fail to hold a platform when they meet at a point, which is one condition, so in every slice of the workspace the failures form a curve. Four lines fail only when all four meet at a point, which is two conditions, so the curve becomes isolated points. The fourth leg buys that and more, and it costs a machine that can no longer be assembled from any four motor angles.

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.

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.

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.

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°.

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.

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