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

Serial and parallel kinematics compared on a 3-RRR platform, with the inverse problem solved leg by leg in closed form and the forward problem reduced to a single scalar equation whose roots are counted.

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

The two Jacobians of a parallel mechanism and the two singularities they produce, with the direct-kinematic singularity located by tracing det A through the workspace and read as three constraint lines becoming concurrent.

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

The assembly modes of a 3-RRR platform found by reducing the forward problem to one scalar equation and scanning it exhaustively, with the count surveyed over 6,750 sets of actuator angles and its odd values identified as singularities.

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

The six-legged parallel platform with its inverse kinematics in closed form, its forward kinematics solved by Newton from many starts and reported as a lower bound, and its singularity read as six leg lines becoming linearly dependent.

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

The delta robot's translation-only motion derived from the couple constraints its three parallelograms impose, with the redundancy measured and the platform shown to gain a rotation when the three bar directions are made parallel.

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

The workspace of a 3-RRR platform computed by sampling, with the direct singularity locus traced through it as a contour, and the usable region shown to be a connected component of the reachable set rather than the reachable set.

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 in these essays: 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

Parallel platforms as intersections of displacement subgroups, with the platform's motion type independent of every length and the degenerate arrangements as the only failure.

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 parallel platform calibrated from its own joint readings: the pose-free residual, what three sensing arrangements determine, and the collapse an angle-sensed fit runs to.

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 cusp of the 3-RRR platform's singular curve located as a triple root of its closure equation, a loop of motor angles round it tracked from every starting assembly, and the assembly that comes back as another without passing a singularity.

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

The planar five-bar as a parallel robot: two assemblies, four working modes, the direct singularity where the distal links line up, and the measurements showing that the sign of det A names its assembly so no loop of the motors can change it quietly.

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

A redundant fourth leg added to the planar 3-RRR platform: the singular curve of a slice reduced to isolated points, how much better the platform is held, where the fourth leg should be attached, the misfit a fourth motor out of step produces, and the fourth encoder that names the assembly.

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

The Gough–Stewart platform's level poses swept through a full turn of yaw at many positions, the yaw at which all of them are singular, the moment condition and the free screw that explain it, and the spread of anchor points at which every yaw is singular.

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

The planar 3-RPR platform with similar base and platform triangles: the two orientations at which its leg lines meet at the centre of similarity for every position, measured over a square of positions and by the meeting point, how they move with the attachment points, how the platform recovers away from them, the proportional law for nearly similar triangles, and why the 3-RRR on the same attachment points is singular only along a curve.

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 fourth leg on a 3-RPR with similar base and platform triangles: the single placement of its attachment that leaves the two everywhere-singular orientations untouched, found by sweeping the attachment round the platform and predicted without the sweep; the linear law relating the holding to the offset from it; and the whole turn of orientation with three legs and with four.

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

The rotation between a Gough platform's base and platform anchor pairs treated as a design variable; the yaw singular at every level position measured at thirteen rotations and shown to be 90° − ρ; the equal-moment condition and its free screw at a rotation of nought, identical in pitch to the 60° design's; the smallest singular value at home and the usable range of yaw about home charted against the rotation; and what the chart leaves out.

Tilted, near the dead yaw

The smallest singular value at the dead yaw against tilt, rising as tilt² with a fitted exponent of 1.987; a workspace that reads uniformly at the solver's floor when level and spans three decades at three degrees of tilt; and a descent from forty-eight starts reaching exactly nought at every tilt tried.

The dead yaw opens into a wedge

The yaw at which each position of a tilted paired Gough platform's workspace is singular, as a plane in the position's offset over its height; which orientations near the dead yaw leave a singular surface in the workspace, level and tilted; the band of such yaws against tilt about three axes, opening linearly; a section through the set of such orientations; the measured singular yaws against the plane law at three heights; and the band at one degree of tilt for five rotations between the pairs.

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 crossing its dead yaw: the leg-screw determinant's sign on a slice of pose space; the determinant against yaw at every position, tilted; two hundred random paths through position, tilt and yaw and where each crosses; the smallest singular value over a whole turn; seven ways to cross and the time each spends near the singular set, against 2ε/(k·|r|).

With a seventh leg the dead yaw is not a dip at all. The paired Gough platform held level at home, turned through sixty degrees of yaw centred on its dead yaw: the smallest singular value of its six leg screws, which falls to 8.9e-9 at the dead yaw, and of the seven screws with a seventh leg from the base circle to the platform circle at base 60°, platform 255°. With it the smallest singular value is 0.0288 at the dead yaw and no less than 0.0287 anywhere in the sweep, against the six legs' 0.0141 ten degrees off the dead yaw and 0.0320 thirty degrees off. The best placement of the 576 tried is the first the slider offers.

A seventh leg opens the dead yaw

A paired Gough platform with a seventh leg: the smallest singular value against yaw at home with six legs and with seven, for three placements; every placement of the seventh leg on a 15° grid by its worst value over the workspace at the dead yaw; that value against the two caps it cannot exceed; a hundred random paths across the dead yaw, six legs and seven; and the legs' internal force at the dead yaw and ten degrees off.

A seventh leg does no work on the free twist only when it leans as the six do. At home, level at the dead yaw, the seventh leg's signed work on the twist the six legs cannot resist, against how far round the axis its platform attachment sits from its base attachment. It is nought at exactly two leans, 82.5° and 97.5°, the two leans of the six legs themselves (ticks), positive between them and negative everywhere else, most negative at 278°. The best leg of the earlier placement map leans 285° and does work −0.506, near that extreme of −0.507.

A seventh leg must lean its own way

Where a paired Gough platform with a seventh leg is singular near its dead yaw: the platform from above with the six legs and a seventh at five leans; the seventh leg's work on the six's free twist at home against its lean; the band of leans whose seven-leg singular set enters the workspace, for tilts up to 1°, 2°, 4° and 8°; one tilted slice of the workspace, cut by that set; and six seventh legs by lean, level behaviour, the tilt at which they fail and their least singular value.

Half a turn on, the six legs lean alike again, and the seventh still the other way. The paired Gough platform from above, level, turned half a turn from its dead yaw: the base circle of radius 2.2, the platform circle of radius 1.1 (dashed), and the six legs, each now leaning 262.5° or 277.5° round the axis, two angles with the same sine, so the six are singular here at every position. The seventh leg (heavy) leans 105°, with the opposite sine, and does work 0.373 to 0.560 in size on the six legs' free twist across the level workspace: never nought.

A seventh leg has no dead yaw of its own

Where a seven-legged paired Gough platform can be singular: the platform from above at both of the six legs' dead yaws with the best seventh leg; the six legs' and the seven legs' least singular value over a whole turn of yaw, level; a seventh leg's work on the six legs' free twist against its lean at both dead yaws; the six legs' singular curves and the seven legs' singular poses in the plane of yaw and tilt at one position; the least tilt at which the best leg's seven-leg singular set appears, by position and tilt axis; and the least tilt against the seventh leg's lean.

All essays