Machine Kinematics
solved, then drawn
Concept

Reciprocal screw — where it appears

A wrench that does no work against any of a mechanism's joint screws, which is what a constraint is and what a singularity leaves behind. It is the object that turns a mobility question into linear algebra: a loop's constraints are the screws reciprocal to all of its joint screws, and the count is a dimension.

Named by 17 essays across 5 fields — each of them below, with the objects they name alongside it.

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.

parallel · Parallel What each loop's constraint system is made of. For each mechanism: the order of the screw system its joints span, the order of the reciprocal system — the wrenches it carries without moving, which is always six minus the first — and what those wrenches are. A planar four-bar carries one force and two couples; a mechanism whose motion lies in no subgroup carries screws of finite pitch instead, and 2 of these 6 do.

What a mechanism cannot do

A mechanism's freedoms are a subspace of screw space. Everything orthogonal to that subspace under the reciprocal product is a force the mechanism carries without moving — so the constraints are not a separate thing to be worked out, they are what is left, and one matrix gives both.

spatial · Spatial Kutzbach, plus the constraints counted twice. Each row is 6(n − j − 1) + Σf, then the redundant constraints ν measured from the two legs' constraint systems, then the mobility measured from the rank of the whole loop's screw system. The first column is wrong for 5 of 6 of these mechanisms; adding ν repairs every one. The catch is that ν is not a property of the joint graph — Bennett's linkage and a spatial four-bar with one twist changed have the same graph and different ν — so the corrected formula needs the measurement it was supposed to replace.

The formula is repaired by the thing it replaced

Kutzbach's count is wrong about most of the mechanisms worth building, and every textbook gives the same repair — add back the constraints that were imposed twice. The repair works on every loop in these essays. It is also not a formula, because the number it adds cannot be read off the joint graph.

constraint · Spatial Three singularities, and where each one is. The elbow singularity is at the edge of the workspace, where the arm is straight and cannot reach further — nothing is lost that could have been used. The shoulder singularity is where the two ways of facing the target merge, on a cylinder of radius 0.18 m about the base axis which is also the boundary of a hole the arm cannot reach into at all. The wrist singularity is the one that stops real machines: axes four and six in line, one rotation gone, in the middle of an ordinary working volume with nothing about the tool's position to suggest it. The rank falls by exactly one in all three — and what is lost differs. The elbow and the shoulder are constrained by a pure force, along the arm and across it; the wrist is constrained by a screw of pitch −0.629, which is a force and a couple together. Those are reciprocal screws rather than singular vectors, because a singular vector's direction depends on whether the arm was written in metres or millimetres and a screw does not.

Where the arm loses a direction

An arm has three singularities and they are three different events. Two of them are inside an ordinary working volume, all three drop the rank by exactly one, and what each one takes away is a screw — a pure force along the arm at the elbow, a pure force across it at the shoulder, and at the wrist a screw of pitch −0.629 that is a force and a couple together.

serial · Serial The same error, twice, in two directions. A Sarrus linkage with one axis of one chain tilted off true, and the motion range that survives. Tilted within the plane the chain works in, it does not care: at 0.2 radians — eleven and a half degrees, which is not a manufacturing error by any standard — it still drives through a full turn. Tilted out of that plane, 0.001 radians stops it dead. Two hundred times the error, in the other direction, for no cost at all. What separates them is whether the perturbation lies in the screw system the mechanism leaves unconstrained — so "an overconstrained mechanism must be exact" is not merely crude, it is wrong about the case it is usually said of.

Fragility has a direction

Tilt one axis of a Sarrus linkage out of true by a thousandth of a radian and it stops dead. Tilt the same axis of the same mechanism by two hundred times as much, in the other direction, and it drives through a full turn with nothing measurably wrong. Three orders of magnitude between two errors of the same size — and the direction that matters is the one the reciprocal screw system names.

practice · Overconstraint Arms with their tools pinned down. Pin an arm's tool to the ground and the open chain is a closed loop, which the mobility count of a planar loop already handles. Kutzbach gives 6(n − 1) − 5n = n − 6 for a loop of n revolutes, and the measurement is n minus the rank of its screw system — the columns of the arm's own Jacobian, read as constraints rather than as velocities. The two agree on every row but one, and the one is the arm at a wrist singularity: the formula says the pinned arm is a structure and the mechanism has a freedom. That is the finding this site opened with, arrived at from the far end of its subject.

Pin the tool and it is a loop

Hold an arm's tool still and the open chain becomes a closed one, which these essays have known how to count since their first field. Kutzbach's criterion says a pinned six-joint arm is a structure. At each of its three singularities the measurement says it can still move — the site's founding finding, arrived at from the far end of its own subject.

serial · Spatial 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 Three joints, and the three screws that describe them. A leg of three revolute joints, drawn as its three axes, and the principal screws of the three-system they span, drawn through the system's own centre. The three principal axes are mutually perpendicular — worst cosine 1.1e-16 — and they meet at one point, missing it by 2.6e-16. Their pitches are -0.3766, -0.0338, 0.8075, and every screw the leg leaves free has a pitch the three of them give by h₁l² + h₂m² + h₃n². Nothing in the three joint axes looks like a right angle and the system's own frame is one.

What a leg of three joints leaves free

Five essays of this field have computed the order of a screw system and drawn none of them. A three-joint leg spans a three-system; its three principal axes are mutually perpendicular and meet at a point, six numbers price every screw in the family, and the directions of the lines it contains form a cone.

spatial · Spatial What a leg of three joints permits and what it resists, on one frame. The leg of three revolute joints in grey, and the principal screws of two three-systems drawn through one centre: the twists the leg lets its platform make, and the wrenches it holds without any joint turning. They are computed separately — the second is the reciprocal complement of the first, found by its own eigenproblem — and they come out with the same centre, 5e-16 apart, and the same three axes, parallel to 0e+0. Only the pitches differ, and each wrench pitch is its twist's with the sign reversed, to 1e-15: -0.3766 against 0.3766, -0.0338 against 0.0338, 0.8075 against -0.8075.

The lines a leg turns about and the lines it is pushed along

A leg of three revolute joints permits a three-system of motions and resists a three-system of loads. The two share a centre and three axes and differ only in the sign of every pitch — and the revolute axes of the first and the lines of force of the second are the two rulings of one hyperboloid, every line of one meeting every line of the other.

spatial · Spatial 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 Two legs that share a line of force. Two legs of three revolute joints each under one platform, drawn about their common centre: the first leg's joints in one colour and the second's in another. The second leg was built so that each of its joints passes through a point of one line, the dark one, which also crosses all three joints of the first leg — the six marked points, each met to 1e-16. A force along that line does no work on any joint of either leg, so both legs resist it: it is a load path the platform has twice over. The dashed line is what comes with it: the axis of the one twist both legs permit, a screw of pitch -0.049 that passes 0.032 from the shared line at 57.2° to it. Dragging moves the shared line along the first leg's first joint and builds the second leg afresh through each position.

A shared load is a shared motion

Put two legs of three revolute joints under one platform and let them share a line of force, so that both carry the same load. The platform does not merely gain a redundant load path; it gains a motion in the same act, because three constraints and three make six. The motion is a screw about a different line, tied to the shared one by h cos φ = d sin φ on two hundred constructed pairs to 4 × 10⁻¹⁵. Pushed along it, the two legs part as the square of the push: the platform shakes and does not move.

spatial · Spatial 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 A third leg that keeps the platform's shake. The two legs of the earlier construction — the second built through a line of force of the first, the dark line — share one twist, whose axis is the dashed line: a screw of pitch -0.049. A third leg is drawn in a third colour, built so that its three joints span a family of motions containing that twist: two joints chosen freely and the third solved for. The platform on all three legs keeps its one freedom. The dotted line is the axis of the one wrench the third leg shares with the first: a screw of pitch 0.201, not a line of force, and 0.244 from the pair's shared line. Dragging picks another third leg from the same family.

A third leg that keeps the shake

Two three-joint legs that share a line of force share one twist as well, and a third leg in general position takes it away. A third leg whose own freedom contains that twist does not, and building one is three conditions on twelve numbers: two joints anywhere and the third solved for. Such a leg must share a load with each of the first two — but not the line they share. What it shares with each is a screw with a pitch of its own, reciprocal to the twist through the sum of their two pitches, and only a third leg built through the pair's line shares the line itself. With any of them the platform still shakes rather than moves.

spatial · Spatial Turning one joint about the shared line tunes the shake's first coefficient to nought. Two legs of three revolute joints each under one platform, sharing a line of force, the dark line, that crosses every joint of both — so they share one twist and the platform shakes along it. The second leg's first joint is turned about the shared line by 0°, which keeps it crossing the line and keeps the shared twist. The shake's second-order coefficient, the reciprocal product of the shared wrench with half the difference of the two legs' bracket terms, is 2.62e-2 here. It passes through nought at 36.488° and again at 109.612°; at those turns the platform still shakes, one order higher.

A shake has an order

Two legs of three joints that share a line of force let a platform shake: pushed along the one twist they share, the legs part as the square of the push. The square's coefficient is a bracket of the legs' own joints, computable without pushing at all, and it can be tuned to nought by turning one joint. The shake does not become a motion. It moves up an order, to the cube, and a second tuned joint moves it to the fourth power. Each order is one more condition, and only legs whose freedoms are groups meet them all.

spatial · Spatial A pair moves until its gap reaches what its joints' play can take up. Two legs of three revolute joints each, sharing one twist, pushed along it: the gap left between their nearest poses, along the one wrench they share, for the pair built through the shared line, the pair with one joint tuned and the pair with two. The horizontal line is the gap the six joints' play can take up, 4.52 times a radial play of 10⁻⁶. Each pair travels until its curve meets the line: built through the shared line, 1.31e-2; one joint tuned, 6.63e-2; two joints tuned, 1.39e-1.

A shake travels as a root of the play

Two three-revolute legs that share a twist let their platform shake along it: the gap between their nearest poses grows as the square of the push, or the cube once one joint is tuned, or the fourth power once two are. With play in the joints, the pair moves until that gap uses up what the play can take, and that is a root: the square, cube or fourth root of the play. At a play of a millionth the three pairs travel 0.013, 0.066 and 0.139 along their twist. A tuned joint is worth more the tighter the joints are, five times the travel at a millionth and eleven at a hundred-millionth.

spatial · Spatial Where two coefficients vanish, a curve; where three do, two points. The second leg's three joints each turned about the shared line by ψ₁, ψ₂ and ψ₃. Every point of the curve makes the shake's second- and third-order coefficients nought, σ₂ = σ₃ = 0, found by continuation from the earlier two-joint tuning (black) both ways, over an arc of 7.25 rad, until it reaches configurations where the two legs come close to sharing a second twist (crosses), where every coefficient runs off to infinity. Left: ψ₂ against ψ₁; right: ψ₃ against ψ₁. The curve is drawn in one colour where σ₄ is negative and another where it is positive; it changes sign at the two dots, (106.31°, 140.35°, 24.66°) and (136.39°, 57.24°, 18.34°), where σ₄ is nought too. Marked: 50% of the way along, at (107.0°, 141.4°, 25.2°), where σ₄ is positive.

Three tuned joints stop at the fifth order

Two legs of three revolute joints that share a line of force let their platform shake, and every joint of the second leg turned about that line clears one more order: the square of the push, then the cube, then the fourth power. A third tuned joint was expected to buy the fifth. Read off the two legs' own Taylor series instead of a push test, the set where the second- and third-order coefficients vanish is a curve in the three tuning angles, and the fourth-order coefficient crosses nought on it twice. At both points the gap falls as the fifth power of the push, confirmed independently with slopes of 4.97 and 4.89. The fifth-order coefficient is −0.0042 at one and 0.109 at the other, and with three angles spent there is nothing left to clear it: three tuned joints stop at the fifth order. And a higher order is not always more travel: in a millionth of play the second of them moves less than the fourth-order pair.

spatial · Spatial

Named alongside it

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

Screw systemPlatformScrewSingularityTwistConstraintWrenchLegPitchArchitecture singularityMobilityRank

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