Machine Kinematics
solved, then drawn
Series

Spatial — the series

22 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Kutzbach's count against the measured mobility. Five closed loops of revolute joints. The count is 6(L − 1) − 5j, a statement about how many links and joints there are; the measurement is the number of joints minus the rank of the loop's screw system, which knows only where the axes point. They disagree for four of the five, and the one they agree on is the generic seven-joint loop — so the formula is not broken, it is blind to the special geometry that makes the other four work. The universal joint is counted at -2 degrees of freedom and is in every car built.

    Six freedoms, not three

    Every mechanism in these essays so far has been flat, and flatness is not a simplification made for teaching — it is a special case that hides the most interesting thing constraint counting does, which is get the answer wrong about mechanisms that are in daily use.

    part 1 · spatial
  2. A universal joint at 40° input, shafts 25° apart. Two shafts meeting at 25°, joined by a cross whose two pins are at right angles to each other and each at right angles to the shaft it carries. That is the whole geometry, and everything else follows from it. This is a four-joint spatial loop with all four axes through one point: Kutzbach counts −2 and the screw system has rank 3, so it has one degree of freedom and turns. At this instant the output shaft is at 42.79° while the input is at 40°, and the output is turning 1.0124 times as fast — which is not 1, and never is except at the four points of each turn where the curves cross.

    The joint that is not constant velocity

    A universal joint is the spatial mechanism everybody has met and almost nobody has been told the truth about. Its output shaft runs fast, then slow, twice per revolution, and the amount depends only on the angle between the shafts — which is why cars have two of them and why the second one has to be fitted the right way round.

    part 2 · spatial
  3. The Sarrus linkage at 0°. Two three-joint chains in perpendicular planes, joining a fixed plate to a moving one. The left chain's three axes are all parallel, so it allows the plate to move in its plane; the right chain's are parallel to a perpendicular direction and allow the plate to move in that one. What both permit is a straight line, and only a straight line. Six revolute joints in a single loop: Kutzbach says 0 degrees of freedom, the screw system has rank 5 and says 1, and the plate rises. Measured over 24 positions, its tilt never exceeds 2.5e-14 radians and it never leaves the axis by more than 6.8e-14 — exact, from pin joints, with no approximation anywhere in it.

    Sarrus, and the straight line that is exact

    The planar answer to the straight-line problem took two hundred years and arrived as an inversion cell with eight bars. There is a six-bar answer that is also exact, that was published eleven years before Peaucellier's, and that works for a reason with nothing to do with inversion — it leaves the plane.

    part 2 · spatial
  4. Bennett's four-bar at 40°. Four bars, four revolute joints, and axes that are not parallel — a spatial four-bar, which Kutzbach counts at -2 degrees of freedom. Bennett's condition, sin α / a = sin β / b, makes the screw system rank 3 instead of 4, so the mechanism has 1. Driving the first joint through a full turn, 48 of 48 positions assemble. Orthographic projection, viewed from 40° azimuth and 24° elevation; dashed stubs mark the joint axes.

    Bennett, and the condition that moves it

    A spatial four-bar is immobile by every count there is, and generically it cannot even be assembled at more than isolated configurations. Bennett found the one relation between four lengths and two twists that makes it turn through a full revolution — and break the relation by two parts in a thousand and most of the travel is gone.

    part 3 · spatial
  5. A screw of pitch 0.25. A single screw drawn from its own decomposition. The six numbers of the twist go into a decomposition that returns a direction, a point on the axis and a pitch of 0.2500; the helix is then the path of a point at radius 0.5 about that axis, advancing 0.250 along it per radian turned. Over 1.15 turns the point advances 1.806 — the pitch times the angle, which is what pitch means. This screw is a screw.

    Every motion is a screw

    Chasles showed that any rigid displacement whatever is a turn about some line together with a slide along that same line. Not approximately, and not usually — always, with the line and the amount of slide computable from the motion. It is the fact that makes spatial kinematics a subject rather than a pile of special cases.

    part 4 · spatial
  6. A spherical four-bar: 90°, 40°, 100°, 80°. Four revolute axes, all through one point, so the linkage lives on a sphere. Its link "lengths" are the angles between consecutive axes — 90°, 40°, 100°, 80° — and the arcs drawn between the axis directions measure 40.00°, 100.00°, 80.00°, 90.00°. The whole planar four-bar theory carries over with sines where lengths were, including Grashof's condition: sorted, the arcs give s + l = 140° against p + q = 170°, so the shortest arc does turn all the way round — and swept, the input reaches 36 of 36 positions over a driveable range of 360°. This is the mechanism a universal joint is a special case of, with two of the four arcs at 90°.

    When the link lengths are angles

    Put every axis of a four-bar through one point and the mechanism lives on a sphere. Its bars become arcs, its lengths become angles, and every planar result carries over with a sine where a length used to be — including Grashof's condition, which still predicts exactly which link goes all the way round.

    part 4 · spatial
  7. 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.

    part 5 · spatial
  8. The cylindroid at 75° and 1.00 apart. Every screw in the two-system spanned by two revolute axes 1.00 apart along their common perpendicular and 75° out of parallel, drawn as its own axis. The axes sweep a ruled surface — the cylindroid — and each generator carries a pitch, running from -0.384 to 0.652 and reaching its two extremes on the two principal screws, which cross at a right angle at the centre. The surface is 1.035 long along its own axis, which is exactly the spread of the pitches: a cylindroid is as long as its pitches are far apart.

    The smallest screw system has a shape

    Add two screws together in every proportion. The results do not scatter — their axes sweep a ruled surface, with the pitch varying along it between two extremes reached at right angles to each other. It is a picture nobody would guess from the algebra, and it is the object that says what two joints between two bodies leave free.

    part 5 · spatial
  9. 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.

    part 6 · constraint
  10. Whose constraints stay put. The largest principal angle between a mechanism's screw system at the start of its motion and at each later position. The 3 mechanisms whose motion lies in a subgroup of the rigid displacements — planar, spherical, translational — never leave the same subspace, and read between 1.9e-6 and 3.0e-6 degrees, which is the precision of an arccosine near one rather than a movement. The paradoxical ones turn through 49° and 57°. Every curve runs over its own range of motion, because Bricard's linkage assembles over 120 degrees and a shared axis would hide it.

    Two ways to be overconstrained

    A planar four-bar and Bennett's four-bar report the same redundancy, the same rank and the same wrong count. One of them is overconstrained at every set of link lengths; the other at exactly one ratio and nowhere near it. The difference is not in any of the numbers so far — but it is measurable, and the measurement is an angle.

    part 6 · spatial
  11. 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.

    part 7 · serial
  12. The framework Maxwell's count calls a structure. Six joints and twelve bars in space. Three coordinates each gives eighteen unknowns, six rigid motions come off, and twelve bars is exactly twelve constraints — Maxwell's count is 6 against six rigid motions, which is the definition of isostatic: no mechanism, no redundancy, every bar carrying its own share and nothing spare. The rank is 11, not twelve. There is one dependency among the bars and one freedom left over, and the freedom is a genuine finite motion: walked here with every bar held to 4.4e-16 of its own length. The reason is a symmetry — three pairs of joints exchanged by a half turn about one line — and it is built into the coordinates rather than assumed of the result. positioned by solving, not by drawing.

    Twelve bars and a symmetry

    Six joints and twelve bars in space is Maxwell's count exactly: no mechanism, no redundancy, nothing spare. Place three pairs of the joints so that a half turn about one line exchanges them and it moves — a finite motion, walked with every bar held to five ten-thousand-billionths of its own length, on a framework the arithmetic calls a structure.

    part 8 · spatial
  13. In space the arithmetic allows almost nothing. A body in space has six freedoms and a revolute joint takes five, so a mobility of one needs (6n − 7)/5 joints — and that is an integer only when the link count leaves a remainder of two on division by five. The whole table is this: 7, 12, 17, 22 links, and nothing else. At seven links the degrees must sum to fourteen across seven links with none below two, so every link is binary and the graph is a single seven-cycle: there is exactly one spatial chain, and it is a loop. That is the census explanation for something the spatial field has lived with since it was written — every spatial mechanism in these essays is one closed loop — and it had never been stated as a count. The next admissible size is twelve links and thirteen joints, where two assortments are arithmetically possible, 157 candidates give 33 graphs, and 5 of them are chains — every one with ten binary links and two ternary, so the assortment with a quaternary link is empty exactly as four of the planar ones are.

    In space there is one chain

    A body in space has six freedoms and a revolute joint takes five, so a mobility of one needs (6n−7)/5 joints — an integer only when the link count leaves a remainder of two on division by five. At seven links every link is binary, the graph is a single seven-cycle, and there is exactly one spatial chain.

    part 9 · spatial
  14. Four instruments, and only the last one names the group. Every instrument these essays have for an overconstrained loop, on the same six mechanisms. Kutzbach's count gives −2 for a planar four-bar and −2 for Bennett's. The rank of the constraint Jacobian gives three and three. Both are right and neither separates them. The last two columns are this field's: the span is how many dimensions the logarithms of the displacements the moving link actually reaches occupy, and closes at is the dimension after those are closed under the bracket. A planar four-bar closes at three and the three are planar motion; Sarrus closes at one, a translation, which is the exact straight line the spatial field measured by solving the mechanism sixty times. Bennett closes at six: its displacements occupy four dimensions and no group smaller than all of them contains those four. That is what "paradoxical" has meant in the earlier essays, stated as an integer.

    A name for each overconstraint

    The spatial field separated subgroup overconstraint from paradoxical by measuring how far a mechanism's screw system turns: 2 × 10⁻⁶ degrees against 89. That is a verdict without a name. Closing the logarithms of the reached displacements under the bracket gives the same verdict and says which group — planar, spherical, a translation — and for Bennett's linkage it says six.

    part 10 · spatial
  15. 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.

    part 11 · spatial
  16. 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.

    part 12 · spatial
  17. 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.

    part 13 · spatial
  18. 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.

    part 14 · spatial
  19. How much of Bennett's turn survives a bar being wrong. The same four bars and the same four twists, with one bar's length changed by the amount on the left and nothing else touched. The bar shows the fraction of 48 sampled positions of the first joint at which the loop closes to within 10⁻⁹. Two parts in a thousand already costs most of the travel. This is what it means for a mechanism to work only on a condition rather than approximately near one — and it is why Bennett's linkage was a curiosity for eighty years before anyone could machine to it.

    Bennett's condition is a ratio

    A spatial loop's parameters are lengths and angles together, so a scaling touches only half of them. Bennett's condition — a over sine alpha equals b over sine beta — is a relation between the two halves, and what it demands of a machine is a relation between its lengths and its twists rather than a property of either.

    part 15 · spatial
  20. 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.

    part 15 · spatial
  21. 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.

    part 16 · spatial
  22. 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.

    part 17 · spatial

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