Machine Kinematics
solved, then drawn
Series

Body — the series

21 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. a crank rocker with a post: the closest pair at one position. The same four-bar with a post bolted to the frame, just clear of the coupler's path. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.2561 here, between rocker · post. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

    A link that takes up room

    For twenty-three fields a link in these essays has been a distance between two points, and a distance cannot collide with anything, because it is not anywhere. Give every link a body and a question arrives that none of the constraint equations can ask.

    part 1 · bodies
  2. Peaucellier's cell: the closest pair at one position. Eight links, ten pins and an exact straight line — the site's densest planar loop. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.1983 here, between long arm A · crank. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

    A gap is a number

    A collision test that answers yes or no cannot say by how much, and therefore cannot say what would fix it. The quantity this field is built on is one signed number: positive is a gap, negative is how far the parts would have to be moved to stop overlapping.

    part 2 · bodies
  3. A dented shape, cut into 2 convex pieces. Ear clipping cuts the outline into triangles and Hertel–Mehlhorn then deletes every diagonal whose removal leaves both sides convex, which takes this crank to 2 pieces rather than the 4 the triangulation produced. The pieces tile the polygon to 1.8e-16 of its area, and — the part the areas cannot check — a point is inside the pieces exactly when it is inside the polygon, tested at 4,000 random points per shape with 0 disagreements. The gap between two parts is then the best answer over the pairs of pieces, which is why the decomposition has to be right rather than merely plausible.

    A shape with a dent in it

    The separating-axis theorem is not approximately right about a non-convex shape; it is wrong, and it returns a confident number while being wrong. The repair is to cut the shape into convex pieces — and the shortcut everybody takes instead adds a hundred per cent more material.

    part 2 · bodies
  4. The gap is a function, and it has corners. The smallest gap over every tested pair, at each of 360 solved positions of the crank. Two things are visible that a check at the ends could not report. The minimum, 0.1041, is in the middle of the travel and not at either end. And the curve has 4 corners, each one a change in which pair is closest — the colours — so the function is piecewise smooth rather than smooth, and its minimum is not where a derivative vanishes. Refining off the sample grid by golden section moves the answer by 1.4e-5, which is what a corner rather than a smooth minimum looks like.

    A gap with corners in it

    The clearance between two parts is a function of the crank angle, and it is not a smooth one. It has a corner wherever the closest pair of features changes hands, so its minimum is not where a derivative vanishes and is not at either end of the travel.

    part 3 · bodies
  5. The bound that says the sweep missed nothing. Every corner of every body is an affine function of its link's two joints, so its speed is the same combination of the joint velocities the mechanism already solves for — and the largest corner speed anywhere on this drive is V = 1.255 per radian. A distance between point sets is 1-Lipschitz in those points, so from each sample the gap can fall no faster than V: the fine lines are those cones. Where two cones cross is the least the gap can be between the samples, and at 60 samples that is 0.0383 — positive, so nothing was missed. The same bound refuses the twelve-sample sweep of the stud machine, where the bound is -0.281.

    A sweep that missed nothing

    A swept clearance check looks at finitely many positions of a machine that has infinitely many, and cannot report what it did not look at. Here is a twelve-sample sweep declaring a machine clear by 0.007 while it is 0.010 inside a stud — and the bound that refuses to certify it.

    part 3 · bodies
  6. crank rocker: the closest pair at one position. The site's standard four-bar: ground 4, crank 1, coupler 3.5, rocker 3. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.5438 here, between coupler · frame. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

    Two bars that have to cross

    The site's own four-bar has its coupler inside its frame by a full link width for the whole of a turn. It is not an impossible mechanism; it is a mechanism that cannot be built in one plane — and the plane it has been drawn in for twenty-three fields was a convenience nobody had to pay for.

    part 4 · bodies
  7. A colouring is not an assignment. A pin joining links in planes 1 and 3 has to pass through plane 2, and anything in plane 2 whose material covers that pin is pierced by it. That condition involves three links at once where a colouring's conditions involve two, and it is invisible to a colouring because permuting colours preserves a colouring and destroys betweenness. It does not change the number of planes on any machine here. What it removes is arrangements: of the 6 three-plane colourings of a crank rocker, 2 can be built, and of Peaucellier's 192 exactly 96 can.

    A plane is a colour

    Assigning links to parallel planes so that no two conflicting parts share one is a graph colouring, and the answer for a four-bar is three. Then the pins have to get through, and the problem stops being a colouring: of the six proper three-plane colourings, two can be built.

    part 4 · bodies
  8. The room crank rocker needs. The shaded region is every point any part of this machine occupies at some position of its drive, computed as an occupancy grid over 150 solved configurations and outlined by marching squares — so the outline drawn is the boundary of the set the area was counted from rather than a second object that agrees with it. Its area is 12.78 square units, filling 54% of the box that contains it. The dashed rectangle is the box the joints need, which is what every figure that treats links as lines could have told you; the material needs a box 20% larger in area.

    The room a machine sweeps

    Every point any part of a machine occupies at some position of its drive. It is a region rather than a curve, its area is an integral computed two ways, and the one shape in the field with a closed form is what the grid is calibrated against.

    part 5 · bodies
  9. The box the joints need, and the box the machine needs. Every machine in the catalogue, measured twice over the same drive: the extent of its joint positions, which is what a site drawing links as lines can report, and the extent of its material, which is what has to fit in something. The ratio runs from 1.13 to 1.36 — between thirteen and 36 per cent more area than the skeleton suggests, on machines whose links are a twentieth of their length wide. The last column is how much of the material's box the swept region actually fills, and it is where the packaging argument really is: a machine at 50% is a machine with a great deal of room inside its own envelope that nothing may be put in.

    The hole the machine needs

    A four-bar's joints fit in a box five units by four. Its material needs a box twenty per cent larger in area, and fills barely half of it. Both numbers are design quantities, and until a link had a width neither could be stated.

    part 5 · bodies
  10. Turning all the way round, against being made of something. Seven four-bars, classified by Grashof's inequality on their four lengths and then asked a question Grashof cannot answer: with a bearing pedestal at each ground pivot, how wide may the links be? The two instruments have nothing in common — one is an inequality on four numbers, the other counts sign changes of (B − A) × (G − A) over a sweep — and they agree about something Grashof was not for. Every four-bar that turns all the way round sweeps a link straight over a ground pivot, so its closest approach is exactly zero and no positive width is admissible; not one of the rockers does, and they take widths up to 0.20 of their shortest link.

    The crank that cannot turn all the way

    Grashof's inequality says which four-bars turn fully. Ask instead how wide their links may be with a bearing at each ground pivot, and the same inequality answers the opposite question: every four-bar that turns all the way round sweeps a link straight over one of its own pivots.

    part 6 · bodies
  11. Free space, in pieces. The driving angle round the circle, with the arcs at which the machine is both assembled and clear drawn heavy. One stud in the way takes a bite out of the turn and leaves 1 arc: the crank can still reach every remaining angle by going the other way. Two studs leave 2, covering 74% of the turn — and every configuration in both arcs is a perfectly good solution of the same constraint equations, on the same assembly branch, at the same mobility. Nothing a solver computes distinguishes an angle in one arc from an angle in the other; what separates them is that the machine cannot be driven from one to the other.

    Free space comes in pieces

    Every arc in these essays has ended at a configuration the mechanism cannot reach. Put two studs in a four-bar's way and its drive falls into two arcs whose ends are configurations it reaches perfectly well and cannot occupy — and no quantity the solver computes tells one arc from the other.

    part 6 · bodies
  12. How close a synthesis puts two pins. The distribution of the smallest pin-to-pin distance on any one link, over all 1,176 exact syntheses. Burmester's construction returns points, and points can be arbitrarily close together: the shortest here is 0.103, on a mechanism whose poses span more than two units. The shaded band is what a boss of radius 0.25 forbids — 148 of them, 12.6%. The link that is worst is most often the crank, which is not where a designer looks: the frame's two ground pivots are the pair everybody checks by eye.

    A pin is not a point

    A joint in the fields before this one is a name and two coordinates. A pin is a cylinder with material round it, a length through the stack of plates, and a head — and every one of those turns some construction that returns points into a construction that may return nothing buildable.

    part 7 · bodies
  13. A link, as a distance and as a body. The same two links every field before this one has drawn as lines, drawn as the material they are made of. A bar is a rectangle with its ends rounded off to the bosses that surround its pins, and it is convex; a bell crank is two arms meeting at a shared pin, and it is not — its inner corner turns the wrong way by 0.52, which is the single fact that puts it outside every separating-axis test here. The pins are marked because they are what has not changed: the constraint equations are the same, the solve is the same, and the positions are the same. What is new is everything between the pins.

    A body is all size

    Twenty-five of the fields before this one compute quantities that are mostly shapes, recoverable from an angle sensor and transferable between machines of any size. This one computes clearances, footprints and swept areas, and not one of them is a shape — which makes it the only field whose whole output needs a ruler.

    part 8 · bodies
  14. The gap is a straight line in the width. Three machines, four widths each, every width a fraction of that machine's own limit. Each set of points is collinear to the last bit of a double — the slopes wander by less than 10⁻¹³ across the range — because a feature-to-feature distance is linear in the corners of the two bodies and a bar's corners are linear in its width. The slope reads the contact: Chebyshev's linkage at -2.692, a crank rocker with a post at -1.350, a crank passing a stud at -1.350. A bar's boss grows 1.35 times as fast as its side, so −1.35 is a boss against something that is not growing and −2.70 is boss against boss. Extending each line to zero gives the widest link the machine will take, and the bisection that finds it the hard way agrees — 0.30113 against 0.30112, inside the bisection's own residual.

    The gap is a straight line in the metal

    Thickening every link by the same amount subtracts the same amount from every clearance, exactly, and moves the angle at which the worst one occurs by nothing at all. So a whole swept check can be done once on bars of any width and every other width read off by subtraction — until the closest pair changes hands, and never past zero.

    part 9 · bodies
  15. 36% shared, and they never touch. The region left arm visits over a whole drive, the region right arm visits, and — in the third colour — the part of the plane both of them visit. The shared area is 4.652 square units, 35.9 per cent of the smaller of the two regions. The two parts are drawn at the configuration where they come closest, and at that configuration the gap between them is 0.3799 — more than twice a link's width, and positive everywhere else on the drive. A test that asks whether the regions intersect has reported a collision between two parts that are never in the same place at the same time.

    The regions overlap and the parts never meet

    A swept region is a projection along time, and a projection cannot be undone. Chebyshev's two arms share thirty-six per cent of the ground the smaller of them covers and never come within twice a link's width of each other — a false alarm the region test cannot avoid, and one it cannot make at all against anything that stands still.

    part 9 · bodies
  16. One link, five bodies. Five bars with the same two pins, offset by -0.24, -0.12, 0, 0.12, 0.24 of the link's length, drawn to a common scale. Every one of them holds its two pins exactly the same distance apart, so every one of them is the same link: put any of them into a mechanism and the mechanism solves to the same joint positions at every configuration. Nothing in the kinematics of this collection — no loop equation, no velocity, no coupler curve, no mobility count — can tell them apart. What they do not have in common is which ground they occupy on the way from one pin to the other.

    A link may be bent

    A link is two pins at a fixed distance and the metal between them is a free choice. Bending it moves no joint of the mechanism by more than 10⁻¹³ and moves the clearance by a tenth of a link length — enough to build a machine that a straight bar refuses, and worth exactly nothing against a bearing pedestal the link sweeps over.

    part 9 · bodies
  17. The drive in 2 arcs. Chebyshev's two arms, whose regions over the whole drive overlap though the parts never come within 0.380 of each other. The outlines are each part's region over the whole drive. The filled ground is what a test that sweeps the drive in 2 equal arcs still flags: the points some arc's sweep of one part shares with the same arc's sweep of the other. With 2 arcs that is 0.859 square units. The parts are drawn at their closest approach.

    Four arcs and the false alarm is gone

    A swept region forgets when, so two parts whose regions overlap may never meet: Chebyshev's two arms share 4.66 square units of ground and never come within 0.38. Cut the drive into arcs, sweep each part over each arc, and compare regions only arc against the same arc. The test stays sound and costs the same sweep, and it forgets less with every cut. Four arcs clear Chebyshev's arms and eight clear Peaucellier's worst pair. A certificate that needs no grid asks for three times as many, and the same arms made wide enough to touch are refused at every number of arcs.

    part 10 · bodies
  18. Every shared cell asks for a cut between two visits; 2 cuts answer all of them. Chebyshev's two arms: each of the 5,510 cells of ground both parts occupy at some point of the drive, as the stretch of drive angle between one part's last visit to it and the other part's first — every such cell is visited by each part in one unbroken stretch, so each gives exactly one interval. Sorted by where they end, one in 21 drawn. A partition of the drive clears the pair exactly when every interval holds a cut. The vertical lines are the fewest cuts that do, placed greedily at the earliest unstabbed end: 1.072 and 1.428, making 3 arcs of 0.360, 0.356, 0.274 radians. The thicker intervals are the ones the rule cut at; they do not overlap, so no partition clears the pair with fewer cuts.

    Three arcs, cut in the right places

    Equal arcs clear Chebyshev's two arms at four and Peaucellier's long arm and rhombus bar at eight. Each cell of ground the two parts share is visited by each part in one unbroken stretch of the drive, so each asks for exactly one cut, somewhere between the two visits — and the fewest cuts that answer every cell are found greedily from one sweep and proved least by the intervals that forced them. Three arcs clear Chebyshev's arms and five clear Peaucellier's pair, and halving only the arcs that still flag finds four and five at three and four times the sweeping.

    part 11 · bodies
  19. 6 pairs, 16 cuts if each is cut alone, 4 for all of them together. Peaucellier's cell's pairs of parts that never touch but whose whole-drive regions share ground, each in its own lane: its intervals between visits to shared cells (thin, one in several drawn) and the fewest cuts that pair needs alone (solid ticks). long arm A / rhombus QB: 5 arcs; long arm A / rhombus PB: 3 arcs; long arm B / rhombus AQ: 5 arcs; long arm B / rhombus AP: 3 arcs; rhombus AQ / rhombus PB: 3 arcs; rhombus QB / rhombus AP: 3 arcs. Cut separately they need 16 cuts. The dashed lines are the fewest cuts for every pair at once, 4, at −1.132, −0.290, 0.658, 1.360 radians — as few as the worst pair alone needs — and the intervals they were placed at do not overlap, so no partition does with fewer.

    Four cuts serve the whole cell

    A region test between moving parts needs its drive cut where each pair's two visits to shared ground separate, and one pair of Peaucellier's arms needs four cuts. The cell has six such pairs, and cut one at a time they would need sixteen. Cut together they need four, the same four moments, because every pair crosses its shared ground on either side of them. Built from a whole-drive region test the cell would take six planes; the four cuts give back the true four, where equal arcs need eight to do it.

    part 12 · bodies
  20. 3 handovers, and 3 cuts where the shared ground stands empty. Four in turn: each tool's lift against the cam angle, from its own cam, with a thick bar under it while its front is inside the shared ground. The dashed lines are the fewest cuts that clear every pair of tools at once, 3, at 114.5°, 204.5°, 294.5°, and the shaded bands are the windows each may move in, 65° to 115°, 155° to 205°, 245° to 295°: the times when no tool is inside the shared ground. The pairs need 1 cut at worst and 6 between them; the station changes hands 3 times after the turn's seam.

    A cam machine is cut at its handovers

    In one symmetric linkage, every pair of parts crosses its shared ground on either side of the same few moments, so the whole machine needs no more cuts of its drive than its worst pair. Four tools driven by four cams into one work station do not share their moments. Each pair needs one cut, the pairs need six between them, and the machine needs three: one for each time the station changes hands. Each cut can move only within the time the station stands empty, which is the cam diagram's own timing margin, and when a schedule is squeezed until that margin closes, the tools collide.

    part 13 · bodies
  21. Two stations, each shared by its own tools, and an arm that visits both. A cell with two work stations 3 apart. Station A is fed from the left and pressed from above; station B is pressed from above and emptied to the right; an arm pivoted below and between them picks a part up at A, swings it to B and parks midway. Every tool is 0.3 wide and reaches 0.15 past its station's centre, so the shaded squares are the only ground any two tools share, each square shared by its own station's tools and the arm. At 230° station A holds nothing and station B nothing.

    Two stations share a cut when they hand over together

    A cam machine's region test needs one cut of its drive for every time its shared ground changes hands. A cell with two work stations has two pieces of shared ground, each with its own handovers, and a single cut can serve both if it falls when both stations stand empty. So the cell needs somewhere between its busier station's count and the two stations' sum, and its cams decide where. Out of step, a press-and-transfer cell needs four cuts; with the same tools and strokes put in step, two. The transfer arm that carries the part between stations makes one shared cut for free, because while it swings it is at neither.

    part 14 · bodies

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