Machine Kinematics
solved, then drawn
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Synthesis — the series

23 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Three prescribed positions of a rigid body. The whole of the design problem, before any mechanism exists. A body has to occupy these three positions — each one a place and an angle, three numbers — and what carries it between them is not yet decided. A forward analysis starts from link lengths and finds the motion. This starts from the motion, and the lengths are what has to be found. The marked points are the poles: any planar displacement is a rotation about one point, so each pair of poses has one, and the arcs show the turn each represents through the body's own origin. A pole is a property of the displacement and not a mechanism — nothing has been chosen yet. 1 of the 3 poles lies outside this frame and is not drawn; near-parallel displacements push their pole a long way off.

    The problem the other way round

    Every essay before this one starts from link lengths and finds the motion. That is the reader's problem, because lengths are what a drawing shows. It is not the designer's problem, which is the reverse — and the reverse is hard enough that for a century the practical method was to look the answer up in a book.

    part 1 · synthesis
  2. One moving pin, and the pivot it turns about. Choose any point of the moving body — this one at (-0.55, 0.5) in the body's own frame. In the three prescribed poses it lands in three places, and three points that are not in a line lie on exactly one circle. That circle's centre is where the fixed pivot has to be and its radius is how long the link has to be: here 1.0860, and all three images sit at that distance to within 10⁻¹². There is no iteration and no tolerance in the construction, because three points determine a circle exactly. The freedom is entirely in which point of the body to pick.

    Three positions, and a circumcentre

    The whole of three-position synthesis is one observation: a moving point occupies three places, three points that are not in a line lie on exactly one circle, and that circle's centre is where the fixed pivot has to be. No iteration, no tolerance, and every point of the coupler is a candidate.

    part 2 · synthesis
  3. Burmester's curves, contoured rather than drawn. With three poses, every point of the moving body works: three images, one circumcircle. With four, a point's four images are concyclic only if it lies on a particular cubic — the circle-point curve — and the fixed pivots those points want lie on a second cubic, the centre-point curve. Both are drawn here as contours of a measured quantity: at each point of a 150×150 grid, how far the fourth image misses the circle through the other three, contoured at zero. Points refined onto the contour are concyclic to 3.8e-15; points 0.47 away from it miss by at least 2.0e-1. The two curves are keyed in the legend and the four prescribed poses are drawn faintly for scale. Three poses leave a designer the whole plane; four leave a curve. The four prescribed poses are outlined faintly for scale, and both curves are clipped to the frame: the centre-point curve is a cubic with unbounded branches that reach 360 units on a mechanism three units across, and the part worth looking at is the part near the machine.

    What the fourth position costs

    With three prescribed positions every point of the coupler will do, and a designer is spoilt for choice. Add a fourth and the whole plane collapses to a curve — only points on a particular cubic have four images that lie on a circle, and the cubic is Burmester's.

    part 3 · synthesis
  4. 3 four-bars, one coupler curve. Three different four-bars, with different ground pivots, different link lengths and different proportions — 0.692 and 0.799 times the size of the first. Every one of them draws this same curve. Roberts's theorem says there are always exactly three, and the construction is one complex multiplication: write the coupler point as λ = (P − A)/(B − A), put the third fixed pivot at O₂ + λ(O₄ − O₂), and the other two linkages fall out with their bars' roles permuted — what is a coupler in one is a crank in another. The curves here were traced separately, each from its own solver runs, and agree to 1.3e-5 against a sampling resolution of 1.4e-5 — which is to say, as closely as the comparison can tell.

    Three linkages, one curve

    Every coupler curve is drawn by three different four-bars, not one. The other two can be constructed from the first with a single complex multiplication, they have different proportions and different ground pivots, and the roles of their bars are permuted — what is a coupler in one is a crank in another.

    part 3 · synthesis
  5. The body points whose three images are in a line. The three prescribed poses, faint, and a curve through them. At every point of a 110×110 grid over the moving body, the point is placed in all three poses and the signed height of its image triangle is measured; the curve is where that length is zero. Those are the points whose three images lie on a line, so the dyad they want is a slide rather than a crank. The curve is a circle. Refined onto the contour, its points lie on the circle through the three image poles to 2e-15, where the pole triangle's own circle misses them by up to 1.98. Its radius is 8.767, so inside a window three units across it reads as a gentle arc; 2 of the three image poles are in the frame and the third, P₁₃, is 6.8 units away.

    Where a pin becomes a slide

    Three-position synthesis gives every point of the moving body a fixed pivot, except the points whose three images fall in a line. Those want a slide, and they are not scattered: they lie on one circle, the circle through the three image poles, which a single line of algebra predicts and a contour of a measured length draws to 10⁻¹⁵.

    part 3 · synthesis
  6. Five positions, and what is left of the curve. Five prescribed poses of a moving body. With four of them, every point of the pale curve is a usable fixed pivot — a one-parameter family. The fifth pose is one more equation, and it leaves 4 points. Bézout's number for the system is 16; 4 paths arrive; 4 of those are real. Every pair of the 4 is a four-bar, so there are 6 candidate linkages and 2 of them reach all five poses in one piece and in order. 1 of the 4 pivots is too far away to draw in frame and is marked at the edge with its true distance — which is why some of the linkages have a bar twenty times the size of the body.

    Five positions, and what is left

    Three prescribed poses leave a whole plane of choices. Four leave a curve. Five leave four points, and finding them is the first thing in the synthesis essays that a compass cannot do — it needs two cubics intersected, which is algebra rather than construction. Four points give six four-bars, and two of them can be built.

    part 4 · synthesis
  7. A four-bar that computes log₁₀. The output rocker's angle, mapped back into y, against the input crank's angle mapped back into x. The pale curve is log₁₀ x and the solid one is what the linkage does. They agree exactly at the 3 precision points and nowhere else; the largest disagreement over the range is 0.2232° of rocker, which is 7.47e-4 in y. The chebyshev spacing put the precision points at 1.0670, 1.5000, 1.9330.

    Three problems called synthesis

    Prescribing a path, prescribing a whole pose, and prescribing a relation between two angles are three different problems with three different counts, and the word synthesis covers all of them. The third has a property the others do not — eliminate the coupler angle and the design equation becomes linear, so a four-bar that computes a logarithm falls out of a 3 × 3 solve with no iteration at all.

    part 4 · synthesis
  8. Four poses: four slider circles through one body point, and its four images on one line. Left, the moving body's own frame: for each of the four ways of choosing three of the four prescribed poses, the circle of body points whose three images are in a line. All four pass through one point, (1.006, 0.560), the slider point. The first two also share the body point that sits on the pole P₁₂, where two images coincide; the other two circles do not pass through it. Right, the fixed frame: the body in its four poses, and the slider point's four images, which lie on one line to 3e-16 of its length. A block on the body at that point, running in a straight guide along that line, carries the body through all four poses. There is no other.

    The slide a fourth pose leaves

    Three prescribed poses leave a whole circle of body points that want a slide. A fourth leaves exactly one, found where two of those circles meet a second time. It had been predicted as one or three, because a real cubic meets the line at infinity in one real point or three. But both of Burmester's cubics pass through the two circular points, so only one real point is left. Four hundred random four-pose problems each have one slide and one slot pin, both found with compasses.

    part 4 · synthesis
  9. The error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; uniform: worst 0.3415°, RMS 0.2296°. The error is zero at each precision point by construction and nowhere else. chebyshev has the smallest maximum here, and which curve is best depends entirely on which measure is asked for.

    Where the precision points go

    A linkage that matches a function at three points is wrong between them, and where the three points are put decides how wrong. Chebyshev spacing cuts the worst error by a third against even spacing, for free — and the reason has nothing to do with mechanisms. It is a fact about a polynomial the linkage has never heard of.

    part 5 · synthesis
  10. The error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; least-squares: worst 0.3900°, RMS 0.1203°; minimax: worst 0.2105°, RMS 0.1463°. The error is zero at each precision point by construction and nowhere else. minimax has the smallest maximum here, and which curve is best depends entirely on which measure is asked for.

    The linkage that is only nearly right

    Stop demanding that a linkage pass exactly through three points, and ask instead that it be close everywhere. Three linkages result, none of them passing exactly through anything, and each is the best by a different measure — the least-squares fit beats the interpolant on average error and loses to it on the worst case. An optimiser gives exactly what it was asked for and nothing else.

    part 5 · synthesis
  11. The error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; least-squares: worst 0.3900°, RMS 0.1203°. The error is zero at each precision point by construction and nowhere else. chebyshev has the smallest maximum here, and which curve is best depends entirely on which measure is asked for.

    Where an optimiser starts

    An approximate synthesis is a local search on an objective that is non-convex, disconnected and not everywhere defined, so the answer depends on where the search began. Nothing in the optimisation supplies that. What supplies it is the exact constructions of four earlier essays, and an atlas of coupler curves — which is why a method superseded by computers is still the thing that feeds them.

    part 5 · synthesis
  12. Five poses, five four-pose slides, and the compromise between them. Left, the moving body's frame with the slider point of each four-pose subproblem — the point whose four images are in a line when one pose is left out — and the compromise point whose five images all miss one guide by the same amount. Right, the fixed frame: the body in its five poses and the images of the chosen point, with its guide. Leaving out pose 1 to 5 misses that pose by 0.5304, 0.0836, 0.0383, 0.0422, 0.1288 units; the compromise misses every pose by 0.0140, on alternate sides. The body is about two units long.

    The slide five poses almost have

    Five prescribed poses leave no slide: a body point would need five images in a line, three conditions on two coordinates. But each of the five four-pose subproblems has exactly one slider point, and each misses the pose it left out by a distance a pair of compasses can find. Those five misses price the best compromise without solving for it. The slide that misses every pose by the same amount misses by one over the sum of their reciprocals — the misses add like resistances in parallel — and giving up the best single pose instead is always between two and four times worse.

    part 5 · synthesis
  13. Watt's chain. Six links and 7 pin joints, so Grübler counts 3(6 − 1) − 2(7) = 1 and the rank of the Jacobian measures 1. Two of the six links must carry three joints; they are shaded. Here they share a joint, which makes this a Watt chain. That adjacency is the entire classification, and it is read off the graph rather than off the picture.

    Six bars, and what the extra dyad buys

    Every linkage so far has had four bars, because four is the smallest closed chain that moves. The next one up is six, not five, and six is where the subject stops being one family and becomes a taxonomy — two chains, distinguished entirely by whether the two links carrying three joints happen to touch.

    part 6 · linkages
  14. A dwell is a measurement, not a stop. The Stephenson six-bar's output against a full turn of the crank, with the four-bar it is built on for comparison. Inside a band of ±1° the six-bar's output holds still for 145.8° of crank and the four-bar's for 41.9°. The dwell comes from a stretch of the coupler curve that fits a circle of radius 3.006 to within 2.96e-3, and the arm from the coupler point is that radius. Nothing here stops; it moves less than the band.

    A dwell made from a curve

    Parts of a coupler curve are very nearly circular arcs. Put a link of the arc's own radius on the coupler point and its far end stands almost still while the point runs along it, so the output dwells — 146° of crank inside a one-degree band, against 42° for the four-bar it is built on. A dwell linkage does not stop. It moves less than the tolerance, and how much less is a number.

    part 6 · linkages
  15. The compromise slide needs its block on alternate walls; a working load chooses otherwise. The five prescribed poses of a body about two units long, driven by the one crank of Burmester's four that turns through them in order — centred at (1.531, -0.730), radius 1.492, turning 149° — and guided by the compromise slide, whose block has play 0.0140 either side of the guide's centre line; the two walls are drawn ten times further apart than they are. At pose 3 the compromise needs its block on the + wall (hollow ring). A working load along the guide, opposing the block's motion, presses it onto the − wall (filled). The walls needed at the five poses are + − + − +; the walls that load presses are + − − − −, and the body's own weight presses + − − + +.

    The load picks the wall

    The best slide for five poses misses each by 0.0140, on alternate sides, so a block with that much play either side reaches all five — if it bears on the right wall at each. Nothing in a slot says which wall. The load does: the block presses whichever wall the load's moment about the crank pin sends it to, and changes walls only where that moment changes sign. Four changes are needed. The body's own weight gives two, in the wrong places, and play then makes the slide worse; a working load gives one, and needs nineteen times the play.

    part 6 · synthesis
  16. How many points a linkage will take. function generation: 5 free numbers, 1 net constraint per point, 5 precision points; motion generation: 4 free numbers, 1 net constraint per point, 5 precision points; path generation: 9 free numbers, 1 net constraint per point, 9 precision points. Every count in the literature that differs from these differs about what is being counted as free, not about the geometry.

    How many points may be prescribed

    Five poses, five angle pairs, nine points — three numbers that get quoted as properties of a four-bar and are properties of what somebody decided to count as free. Derive them instead, and the fifth precision point turns a linear solve into a system with 128 paths, twelve finite solutions, four real ones, and exactly one linkage anybody could build.

    part 7 · synthesis
  17. A spring hung below the body puts the block on alternate walls. The five prescribed poses, the crank that drives them and the compromise slide, with one spring of no free length from a fixed point on the frame at (3.41, 1.16) to a point (-0.60, -1.20) of the body's frame, 0.92 below its lower edge on an arm, stiffness 2.51 per unit length against a unit working load along the guide. The spring's pull is at most 11.7 times the load. With it the block bears on the walls + − + − + at the five poses, exactly the compromise's, and crosses its slot 4 times on the way. At pose 2 the spring pulls with 10.1 and the block needs, and gets, the − wall.

    A spring balanced against the load

    The best slide for five poses reaches all five with play in its slot only if its block bears on alternate walls, and neither the body's weight nor a working load puts it there. A spring can, and whether one does is a linear program. No point of the body can carry it: for every such point there are weights on the poses under which any spring's pull cancels. A point on an arm below the body can, and a spring from it pulling at most 11.7 times the load gives + − + − + exactly. The spring alone presses one wall throughout; it works by balancing the load, and only for loads within nine per cent of the one it was chosen for.

    part 7 · synthesis
  18. A catalogue is a search space, and a requirement is a filter on it. What a census is for. Four requirements applied in turn to the 230 ten-link chains, each of them a statement about the graph alone: a link carrying four pins, a link none of whose neighbours is binary, and a way of driving it that comes apart into dyads. 26 chains survive all of them. None of this is dimensional synthesis and none of it can be — no requirement here mentions a length, an angle or a position, and every one of them can be checked before a single dimension is chosen. That is the argument for having the census at all: the design problem is a search over shapes within a topology, and knowing which topologies there are turns an open question into 26 closed ones.

    Choosing the chain before the lengths

    Every synthesis method in these essays starts by assuming a topology, and the assumption is usually a habit. What the graph fixes before any dimension is chosen is the number of free parameters — two per pin less four — and therefore how many positions can be prescribed at all.

    part 8 · synthesis
  19. One body point, one stiffness, and an attachment point for each load. The five prescribed poses with the lightest spring, from a point within a unit of the body's outline, that puts the rocking block on + − + − + against the body's weight: from the body point (-0.60, -1.20), on an arm 0.92 below the body, to an attachment point at (2.93, 1.97), stiffness 2.513, pulling at most 8.97. Designed against no load, the weight, the working load or both, the program picks the same body point and the same stiffness every time and moves only the attachment point: no load (3.04, 2.10), the body's weight (2.93, 1.97), the working load (3.41, 1.16), weight and working load (3.29, 1.03) (all four marked).

    The arm belongs to the poses

    A spring from the frame can put a rocking slide's block on the alternate walls its five poses need, but only from points on an arm below the body, never from the body itself. That looked like a fact about the working load the spring was balancing, and the body's own weight, which never changes, might have moved the reachable points onto the body. It moves them nowhere. Under no load, the weight, the working load and both, the program finds the same 155 points of the body's frame, the same lightest body point and the same stiffness, and only the spring's attachment point moves. The arm belongs to the five poses. And the earlier spring's band of nine per cent on the working load was its design margin: the band grows with the spring's pull, about one unit of load per hundred of pull.

    part 8 · synthesis
  20. What each curve costs, in cosines. Every polynomial in the two arm angles is a constant plus a sum of terms A cos(mα + nβ + φ) with whole-number m and n, and the number of those terms is what a machine has to build. The count is not the degree and not the monomial count: a circle costs one term, a general line two, and a lemniscate — degree four — costs five, fewer than the cubic above it, because its symmetry cancels frequency pairs the cubic keeps. Each row's expansion was checked against a direct evaluation of its own polynomial at random angles, worst disagreement 1.8e-14.

    Prescribing a curve rather than points

    A four-bar can be made to pass through nine prescribed points and no more; past nine the problem is over-determined and the answer is an optimiser's. Prescribe the whole curve as an equation instead and there is no counting to do — but the mechanism that comes back has four hundred bars where the four-bar had four.

    part 9 · synthesis
  21. What a pin's size costs a synthesis. Every one of the 1,176 exactly correct three-position syntheses the synthesis field's survey produces, asked a question its three verdicts cannot ask: are any two of its pins closer together than the material round them? The poses span about two units, so the horizontal axis is a pin as a fraction of the mechanism. At a boss radius of 0.05 nothing is lost; at 0.4 — a pin nearly a fifth of the pose span — 27% of the exact solutions cannot be built, and 15% of the ones that had already passed the branch, circuit and order tests go with them. The defect is real, it is rarer than the kinematic ones, and it is not correlated with them.

    A defect that is not kinematic

    The synthesis field's survey ends in three verdicts and all three are about which solutions a sweep visits. Here is a fourth, found by asking how much room two pins need — and it is the first defect here that a simulation cannot find, because in the equations a pin is a point.

    part 10 · synthesis
  22. Three prescribed positions of a rigid body. The whole of the design problem, before any mechanism exists. A body has to occupy these three positions — each one a place and an angle, three numbers — and what carries it between them is not yet decided. A forward analysis starts from link lengths and finds the motion. This starts from the motion, and the lengths are what has to be found. The marked points are the poles: any planar displacement is a rotation about one point, so each pair of poses has one, and the arcs show the turn each represents through the body's own origin. A pole is a property of the displacement and not a mechanism — nothing has been chosen yet. 1 of the 3 poles lies outside this frame and is not drawn; near-parallel displacements push their pole a long way off.

    What a synthesis assumes it knows

    Every construction in this field is handed a demand in absolute coordinates — three positions of a coupler plane, at stated places — and returns a linkage in the same coordinates. Scale the demand and the answer scales, which means the construction's whole content is about shape and its answer carries a size it was given.

    part 11 · synthesis
  23. A spring that pulls the same at every pose holds the band for less. The five prescribed poses of the rocking slide's body, with the arm point the earlier essays found lightest, (-0.60, -1.20) in the body's frame, and the spring that keeps the block on + − + − + for every working load from nought to the design load. The shaded sliver is every attachment point from which any spring that pulls can choose those walls at all. A constant pull of 44.6 from (3.10, 2.05), no stiffness, pulls at most 44.6 times the unit load.

    A constant pull holds the band for less

    A spring with free length was the obvious way to make the rocking slide's spring cheaper, and it does nothing: offered one, the lightest design takes none, at every arm point tried. The spring's law never mattered to where it may hang, because the walls depend only on which side of each crank pin the spring's line passes, and that cuts the attachment points to a polygon whatever the spring is. What the law decides is the price. Three poses bind at the lightest attachment point, and a spring whose tension follows its length must pay for the longest of them. A pull that is the same at every pose pays for none of them, and holds the band from nought to the design load for 44.6 units against 52.0. Designing for the band directly halves the hundred units the earlier essay read off by scaling.

    part 12 · synthesis

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