Field

The problem backwards

Given the lengths, find the motion is the reader's problem. Given the motion, find the lengths is the designer's, and it is why coupler-curve atlases were printed and sold. The constructions are exact; whether what they produce can be built is a separate question.

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

Why kinematic synthesis is the inverse of everything else on this site, what a coupler-curve atlas was for, and the three classes of synthesis problem a designer actually faces.

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

Burmester's three-position construction for rigid-body guidance, the two-parameter family of solutions it produces, and each one verified by driving the forward solver to every prescribed pose.

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

Four-position synthesis, the circle-point and centre-point curves drawn as contours of a measured circle-fit error, and why five positions leave finitely many solutions and six generally leave none.

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

The Roberts–Chebyshev cognate theorem, the construction that produces the other two linkages, and all three curves traced independently and compared against the resolution of the comparison.

Five positions, and what is left

Five-position Burmester synthesis, where the construction stops being a construction, and what the six candidate linkages turn out to be worth.

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

Function, path and motion generation as three distinct problems, and Freudenstein's equation — the one place in this subject where the design problem is linear.

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

Chebyshev spacing against even spacing for a four-bar function generator, why the spacing is a statement about interpolation rather than about linkages, and how to tell a good choice from a lucky one.

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

Approximate synthesis by optimisation, and a measurement showing that each objective wins only on the measure it was given.

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

Why approximate synthesis needs an initial guess it cannot generate, and what the classical constructions and coupler-curve atlases are actually for now.

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

Precision-point counts derived from the free parameters rather than quoted, and the five-point function generator computed by tracking every path.

Choosing the chain before the lengths

Type synthesis as the stage before dimensional synthesis: how many dimensional parameters a chain has, why both six-bars have ten, and what the graph decides about a synthesis problem.

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

Where synthesis stops and compilation starts: what changes when a specification is a polynomial rather than a finite list of positions, and what each approach can and cannot promise.

What a synthesis assumes it knows

What a scaling does to a synthesis, why the constructions are similarities, and which parts of a design specification are actually deciding anything.

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

A defect that is not kinematic

The size defect: how many exactly correct three-position syntheses put two pins closer together than the material round them, and how the count grows with the pin.

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

The slider points of three-position synthesis found as the zero set of a measured collinearity, shown to be the circle through the image poles, its fixed-plane twin for a pin in a slot, the one-over-the-miss law for cranks near it, a slider-crank synthesised and solved, and the limit in which the circle becomes the inflection circle.

The slide a fourth pose leaves

Four-position synthesis of a slide: the slider point as the second intersection of two three-pose slider circles through a common pole point, its four images collinear to 10⁻¹⁵, the dual slot pin in the fixed frame, both Burmester cubics shown circular by dividing their cubic parts by x² + y², the centre-point cubic's single real direction at infinity square to the slide, and a census of 400 random problems.

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-position synthesis of a slide: the five four-pose slider points and the pose each gives up; the one-pose misses, their reciprocals and the weights of the problem's single obstruction; each slide's five residuals, with the compromise's five equal and alternating; 560 five-pose problems, near a slide and random, against the parallel law; the gain of the compromise over giving up a pose, bounded between two and four; and five poses of one coupler brought together, where the miss falls as the fourth power of the spacing with weights 1 : 4 : 6 : 4 : 1.

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 compromise slide for five prescribed poses, driven by a Burmester crank with play in its slot: the four cranks and which one drives the poses in order; the force across the slot along the travel under the body's weight and under a working load; the obstruction's weights and the walls each load gets wrong; the play each of fifteen patterns of walls needs, by F/Σλs and solved; and the best worst miss against play for three patterns.

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

One spring of no free length from the frame to a body point, chosen so that the rocking slide's block bears on + − + − + at five prescribed poses: the five poses with the spring on an arm below the body; where in the body's frame a spring's point can be, and how hard it must pull; the moment about the crank pin along the travel from the load, the spring and both; the least pull against the arm's reach; the band of loads over which the walls stay right; and the pairs of springs on the body that work.

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

One spring of no free length chosen to put the rocking slide's block on + − + − + at five poses, designed against no load, the body's weight, the working load and both: the lightest spring for each, one body point and four attachment points; the body's frame under each load, with the points a spring can hang from; the band of working loads a spring keeps against its pull; the certificate that shuts the body out under the weight; and a ledger of the four designs.

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

The rocking slide's spring with a constant term along its line added to its stiffness: the five poses with the lightest arm point, its attachment polygon and the two designs for the band from nought to the design load; the five half-planes, one a pose, that cut out the polygon; the price of a band against its width for the scaled no-load design, the direct design and a constant pull; the tension each pose needs against the tension each spring gives; the saving across the wedge of arm points; and a ledger of the six designs at the lightest point.

All essays