Machine Kinematics
solved, then drawn
Concept

Approximate synthesis — where it appears

Finding link lengths that come as close as possible to a wanted motion, when no exact solution exists or none is usable. The measure of closeness is a choice, and a least-squares fit and a minimax fit give different linkages from the same specification.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

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.

synthesis · Synthesis 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supplies it is the exact constructions of four earlier essays, and an atlas of coupler curves — which is why a method superseded by computers is still the thing that feeds them.

synthesis · Synthesis 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.

synthesis · Synthesis 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.

synthesis · Synthesis 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.

synthesis · Synthesis 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.

synthesis · Synthesis 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.

synthesis · Synthesis

Named alongside it

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

MinimaxPrecision positionClearancePrismaticSynthesisObjective functionKinematic synthesisLeast-squaresNelder meadSlider-crankAtlasBranch defect

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