Machine Kinematics
solved, then drawn
Concept

Nelder mead — where it appears

A direct-search optimiser that walks a simplex downhill without derivatives. It is what this site uses where an objective is cheap and its gradient is awkward, and it finds a local optimum with no claim about any other.

Named by 2 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it is the exact constructions the field spent four rungs on, and an atlas of coupler curves — which is why a method superseded by computers is still the thing that feeds them.

synthesis · Synthesis

Named alongside it

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

Approximate synthesisObjective functionAtlasBranch defectCoupler curveFunction generationInitial guessKinematic synthesisLeast-squaresMinimaxPrecision-pointStructural error

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