Machine Kinematics
solved, then drawn
Concept

Path tracking — where it appears

Following a solution as a parameter changes, carrying the previous answer forward as the next guess. It keeps a sweep on one assembly branch, and it is what stops a figure jumping between two configurations a built mechanism could not pass between.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Every path, in the plane of one unknown. The 16 tracked paths of the 3-RPR platform, projected onto the complex plane of x. Each curve starts at a solution of the start system and ends at a solution of the target or leaves the frame on its way to infinity. This run is γ random, start constants complex, and it found 6 solutions.

Following a root from a problem already solved

Homotopy continuation solves a system nobody can solve by deforming one that anybody can, and following every root as it moves. The whole method rests on the deformation being generic, and the folklore says that is what the γ-trick is for. Running all four combinations says the folklore names one of two places the randomness can live, and either will do.

algebra · Algebra What became of Bézout's paths. four-bar coupler pin: 2 of 4 paths arrived at a solution and 2 went to infinity; 3-RPR platform: 6 of 16 paths arrived at a solution and 10 went to infinity; Gough, generic: 80 of 1458 paths arrived at a solution and 1378 went to infinity. The surplus is not merely wasted — it is cheap: a path on its way to infinity is abandoned in a handful of steps, while every path that arrives is tracked in full.

The paths that leave

Bézout's number over-counts, and the over-count is enormous — 1,458 tracked paths for 80 solutions. The obvious response is to find a method that tracks only the paths that arrive. That method exists, it was built, and it is four times slower, because the surplus paths are not merely surplus. They are cheap.

algebra · Algebra

Named alongside it

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

BezoutHomotopy continuationPolynomial systemComplex solutionComputational costGamma trickGough platformMonodromyNewton–RaphsonPredictor correctorSolutions at infinity

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