Machine Kinematics
solved, then drawn
Concept

Monodromy — where it appears

Moving a polynomial system's coefficients around a loop and seeing which solutions have swapped places when it returns. It finds solutions a single homotopy missed, and it is the site's route to a count of assemblies that is a claim rather than a tally of what a search happened to reach.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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 A lower bound that happened to be tight. The site's own Gough platform at its home pose. The search starts Newton from a spread of guesses and reports what it lands on: 16 from 400, 16 from 1200, 16 from 4000. Tracking every one of the 1458 Bézout paths says there are 28 poses in the complex numbers and 16 of them are real. The search was right. Nothing available to the search could have said so.

The search that was right

This site has reported sixteen assemblies for its Gough platform and labelled the number a lower bound found by search, everywhere it appears. Tracking every path says there are twenty-eight poses and sixteen of them are real. The lower bound was tight. Nothing available to the search could have said so, and a second method that was supposed to settle it turns out to have the same defect one level up.

algebra · Algebra A line across the four-bar's coupler curve. The curve traced by a point on the four-bar's coupler, over every real configuration, with the machine drawn faintly at one of them. A real line crosses it at 4 marked points. Written as polynomials, the machine and the line have 8 paths to track; 6 arrive, 2 leave for infinity, and the arrivals draw 6 distinct points — 4 real and 2 complex. A random complex line gives 6 as well, which is the curve's degree.

A degree counted on a line

A curve of degree six meets a general line in six points, and that sentence is a way to measure the degree with no equation in it. Written as polynomials, a four-bar and a random complex line have eight paths to track; six arrive on every line tried, the other two run off towards the circular points, and a sum of the six stays straight to fifteen figures only when none is missing.

algebra · Algebra Every body of the Stephenson six-bar, sliced by one line. One random complex line, and a point on each body of the Stephenson six-bar required to lie on it. Every system has Bézout number 32. crank: 8 configurations drawing 2 distinct points, each 4 times, so degree 2; ternary coupler: 12 configurations drawing 6 distinct points, each 2 times, so degree 6; rocker: 8 configurations drawing 2 distinct points, each 4 times, so degree 2; arm: 18 configurations drawing 18 distinct points, each 1 time, so degree 18; output: 12 configurations drawing 2 distinct points, each 6 times, so degree 2. The configurations are what the tracker counts; the points are what the curve has.

The curve nobody eliminates

A point on the arm of the site's dwell six-bar draws a curve whose equation nobody writes down and no fit can find: at degree eighteen a fit needs a hundred and ninety coefficients, and its singular values have no gap to decide by. Sliced by a random line, the same machine has thirty-two paths to track and eighteen arrive, on every line tried, all eighteen distinct and all on one curve.

algebra · Algebra

Named alongside it

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

Homotopy continuationAlgebraic curveDegreeGough platformImplicit equationTrace testWitness setAssembly branchBezoutComputational costCoupler curveDirect kinematics

All concepts