Machine Kinematics
solved, then drawn
Concept

Direct kinematics — where it appears

Finding where a mechanism's output goes given its actuated inputs. For a serial arm it is a product of transforms and is easy; for a parallel platform it is a system with many roots and is the hard direction.

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

The twelve that were at infinity, coming back. Every solution of the platform's direct kinematics, plotted by how far from the origin it sits, as the six anchors are jittered. At no jitter the site's own platform has 56 solutions and the largest is at 25.8. At a jitter of 0.2 there are 80, and the extra ones arrive from far out — they were never missing, they were at infinity.

Twenty-eight, not forty

The general six-legged platform has forty poses for a given set of leg lengths, and this site has quoted that number beside a picture of a platform that has twenty-eight. Its anchors are arranged symmetrically, which makes it a special architecture, and the missing twelve poses are not missing. They are at infinity, and perturbing the anchors brings them back.

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

Named alongside it

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

Gough platformAssembly branchBezoutComplex solutionHomotopy continuationLower boundMonodromyNewton–RaphsonPolynomial systemQuaternionReal solutionSolutions at infinity

All concepts