Machine Kinematics
solved, then drawn
Concept

Real solution — where it appears

A root of a mechanism's closure equations with real coordinates, which is an assembly that can be built. It is a subset of the complex solutions and a much less stable one: the real count changes as link lengths vary while the complex count does not.

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

The count that moves, and the one that does not. 676 sets of leg lengths for one 3-RPR platform, the third leg held at 2.2. Every one of them has exactly 6 complex solutions. The number that is real runs 0, 2 — 141 cells at 0, 535 cells at 2 — and that number is what a machine shop would call the assembly modes.

The count that does not move

A mechanism does not have a number of assembly modes. Its family has a complex solution count that never changes, and each member has a real count that does — 676 sets of leg lengths for one platform, all with six complex solutions, and nought, two or four of them real. The number a machine shop cares about is the one that is not a property of the machine.

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 Every boundary touches the circle on the rod. The coupler's plane in the rod's units, crank pin at (0, 0) and rocker pin at (1, 0). Shaded: the coupler points whose three departing double points are all real at a ratio of rocker to crank of 1, the region where R = λ⁴A + λ²B + C is negative; 8.1% of a window from −1.4 to 2.4 along the rod and ±1.5 across it. Dotted: the circles one rod from each pin, the boundary at the family's two ends. Dashed: the circle with the rod as its diameter. Every boundary in the family touches that circle and never crosses it; at this ratio it touches at (0.50, ±0.50), the points where −B/2A = λ². Inside it all three are real at every ratio.

Three discs decide the real double points

A four-bar growing into the elliptic trammel loses three double points, and how many of them are real depends on the coupler point and on the ratio λ of rocker to crank. The boundary of the region where all three are real is one polynomial, R = λ⁴A + λ²B + C, quadratic in λ² and short when written in the squared distances from the two pins. Its sign is the count at every one of 23,100 cells tried. As a quadratic in λ² its discriminant is the circle with the rod as diameter times a figure of eight, so that circle is touched by every boundary and crossed by none, and a coupler point's whole history along the family is settled by three discs.

algebra · Algebra A growing four-bar walks away from its own double points. A four-bar with a rod of 3, its crank and rocker both of length t, pivoted so that both pivot circles pass through the slides' crossing, with the coupler point at (0.3, 0.2) in the rod's units. Drawn in units of t: the two pivot circles (dotted), their second meeting (ringed), the real coupler curve the machine draws (solid), and its real double points, filled where two real configurations meet and hollow where none reaches. At size t = 7 the curve's real double points are isolated (2.02), isolated (3.32), crossing (0.93); the number in brackets is how far the line of rod orientations through each passes from the unit circle's centre, a crossing below one and isolated above. As t grows the curve shrinks onto the crossing and the second meeting and the double points stay where they are.

A growing four-bar reaches none of its departing double points

A four-bar growing into the elliptic trammel loses three double points, and wherever they are real the question is whether the machine passes through them. In the limit it passes through none: a configuration that reaches a point a crank's length away needs its rod along an isotropic direction, which no real rod has. How fast each point leaves the machine's reach is a ratio of four distances, √(1 + λ²)·|Z|·|Z − Zₛ| / (2b·|Z + 1|·|Z + iλ|), and it vanishes only at the pivot circles' second meeting. Every crossing is lost at a finite size: through a cusp, when the law predicts it, or by meeting another crossing and leaving as a complex pair, when it does not.

algebra · Algebra

Named alongside it

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

Assembly branchDouble pointElliptic trammelAlgebraic curveBezoutCircular pointsComplex solutionConfiguration spaceCoupler curveCuspDirect kinematicsDirect singularity

All concepts