Machine Kinematics
solved, then drawn
Concept

Trigonometric polynomial — where it appears

A finite sum of cosines of whole-number combinations of two angles, plus a constant. Any polynomial in the coordinates of a two-link arm's tip is one, exactly, which is the identity a linkage compiled from a curve is built on.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Two routes to the same number, along one sweep. The polynomial x^4 + 2x^2y^2 + y^4 − 1.2x^2 + 1.2y^2 evaluated at the tip of the arm as the second angle goes right round, by two routes that share nothing: once by putting the tip's coordinates into the polynomial, and once by adding up 5 cosines of whole-number combinations of the two angles. The two curves are drawn on top of each other and the worst gap between them is 8.9e-15. Where the line crosses zero is where the machine may stand: 2 crossings at this α, which is how many assemblies the compiled machine has at this driving angle.

Every curve is a sum of cosines

Put the two angles of a two-link arm into any polynomial in x and y and what comes out is a constant plus a finite sum of cosines of whole-number combinations of them. Nine curves, three hundred random angle pairs each, and the two routes agree to 1.8 × 10⁻¹⁴.

computing · Compute What each curve costs, in cosines. Every polynomial in the two arm angles is a constant plus a sum of terms A cos(mα + nβ + φ) with whole-number m and n, and the number of those terms is what a machine has to build. The count is not the degree and not the monomial count: a circle costs one term, a general line two, and a lemniscate — degree four — costs five, fewer than the cubic above it, because its symmetry cancels frequency pairs the cubic keeps. Each row's expansion was checked against a direct evaluation of its own polynomial at random angles, worst disagreement 1.8e-14.

A circle costs one term

A line expands to two cosines, a circle to one, a general conic to six. A lemniscate is degree four and costs five; a general cubic is degree three and costs eight. What a curve costs is not its degree — it is how many frequency pairs its own symmetry fails to cancel.

computing · Compute Two circles, one term. The curve r² = 1.44 together with r² = 2.56, whose squared radii sum to 4.00 — four times the square of the arm's link length. Their product equation expands to 1 term, which is what either circle costs on its own, so the second circle is free. The machine compiled from it has 17 bars against 11 for the single circle, runs over 5.200 radians against 1.560, and stays on the outer component throughout: its radius varies by 4.15e-13 over 240 solved positions. A mechanism moves continuously and the two circles are disjoint, so no assembly of it reaches both.

Two circles for the price of one

Search every multiple of a curve's equation by a polynomial of degree two and the cheapest is the curve's own equation, on four curves and by exhaustion. On the circle a second multiplier ties — and what it describes is two concentric circles, whose squared radii sum to four times the arm's link length squared, at exactly the cost of one.

computing · Compute The cheapest curve of each degree. The curve of degree 6 whose equation expands into the fewest cosines of the arm's two angles: cos 3θ = 0.3: 3 concentric circles about the pivot. It costs 1 term — cos(3α − 3β + …) — and the machine compiled from it has 27 bars. The dashed circle is the arm's reach. The dots are the machine's tracing point over 140 solved positions; it draws one component and stays on it. At every even degree the cheapest curve costs one term and at every odd degree two, whatever the degree.

The cheapest curves at each degree

Search every equation of degree up to six, not the multiples of one curve, for the fewest cosines a compiled machine has to build, and the answer does not depend on the degree at all: one term at every even degree, two at every odd one, each reached by exactly one set of frequencies. The cheap curves are concentric circles placed where Chebyshev's polynomials put them — k of them for the price of one — and, at odd degree, a line through the pivot with circles that come free. One symmetry of the arm forbids anything cheaper: its two links can be swapped without moving the tip.

computing · Compute A quintic of eighteen terms, and the nearest curve that costs a few. The quintic x^5 + 0.4x^2y − y = 0, whose equation expands into 18 cosines of the arm's two angles, inside the arm's reach (dashed). Drawn over it, the curve of the same degree with at most 4 terms that comes nearest to it inside the reach, found by trying every set of terms of that size: it keeps 0,1|1,0, 1,-2|2,-1 and is at most 2.77e-2 from the quintic and the quintic at most that from it — 1.4% of the reach. The slider steps through the budgets at which the nearest curve improves.

Four terms of eighteen draw the quintic

A quintic whose equation costs a compiled machine eighteen terms is drawn to 1.4% of the arm's reach by four of them and to 0.02% by ten, found by trying every set. The four are not its four largest. They are a line through the pivot bent by the square of the radius, the family the cheapest curves come from, and a curve close to that family is nearly free. An ellipse, a parabola, a lemniscate and a folium are not close to it, and each costs its whole equation to come within four per cent.

computing · Compute

Named alongside it

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

Algebraic curveFrequency pairTwo-link armCost modelDegreeImplicit equationSparsitySymmetryCompiled machineApproximationCancellationConjugate pair

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