Machine Kinematics
solved, then drawn
Concept

Locating scheme — where it appears

A set of contacts that determines where a part sits, given that something presses it against them. The machinist's three-two-one arrangement is the canonical one: six pads, rank six, a unique pose — and no restraint whatever, since six unilateral contacts can never hold a body in space.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

6 contacts, and the box lifts straight off. 3-2-1, six contacts. Each pad is a contact and each arrow the direction the box is free to move there — the inward normal, and the whole of what the contact contributes. The rank is 6, which is full: these 6 contacts are 6 independent constraints and a bilateral version of them would fix the box completely. The margin is nought, so the box is free to leave, and no amount of tightening the tolerances on where the pads are would change that. Rank and hold are different questions and this is the pair of pictures that separates them. positioned by solving, not by drawing.

Four in the plane and seven in space

Six independent constraints fix a body in space and six contacts fix nothing, because d vectors can span d dimensions and can never positively span them. The minimum is one more than the dimension — and it is a floor rather than an answer: six contacts on a box held it in none of four thousand random arrangements and seven held it in twenty-one.

holding · Restraint A 4-sided bar, and the 13.6% it is out by. 3 flat jaws advancing together on a regular 4-sided bar of unit circumradius, with the bar turned 10.3° from square. The dashed circle is the axis the chuck is turning about and the marked point is where the bar's own centre has ended up: 0.13567 of a circumradius away. The arithmetic is one line — the jaws touch when c·u_k + h(u_k) = d, three unit vectors at 120° satisfy Σ u u ᵀ = 3/2 I, and so c = −⅔ Σ h(u_k) u_k — and it says that the offset vanishes exactly when the bar's own support function is unchanged by a 120° turn. Round, triangular, hexagonal, nine- and twelve-sided bars centre at any orientation; everything else does not, and by an amount that depends on how it happened to go in. Checked here against a linear program that closes the jaws without knowing the identity. positioned by solving, not by drawing.

Where the jaws put it

Three jaws closing on a bar put its axis at −⅔ Σ h(u_k) u_k, which vanishes exactly when the section's support function is unchanged by a 120° turn. So a three-jaw chuck centres round, triangular and hexagonal stock perfectly and a square bar by up to 17.3 per cent of its own circumradius — and the workshop rule about symmetry that predicts this is wrong, because a six-jaw chuck centres a square.

holding · Restraint Seven arrangements, one routine, and the two that hold. Every row is the same three steps: write down one row per contact — the moment of its normal about the origin, then the normal itself — take the convex hull of those rows, and ask whether the origin is inside it. The parts differ, the numbers of contacts differ, and the routine does not. Two of the seven hold. The other five leave the part something, and the interesting column is what: four rays of rotation for the pinwheel, a translation straight out of the vee, and for the last two a whole line rather than any number of rays, which is what a rank below three means and is the case a reader has to be warned about. Note that the four contacts of the second row are the four of the first row, on the same four edges of the same square, at the same distance along each. positioned by solving, not by drawing.

Six hold nothing

Every exact-constraint coupling on this site — Kelvin, Maxwell, three-two-one, and a Kelvin clamp with a seventh pad added — has rank six and holds the part not at all. The escape a Maxwell coupling leaves is a pure vertical translation with nothing else in it, which is not a defect: it is what a coupling is, and gravity is the seventh contact nobody draws.

holding · Mobility

Named alongside it

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

Form closureDegrees of freedomExact-constraintFixtureMobilityRankUnilateral constraintChuckContact normalConvex hullEscape coneLinear program

All concepts