Machine Kinematics
solved, then drawn
Concept

Closure margin — where it appears

How far the origin sits inside the convex hull of a set of contact rows, taken as unit vectors. It is a distance rather than a share, so it falls continuously to nought as an arrangement approaches one that lets go, and a positive value is what makes a hold a measurement rather than a verdict.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

How far inside the hull the origin actually is. The same seven arrangements with their margins drawn rather than tabulated, because the shape of this chart is the argument: the quantity is not a probability and not a percentage, it is a distance — how far the origin sits from the nearest face of the hull of the contact rows, with every row a unit vector so the number is comparable across arrangements. The two that hold come in at 0.211 and 0.091; the five that do not come in at exactly nought, and they are drawn at nought rather than left off. A margin that falls smoothly to nothing is what makes this a measurement: an arrangement approaching one that lets go says so before it does.

The test is a program, not a rank

Three independent routes to one yes-or-no: enumerate the escape cone's extreme rays by cross products, take the convex hull of the contact rows and ask where the origin is, or hand the whole thing to a simplex. They agree on every arrangement — and the first version of the third one reported a disc as held, which is the one part in the field that no number of contacts holds.

holding · Restraint 5 contacts, and 1 of them free not to touch. A hexagon on five contacts. Each contact is removed in turn and the hold recomputed; the ones drawn in the warning colour are those whose removal leaves the part still held, which is to say the ones that are constraining nothing the others were not already constraining. There is one here, and the margin without it is 0.091 — unchanged, to every figure. That is the unilateral form of what a redundant constraint costs, and it costs something different from the bilateral form: a redundant bilateral constraint has to be satisfied and cannot be, so it leaves a gap somewhere; a redundant contact is simply free not to touch, and whether it does is decided by errors nobody controls. positioned by solving, not by drawing.

The contact that is free not to touch

A hexagon on five contacts holds, and taking one of the five away leaves the margin at 0.0914 — unchanged, to every figure. That contact constrains nothing the others were not already constraining, and what it actually does is become the one member of the set that is free not to touch, with the decision made by errors nobody controls.

holding · Restraint The pose set is finite exactly when the part is held. Two arrangements, both with 0.06 of clearance on every contact, with the set of positions the part's centre may occupy drawn to scale. The one on the left holds: its pose set is a small bounded polyhedron, and every dimension of it is proportional to the clearance. The one on the right does not: its pose set runs off the page in the direction the part slides out of the vee, and giving the contacts a tighter tolerance narrows the box without ever closing that direction. A tolerance cannot buy a hold. The clearance decides how big a finite pose set is and the arrangement decides whether it is finite, and the second question has to be settled first because no amount of the first will settle it.

Held is not located

Back every obstacle off by a clearance and the permitted poses become a polyhedron — bounded exactly when the arrangement is a hold, since an unbounded direction of it would be a ray of the escape cone. So whether a part is held is whether its pose set is finite, the clearance is what gives that set a size, and the two questions have to be settled in that order because no tolerance settles the first.

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 things a hold is not

A rank read as a restraint, a count read as an answer, four contacts placed the wrong way round, a nullity taken for a spin, a part free in every direction and unable to leave, and a tolerance offered as a cure for an arrangement that was never a hold. Six claims, each with the number that kills it.

wrong · Misconception Two jaws on a polygon, and the cones they can push within. A regular polygon of 5 sides and circumradius 1, gripped by two contacts with a friction coefficient of 0.3. Each wedge is the cone of directions its contact can push in, half-angle atan 0.3 = 16.7° about its normal. Two contacts hold the part — resist every load, given a squeeze — exactly when the line between them lies inside both cones. An odd polygon has no parallel faces, and its most nearly opposed faces are 36.0° short of parallel, so the line can be inside both cones only if their half-angle is at least half that: a friction of tan(π/10) = 0.3249. Here the part is not held. Dragging changes the number of sides.

Two jaws and a coefficient

Every hold in this field has been frictionless, and two of its essays say what that leaves out. Put friction back and a part is held by two contacts in the plane instead of four and by three in space instead of seven; a disc, which no frictionless fixture can hold at all, is held by two jaws at any friction above nought; and an odd polygon needs a friction of exactly tan(π/2n). What friction takes away is the thing that made a frictionless hold a fact about geometry: a frictional hold is a hold per unit of squeeze.

holding · Restraint

Named alongside it

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

Form closurePositive spanClearanceEscape coneFixtureLinear programConvex hullRankToleranceUnilateral constraintCagingConditioning

All concepts