Machine Kinematics
solved, then drawn
Concept

Escape cone — where it appears

The set of velocities a part's contacts permit at one configuration: everything satisfying one linear inequality per contact. It is a cone rather than a subspace, so a freedom in it need not be two-sided, and read in the plane it becomes a region of rotation centres with a sense attached to each.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

A constraint that only pushes

Every constraint on this site so far has been an equation: a pin holds two points together, a bar holds two apart, a mesh holds a ratio. A part resting against another part says only *do not come closer* — so what it may do is a cone rather than a subspace, and whether it can move at all stops being a rank.

holding · Restraint What one contact forbids, drawn as a place. A single contact on one edge of a square, and the whole plane coloured by what it permits. A rotation about a point is a twist, and a twist is permitted when it does not drive the part into the obstacle; because a rotation about (x, y) is affine in the point, the condition is a half-plane and the boundary is a straight line — the line through the contact along its own surface. On one side of it only anticlockwise rotations are permitted, on the other only clockwise, and the two together are the whole plane bar the line itself. So one contact rules out exactly half of what the part could do and leaves the other half untouched, which is why the count of contacts a hold needs is one more than the dimension rather than equal to it: the first 1 of them cannot leave nothing over. The picture is exact — the regions are clipped polygons, not a sampled grid.

What one contact forbids

A rotation about a point is a twist, and a twist is affine in the point — so what a single contact permits is a half-plane of centres, with the boundary being the contact surface's own line. Reuleaux drew it in 1875 and it is exact rather than sampled, which is why every figure in this field is a picture of the plane rather than of a cone.

holding · Restraint 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 4 contacts, and the centres they still allow. The same four, placed pinwheel. The same square, the same four edges, the same distance along each — and taken the same way round rather than alternately. Every row's moment then has the same sign, so no positive combination can cancel it, and the part turns. Each contact contributes one half-plane of permitted centres per sense, and the shaded regions are what survives all 4 of them: the darker one is where an anticlockwise rotation is still permitted and the lighter one where a clockwise one is. What is left is the escape, and it is a region rather than a direction: any point inside it will do as a centre. The enumeration finds 4 extreme rays, of which 4 are rotations and the rest are translations — the corners of the region and its unbounded directions respectively. positioned by solving, not by drawing.

The escape is a place

A part that is not held escapes, and the useful thing is not that it escapes but where. The extreme rays of the cone are the corners of a region of the plane and its unbounded directions are translations — so the answer to 'this does not hold' is a picture with a shape, and the shape says where the next contact has to go.

holding · Restraint Free to spin, and it cannot turn at all. An ellipse of semi-axes 1.4 and 0.9 inside four flat walls that touch it at the ends of its own axes. Every normal points at the centre, so every row's moment is nought and the four rows span two dimensions rather than three: the cone of permitted twists is the whole spin axis, a line through the origin, and the first-order answer is that the part is free to turn either way. Drag the angle and watch what happens. The ellipse's reach in the direction of the top and bottom walls is √(a²sin²θ + b²cos²θ), which is smallest at θ = 0 and grows from there, so any rotation whatever drives it into both of them — by 0.016 at this angle. A nullity is a candidate and not a motion, and this is the shape of case the fields before this one could not produce: not a mechanism at a singularity, but an ordinary part in an ordinary pocket. positioned by solving, not by drawing.

Free to turn and unable to

An ellipse in a pocket the size of its own bounding box has four contacts whose rows span two dimensions, so the cone of permitted twists is a whole line and the first-order answer is that it spins both ways. It cannot turn by any amount whatever: the penetration grows as the square of the angle, with a fitted exponent of 1.9944, and a circle in the same pocket turns for ever.

holding · Restraint Free in every direction, and it cannot get out. A disc of radius 1 among 3 point obstacles on a circle of radius 1.100. The shaded discs are the obstacles grown by the part's own radius, which is what the part's centre may not enter — the configuration space, and for a round part it is the plane itself. The part is caged when those grown discs overlap enough to close a ring around it, which happens below R = 1/sin(π/3) = 1.154701, and here it does. At every configuration inside the cage the part is free. The three normals all point at its centre, the rank of its rows is two, the escape cone is a whole line, and none of that has anything to do with whether it can leave. A hold is a statement about velocities at one configuration; a cage is a statement about where a finite motion can go, and the second does not follow from the first in either direction. positioned by solving, not by drawing.

Free at every instant and going nowhere

Three points on a circle of 1.1 radii around a unit disc leave it free in every direction at every configuration — rank two, margin nought, the whole plane of centres shaded — and it cannot get out. The threshold is 1/sin(π/n), which is 1.154701 for three, and a flood fill of the free space agrees with the formula at every radius sampled.

holding · Restraint A joint with no way out in the plane it is drawn in. A dovetail. The tail is wider at its far end than at the mouth it went in through, so every direction out of the mouth is blocked by a slanted face and every direction further in is blocked by the floor. In the plane of this drawing the joint cannot be taken apart at all, and the direction it does come apart in is the one the drawing does not show. The moving part touches the rest at 3 faces, each contributing one inequality on the direction it may be translated in — the direction must not have a negative component along that face's inward normal — and the set of directions that satisfy all of them is a cone in two dimensions rather than three, because a translation has no moment term. That is why a removal cone can be drawn as an angle where a mobility cone cannot. Here it is empty: every direction is refused by one face or another, so the part cannot be taken out by any translation and cannot have been put in by one either. That is a statement about the assembly and not about the part, and the direction the joint does come apart in is perpendicular to this drawing. positioned by solving, not by drawing.

Which way it comes out

Drop the rotation from the inequalities and the cone lives in two dimensions rather than three, so it can be drawn as an angle: a block in a vee has ninety degrees of directions out, a key in a slot has exactly one and no arc around it, and a dovetail has none at all. Three answers, and each of them is a different kind of joint.

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 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 closureUnilateral constraintRankContact normalClosure marginConvex hullEnumerationPositive spanTwistCagingClearanceConditioning

All concepts