Machine Kinematics
solved, then drawn
Contacts that only push

A cone has no size

What a set of contacts permits is a cone of twists, and a cone is closed under positive scaling by definition — so nothing about it changes when the part it holds is made bigger. Except that a twist is a screw, a screw has a pitch, and a pitch is a length.

Assumes A constraint that only pushes.

14 min read 5 figures Two routes to the same numberThe singularities are the point

The restraint field’s object is a cone. A set of contacts that can only push permits a set of instantaneous motions closed under adding and under multiplying by a positive number, and a set with those two properties is a convex cone.

A cone is scale-free by construction. That sounds like the end of the story and it is not.

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.
Fig. 1 The cone of twists a set of unilateral contacts permits, whose own description contains no length.

The cone is closed under scaling by definition

If a twist is permitted, so is twice that twist, and half of it, and any positive multiple. That is what a cone means and it is not a scaling of the fixture — it is a scaling of the motion.

The two get confused and it is worth separating them at once.

Scaling a motion takes a permitted twist to another permitted twist. It says the cone contains rays rather than points, so a cone is described by directions and nothing else.

Scaling the fixture takes a part and its contacts to a bigger part with bigger contacts in corresponding places. That is the operation the survey means, and whether the cone survives it is a separate question with a separate answer.

The answer is yes, mostly, and the exception is the whole content of this essay.

What survives

Scale a held part and its contacts together. Every contact normal points the same way. Every contact sits at the corresponding place. The conditions that decide whether a motion is permitted are conditions on the sign of a wrench-twist product, and both factors scale, so every sign is unchanged.

The step worth checking is the middle one. A contact’s wrench is a normal direction and a moment about the origin, and the moment is the position crossed with the normal — so scaling the fixture multiplies the moment part and leaves the direction part alone. A twist likewise has a rotation part that does not scale and a translation part that does. Their product pairs each scaled half against an unscaled one, so the whole product scales by exactly one factor and its sign is untouched.

That is not automatic and it is the reason the cone survives at all: the pairing between wrenches and twists is arranged so that a scaling multiplies the product uniformly. Had the pairing been anything else, a fixture’s classification would depend on how big it was.

So the cone’s combinatorial structure survives exactly: which contacts constrain which motions, how many faces the cone has, whether it is a single ray or a whole half-space, whether the part is held.

Form closure is a shape. Whether a part is held or free is decided by whether the cone reduces to nothing, and that is a rank-and-sign question with no length in it. Four contacts in the plane and seven in space is a count, and it holds at any size.

And the number of contacts that are redundant is a count, so the ledger’s own tallies all transfer.

And the number of contacts that are redundant is a count, so the ledger’s own tallies all transfer.

Where the length is

The cone’s rays are twists, and a twist is a screw: a rotation about an axis together with a translation along it. The pitch is the ratio of the translation to the rotation, and it is a length.

Worth pausing on why. A rotation is measured in radians, which are dimensionless; the translation that accompanies it is a distance. Divide one by the other and the units of the quotient are a length per radian, which is a length. So a screw’s pitch has a dimension for the same reason a wheel’s radius does — it converts an angle into a distance.

So a ray of the cone is labelled by an axis — a line in space, which is a position and a direction — and a pitch, which is a length. Scale the fixture and every axis moves outward and every pitch scales.

That is the one place a size enters, and it enters in the field’s central object rather than at its edges.

The consequence is precise. The cone’s shape is a shape and the screws labelling its boundary rays are not. Two fixtures related by a scaling permit motions that are the same motions described in a scaled frame, and a table of their permitted screws’ pitches differs by the factor.

That is the one place a size enters, and it enters in the field’s central object rather than at its edges.

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.
Fig. 2 The ledger a fixture produces: counts, which transfer between sizes, and the geometry they are computed from, which does not.
Free in every direction, and it cannot get outA 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.R 1.260 of a threshold 1.1547it leaves
Fig. 3 And a caged part, where the question is whether a motion exists at all rather than what its pitch is.

Second-order restraint is a shape too

The field’s subtler result deserves the same treatment because it is where a reader might expect a size to appear.

A part can be free at every instant and still unable to escape: every permitted twist, followed for any finite distance, runs into a contact. That is second-order restraint, and it depends on the curvatures of the contacting surfaces rather than only on their normals.

A curvature is a reciprocal length, so a reader might expect the second-order condition to bring a size in. It does not, and the reason is the same one that saves Bennett’s condition: the condition is homogeneous.

Scale the fixture and every curvature is divided by the factor while every distance from the contact is multiplied. The second-order term is a product of a curvature and a squared displacement, so it scales as the displacement — and it is being compared against a first-order term that scales the same way. Both sides scale together and the comparison’s sign is unchanged.

Whether a part is caged is a shape, and it survives a scaling for the same reason a cone’s structure does: every condition in it is a comparison of two things that scale alike.

A pitch of zero and a pitch of infinity

The two degenerate pitches are worth noting because they are the ones that behave differently.

A pure rotation has pitch zero. Scaling the fixture leaves it zero — zero times anything is zero — so this permitted motion is a pure rotation is a scale-free statement.

A pure translation has infinite pitch, and the site has an essay on why that is a defect of the representation rather than of the motion. Infinity times anything is infinity, so that too is scale-free.

So the two ends of the pitch range are invariant and everything between them scales. The classification of a permitted motion — rotation, translation, or a screw of finite pitch — is a shape, and the pitch of the screws in the middle is a size.

That mirrors the pattern the whole survey keeps producing: the classification transfers and the number does not.

The mixed-unit problem is real

The pitch’s presence in the cone creates a difficulty this field has and no other on the site does, and it is worth setting out because it is why the field’s code carries a units assertion.

A twist has three rotational components and three translational ones. The first are dimensionless rates and the second are lengths per unit time. Stack them in a six-vector and the vector’s norm has no meaning: adding the square of a radian to the square of a millimetre is arithmetic with no content.

Every rank, every angle between twists and every condition number computed on such a matrix therefore depends on a choice of length scale, made implicitly by whatever units the numbers happen to be in.

The site’s own guard is a check that the answer survives a change of units, and what it protects is the answers that ought to be invariant — whether the part is held, how many contacts are redundant. Those are decided by signs and ranks that a diagonal rescaling cannot change.

What genuinely does depend on a length scale is any quantity that compares a rotation against a translation: the angle between two twists, the condition number of a grasp matrix, the “distance” between two screws. Those are not invariant and cannot be made so, and the honest treatment is to state the scale — which is what a characteristic length of the part is for, and is the same device the serial field uses to compare a position error against an orientation error.

Pushing on a part, and what pushing tells

Applying the field’s own question to a fixture rather than to a mechanism.

Suppose the observable is which motions the part can make — determined by pushing it and seeing what happens, which is how a fixture is actually tested. Those observations are directions of motion.

A direction is dimensionless. So an observation of directions determines the cone’s shape exactly and cannot determine the fixture’s size, which is the standing result arriving in a field where the object being measured is a set rather than a machine.

What it also cannot determine is the pitches, since a pitch is a length and the observation carries none. A fixture tested by pushing recovers what it holds against and not the geometry of the screws it permits.

Measuring where the contacts are — one length — closes it, exactly as everywhere else.

What one contact forbids, drawn as a placeA 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.anticlockwise permittedclockwise permittedone contact · one half-planethe boundary is the surface's own line
Fig. 4 The regions a set of contacts carves the motion space into, whose boundaries are decided by signs rather than by sizes.
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.
Fig. 5 And how the work of holding is shared between contacts, which is a set of ratios.

The redundancy count is a count

The field’s most useful result is a count and the scaling probe files it in the third class.

A redundant contact is one whose removal leaves the part held. That is a property of a set of inequalities, decided by whether a linear program is feasible, and it is an integer.

The probe returns nothing for it, correctly: it does not move under a scaling and it does not move under a perturbation of one contact position either, until the perturbation is large enough to change the answer entirely.

So it is a count with all a count’s properties, and the recommendation follows: report the margin. Here the margin is how far a contact can be moved before the redundancy changes, which is a length and which the field computes as part of deciding the count.

A count and its margin, again, in the fourth field of this survey to produce the pair.

It is worth saying why the margin is the interesting quantity here and not merely a formality. A fixture with one redundant contact and a margin of a millimetre is a fixture whose redundancy will not survive a production run; one with a margin of ten millimetres is a design decision. Both report the same integer, and a designer choosing between two layouts on the basis of the integer alone is choosing between two claims with very different evidence behind them.

A change of units, already checked

Worth noting that this field is the one place on the site where an invariance of this kind was already asserted and tested.

The form-closure field carries an assertion named for it: the answer survives a change of units. It exists because a cone’s description involves both forces and moments, or both translations and rotations, and the two have different dimensions — so a numerical routine that mixes them is implicitly choosing a length scale, and its answers can depend on that choice.

That assertion is a narrower statement than this essay’s and it is the important one for the arithmetic’s correctness. A rank computed on a matrix whose rows are in mixed units can change when the units change, silently, and the field guards against it.

A change of units and a scaling of the machine are different operations with the same guard. The first changes the numbers and not the object; the second changes the object. The site checks the first and this essay measures the second, and the two agreeing is a small piece of evidence that both are right.

Held is not located, and neither is scale-free

The field’s own sharpest distinction interacts with this in a way worth working out.

Held is not located: a part can be prevented from moving and still sit anywhere within a region, because contacts that block every motion do not thereby fix a position. The field measures the size of that region.

That region is a set of positions, so its extent is a length, exponent one. Scale the fixture and the part’s freedom to sit anywhere scales with it.

But the region’s shape, and the ratio of its extent to the part’s own size, are shapes. So the same object splits the same way: a fixture that leaves a part free to wander by a tenth of its own length does so at any scale, and by more millimetres at a bigger one.

That is the practically important form. A fixture’s positional accuracy relative to the part is a shape and its accuracy in millimetres is not, so a fixture design transfers between part sizes and its absolute performance does not — which is exactly what a production engineer scaling a jig needs to know and is not what a jig drawing says.

The field that had already noticed

The survey found less to say here than anywhere else, and the reason is worth more than the findings.

Everything this field computes about a cone survives a change of size. Whether a part is held, how many contacts hold it, which are redundant, whether it is caged, what escapes and in which direction — every one of those is a statement about a set of directions, and directions have no length in them. A fixture drawn at twice the size holds the same way, and the figures here are figures of families whether or not they were drawn as such.

What does not survive is anything with a pitch in it. A screw’s pitch is a length per radian, the boundary of a restraint cone is made of screws, and a cone described by the pitches on its boundary is a description that moves when the units do. The classification transfers and the numbers on the boundary do not, which is the finest-grained version of the split this survey keeps finding, because for once both halves live inside one object rather than in two.

And that is exactly the hazard this field had already written a check against, for a reason that had nothing to do with any of this. A restraint calculation forms a matrix whose rows mix forces with moments, or translations with rotations, and those have different dimensions. Its rank therefore depends on the relative scaling of the two halves — which is to say on a length scale chosen implicitly by whoever wrote the matrix down. A rank is an integer and an integer looks like a fact, so a rank that moves when the units do is the most dangerous shape of wrong answer available: an integer with a hidden parameter in it.

The field’s own assertion says the answer survives a change of units, and it was written to catch that and nothing broader.

A discipline adopted for a narrow reason turned out to be the general one. That is the useful thing here, and it generalises past cones. Any quantity computed from a matrix whose rows carry different dimensions has a scale in it whether or not anybody chose one, and the check is cheap: recompute in different units and see whether the answer moves. Every other field on this site could run it and none of them does, because none of them mixes units inside one matrix quite so obviously — and quite so obviously is doing a great deal of work in that sentence. A tolerance Jacobian whose columns are lengths and whose one dimensionless column is a coupler fraction has the same problem in a quieter form, and nothing anywhere tests for it.

The field where the scaling question produced the fewest surprises is the field that had already asked half of it. That is not a coincidence and it is not luck; it is what a check earns.

Where this goes next

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. A screw of pitch 0.25. A single screw drawn from its own decomposition. The six numbers of the twist go into a decomposition that returns a direction, a point on the axis and a pitch of 0.2500; the helix is then the path of a point at radius 0.5 about that axis, advancing 0.250 along it per radian turned. Over 1.15 turns the point advances 1.806 — the pitch times the angle, which is what pitch means. This screw is a screw. Every motion is a screw Chasles showed that any rigid displacement whatever is a turn about some line together with a slide along that same line. Not approximately, and not usually — always, with the line and the amount of slide computable from the motion. It is the fact that makes spatial kinematics a subject rather than a pile of special cases. Which numbers have a size. Twelve quantities from eight fields of mechanism kinematics, each scaled by taking every length in its mechanism up and down together and fitting the power its value follows. A shape lands on zero, a length on one, a curvature on minus one, an area on two, and every row lands on an integer to 3.3e-11. The sorting is not a units argument — it is measured from each field's own library, by asking that library the same question. What it says is that most of what kinematics computes is a shape: a mobility, a Grashof class, a transmission angle, a contact ratio and a velocity ratio are all unchanged by making the machine bigger, so all of them are recoverable from a measurement that cannot recover a single length. The row worth stopping at is the tolerance band: held to a fixed ±0.01 it lands on minus one, because ±0.01 is a length and a bigger machine held to the same absolute tolerance is a better machine. Written as a percentage the same band lands on zero. One quantity, two drawing conventions, two different kinds of number. Which numbers have a size Take every length in a mechanism up and down together and fit the power each computed quantity follows. A transmission angle lands on zero, a coupler point's speed on one, a path curvature on minus one, an enclosed area on two — and a tolerance band held to a fixed ±0.01 lands on minus one, which nobody would guess.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Contact normalForm closureIdentifiablePitchRestraintScale invarianceScrewTwist cone