One character apart

A wheel obeys

$$\dot{x}\sin\theta - \dot{y}\cos\theta = 0,$$

and the previous essay claimed, without proving it, that this forbids a direction without costing a dimension. Here is a mechanism that obeys

$$\dot{x}\sin\alpha - \dot{y}\cos\alpha = 0$$

for a fixed angle $\alpha$, and it is a trolley clamped to a straight rail, free to spin about its own axis but not to leave the rail. Three coordinates, one row, two controls — spin, and run along the rail. Every count that can be taken of the two mechanisms gives the same answer for both.

One of them can be brought to any configuration whatever. The other is stuck on a line for the rest of its life, and no sequence of permitted motions will ever take it off. The difference between the two rows is that the first has a coordinate in it and the second has a constant.

Why the obvious approach does not work #

The trolley is stuck because it has a conserved quantity. Along any permitted motion,

$$\frac{d}{dt}\big(x\sin\alpha - y\cos\alpha\big) = \dot{x}\sin\alpha - \dot{y}\cos\alpha = 0,$$

so $x\sin\alpha - y\cos\alpha$ never changes. Whatever value it had at the start it has forever, and the trolley lives on that one line — a two-dimensional set inside a three-dimensional space, since the spin is still free. The velocity constraint was a position constraint wearing a velocity constraint’s clothes, and one integration takes the disguise off.

So the natural test is: look for a conserved quantity. If one exists the constraint is really about positions; if none exists it is not.

That is not a test. Finding a conserved quantity settles the question one way, and failing to find one settles nothing at all — it might mean there is none, or it might mean nobody looked hard enough. Half a test is worse than none here, because the failing half is the one that would be used most often: the interesting mechanisms are the ones with no invariant, and no invariant was found is exactly what an unsuccessful search reports.

The test that is a test #

Frobenius’ theorem gives the missing half, and it does so by asking a question about the permitted directions rather than about the forbidden one.

The permitted velocities at a configuration form a subspace — a plane, for both mechanisms here. As the configuration moves, that plane tilts, and the theorem is about whether the tilting is consistent. Take two permitted directions $F$ and $G$. Move a little way along $F$ and ask how $G$ has changed; move a little way along $G$ and ask how $F$ has changed; subtract. What comes out is another direction, the Lie bracket $[F, G]$, and the theorem says:

The constraint is a position constraint in disguise exactly when every such bracket is itself a permitted direction.

If the brackets stay inside the plane, the planes fit together into a family of surfaces and the mechanism is confined to one of them. If a bracket points out of the plane, no such family exists, and — this is the part that is not obvious and is the whole of the next essay — the direction it points in is one the mechanism can actually be driven along.

The test is a computation on the two fields at one configuration. It needs no integration, no search, and no cleverness, and it returns a number rather than a verdict: how much of the bracket lies outside the plane, as a fraction of the bracket’s own length. That is a quantity between zero and one with no units, so mechanisms of different sizes and different coordinates are directly comparable.

What the two mechanisms return #

For the trolley, both fields are constant — the direction along the rail does not depend on where the trolley is or which way it is pointing, and neither does the spin. Two constant fields have a bracket of exactly zero. Not a small number: zero, because every derivative that goes into it is identically zero. The measured value is $0$, and the trolley is integrable.

For the wheel, the drive field turns as the wheel turns, and the bracket comes out as

$$[F_{\text{drive}}, F_{\text{turn}}] = (\sin\theta,\ -\cos\theta,\ 0),$$

which is the constraint row itself, written as a direction. The bracket of the two permitted directions is the forbidden direction, exactly, and the measured fraction outside the plane is $1.000$. Not merely nonzero: the bracket lies entirely outside.

That is worth pausing on, because it is a coincidence of the plainest case rather than a general rule, and it is the cleanest possible statement of what the field is about. The one thing a wheel may not do is the one thing that is left over when the two things it may do are done in the wrong order.

The invariant, checked rather than asserted #

The claim that the trolley cannot leave its line is exactly the kind of claim this site does not make without a test that could fail it. So the trolley is given a control history designed to be awkward — run out, spin, run back part way, spin the other way, run out again — and $x\sin\alpha - y\cos\alpha$ is evaluated at every step of every leg.

It moves by $2 \times 10^{-15}$ over the whole history, which is the accumulated rounding of a few thousand fourth-order integration steps and is not a motion. The trolley is on its rail at the end as exactly as it was at the beginning.

The same measurement run on the wheel is the reachability cloud of the previous essay: nine hundred control histories, and the resulting positions fill a region rather than a line. Neither of those is a proof, and neither is meant to be — Frobenius is the proof. They are the check that the proof was applied to the mechanism actually drawn, which is a different question and the one this site has been caught out by before.

The word, and why it is worth less than the number #

A constraint that integrates is called holonomic and one that does not is nonholonomic. The words are useful for talking and are close to useless for deciding, because the property they name is not visible in the constraint’s appearance. Both of the rows at the top of this essay are one linear condition on three velocities with trigonometric coefficients. They look the same because they are the same, up to what the angle is.

The literature that a mechanical engineer meets tends to present a short list of nonholonomic examples — the rolling wheel, the rolling ball, the skate — and leave the impression that the property is a feature of a small named family. It is not. It is a property of a particular set of fields at a particular configuration, it is decidable in a few lines of arithmetic, and the decision comes back seventeen orders of magnitude away from the boundary on every mechanism in this field.

Constraints that look nonholonomic and are not #

The reason to have a test rather than a list is that the list gets things wrong in both directions. Two cases from elsewhere on this site are worth walking through.

A gear train. A mesh imposes $z_1\dot\theta_1 + z_2\dot\theta_2 = 0$, which is a linear condition on velocities with no position in it at all — the same shape as a rolling constraint, and physically it is a rolling constraint, since conjugate tooth flanks roll and slide against each other and the pitch circles roll without slipping. Yet a gearbox obviously has a ratio, and a ratio is a statement about angles rather than about rates.

The reason is that the coefficients are constants. $z_1\dot\theta_1 + z_2\dot\theta_2$ is the derivative of $z_1\theta_1 + z_2\theta_2$, so the combination is conserved and the train’s configuration space is a line inside its coordinate space. The test returns zero. That is why a ratio is a number and why a gear train is the one mechanism on this site whose constraint rank cannot change as it moves: there is nothing in the rows for a configuration to change.

A gear train and a rolling wheel are both rolling contacts and they are on opposite sides of this test. What separates them is not the contact; it is that a gear’s contact is between two bodies whose relative geometry never changes, and a wheel’s is between a body and a surface it is free to wander over.

A car with the steering held. Clamp a car’s steering wheel and the mechanism has two rows — no sideways motion at the rear axle, and a heading rate tied to the steering angle — plus the clamp, which is a third. What is left is a mechanism that runs along one arc of one circle. Two of the three conditions are position constraints in disguise; the sideways one is not, and it does not matter, because with the steering clamped there is only one control and there are no pairs of fields to take a bracket of. A mechanism with one control is integrable whatever its constraint says, since a single direction has no bracket with anything, and the reachable set is a curve.

That is the honest reason a car is hard to park rather than impossible: not the sideways constraint on its own, but the sideways constraint together with a second control that changes what sideways means.

What integrability costs, in one number #

The clean way to state the difference is with the two columns of the field’s ledger. The velocity freedoms are what the rows leave; the reachable dimension is what the brackets fill. Frobenius’ theorem is precisely the statement that they are equal when the test returns zero.

Note what the table does not say. It does not say the wheel has three freedoms; it has two, and at any instant it can only do two things. It says the wheel can reach a three-dimensional set of configurations using its two freedoms, over time. Those are compatible statements and confusing them is the mistake an essay of its own is given to.

The bracket, before it is measured #

The bracket has been used above as a computation and it has a physical reading which the next essay is built on, so it is worth stating here in the form that makes the following pages read as a consequence rather than as a new idea.

Two motions done in one order and then undone in the other do not cancel. Drive forward, turn, drive backward the same distance, turn back the same angle: a wheel does not come home. What is left over is small — second order in how far each leg went — and it points along the bracket.

So the bracket is not merely a test of whether the mechanism is confined. It is the manoeuvre, and its size is what the manoeuvre wins. A mechanism whose brackets stay inside the distribution gains nothing from any manoeuvre, however long, because there is nothing outside to gain. A mechanism whose brackets point out gains a little, and can repeat.

The test is pointwise, and the answer can vary #

One property of Frobenius’ test is easy to pass over and matters for how the results above should be read: it is a statement about a configuration, not about a mechanism. The bracket is computed from the fields at a point, and nothing forbids it from lying inside the permitted plane at some configurations and outside it at others.

So the three verdicts are really three cases rather than two. A mechanism whose brackets stay inside everywhere is confined everywhere, and a conserved quantity exists globally. A mechanism whose brackets escape everywhere is manoeuvrable everywhere. And a mechanism whose brackets escape at most configurations and vanish on some subset has regions where the manoeuvring works and a set of configurations where the leading-order manoeuvre wins nothing — which is not a third kind of mechanism so much as a place where the first estimate of what a manoeuvre buys goes to zero and the next term has to be found.

Both mechanisms in this essay are uniform, which is worth checking rather than assuming. The trolley’s fields are constant, so its bracket is identically zero at every configuration and its confinement is global. The wheel’s bracket comes out as the constraint row written as a direction, whose length is one wherever the wheel is and however it is pointing — so the escape from the plane never weakens, and the wheel is nonholonomic uniformly rather than generically. That is why the field’s later measurements can quote a single exponent and a single constant for the whole configuration space, and it is a fact about this mechanism rather than a licence.

Where the uniformity fails is exactly where the constraint rows themselves degenerate, and this field has such a case: a car and trailer folded to a right angle is the configuration where the geometry is most extreme, and the honest way to ask whether anything has changed there is to recompute the brackets rather than to reason from the picture. The measurement in that essay says nothing has, which is a result and not a foregone conclusion — a mechanism can perfectly well have a configuration at which its growth vector changes, and the only way to know is to evaluate the test where the suspicion is.

That also explains why the test is written as a computation rather than looked up. A list of nonholonomic mechanisms is a list of mechanisms, and the property being listed belongs to configurations. A rolling wheel is on every such list and deserves to be; the general form of the claim is that its bracket escapes at every configuration, and that sentence has to be checked and not remembered.

Where the test is used in this field #

Three times, and each is worth naming so that the test does not read as a formality performed once at the start.

On the mechanisms themselves, above, to establish that there is something to talk about.

On the depth of the manoeuvring. The bracket of two fields may point out of the plane; the bracket of that with a field may point out of the larger space it spans; and so on. How far the process has to go before it stops growing is an integer, one per mechanism, and it is the subject of how many wiggles. Frobenius’ theorem is the statement that the integer is 1 when the mechanism is confined.

On the mechanisms that fail it in an interesting place. A car and trailer folded to a right angle is the case everybody assumes is special; the test says it is not, and the assumption turns out to be about the room the rig has rather than about its constraints.

The character that differs #

It is worth ending where the essay started, because the whole argument fits in one substitution and it is easy to lose that in the machinery.

$$\dot{x}\sin\theta - \dot{y}\cos\theta = 0 \qquad\text{against}\qquad \dot{x}\sin\alpha - \dot{y}\cos\alpha = 0.$$

In the first, the angle in the row is one of the mechanism’s own coordinates, so the forbidden direction turns as the mechanism turns, and a mechanism that can turn can therefore change which direction is forbidden. In the second the angle is a number fixed when the rail was bolted down, the forbidden direction never moves, and the trolley is trapped by it forever.

Everything else in this field follows from that. The forbidden direction is not a fixed thing to be got around; it is attached to the mechanism, and moving the mechanism moves it. What a manoeuvre does is move the forbidden direction out of the way and then take the opportunity, and what the bracket measures is how much opportunity there was.

A last remark on the economy of the test, since it is what makes the pointwise character bearable. Evaluating it costs two directional derivatives and a subtraction at one configuration, which is microseconds, so checking everywhere means checking on a grid and is entirely affordable on a mechanism with three coordinates. What it is not affordable on is a mechanism with twenty, and there the test reverts to what it is on paper: an argument made once at a generic configuration, with the degenerate set identified by hand from where the constraint rows lose rank. That is the ordinary trade between a computation and a proof, and it is worth knowing which of the two is standing behind any particular claim of nonholonomy.