Not unreachable, only expensive

Six sentences, each of which is said confidently about wheeled mechanisms, and each of which is either false or is a true statement about something else.

They are worth taking together rather than one at a time, because they share a structure: a correct fact about what a mechanism can do at an instant is read as a fact about what it can do over an interval, or the reverse. The two questions have had the same answer on every mechanism this site drew before wheels appeared, which is why the habit of not distinguishing them is so easy to acquire and so hard to notice.

“A constraint means a direction that cannot be reached” #

False, of every mechanism in this field except one.

A rolling wheel cannot move sideways at any instant and can be brought to any position at any heading. The two statements are compatible because the forbidden direction is attached to the mechanism rather than to the plane: turning the wheel turns the forbidden direction, so a mechanism that can turn can change which direction is forbidden, and the leftover of doing the two in the wrong order points exactly where the constraint said it could not go.

The one exception is the trolley on a rail, whose constraint row has a fixed angle in it rather than a coordinate. Its forbidden direction never moves, it has an invariant that stays constant to $2\times10^{-15}$ over any control history, and it cannot leave its line. Same count, same shape of row, opposite answer.

So the sentence is not merely usually wrong; it is wrong in a way that no amount of counting could correct, because the mechanism it is true of and the mechanisms it is false of are indistinguishable to every count.

“A car has two degrees of freedom, so its configurations form a two-dimensional set” #

The premise is true and the conclusion does not follow.

A car has two velocity freedoms; there are two things a driver controls, and at any instant the mechanism can do a two-parameter family of things. Its configurations form a four-dimensional set: two for position, one for heading, one for the steering angle, and all four are reachable.

The inference from the first to the second is valid whenever the constraints integrate, which is a theorem rather than an assumption — and it is exactly the theorem that fails here. Every essay on this site before this field was entitled to make that inference silently, and an essay of its own is given to what has to change now.

“A jackknifed lorry is stuck because the mechanism has gone singular” #

Misapplied. There is no singularity there.

The growth vector of a car and trailer is $2, 3, 4, 5$ at a hitch angle of zero, of forty-five degrees, of ninety, of a hundred and thirty-five and of a hundred and eighty. Nothing about the reachable set changes as the rig folds; every configuration remains reachable from every other; and a rig at 180° is as manoeuvrable, in the sense this field measures, as one in line.

What a jackknifed rig has run out of is room to drive forwards, which is where a hitch angle is undone — at a rate of $e^{-s/d}$ in the distance driven. The situation is a configuration plus an absence of space, and the space is not in any of this field’s arithmetic.

The reason the wrong explanation is attractive is that this site has plenty of genuine singularities and they behave the way the word suggests: a four-bar at a toggle really does lose the ability to be driven, a parallel platform inside its own workspace really does gain a motion nobody commanded, and an arm at a wrist singularity really does lose a direction of tool velocity. All three show as a rank drop in a matrix. The jackknife shows as nothing at all, and the measurement is what says so.

“Since a car can be parked, the constraint does not really restrict anything” #

The overcorrection, and it is as wrong as the claim it corrects.

The constraint costs an exponent, which is the most expensive thing a constraint can cost short of forbidding something outright. A car with 0.6 m of room to shuffle in gains 144 mm per cycle; with 0.3 m it gains 36 mm; with 0.15 m it gains 9 mm. The gain is quadratic in the room, so the number of cycles is quadratic in its reciprocal, and the distance driven is linear in it — which goes to infinity as the room goes to zero.

So a constrained mechanism reaches everywhere and pays for it at a rate that has no upper bound. Both halves of that are needed and dropping either produces one of the two misconceptions on this page.

The exponent itself is where the interesting variation is. A trolley’s sideways direction is second order in the amplitude and a car’s is third, and one integer is the difference between a manoeuvre that is a nuisance and one that is a job.

“Nonholonomic mechanisms are an exotic special case” #

False, and the belief is maintained by the way the subject is usually introduced.

The standard presentation gives a short list of examples — the rolling wheel, the rolling ball, the ice skate — which reads as a list of curiosities. It is not a list of curiosities; it is a list of the simplest instances of a property that every wheel has. A shopping trolley, a bicycle, a wheelbarrow, a pram, a car, a lorry, a wheelchair, a hospital bed, a warehouse robot and a suitcase on castors are all mechanisms whose constraints do not integrate. The exotic case is the mechanism that has an integrable constraint and rolls — a train, which is the trolley on a rail with a real rail under it.

The consequence of thinking of the property as rare is that its arithmetic gets treated as a special technique rather than as the ordinary description of an ordinary machine. Every wheeled machine anybody designs has a growth vector, and reading it is how many nested manoeuvres that machine will need.

“A trailer is hard to reverse because it is unstable” #

True, and it is only half the reason, and the two halves are independent.

The instability is real: a hitch angle grows as $e^{s/d}$ while reversing, doubling every 4.16 m for a six-metre trailer. That is one difficulty and it is about keeping the rig straight.

The other is that the trailer’s position is one bracket deeper than the car’s — a growth vector of $2, 3, 4, 5$ against a car’s $2, 3, 4$ — so putting the trailer somewhere particular costs a higher power of the room available. That is a difficulty about getting the rig somewhere, and it would be there even if the mechanism were perfectly stable.

Separating them matters because they have different remedies. The instability is fixed by feedback: watching the trailer and correcting, which is what a driver does and what a reversing assistant does electronically. The depth is not fixed by anything — it is a property of the mechanism’s geometry, and no amount of skill or automation removes a power from an exponent.

What makes them the same mistake #

Four of the six confuse two questions that had one answer everywhere else:

• what the mechanism can do now, which is a rank computation on the constraint rows;
• what the mechanism can be brought to eventually, which is a question about brackets.

The first three take the first answer for the second. The fourth takes the second for the first — noticing that everything is reachable and concluding that the instantaneous restriction is unimportant. The remaining two are of a different kind: one is a fact about how the subject is taught, and one is a real cause mistaken for the whole cause.

The pattern is one this site’s misconceptions keep having. A ratio quoted as a number is an instantaneous statement read as a statement about a cycle. Two things called jamming are two configurations 222° apart on the same linkage, given one name. A gearset’s ratio is a plane of permitted motions read as a single number. In each case the wrong sentence is not invented; it is a true sentence about an adjacent object.

Why the confusion is structural rather than careless #

It is worth saying plainly that none of the six sentences is a foolish thing to believe, because the alternative reading — that people are careless about mechanisms — is both unkind and wrong.

Every one of them is the correct reading of a mechanism with an integrable constraint, and until this field, every mechanism on this site had one. A four-bar’s velocity freedoms and its configuration dimension are both 1. A Gough platform’s are both 6. Six fields of essays were written in which the distinction could not arise, and the vocabulary that grew up in them has one word where two are needed.

That is the same situation the transmission field found with the word ratio: it means one thing on a mechanism with one input and something else on a mechanism with two, and nobody had needed to say which because until then every gearbox anybody wrote about had one input at a time. The word was not being used carelessly; it had been adequate for as long as the objects were.

The remedy in both cases is the same and it is not vigilance. It is to compute both quantities, print both, and let the gap between them be visible on the page — which is what the ledger in this field is for.

What would settle each of them #

The site’s habit is that a claim is worth what the test that could refute it is worth, so it is worth saying what each of these turns on.

The first turns on the involutivity test — a number between zero and one, computed at a configuration, that comes back at $1.000$ for every mechanism in this field and $0$ for the rail. Seventeen orders apart, with nothing in between.

The second turns on the growth vector, whose first and last entries are the two quantities being confused. It is a sequence of ranks and a mismatch in either end would show as an integer that does not match the exponents.

The third turns on the same vector computed at five hitch angles. Any one of them differing would have made the word singular correct.

The fourth turns on a fitted exponent, measured over a decade of amplitudes and coming out at 1.999 for a car’s shuffle against a closed form that predicts 2.

Each of those is a number that could have come out differently, and the discipline is that they were computed before the sentences above were written rather than after.

How to tell, on a mechanism nobody has analysed #

The four tests above are the ones this field runs, and it is worth reducing them to something a reader can apply to a machine in front of them, since the point of a field is not the mechanisms in it.

Count the controls. One control means a curve, always, with no further work. That is the only case where the answer needs no computation.

Ask whether the forbidden direction moves with the mechanism. If the constraint’s coefficients contain one of the mechanism’s own coordinates, the forbidden direction turns as the mechanism turns and the constraint will not integrate. If they contain only fixed numbers, it will. That is not a proof — a mechanism can have coordinates in its coefficients and still be integrable — but it is where to look, and it is the difference the second essay is built on.

Count coordinates against controls. The gap is how many entries the growth vector has to add, and each entry is another power of the amplitude. A machine with five coordinates and two controls will need at least three nested manoeuvres and its worst direction will cost the cube of whatever room it has.

Then measure. The bracket is a few lines of finite differences and the involutivity test returns a number between zero and one. On the mechanisms in this field it has never returned anything but an endpoint.

Which direction each of them errs in #

The six sort into two groups by the direction of the mistake, and the sorting says something about who makes which — which is more useful than the individual corrections, because it says where to expect the next one.

Three of them overstate the constraint. A direction that cannot be reached, a two-dimensional set of configurations, a singular configuration: each takes a correct statement about an instant and extends it to an interval, and each therefore credits the constraint with more power than it has. All three are the natural reading for somebody who has learned to compute a rank and has not yet been given a reason to want a second number.

One of them understates it. Since a car can be parked, the constraint does not really restrict anything is the overcorrection, and it is worth separating out because of who makes it. Nobody arrives at that sentence first. It is said by somebody who has already learned that the first three are wrong, has drawn the correct conclusion that reachability is unrestricted, and has drawn the incorrect one that nothing is therefore lost. The second lesson undoes the first if it arrives without the exponent attached.

That is the pattern worth carrying, and it is not specific to wheels. A correction that removes a false limit tends to remove the true cost with it, because both were being carried by the same wrong sentence. The remedy is that the corrected claim has to carry a number: not the constraint does not forbid the direction but the constraint does not forbid the direction and charges $\varepsilon^2$ per manoeuvre for it, which is a sentence nobody can overcorrect from.

The remaining two are of different kinds again. Nonholonomic mechanisms are exotic is not a mistake about a mechanism at all; it is a mistake about a population, produced by a teaching tradition that lists examples rather than stating a test. And a trailer is hard to reverse because it is unstable is the only one that is true as far as it goes, and its fault is incompleteness rather than error.

So the six are one over-reading made three times, one overcorrection, one sampling error and one incomplete truth — which is a more interesting distribution than six wrong sentences, and it says the field’s exposure is not uniform. The over-readings are caught by the growth vector, the overcorrection by the exponent, the sampling error by the involutivity test applied widely, and the incomplete truth only by having both halves computed.

Which is why the field computes all of them on every mechanism it draws rather than reaching for whichever one the argument at hand needs. Four numbers, always, is the cheapest defence against a list of six mistakes with four different shapes.

The one that is true #

It is worth ending on the claim that survives, because a page of refutations reads as though nothing can be said.

A mechanism with one control cannot reach anything but a curve. That is true, it is not a matter of degree, and it does not need a test — a single vector field has no bracket with anything, so the sequence stops at its first entry and the reachable set is one-dimensional. A car with its steering clamped runs along one arc of one circle and stays there forever.

That is the correct version of there is somewhere it cannot get to, and it is why every mechanism in this field has at least two controls. What makes a wheeled mechanism go anywhere is not the wheel; it is having two things to do and the wit to do them in the wrong order.