A constraint that takes nothing away

Every mechanism on this site so far has been described the same way. There are coordinates; there are equations in those coordinates; a configuration is a solution of the equations, and the set of solutions is what the mechanism can do. A four-bar’s loop closure is two equations in four joint angles, so its configurations form a curve. A Gough platform’s is six equations in twelve, so its configurations form a six-dimensional set. Solve, and the mechanism is somewhere.

A rolling wheel is not describable that way, and the reason is worth being precise about before anything else in this field is attempted.

What a wheel forbids #

Put a wheel on a plane, upright, free to roll and free to be steered. Three numbers say where it is: the contact point $(x, y)$ and the heading $\theta$ the wheel is pointing.

Now say what rolling without sliding means. It means the material point of the wheel that is touching the ground is instantaneously stationary — the wheel is not scrubbing across the floor. For an upright wheel that reduces to a single statement about the contact point’s velocity: it must lie along the wheel, not across it.

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

That is the entire physical content of the wheel, and the thing to notice is what kind of statement it is. It is a statement about $\dot{x}$, $\dot{y}$ and $\theta$ — velocities and a coordinate — and it is not a statement about $x$, $y$ and $\theta$. Nothing in it says where the wheel may be. It says how the wheel may move from wherever it is.

The two things a wheel can be told to do are drive and turn, and each is a direction in the three-dimensional space of configurations:

$$F_{\text{drive}} = (\cos\theta,\ \sin\theta,\ 0), \qquad F_{\text{turn}} = (0,\ 0,\ 1).$$

Both satisfy the row: driving moves along the heading, and turning moves neither $x$ nor $y$ at all. Any combination of them satisfies it too, and nothing else does. So the permitted velocities at any configuration form a plane inside a three-dimensional space, and the wheel has two freedoms in exactly the sense the mobility field established: three coordinates, one independent constraint, two left.

The thing that is new #

Here is the sentence that separates this field from every other one on this site.

The constraint removes a direction of motion and removes no dimension from the set of reachable configurations.

A wheel may not go sideways. A wheel can also be brought from any position at any heading to any other position at any other heading — parked in a bay it did not start in front of, facing a way it did not start facing. Nothing is out of reach. The set it can reach is the whole three-dimensional space, and the constraint that forbids one of the three directions has not made any part of that space unreachable.

That is not obvious and it is not universal, which is why it is the subject of a field rather than a remark. There are constraints of exactly the same shape — one linear condition on the velocities, with coefficients that depend on the configuration — that do remove a dimension. The next essay is about a mechanism whose constraint differs from this one by a single character and which cannot leave a line.

Why the usual reasoning fails #

The reason a constraint normally costs a dimension is that it can be integrated. If a mechanism obeys $\dot{f}(q) = 0$ for some function $f$, then $f(q)$ is constant along every motion, and the mechanism is stuck on the level set $f = f_0$ forever. It began somewhere and it can never get to a configuration where $f$ has a different value. One equation, one dimension gone.

The wheel’s row is not the derivative of anything. There is no function of $(x, y, \theta)$ whose rate is $\dot{x}\sin\theta - \dot{y}\cos\theta$, and therefore no quantity the wheel has to conserve, and therefore no level set it is confined to. A constraint of that kind has a name — nonholonomic — and the name is worth less than the test, which is Frobenius’ theorem and which this field measures rather than invokes.

The everyday version of the same fact is that a car can be parked. The constraint is real: the car genuinely cannot translate sideways, and anyone who has tried to slide into a tight space knows it. What the constraint does not do is make the space unreachable. It makes it expensive, and the price has an exponent, which is the fifth essay in this field.

What the count means now #

The mobility field’s whole argument was that a mechanism’s freedoms are measured — as the dimension of the null space of the constraint Jacobian — rather than counted from a formula, and that the measurement is right whenever the formula and it disagree. That argument still holds here. What changes is that the measured number no longer answers the question anybody was asking it.

There are now two numbers:

• The velocity freedoms. The dimension of the space of permitted velocities at a configuration: three coordinates minus the rank of the constraint rows. Two, for a wheel.
• The reachable dimension. The dimension of the set of configurations the mechanism can be driven to. Three, for a wheel.

On every mechanism drawn on this site before this field the two were the same number, which is why nobody had to say which one mobility meant. A four-bar has one velocity freedom and a one-dimensional set of configurations. A Gough platform has six and six. The word was unambiguous because the ambiguity had never been tested.

The gap between the columns is not an error in either of them and it is not a defect in the wheel. It is what a velocity constraint is. The velocity count is exactly right about velocities: at any instant, a wheel really can only do two things. The reachable count is exactly right about reachability: over any interval, a wheel can get anywhere. Both are true and they are answers to different questions, and the reason this site had never needed to distinguish them is that until a wheel appeared, no mechanism it had drawn could tell them apart.

The mechanisms this field holds #

Six objects, and they differ from one another only in their coordinates and their fields. Nothing else in this field’s machinery knows what any coordinate means.

The wheel, above. Three coordinates, one row, two controls.

The differential drive — two wheels on a common axle, each with its own speed. Kinematically it is the wheel: the same three coordinates and the same forbidden direction. What is different is which pair of controls a real machine has. Driving and turning are combinations of the two wheel speeds rather than things anybody commands directly, and that is what makes its error budget computable: the wheels have radii, the radii have tolerances, and every one of them lands on the heading.

The car. Four coordinates, because the steering angle is part of the configuration rather than a control: what a driver holds is the steering rate. Two rows, because a car’s rear axle may not slide sideways and its heading is tied to its steering angle by the wheelbase. Two controls, drive and steer.

The car and trailer. Five coordinates, three rows, and still two controls. The trailer’s heading is dragged rather than driven, at a rate proportional to the sine of the angle between the two — one sine, from which the jackknife and the reversing instability both follow.

The ball. Five coordinates: a contact point and an orientation, an orientation being three numbers. Two rows, because the contact point’s velocity is fixed by the spin. Three controls if the ball may be twisted about the vertical through its contact — which is allowed, and is why a ball has three freedoms where a wheel has two — and only two if it may not.

A figure here is reached, not solved #

Every family of figures on this site draws mechanisms at configurations that are roots of something. A linkage is placed by Newton–Raphson on its loop closure; a spatial loop by Newton on the logarithm of its closure transform; a gearset’s speeds by the null space of an integer matrix. In each case a figure may not show a configuration the equations refuse, because there are no coordinates to draw it from.

There are no equations to solve here, and the premise survives in a stronger form rather than a weaker one. The figures in this field are drawn by integrating: from a configuration already reached, along a velocity the rows permit, for a stated time. The coordinates in the picture are the integrator’s output and nothing else, and there is no way to write down a configuration by hand even if somebody wanted to — the only route into the space is a history.

That makes the drawn residual a different quantity from the one the rest of the site reports. Elsewhere it is how far from a root; here it is how far off the constraint surface a trajectory has drifted, which is an integration error rather than a modelling one, and the fourth-order integrator used throughout keeps it at the floor of double precision over the whole of every history drawn.

Two routes, kept apart on purpose #

The site’s standing habit is that every claim gets a second, independent computation that could disagree with it. In the mobility field it was Grübler’s formula against the rank of a Jacobian; in the gear field it was a closed-form ratio against an elimination in exact rationals.

Here it is the rows against the fields. The rows are written from what the mechanism may not do; the fields are written from what it may; and the two are separate pieces of arithmetic derived from the same picture by different reasoning. The check is that every field is annihilated by every row — that the permitted directions are exactly the forbidden direction’s complement — and it is made at two dozen configurations for every mechanism in the field. The largest product between any row and any field, over all of them, is at the level of rounding.

That is worth more than it looks. A sign error in a constraint row produces a mechanism that is some mechanism, drawn perfectly, obeying a condition nobody intended — the kind of failure this site has met before, where a velocity solve came out reversed and every figure still drew. Writing the fields from the geometry rather than deriving them from the rows means an error in either shows up as a product that is not zero.

What a wheel is, as a piece of vocabulary #

One more thing is worth settling before the field proceeds, because the word rolling has already appeared on this site in a different sense.

Where the coupler is turning says that a coupler’s motion is the rolling of one curve on another — the moving centrode on the fixed one — and calls that rolling without slipping. That is a description of a planar motion, arrived at after the motion is known: given a four-bar and its solved sweep, the two centrodes can be computed and they touch at the instant centre and they do not slide against each other. It is a theorem about a motion that was already determined by a loop-closure equation.

The rolling here is the opposite kind of statement. It is a constraint imposed in advance, which decides what motions are available rather than describing one that already exists. The four-bar’s centrodes roll because the four-bar moves the way its bars make it move; the wheel moves the way it does because rolling is what it is required to do.

The two meet in one place, and it is worth saying where: the instant centre. A rolling wheel turns about some point on its own axle line, which is the instant centre of the body it is attached to, and when several wheels are attached to one body every one of their axle lines has to pass through the same point. That is the whole of steering geometry, and it is a rank condition on a matrix rather than a formula anybody has to remember.

What is outside this field #

Two things, stated here because eight essays in this field would otherwise have to keep saying them.

Force. A rolling constraint is maintained by friction, and friction has a limit; past it the wheel slides and the row is simply not true any more. None of that is here. Every result in this field holds with every force in the mechanism unknown, and the separating test is the one the site has used since the transmission field’s ruling on torque: if an argument needs to know what is pushing, it is somebody else’s argument. What a locked axle scrubs in a turn is computable and is computed; which wheel spins on ice is not.

Planning. Given that a wheeled mechanism can reach everywhere, the question of how to choose a route — through a car park, around obstacles, under a time budget — is a search problem and belongs to whoever owns searches. What is claimed here is the set itself: its dimension, how many nested manoeuvres are needed to fill it, and the closed-form shortest path under a curvature bound, which is a classification of six shapes rather than a search through anything. The same boundary was drawn when the configuration space of an arm was taken as a set rather than as something to search.

The shape of what follows #

The field is short and it is ordered by a single question: given that the forbidden direction can be reached, what does reaching it cost?

The answer turns out to be an integer — the number of nested manoeuvres required — and every mechanism here has one. A trolley’s is one, a car’s is two, a car and trailer’s is three, and a ball that may not be twisted has two despite having five coordinates and only two controls. The integer is computable without ever attempting a manoeuvre, and it is also measurable by attempting one and watching how the gain shrinks as the manoeuvre does. Those are the two routes, they are computed from nothing in common, and where they meet is where this field’s arguments live.

The rest is consequences. A towed axle obeys one line of algebra and it is why a lorry needs a wide turn. Two tyre tracks in mud contain enough information to say which way the bicycle went, and the test is a measurement rather than an eye. A car that may not reverse pays a whole circle for its first millimetre sideways, and pays exactly the same for four turning radii. A ball rolled round a closed loop comes back turned by the loop’s area. And a machine with rollers in its wheels has no forbidden direction at all, which is worth a section of its own because it shows what the constraint was doing by removing it.

There is a compact way to say what makes this field different from every other one on the site, and it is worth having because it is the sentence a reader should carry into all the essays that follow. Everywhere else, a constraint removes a coordinate: the mechanism’s configuration space is smaller than its coordinate list, and the count of freedoms and the dimension of the reachable set are the same number. Here a constraint removes a direction and leaves every coordinate in place, so those two numbers come apart — and every essay in the field is a consequence of the gap between them. The gap is why parking takes a manoeuvre, why the manoeuvre costs an exponent, why a growth vector is needed at all, why a trailer is harder than a car, and why a jackknife is not a singularity. One distinction, made once, and the whole field falls out of it. That is unusual enough to be worth saying plainly: most fields on this site are a subject with several ideas in it, and this one is a single idea with several consequences.