A wheel that cannot report its radius

Every field so far has divided into readings that carry a length and readings that do not. A wheel encoder sits awkwardly across that division and the awkwardness is the subject.

The reading is an angle and it means a distance #

A wheel encoder reads the wheel’s rotation, which is an angle and is dimensionless.

The rolling condition says that the contact point does not slip, so an angle φ of wheel rotation corresponds to a distance rφ travelled, where r is the wheel’s radius. The reading is dimensionless and its interpretation has a length in it.

That is different from every other angle sensor on this site. A protractor on a four-bar’s output link reads an angle that is the quantity of interest; a wheel encoder reads an angle that stands for a distance, and the standing-for involves a parameter.

So the standing argument — an angle-only measurement cannot recover a size — needs re-running rather than quoting, because the premise about what the readings are is different.

Running the argument again #

The vehicle’s parameters are the wheel radius r and the wheelbase L, both lengths. Its readings are two wheel rotations, both dimensionless.

Scale the vehicle: double r and double L. Drive it through the same sequence of wheel rotations. The path it traces is the original path doubled — same shape, twice the size — and the two encoder readings are identical.

That is worth checking rather than asserting, because the claim is about a physical operation. Doubling the radius doubles the distance each wheel rotation carries the vehicle. Doubling the wheelbase halves the heading change each unit of wheel difference produces. The two effects are separate and both are needed: a vehicle with a doubled radius and an unchanged wheelbase turns differently, and one with a doubled wheelbase and an unchanged radius does too. Only doubling both leaves the readings alone.

So the readings cannot distinguish a vehicle from a scaled copy of it, exactly as before. One direction of the two-dimensional parameter space is invisible, and what is recoverable is one combination: the ratio L/r.

That is the wheelbase measured in wheel radii, and it is recovered exactly. What is not recovered is either length in metres.

The vehicle’s parameters are the wheel radius r and the wheelbase L, both lengths. Its readings are two wheel rotations, both dimensionless.

That is the wheelbase measured in wheel radii, and it is recovered exactly. What is not recovered is either length in metres.

Where the length hides #

It is worth being exact about where the length is in the reading, because the sentence “an angle that means a distance” can be read as loose talk.

A wheel encoder produces a count. The count is converted to a rotation by dividing by the counts per revolution, which is a pure number written on the encoder. The rotation is converted to a distance by multiplying by the radius, which is a length nobody wrote on anything.

So the chain from raw data to a distance has exactly one dimensional step in it, and that step’s parameter is the wheel’s radius. Everything else in the chain is integers and angles.

That is why the odometry’s scale error is a pure multiplier: it enters at one point, multiplies everything downstream, and is common to every reading the wheel ever produces. An encoder that miscounts would be a different problem with different symptoms; a radius that is wrong is one number wrong everywhere.

One dimensional conversion means one scale factor, and the practical consequence is that the whole of an odometry system’s size calibration is one number to determine and one number to get wrong.

Which is the practically important half #

Here the answer differs from the four-bar’s, and it differs in the direction that matters.

For a four-bar, what a designer wants is mostly ratios and the missing size is a nuisance rather than a problem. For a vehicle, what is wanted is a distance travelled in metres, and that needs the wheel radius in metres, which the odometry cannot supply.

So dead reckoning from wheel encoders alone gives a path whose shape is exact — every turn, every heading change, every relative position — and whose scale is whatever the assumed wheel radius says. A vehicle whose tyres are five per cent worn believes it has travelled five per cent further than it has, consistently, for ever.

The error is a scale factor rather than a drift, which is worth distinguishing because the two are usually lumped together as odometry error. A scale factor accumulates proportionally with distance and never averages out; a random drift accumulates as a square root and does.

The ratio is what a vehicle can calibrate #

The recoverable combination L/r is exactly what a vehicle’s own manoeuvres determine, and there is a classical procedure that finds it.

Drive a closed loop and return to the starting heading. The heading change is the integral of the difference between the two wheels’ rotations, divided by the wheelbase in wheel radii — so a manoeuvre that ends where it started in heading gives one equation in L/r.

Drive several and the ratio comes back well determined. Nothing external is needed, no fixture and no reference: this is self-calibration with the same structure as a four-bar’s two encoders, and the same limitation, arriving at a different-looking quantity.

A vehicle can measure its own shape and not its own size, which is the sentence this whole survey keeps producing, and here the shape is one number.

Two wheels and what they separate #

The differential-drive case repays going through in full, because the two readings separate into two quantities with different fates.

Call the two wheel rotations φ_L and φ_R. Their sum determines the distance travelled: (r/2)(φ_L + φ_R), which has the radius in it and is therefore known only up to the scale.

Their difference determines the heading change: (r/L)(φ_R − φ_L), which has the ratio r/L in it — and that is dimensionless.

So one of the two combinations a differential drive produces is dimensionless and one is not, and the dimensionless one is the heading. A vehicle’s belief about which way it is pointing is therefore recoverable exactly from its own sensors, and its belief about how far it has gone is not.

That is a sharp and practically important split. Heading is a shape and distance is a size, on a machine where both come from the same two encoders, and it explains why a robot that has calibrated its wheelbase-to-radius ratio navigates in the right shape while drifting in scale.

It also says which errors compound. A heading error rotates every subsequent displacement and compounds badly; a scale error multiplies them and compounds benignly. The vehicle can fix the first from its own data and cannot fix the second.

What is scale-free here #

Running the field’s own quantities through the probe gives the usual two columns and one entry that belongs to neither.

The constraint itself is a statement that a velocity component is zero. Scaling every length scales every velocity, so a zero stays zero: the constraint is scale-free, and a scaled vehicle is subject to the identical constraint.

Whether the constraint integrates — the field’s central result, that a rolling condition does not reduce to a condition on positions — is a statement about the Lie bracket of two vector fields being outside their span. That is a rank condition, dimensionless, and it holds or fails independently of size.

The holonomy — how much a vehicle’s configuration shifts after a closed manoeuvre — is a length, exponent one, since it is a displacement.

And off-tracking, how far a trailer’s path falls inside its tractor’s, is a length whose ratio to the wheelbase is a shape.

So the field’s structural results are scale-free and its measured displacements are lengths, which is the ordinary pattern. What is unusual is only the sensor.

What a trailer adds #

A tractor and trailer is the field’s richer object and its parameters sort the same way with one extra number.

The trailer’s hitch offset and its own wheelbase are two more lengths. The articulation angle between the two bodies is a configuration, dimensionless, and often sensed.

So a tractor-trailer rig with wheel encoders and an articulation sensor has four lengths and readings that are all dimensionless. One direction invisible, three combinations recoverable — the same one-dimensional loss on a bigger parameter space, which is the pattern the spatial field also shows.

And off-tracking, the field’s own signature quantity, is a length: how far inside the tractor’s path the trailer’s falls. Its ratio to the trailer’s wheelbase is a shape, and quoting it that way is what makes an off-tracking figure transferable between a car with a caravan and an articulated lorry.

A rig’s own sensors determine its proportions and not its size, and a rig’s proportions are exactly what decide whether it can make a given turn — so the recoverable half is again the half that matters for the geometry and the unrecoverable half is the one that matters for the metres.

A rank condition has no size #

Worth its own paragraph because it is the field’s headline result and it lands in the third class rather than in either column.

The Lie bracket test asks whether two vector fields’ bracket lies in their span. That is a rank, an integer, and it is either two or three — locally constant in the parameters, unchanged by any scaling, and unchanged by a five per cent perturbation of anything.

So it is a count, with all a count’s properties: recoverable from any observation whatever, and with no error bar unless somebody reports the margin.

Its margin is the smallest singular value of the matrix whose rank is being taken, and that quantity has dimensions — it is a rate — so the margin is a size and the count is not. Same structure as the Grashof class and its margin, in a completely different field, which suggests the pattern is general rather than a coincidence of the two.

Two wheels, two lengths, one number #

The counting is worth doing because it says what a richer vehicle recovers.

A differential-drive vehicle has two parameters — a wheel radius and a wheelbase — and its readings determine one combination. One of two.

A vehicle with two differently-sized wheels has three parameters, two radii and a wheelbase, and its readings determine two combinations: the ratio of the radii and the wheelbase in units of one of them. Two of three.

A vehicle with a steering angle sensor adds a dimensionless reading and no parameter, so it determines more combinations without adding to the invisible direction.

In every case the invisible direction is exactly one-dimensional, because a scaling is a one-parameter group whatever the mechanism. The fraction of a vehicle’s parameters it can recover from its own sensors rises with the parameter count, which is the same conclusion the spatial field reaches by a different route.

Wear makes both drift, differently #

A last practical case, because it is the one that keeps a calibration from being a one-off.

A tyre wears and its radius falls. The wheelbase does not change, so both recoverable and unrecoverable quantities move: the scale factor changes because r changed, and the ratio L/r changes because r changed and L did not.

The vehicle can detect the second from its own closed manoeuvres, and detecting it is diagnostic of the first — a ratio that has drifted by one per cent says the radius has fallen by one per cent, so the scale factor has moved by one per cent too.

That is a pleasant piece of leverage. The recoverable quantity carries information about the unrecoverable one, provided the wheelbase can be assumed constant, which for a rigid chassis it can.

It works only because the two share a parameter. It would not work if the scale factor and the ratio depended on disjoint sets of parameters, and there is nothing general about it — it is a fact about this particular vehicle’s two-parameter model, and worth having precisely because such leverage is rare.

What fixes it #

One measurement with a length in it, as always, and here it has a familiar name.

Drive the vehicle a measured distance — along a marked track, between two surveyed points, past a wheel-revolution counter — and the wheel radius follows from the encoder count. After that everything the odometry says is in metres.

That is exactly the calibration a wheeled robot receives, and it explains why: not because odometry is imprecise, which it is not, but because it is structurally unable to produce a length. A scale factor cannot be estimated from data that contains no scale, and one measured distance supplies it once and for ever.

The practice also explains why the calibration has to be repeated as tyres wear. The ratio L/r drifts too, since r changes and L does not, so the shape of the odometry degrades as well as its scale — which the vehicle can detect for itself from closed manoeuvres and the scale factor it cannot.

The site’s own vehicle figures #

Worth checking the collection against the argument, since the field draws vehicles constantly.

Every rolling figure on this site is drawn at one scale, with the wheelbase and wheel radius set to stated numbers, and every quantity in every caption is quoted in those units. That is correct and it is a statement about one vehicle.

The quantities that would transfer to a vehicle of another size are the dimensionless ones: the ratio of wheelbase to wheel radius, the heading change per unit of encoder difference in wheel radii, the off-tracking as a fraction of the wheelbase, the fraction of the configuration space reachable within a given number of manoeuvres.

None of those is what the captions quote. That is not an error — the figures are about the vehicle they draw — and it does mean a reader wanting to apply a result to a different vehicle has to redo the arithmetic.

The dimensionless form is the transferable one and the site prints the dimensional one, which is the third field in this survey to produce that observation and is beginning to look like a fleet-wide reporting habit rather than a field-specific oversight.

What a vehicle can learn about itself #

A wheel encoder reads a dimensionless number — turns — and means a distance, and the length that converts one to the other is the wheel’s radius. That is a case the site’s standing argument did not cover and it obeys it anyway: one invisible direction, one recoverable combination, and a length needed from outside.

What makes the case worth its own essay is that the recoverable combination is useful on its own, which is not true everywhere. A two-wheeled vehicle’s odometry depends on the wheelbase and the wheel radii, and what its own manoeuvres determine is the wheelbase measured in wheel radii — a pure number, recovered from nothing but the vehicle driving itself in circles, with no rule, no marked track and no surveyor.

That number is most of what a vehicle needs. It is what converts differential wheel counts into a heading change, so a robot that has calibrated it tracks its own orientation correctly however wrong it is about distance. Its position accumulates a scale error and its shape of trajectory is right — a square driven with an uncalibrated radius is a square of the wrong size and not a parallelogram. The error is in the units and not in the geometry, which is exactly the standing result and is unusually easy to see here because a reader can picture the trajectory.

The absolute scale needs one external length and there is no way round it. Roll the vehicle a measured distance, or drive it between two marks, once: everything else follows. That is the same measure one length once the four-bar and the platform both need, and the vehicle case is the one where it is most obviously cheap.

There is a complication the other cases do not have, and it is the reason odometry calibration is a maintenance task rather than a commissioning one. Both quantities drift, and they drift differently. A wheel wears and its radius falls slowly; the wheelbase does not change at all unless something is bent. So the dimensionless combination stays valid for a long time and the absolute scale does not, which means the two want re-measuring on completely different schedules — and a procedure that re-runs the whole calibration when only the scale has moved is doing several hours of work to correct one number.

The field’s structural results are untouched by any of it. Whether a constraint integrates, what the reachable set looks like, how many freedoms a vehicle has: rank conditions on vector fields, integers, and no scaling touches an integer. That is why a result about a toy car applies to a lorry, and it now has an argument behind it rather than an intuition — the generality is a consequence of the quantity’s class, and the quantities that do not transfer are exactly the ones with an exponent.