Machine Kinematics
solved, then drawn
Concept

Rolling constraint — where it appears

The condition that a wheel's contact point is instantaneously still, which forbids one direction of the body's velocity and removes no coordinate. It is the constraint that makes a wheeled vehicle interesting: the wheel may not go sideways and can still be parked anywhere at any heading, so counting freedoms and counting reachable dimensions give different answers.

Named by 19 essays across 5 fields — each of them below, with the objects they name alongside it.

A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0.

A constraint that takes nothing away

A rolling wheel forbids one direction of motion and removes no coordinate from the mechanism's description. It cannot slide sideways and it can still be brought to any position at any heading — and the gap between those two sentences is the whole of this field, because in every mechanism built of pins and slides the two agree.

rolling · Rolling Which of these constraints is secretly about positions. How much of the bracket of two permitted directions lies outside the permitted directions, as a fraction of its own length. Frobenius' theorem says a distribution is the tangent field of a family of surfaces exactly when this is zero, so the test needs no integration and no recognition. The scale is logarithmic because the answers are seventeen orders apart: the rail returns nothing at all and everything else returns essentially the whole bracket. There is no mechanism in the middle.

One character apart

Two mechanisms with three coordinates, one constraint row of the same shape and two controls each. In one of them the angle in the row is a coordinate; in the other it is a constant. The first can be driven anywhere and the second can never leave a line, and Frobenius' theorem decides which is which without integrating anything.

rolling · Rolling Four legs that do not cancel. Drive forward, turn, drive back, turn back — each leg exactly as long as the one it is undoing. The mechanism does not come home. What is left over is 1149.9 mm at an amplitude of 1.10, and it points along the direction the wheel forbids. The gap and the computed bracket are 31.51° apart here and 3.15° apart at a tenth of this amplitude — the agreement is a leading-order statement and the departure is the third-order remainder, which falls with the manoeuvre rather than staying put. Every point on the path was reached by a permitted velocity, so nothing here cheats; the sideways motion is assembled out of motions that are not sideways.

The motion left over by going nowhere

Drive forward, turn, drive back the same distance, turn back the same angle. Every leg is undone by another leg and the mechanism does not come home — it has moved sideways, in the one direction it is forbidden to move in. The leftover has a name, a formula, and a measured exponent of 1.997.

rolling · Rolling How many wiggles it takes. The growth vector: how many independent directions are available after one bracket, two, three. The first number is what the constraints leave and the last is the dimension of the configuration space, so the length of the row is how deep the manoeuvring has to go. A car needs one bracket more than a trolley and a car with a trailer one more again — and the ball changes by one depending only on whether it may be twisted.

How many wiggles

A bracket of two permitted directions may point somewhere new; the bracket of that with a permitted direction may point somewhere newer still. How deep the process goes before it stops is an integer — 2·3 for a wheel, 2·3·4 for a car, 2·3·4·5 for a car and trailer — and the same integer turns up as the exponent of a manoeuvre nobody told it about.

rolling · Rolling The parking shuffle, three times over. Forward on left lock, forward on right lock, back on left lock, back on right lock — and repeat. The heading comes back to where it started and so does the position along the road, both exactly and at every leg length, because the four legs are a symmetric set. What is left is 255 mm of pure sideways translation per cycle at a leg of 0.80 m, which is exactly 4R sin(φ) tan(φ/2) with φ = s/R. Halve the leg and it quarters.

Parking is an exponent

Four legs — forward on left lock, forward on right lock, back on left lock, back on right lock — return a car to its own heading and to its own place along the road, exactly, and move it sideways by 4R sin φ tan(φ/2). Halve the room and the gain quarters, so the number of shuffles goes up by four and the distance driven doubles.

rolling · Rolling The path a towed wheel takes. The front wheel is given a path; the rear one obeys a single equation — roll along your own heading, and stay attached. The rod is drawn every twelfth sample and is never imposed: the integrator carries the axle's position and heading and nothing else, and the distance from hitch to axle comes out constant to 1.4e-13 m over the whole run. The rear track cuts every corner, which is off-tracking, and it is the reason a long vehicle needs a wide turn.

The path a towed wheel takes

A towed axle obeys one line: roll along your own heading, and stay attached. Nothing tells it to keep its distance from the hitch and it keeps it to 10⁻¹³ anyway, it settles onto a circle of exactly √(R² − L²), and the residual against that is not the integrator — it is the difference between a circle and the polygon it was sampled as, and it falls by four when the sampling doubles.

rolling · Rolling What each of them can reach. Nine hundred control histories of four legs each, from the same starting configuration, with the resulting position plotted. The wheel's cloud is two-dimensional and fills the region; the trolley's is one-dimensional and lies exactly on its rail — the same number of coordinates, the same number of constraints, the same count of freedoms, and a reachable set of a different dimension. Nothing here is a matter of degree.

Not unreachable, only expensive

The sentence is false of every wheeled mechanism in this field and true of exactly one — the trolley bolted to a rail. A rolling constraint forbids a direction and reaches everywhere; the mistake is reading a statement about instants as a statement about intervals, and it is made in both directions.

wrong · Misconception A bevel differential, turning. cage in, hold left, with every member's speed taken from the train's null space and every angular position that speed integrated. The teeth are marked at the pitch points rather than cut as involutes — the flank is the teeth field's subject — but the count is the tooth count and the positions are the solved ones, so what turns and how fast is real. Drag it and watch which way each member goes: left 0.000 · right 2.000 · cage 1.000.

One wheel on ice

A differential with one wheel stopped turns the other at exactly twice the cage, and the relation it imposes is satisfied the whole time — nothing has failed, nothing is confused, and the reason the car does not move is not in this site. What is here is the other half: a locked axle is an overconstrained mechanism, and the sliding it produces is 2π times the track per circle driven, whatever the radius.

transmission · Transmission Which of these two tracks was made by the front wheel. The rear wheel of a bicycle is towed, so it points at the front wheel at every instant: the tangent to the rear track, extended forward by the wheelbase, lands on the front track. Done that way round the tangents land a mean of 0.13 mm off; done the other way round they land 461 mm off, a factor of 3625. It is a measurement rather than an eye for tracks, and it needs neither the wheelbase nor the direction of travel to be known in advance.

Which way did the bicycle go

Two tyre tracks in mud, and a question with a definite answer. The rear wheel is towed, so its tangent extended forward by the wheelbase must land on the front wheel's track — and it does, to 0.13 mm one way round and 447 mm the other. The test needs neither the wheelbase nor the direction of travel, and it returns both.

rolling · Rolling Six words, and the shortest of them. Every path a car that may not reverse and may not turn tighter than R can take between two placements is one of six shapes: three arcs, or two arcs with a straight between. All six are drawn; the shortest is LSR at 2.2557 R and the longest is RSR at 14.788 R. A numerical shooting solve that shares no line of code with the closed forms returns 2.2557 R, which agrees to 4e-16.

A circle for the first millimetre

The shortest path for a car that may not reverse, from here to a point one millimetre to the side at the same heading, is 31.417 m for a five-metre turning radius. The shortest path to a point twenty metres to the side is 31.416 m. The cost of going sideways is not monotonic in how far sideways, and below a crossover at 2.956 R it is exactly 2πR + δ.

rolling · Rolling Two numbers, and where they differ. Every mechanism in this field, with what its constraints leave and what its brackets fill. On every mechanism without a rolling contact the two columns are the same number, which is why nobody had to say which one mobility meant. Here only the rail agrees with itself — and the rail is the one mechanism in the table that cannot go anywhere new.

The count that counts the wrong thing

Mobility has meant one number for six fields, because until a wheel appeared no mechanism could tell two questions apart. A rolling wheel has two velocity freedoms and a three-dimensional reachable set, and the formula that gives 2 is not wrong — it is answering the question about instants when the question anybody asks is about intervals.

constraint · Mobility A ball that remembers the area. Roll a ball round a closed loop on the plane, without ever twisting it about the vertical, and it comes back to the same place turned. The angle is the loop's area divided by r², and the dashed line is that law with nothing fitted to it. The departure at the top is not an integration error: it grows as the square of the angle — a fitted exponent of 1.99 — which is what a leading term's first correction does. The rotation is composed from exact exponentials, so the drawing carries no drift of its own.

The ball that remembers where it has been

Roll a ball round a closed loop on a table without ever twisting it, and it comes back to the same place pointing somewhere else. The angle is the loop's area divided by the square of the radius — 0.0016 radians for a 2 mm square under a 50 mm ball — and it is exact in the limit with a departure that is second order in the angle itself.

rolling · Rolling How much each set of wheels forbids. Every wheel contributes the same row, and whether it is a constraint or a drive is one factor of sin γ in it — γ being the angle the rollers make with the wheel's own axle. At γ = 0 the row says the body may not move across the wheel and the wheel's speed drops out of the statement; at γ = 45° the row says nothing about the body at all and fixes the wheel's speed instead. The whole difference between a machine that shuffles and one that slides sideways is in that factor.

The wheel that forbids nothing

Every wheel contributes the same row to the same matrix, and whether that row is a constraint on the vehicle or a statement about the wheel's own speed is decided by one factor of sin γ. At γ = 0 the vehicle may not move across the wheel; at 45° the row says nothing about the vehicle at all, and sideways costs exactly what forwards costs — to the last digit, and at no other angle.

rolling · Rolling Every axis through one point. Four wheels on one rigid body, each rolling without sliding. Each turns about some point on its own axle line, and a rigid body has one such point, so every axle line has to pass through it. That is the whole of steering geometry, and it is a rank condition rather than a formula: here the four rows have rank 2 of 3, leaving a one-dimensional family of twists, and the centre they agree on is 12.000 m to the side. The scrub is 3.2e-17 m per metre — zero, to the last digit.

Every axis through one point

Bolt several rolling wheels to one rigid body and they impose one condition between them: every axle line must pass through a single point. The familiar steering formula falls out of it as a consequence rather than being quoted — cot δₒ − cot δᵢ = 0.574074 at a turn of six metres, of eight, of twelve and of twenty, on a track of 1.55 m and a wheelbase of 2.7.

rolling · Rolling Forwards it settles, backwards it runs away. A trailer starting a hundredth of a radian out of line, with the steering held straight. Driving forwards the angle decays as e^(−s/d); reversing, the same equation runs the other way and it doubles every 4.16 m. Nothing about forces is involved and nothing about the driver: it is the sign of one exponent, and the length scale is the trailer's own length. The curve flattens at the top because the sine that generates it saturates — the runaway is exponential only while the angle is small.

The angle that doubles

A trailer a hundredth of a radian out of line decays back into line as e^(−s/d) driving forwards and grows as e^(+s/d) reversing — doubling every 4.16 m for a six-metre trailer. And a jackknifed rig is not a rig that has lost anything: its growth vector is 2·3·4·5 at a hitch angle of zero, of ninety degrees and of a hundred and eighty.

rolling · Rolling A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0.

A wheel that cannot report its radius

Rolling relates a wheel's turning to a vehicle's travelling, and the relation has a length in it. So a rolling constraint is the one place on this site where an angle measurement does carry a size — and the size it carries is the one thing a vehicle's own odometry can never separate from its wheelbase.

rolling · Rolling An open belt, whose wraps make one turn. Two pulleys of radius 40 and 24 mm with their centres 160 mm apart, with the strand running the same way round both. The two tangency points on each pulley are marked; the run between them is computed from the signed radius difference and nothing else. The wrap angles are 191.48° and 168.52°, and they add to exactly one turn — the turning number of a simple loop. Length 522.6633 mm, against the textbook formula's 522.6633. positioned by solving, not by drawing.

The road a wheel carries with it

A taut strand on a pulley is a rolling contact: the material at the tangency is at rest against the surface, and the ratio between two bodies on one span is the ratio of their arms. But this rolling constraint integrates, where a wheel's does not — and the difference is that a strand rolls along a line and a wheel rolls across a plane.

rolling · Rolling The square a robot thinks it drove, and the two it drove. A differential drive with wheels one per cent apart in radius, sent round a four-metre square twice — once clockwise and once anticlockwise. What it believes is the square; what it did closes 1.60 m out one way and 1.40 m out the other, and the errors point different ways. A wrong track width instead gives 0.177 m and 0.177 m — the same both ways round. That difference is why the test is run in both directions: one run cannot tell the two errors apart and two runs can.

The error that is an integral

A tolerance on a link length moves an output by a bounded amount. A tolerance on a wheel radius moves a vehicle by an amount that grows with how far it has driven: one per cent of mismatch between two wheels bends a commanded straight line onto a 30 m radius, and a four-metre square comes back 1.60 m from where the machine thinks it is.

practice · Tolerance A ball bearing turned by its inner race. A bearing of pitch diameter 40 with 8 balls of diameter 8, its outer race held and its inner race turned 60°. Rolling without slipping at both contacts leaves the cage — the ring that keeps the balls apart, marked by the dark tick — turned 24.00°, a share of 0.4000 of the race, and each ball spun 120.0° backwards about its own centre, marked by its own tick. The share is (1 − d/D)/2, less than a half by d/2D; the gear-train solver, handed a planetary with a sun of 32 teeth and a ring of 48, returns 2/5 for its carrier. Dragging turns the inner race.

A bearing is a planetary with no teeth

Roll a ball between two races and nothing but the two rolling constraints decides how fast its centre goes round. Solved, they put the cage at (1 − d/D)/2 of the inner race's speed — always less than half — and the gear field's train solver, handed a planetary with a sun of D − d teeth and a ring of D + d, returns the same fraction exactly. And a tapered roller rolls without slipping along its whole line only if its axis meets the bearing's at the apex, the wheel field's concurrency one dimension up.

rolling · Rolling

Named alongside it

The objects these essays reach for when they reach for this one.

NonholonomicReachable setGrowth vectorConfiguration spaceLie bracketConstraint jacobianIntegrabilityMobilityOff-trackingRankSecond-orderTurning radius

All concepts