Machine Kinematics
solved, then drawn
Concept

Growth vector — where it appears

The sequence of dimensions spanned by a mechanism's permitted directions, then their brackets, then the brackets of those, which ends at the dimension it can reach. It is the signature of a wheeled mechanism — (2, 3) for a wheel, (2, 3, 4) for a car, (2, 3, 4, 5) for a car and trailer — and its last entry is the dimension actually reachable.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

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 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 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 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 Two pins, and what their bracket costs. The bracket, on the smallest case there is. Two revolute joints span a two-dimensional set of twists whichever way they are arranged, and no count on this site can tell the two arrangements apart. Their brackets can. Two parallel pins bracket to a translation, which was not in the span, and the span closes at three — planar motion, which is the group the pair of them lives in. Two skew pins bracket to something that closes at six: nothing smaller than the whole of the rigid displacements contains them. The defect column is how far the bracket lies outside the original span as a fraction of its own length, and in both cases it is of order one — which is the ordinary case, and is why a mechanism confined to a subgroup is the exception.

One bracket, two subjects

A wheel can be parked sideways because the forbidden direction is the bracket of two permitted ones. A mechanism's motion is a group when the brackets of its permitted twists are already permitted. Same operation, same two plateaux, and the two fields want opposite answers — which is why the two are named for different things here, with each one saying beside itself that the other exists.

rolling · Rolling

Named alongside it

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

NonholonomicRolling constraintReachable setLie bracketControllabilitySecond-orderConfiguration spaceDistributionIntegrabilityJackknifeMisconceptionMobility

All concepts