Machine Kinematics
solved, then drawn
Concept

Controllability — where it appears

Whether a mechanism's reachable set is the whole of its configuration space, decided by whether its brackets fill that space. For a wheeled vehicle it is usually true and unhelpfully so: everything is reachable, and the interesting question is how much shuffling it takes to get there.

Named by 3 essays across 2 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 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 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

Named alongside it

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

Growth vectorNonholonomicRolling constraintJackknifeMisconceptionReachable setConfiguration spaceDistributionEquilibriumExponential growthIntegrabilityInvariant

All concepts