Machine Kinematics
solved, then drawn
Concept

Arc — where it appears

A piece of a circle, and the only curve a rotation about that circle's centre carries into itself. That property is why arcs appear wherever a mechanism has to hold something still while it moves — a locking face, a dwell, a detent.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as dubins path — the same set of essays touches all of them, so they are one junction rather than several.

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 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

Named alongside it

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

Closed formDubins pathNonholonomicReachable setRolling constraintTurning radiusClassificationCurvature boundGrowth vectorLie bracketManoeuvreSecond-order

All concepts