Machine Kinematics
solved, then drawn
Concept

Towed axle — where it appears

An axle joined to a towing point by a rigid rod and free to roll only along its own heading, so that it points at the towing point at every instant. Its heading is not a coordinate anybody sets: it is determined by where it has been, which is what makes reversing a trailer a control problem rather than a steering one.

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

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 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 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.

Off-trackingRolling constraintMisconceptionSwept pathTractrixWheelbaseClosed formConserved quantityControllabilityEquilibriumExponential growthGrowth vector

All concepts