Machine Kinematics
solved, then drawn
Concept

Swept path — where it appears

The band of ground a vehicle's outermost and innermost parts cover through a turn, which for an articulated vehicle is set by its nested off-tracking. It is the number a road is designed against, and it is larger than the vehicle for a reason the geometry supplies exactly: every unit runs inside the one towing it.

Named by 3 essays across 2 fields — 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 The room crank rocker needs. The shaded region is every point any part of this machine occupies at some position of its drive, computed as an occupancy grid over 150 solved configurations and outlined by marching squares — so the outline drawn is the boundary of the set the area was counted from rather than a second object that agrees with it. Its area is 12.78 square units, filling 54% of the box that contains it. The dashed rectangle is the box the joints need, which is what every figure that treats links as lines could have told you; the material needs a box 20% larger in area.

The room a machine sweeps

Every point any part of a machine occupies at some position of its drive. It is a region rather than a curve, its area is an integral computed two ways, and the one shape in the field with a closed form is what the grid is calibrated against.

bodies · Body 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 constraintTowed axleClosed formConserved quantityControllabilityEquilibriumExponential growthFootprintGrowth vectorIntegration errorInterference

All concepts