Machine Kinematics
solved, then drawn
Concept

Dead reckoning — where it appears

Working out where a vehicle is by integrating what its wheels did, which carries an error that grows with the distance driven rather than settling at a band. Its error is unbounded rather than bounded, which is the whole reason a vehicle needs something else to fix its position — the geometry says the drift cannot settle.

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

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

How much each set of wheels forbids. Every wheel contributes the same row, and whether it is a constraint or a drive is one factor of sin γ in it — γ being the angle the rollers make with the wheel's own axle. At γ = 0 the row says the body may not move across the wheel and the wheel's speed drops out of the statement; at γ = 45° the row says nothing about the body at all and fixes the wheel's speed instead. The whole difference between a machine that shuffles and one that slides sideways is in that factor.

The wheel that forbids nothing

Every wheel contributes the same row to the same matrix, and whether that row is a constraint on the vehicle or a statement about the wheel's own speed is decided by one factor of sin γ. At γ = 0 the vehicle may not move across the wheel; at 45° the row says nothing about the vehicle at all, and sideways costs exactly what forwards costs — to the last digit, and at no other angle.

rolling · Rolling The square a robot thinks it drove, and the two it drove. A differential drive with wheels one per cent apart in radius, sent round a four-metre square twice — once clockwise and once anticlockwise. What it believes is the square; what it did closes 1.60 m out one way and 1.40 m out the other, and the errors point different ways. A wrong track width instead gives 0.177 m and 0.177 m — the same both ways round. That difference is why the test is run in both directions: one run cannot tell the two errors apart and two runs can.

The error that is an integral

A tolerance on a link length moves an output by a bounded amount. A tolerance on a wheel radius moves a vehicle by an amount that grows with how far it has driven: one per cent of mismatch between two wheels bends a commanded straight line onto a 30 m radius, and a four-metre square comes back 1.60 m from where the machine thinks it is.

practice · Tolerance

Named alongside it

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

Rolling constraintSystematic errorAccumulationCalibrationConditioningConstraint jacobianInstantaneous centreNonholonomicOmnidirectional wheelRankRedundant constraintSensitivity

All concepts