Machine Kinematics
solved, then drawn
Concept

Parallel axis defect — where it appears

The failure of the Denavit-Hartenberg parameterisation for consecutive joint axes that are nominally parallel. A hundredth of a degree of unintended twist moves the offset by a thousand link lengths, because the common normal it is measured along jumps to somewhere else entirely.

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

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

The common normal, and where it is. Two joint axes 1.0 unit apart, nominally parallel, 0.50° apart in one plane and 0.05° in the other. The Denavit–Hartenberg convention takes all four of its numbers from the one line that meets both at right angles, and for these two axes that line crosses the first 11.3 units from the joint — off this page by a factor of about 11, which is why the tilt here is drawn at 26° and not at a fraction of one. Along it, the "link length" of a link 1.0 unit long reads 0.9950, and the angle round the first axis reads 5.71°. Nothing has moved by more than 0.50°.

The common normal, and where it is

The Denavit–Hartenberg convention reads all four of its numbers off one line: the common normal between two joint axes. Two parallel axes do not have one — every perpendicular meets both at right angles — and two nearly parallel axes have one that is somewhere else entirely.

metrology · Parameter A number that runs away. The Denavit–Hartenberg offset of a pair of nominally parallel axes, against how far from parallel they actually are, for a fixed out-of-plane tilt of 0.05°. The marks are extracted from the geometry by finding the common normal; the line is the closed form −A cos α cos β sin β / (sin²α cos²β + sin²β), and the two agree to 2.4e-16 relative over three decades. At 30° of twist the offset is 0.0030 of a link length; at 0.01° it is 1102. The dashed line is the worst case over the tilt, which sits at β = α and is exactly A/2α. Nothing about the machine has changed by as much as a degree.

A number that runs away

A hundredth of a degree of unintended twist on a nominally parallel pair of joint axes puts the Denavit–Hartenberg offset at −1,102 link lengths. The extraction from the geometry and the closed form agree to 5 × 10⁻¹⁶ over three decades, and the worst case over the tilt is exactly A/2α.

metrology · Parameter The machine is fine; the description is not. Above: the offset the Denavit–Hartenberg chart assigns to a pair of nominally parallel axes, over three decades of twist, running from 0.0030 to 1102 link lengths. Below: the condition number of the identification Jacobian in a chart that describes the second axis by two small rotations from the first and never asks for a common normal, over the same range. It is 7.5501 at every one of them, flat to 2.4e-9. The machine is the same machine in both rows and it is perfectly well behaved. What breaks is a convention that locates its parameters on a line which, for two parallel axes, does not exist.

The chart breaks, the machine does not

The same two axes, over the same three and a half decades of twist. In one description a parameter runs from 0.003 to 1,102; in another the condition number is 7.5501 and does not move in the fifth figure. A quantity that diverges in one chart and is constant in another is a property of the chart.

metrology · Parameter

Named alongside it

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

CalibrationCommon normalDH parametersIdentifiableParameter chartSerial chainIdentification jacobianJoint axisNoise amplificationSingular value

All concepts