Machine Kinematics
solved, then drawn
Concept

Cylindroid — where it appears

The ruled surface swept by the axes of every screw in a two-system, with a pitch varying along it between two extremes reached at right angles to each other. Its length along its own axis is exactly the spread of those two pitches, so a cylindroid is as long as its pitches are far apart.

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

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

The cylindroid at 75° and 1.00 apart. Every screw in the two-system spanned by two revolute axes 1.00 apart along their common perpendicular and 75° out of parallel, drawn as its own axis. The axes sweep a ruled surface — the cylindroid — and each generator carries a pitch, running from -0.384 to 0.652 and reaching its two extremes on the two principal screws, which cross at a right angle at the centre. The surface is 1.035 long along its own axis, which is exactly the spread of the pitches: a cylindroid is as long as its pitches are far apart.

The smallest screw system has a shape

Add two screws together in every proportion. The results do not scatter — their axes sweep a ruled surface, with the pitch varying along it between two extremes reached at right angles to each other. It is a picture nobody would guess from the algebra, and it is the object that says what two joints between two bodies leave free.

spatial · Spatial Three joints, and the three screws that describe them. A leg of three revolute joints, drawn as its three axes, and the principal screws of the three-system they span, drawn through the system's own centre. The three principal axes are mutually perpendicular — worst cosine 1.1e-16 — and they meet at one point, missing it by 2.6e-16. Their pitches are -0.3766, -0.0338, 0.8075, and every screw the leg leaves free has a pitch the three of them give by h₁l² + h₂m² + h₃n². Nothing in the three joint axes looks like a right angle and the system's own frame is one.

What a leg of three joints leaves free

Five essays of this field have computed the order of a screw system and drawn none of them. A three-joint leg spans a three-system; its three principal axes are mutually perpendicular and meet at a point, six numbers price every screw in the family, and the directions of the lines it contains form a cone.

spatial · Spatial What a leg of three joints permits and what it resists, on one frame. The leg of three revolute joints in grey, and the principal screws of two three-systems drawn through one centre: the twists the leg lets its platform make, and the wrenches it holds without any joint turning. They are computed separately — the second is the reciprocal complement of the first, found by its own eigenproblem — and they come out with the same centre, 5e-16 apart, and the same three axes, parallel to 0e+0. Only the pitches differ, and each wrench pitch is its twist's with the sign reversed, to 1e-15: -0.3766 against 0.3766, -0.0338 against 0.0338, 0.8075 against -0.8075.

The lines a leg turns about and the lines it is pushed along

A leg of three revolute joints permits a three-system of motions and resists a three-system of loads. The two share a centre and three axes and differ only in the sign of every pitch — and the revolute axes of the first and the lines of force of the second are the two rulings of one hyperboloid, every line of one meeting every line of the other.

spatial · Spatial

Named alongside it

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

ConstraintPitchPrincipal screwScrewScrew systemDegrees of freedomPlatformReciprocal screwType synthesisCoupleScrew axisTwist

All concepts