Machine Kinematics
solved, then drawn
Concept

Tangent space — where it appears

The flat that best approximates a set at one of its points, and what any measurement made over a small enough range returns. A mechanism's screw system is one, which is why no amount of care with it answers a question about composing finite displacements: every set equals its tangent space to first order.

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

Why this measurement is not a derivative. The honest failure mode of the field's instrument, drawn rather than hidden. Every set looks like its own tangent space near the identity — that is what a tangent space is — so a chain sampled over a thousandth of a radian reports the dimension of its velocities, which is the number the screw system already gives. Four pins at random has four joints, and sampled over 10⁻⁹ radians its displacements occupy four dimensions; sampled over two radians they occupy six. The step is at 10⁻⁶, which is where the departure from the tangent space falls below the rank tolerance — so the position of the step is a fact about arithmetic and the two plateaux are facts about the mechanism. A group is a statement about displacements you could compose, and no derivative can make it.

The instrument that is not a derivative

Sample a chain of four random pins over a millionth of a radian and its displacements occupy four dimensions, exactly as a SCARA arm's do. Sample the same chain over two radians and they occupy six. The step is at 10⁻⁶, and where it sits is a fact about arithmetic while the two plateaux are facts about the mechanism.

pairs · Pair Two pins, and what their bracket costs. The bracket, on the smallest case there is. Two revolute joints span a two-dimensional set of twists whichever way they are arranged, and no count on this site can tell the two arrangements apart. Their brackets can. Two parallel pins bracket to a translation, which was not in the span, and the span closes at three — planar motion, which is the group the pair of them lives in. Two skew pins bracket to something that closes at six: nothing smaller than the whole of the rigid displacements contains them. The defect column is how far the bracket lies outside the original span as a fraction of its own length, and in both cases it is of order one — which is the ordinary case, and is why a mechanism confined to a subgroup is the exception.

One bracket, two subjects

A wheel can be parked sideways because the forbidden direction is the bracket of two permitted ones. A mechanism's motion is a group when the brackets of its permitted twists are already permitted. Same operation, same two plateaux, and the two fields want opposite answers — which is why the two are named for different things here, with each one saying beside itself that the other exists.

rolling · Rolling

Named alongside it

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

ConstraintDisplacement subgroupLie bracketSubalgebraTwistDistributionGrowth vectorIntegrabilityNonholonomicRankScrew system

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