Machine Kinematics
solved, then drawn
Concept

Pfaffian constraint — where it appears

A constraint written as one linear condition on a mechanism's velocities, with coefficients that may depend on its configuration. It is the form every rolling condition takes, and whether a set of them integrates into conditions on position is decided by the brackets of the directions they permit.

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

A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0.

A constraint that takes nothing away

A rolling wheel forbids one direction of motion and removes no coordinate from the mechanism's description. It cannot slide sideways and it can still be brought to any position at any heading — and the gap between those two sentences is the whole of this field, because in every mechanism built of pins and slides the two agree.

rolling · Rolling Which of these constraints is secretly about positions. How much of the bracket of two permitted directions lies outside the permitted directions, as a fraction of its own length. Frobenius' theorem says a distribution is the tangent field of a family of surfaces exactly when this is zero, so the test needs no integration and no recognition. The scale is logarithmic because the answers are seventeen orders apart: the rail returns nothing at all and everything else returns essentially the whole bracket. There is no mechanism in the middle.

One character apart

Two mechanisms with three coordinates, one constraint row of the same shape and two controls each. In one of them the angle in the row is a coordinate; in the other it is a constant. The first can be driven anywhere and the second can never leave a line, and Frobenius' theorem decides which is which without integrating anything.

rolling · Rolling

Named alongside it

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

Configuration spaceIntegrabilityNonholonomicRolling constraintConserved quantityConstraint jacobianDistributionHolonomicInvariantLie bracketMobilityReachable set

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