Machine Kinematics
solved, then drawn
Concept

Principal angle — where it appears

The angle between two subspaces of the same dimension, taken as the largest turn needed to carry one onto the other. It is how far a mechanism's screw system moves through its own motion — two millionths of a degree for a planar four-bar against eighty-nine for Bennett's.

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

Whose constraints stay put. The largest principal angle between a mechanism's screw system at the start of its motion and at each later position. The 3 mechanisms whose motion lies in a subgroup of the rigid displacements — planar, spherical, translational — never leave the same subspace, and read between 1.9e-6 and 3.0e-6 degrees, which is the precision of an arccosine near one rather than a movement. The paradoxical ones turn through 49° and 57°. Every curve runs over its own range of motion, because Bricard's linkage assembles over 120 degrees and a shared axis would hide it.

Two ways to be overconstrained

A planar four-bar and Bennett's four-bar report the same redundancy, the same rank and the same wrong count. One of them is overconstrained at every set of link lengths; the other at exactly one ratio and nowhere near it. The difference is not in any of the numbers so far — but it is measurable, and the measurement is an angle.

spatial · Spatial Four instruments, and only the last one names the group. Every instrument this site has for an overconstrained loop, on the same six mechanisms. Kutzbach's count gives −2 for a planar four-bar and −2 for Bennett's. The rank of the constraint Jacobian gives three and three. Both are right and neither separates them. The last two columns are this field's: the span is how many dimensions the logarithms of the displacements the moving link actually reaches occupy, and closes at is the dimension after those are closed under the bracket. A planar four-bar closes at three and the three are planar motion; Sarrus closes at one, a translation, which is the exact straight line the spatial field measured by solving the mechanism sixty times. Bennett closes at six: its displacements occupy four dimensions and no group smaller than all of them contains those four. That is what "paradoxical" has meant on this site for six phases, stated as an integer.

A name for each overconstraint

The spatial field separated subgroup overconstraint from paradoxical by measuring how far a mechanism's screw system turns: 2 × 10⁻⁶ degrees against 89. That is a verdict without a name. Closing the logarithms of the reached displacements under the bracket gives the same verdict and says which group — planar, spherical, a translation — and for Bennett's linkage it says six.

spatial · Spatial

Named alongside it

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

Bennett's linkageBricard's linkageDisplacement subgroupMobilityOverconstraintParadoxical mechanismScrew systemConstraintCoupleLie bracketPitchRank

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