Concept

Structure — where it appears

A set of links held so that none of them moves relative to any other, its mobility count coming out at nought. Three links joined by three joints is the smallest one in the plane, and a graph containing any structure is not a mechanism of the link count it appears to have.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Three verdicts, and only one instrument can give all three. Every 10-link graph that satisfies Grübler's count, split by what is actually true of it. 230 are mechanisms with 10 links. 1,165 carry a subchain whose own count is exactly nought — and neither of the two standing routes can see them: the count returns one and the rank returns one, and both are right, because a rigid subchain removes exactly the freedoms it is supposed to. What is false is the description. 483 carry a subchain whose count is below nought, and those the rank does catch: the surplus pins repeat a constraint already imposed, the Jacobian loses rank, and the measured mobility comes out above the count. The third instrument — a count run over every subset of the links — is the only one that answers the question at all.

What a count cannot see

At ten links, 1,878 graphs satisfy Grübler's rule and 230 are mechanisms. The other 1,648 contain a subchain that is already a structure — and on 1,165 of them the count says one degree of freedom, the rank of the constraint Jacobian says one degree of freedom, and both are right about a mechanism that does not have ten links.

topology · Topology

What each instrument returns, on each kind of graph. The 8-link census, three rows, and the same three questions asked of every graph in it. Grübler returns 1 in every row — it has to, because that is what the census selected on. The rank returns 1 in the first two rows and 2 in the third. Only the third column changes across all three rows, and it is the one this site did not have before this field: a mobility computed for every subset of the links rather than for the whole. Read down the middle two columns and the site's standing pair of routes is unanimous about 62 graphs, of which only 16 are what it says they are.

The count was right and the name was wrong

The constraint field has checked Grübler's count against a Jacobian rank since the foundation, and the two disagree only where the geometry is special. Here is an assembly where they agree, where both are correct, and where the mechanism does not have the number of links it is described as having.

constraint · Mobility

Watt chain with 1 slide: 3 chains, 11 mechanisms. The same 6 links and 7 joints with 1 of the joints made a slide instead of a pin, drawn as a block astride the line. There are 7 ways to choose the joint, and the chain's 4 symmetries fold them into 3 that are genuinely different: with the slide at 0–3, 2 mechanisms; with the slide at 0–1, 6 mechanisms; with the slide at 1–2, 3 mechanisms. The mechanism count is the number of orbits of a held link and the slide set together, so a slide breaks symmetry the pin-only chain had, and links that gave one machine between them give two. The pin-only chain gave 2; one slide gives 11.

A slide turns nothing

Make one joint of a chain a slide instead of a pin and the graph has a second decision in it before any length exists. The symmetries that counted mechanisms count these too — Watt's chain with one slide is three chains and eleven machines — and two facts read off the graph say which placements still work: a loop of slides alone is freer than the count, and a pin in a group of links the slides hold at one orientation cannot turn. Across 102 placements on the three smallest chains, both agree with the rank of the constraint Jacobian.

topology · Topology

Named alongside it

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

Grübler's criterion Kinematic chain Constraint Degenerate chain Degrees of freedom Mobility Overconstraint Rank Redundant constraint Automorphism Constraint rank Inversion

All concepts