Given a (bar-and-joint) framework/linkage, I would like to know what are possible ways of showing that the framework is locally rigid. Also, what is known about the computational complexity of checking whether a given framework is locally rigid?
Note that I am talking about local rigidity and not infinitesimal rigidity. Checking infinitesimal rigidity is easy (one just have to compute the rank of the rigidity matrix). Moreover it is known that infinitesimal and local rigidity are equivalent for generic frameworks. Now I was wondering if the following method for showing local rigidity would work. Given a framework, I slightly perturb the nodes to make it generic. Now I check the rigidity (or equivalently infinitesimal rigidity) of this new framework. Does it guarantee the rigidity of the original framework?
I came across a certain criteria (https://en.m.wikipedia.org/wiki/Chebychev%E2%80%93Gr%C3%BCbler%E2%80%93Kutzbach_criterion) which is used by engeeneers to compute the degrees of freedom of a given linkage. Although I don't fully understand it, but it seems to say that a framework in plane is rigid iff $m\geq 2n-3$, where $m$ is the number of edges and $n$ is the number of vertices. But it certainly doesn't work for all (even generic) frameworks. I would like to know for which frameworks this criteria actually works. Please correct me if I misunderstood the criteria.
A related post (awaiting answer) which asks to prove local rigidity of a specific framework: https://math.stackexchange.com/questions/4774712/is-k-3-3-with-vertices-on-a-circle-locally-rigid-in-the-plane