Higher-order rigidity--What is the proper definition?

We show that there is a bar-and-joint frameworkG(p) which has a configurationp in the plane such that the component ofp in the space of all planar configurations ofG has a cusp atp. At the cusp point, the mechanismG(p) turns out to be third-order rigid in the sense that every third-order flex must have a trivial first-order component. The existence of a third-order rigid framework that is not rigid calls into question the whole notion of higher-order rigidity.