No fractional rigidity orders occur before third order, and second-order prestress stability can be checked by explicit linear-algebra criteria, but a general third-order test exists only when the rigidity matrix has one-dimensional null space.
A Proper Definition of Higher Order Rigidity
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[Connelly and Servatius, 1994] shows the difficulty of properly defining n-th order rigidity and flexiblity of a bar-and-joint framework for higher order (n >= 3) through the introduction of a cusp mechanism. The author proposes a "proper" definition of the order of rigidity by the order of elongation of the bars with respect to the arclength along the path in the configuration space. We show that the classic definition using formal n-th derivative of the length constraint is a sufficient condition for the n-th flexiblity in the proposed definition and also a necessary condition only for n = 1, 2.
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Second-order prestress stability and third-order rigidity of polyhedral surfaces
No fractional rigidity orders occur before third order, and second-order prestress stability can be checked by explicit linear-algebra criteria, but a general third-order test exists only when the rigidity matrix has one-dimensional null space.