We extend the mathematical theory of rigidity of frameworks (graphs embedded in $d$-dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call "almost-rigidity"; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost-rigidity. We briefly discuss how the theory may be applied to tensegrities.

Almost-rigidity of frameworks

Snappability and singularity-distance of pin-jointed body-bar frameworks

In a recent article the author presented a method to measure the snapping capability -- shortly called snappability -- of bar-joint frameworks based on the total elastic strain energy by computing the deformation of all bars using Hooke's law and the definition of Cauchy/Engineering strain. Within the paper at hand, we extend this approach to isostatic frameworks composed of bars and triangular plates by using the physical concept of Green-Lagrange strain. An intrinsic pseudometric based on the resulting total elastic strain energy density cannot only be used for evaluating the snappability but also for measuring the distance to the closest singular configuration. The presented methods are demonstrated on the basis of the 3-legged planar parallel manipulator.

On the snappability and singularity-distance of frameworks with bars and triangular plates.

It is well-known that there exist bar-joint frameworks (without continuous flexions) whose physical models can snap between different realizations due to non-destructive elastic deformations of material. We present a method to measure these snapping capability – shortly called snappability – based on the total elastic strain energy of the framework by computing the deformation of all bars using Hooke’s law. The presented theoretical results give further connections between shakiness and snapping beside the well-known technique of averaging and deaveraging.

Evaluating the Snappability of Bar-Joint Frameworks

It is well-known that there exist bar-joint frameworks (without continuous flexions) whose physical models can snap between different realizations due to non-destructive elastic deformations of material. We present a method to measure these snapping capability -- shortly called snappability -- based on the total elastic strain energy of the framework by computing the deformation of all bars using Hook's law. The presented theoretical results give further connections between shakiness and snapping beside the well-known technique of averaging and deaveraging.

Evaluating the snappability of bar-joint frameworks

Polyhedra called Siamese dipyramids are known to be non-flexible, however their physical models behave like physical models of flexible polyhedra. We discuss a simple mathematical method for explaining the model flexibility of the Siamese dipyramids.

I. Fesenko

Papers

On the model flexibility of Siamese dipyramids