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. © 2020 Wiley Periodicals LLC.

中文翻译：

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几乎刚性的框架

我们扩展了框架刚性的数学理论（嵌入在d维空间）考虑非局部刚性和柔性特性。我们在框架上提供了条件，在该框架下（I）随着框架不断弯曲，它必须保持在一个小球内，我们称之为“几乎刚性”的属性；(II) 具有相同边长的任何其他框架必须位于更大的球之外；(III) 如果框架变形一定量，其边长变化最小；(IV) 附近有一个预应力稳定的框架，因此是刚性的。可以使用半定规划有效地测试条件。该测试是框架预应力稳定性测试的一个小延伸，给出了球半径和边长变化的解析表达式。例子说明了理论如何在实践中应用，我们提供了一种算法来测试刚性或几乎刚性。我们简要讨论了该理论如何应用于张拉整体。© 2020 威利期刊有限责任公司。