Catastrophe in Elastic Tensegrity Frameworks

TL;DR

This paper analyzes how elastic tensegrity frameworks can suddenly lose stability due to parameter changes, using nonlinear algebra to identify and compute potential catastrophic shape changes.

Contribution

It introduces a method to characterize and compute the catastrophe set in elastic tensegrity frameworks using nonlinear algebra techniques.

Findings

The catastrophe set can be characterized as a semialgebraic subset of parameters.

Numerical nonlinear algebra tools enable reliable computation of stable equilibria.

The framework predicts large shape changes from small parameter variations.

Abstract

We discuss elastic tensegrity frameworks made from rigid bars and elastic cables, depending on many parameters. For any fixed parameter values, the stable equilibrium position of the framework is determined by minimizing an energy function subject to algebraic constraints. As parameters smoothly change, it can happen that a stable equilibrium disappears. This loss of equilibrium is called `catastrophe' since the framework will experience large-scale shape changes despite small changes of parameters. Using nonlinear algebra we characterize a semialgebraic subset of the parameter space, the catastrophe set, which detects the merging of local extrema from this parametrized family of constrained optimization problems, and hence detects possible catastrophe. Tools from numerical nonlinear algebra allow reliable and efficient computation of all stable equilibrium positions as well as the catastrophe set itself.