  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.
