Linking rigid bodies symmetrically

The mathematical theory of rigidity of body–bar and body–hinge frameworks provides a useful tool for analyzing the rigidity and flexibility of many articulated structures appearing in engineering, robotics and biochemistry. In this paper we develop a symmetric extension of this theory which permits a rigidity analysis of body–bar and body–hinge structures with point group symmetries.The infinitesimal rigidity of body–bar frameworks can naturally be formulated in the language of the exterior (or Grassmann) algebra. Using this algebraic formulation, we derive symmetry-adapted rigidity matrices to analyze the infinitesimal rigidity of body–bar frameworks with Abelian point group symmetries in an arbitrary dimension. In particular, from the patterns of these new matrices, we derive combinatorial characterizations of infinitesimally rigid body–bar frameworks which are generic with respect to a point group of the form Z/2Z×⋯×Z/2ZZ/2Z×⋯×Z/2Z. Our characterizations are given in terms of packings of bases of signed-graphic matroids on quotient graphs. Finally, we also extend our methods and results to body–hinge frameworks with Abelian point group symmetries in an arbitrary dimension.