The infinitesimal rigidity of symmetric (bar-joint) frameworks has been studied extensively for over two decades. The area splits into the research of forced and incidental symmetric rigidity. Whereas forced symmetric rigidity only considers infinitesimal motions which maintain the symmetry of the framework, incidental symmetric rigidity allows infinitesimal motions to break symmetry.
In both settings and for various symmetry groups, combinatorial characterisations have been obtained for ‘symmetry-generic’ frameworks, i.e. frameworks which are as generic as possible allowed by their symmetry. Forced symmetric infinitesimally rigid frameworks have been characterised for all cyclic groups, and for dihedral groups Ckv, where k ≥ 3 is odd. Incidentally symmetric infinitesimally rigid frameworks have been characterised for cyclic groups of order 2,4,6, and of odd order less than 1000.
A limitation of these results is the assumption that the symmetry group acts
freely on the joints of the framework. In this thesis, we fill this mathematical gap.
This is also motivated by problems in applied areas such as structural engineering or formation control, where symmetric frameworks are frequently studied, and frameworks may present joints fixed by the point group (e.g. joints on the symmetry line of a reflection-symmetric framework, or in the centre of rotation of a rotational-symmetric framework).
We consider plane frameworks which are symmetric with respect to cyclic groups or dihedral groups. We give necessary conditions for incidentally infinitesimally rigid frameworks for all cyclic groups and for the dihedral group of order 4, and necessary conditions for cyclic groups of order 2,4,6, or of odd order less than 1000. For cyclic groups of even order, we present counterexamples to show that the expected sparsity count is necessary, but not sufficient. We also give necessary conditions for the forced infinitesimal rigidity of frameworks that are symmetric with respect to dihedral groups of arbitrary finite order.
In order to do so, we introduce a generalisation of tools commonly used in the study of symmetric frameworks, known as ‘orbit matrices’ and ‘gain graphs’. Orbit matrices are symmetry-adapted rigidity matrices, whose underlying combinatorial structures are gain graphs, directed multigraphs whose edges are labelled with group elements. A generalisation of gain graphs, and hence of orbit rigidity matrices, is needed when working with joints which are fixed by the symmetry group. A further generalisation is required if some joints are neither free nor fixed by the symmetry group (when working, say, with dihedral groups). We introduce such a generalisation, and show how some of the properties of usual gain graphs hold for this new definition, whilst others do not. This generalisation of gain graph is useful in future research, for the combinatorial characterisation of infinitesimally rigid dihedral-symmetric frameworks.
