Kinematics, understood as the study of the mobility of mechanical devices, or linkages}, has always interested mathematicians, and in particular algebraic geometers. Since the 19th century, algebraic and geometric techniques have been developed to classify such mechanisms and to investigate the existence of new families. Important results have been achieved so far, as for example the classification of mobile closed chains composed of at most five rods and connected by revolute joints, or several algorithms that are currently used in robotics.
In this work we describe two techniques, called bond theory and Möobius photogrammetry, for the study of a particular class of linkages, namely pods, devices constituted of two rigid bodies, called the base and the platform, connected by several rods, called legs, that are attached to the base and the platform via spherical joints. We show how bond theory provides necessary conditions for the mobility of pods in terms of the geometry of its base and platform, and how Möbius photogrammetry can refine these conditions in the case of pentapods, i.e. pods with five legs, with unexpected mobility. The combined use of these two methods, together with some elementary facts from liaison theory - which describes the properties of two algebraic varieties whose union is a complete intersection - yields a construction for a new family of mobile hexapods, i.e. pods with six legs. By employing some recent results on spectrahedra - objects that arise in the context of semidefinite programming - we show that it is possible to obtain a concrete instance of a mobile icosapod, namely a pod with 20 legs.
The approach we use is mainly geometric: the starting point for bond theory is to associate to every pod a subvariety of a fixed projective variety - that encodes all the possible configurations of the pod - and to study some of its points that are "limits" of configurations; in Möbius photogrammetry, we attach to every tuple of points in real space a complex curve reflecting how the orthogonal projections of such points behave under Möbius transformations. In both situations, the driving principle is to exploit the presence of complex structures in problems that arise from real situations - namely for which the input data can be encoded via real numbers.