Graph theory roots of spatial operators for kinematics and dynamics

Spatial operators have been used to analyze the dynamics of robotic multibody systems and to develop novel computational dynamics algorithms for them. Mass matrix factorization, inversion, diagonalization, linearization are among the several techniques developed using operators. These techniques have been shown to apply broadly to systems ranging from serial, tree, to closed topology systems, as well as to systems with rigid and flexible links/joints. This paper uses concepts from graph theory to obtain a deeper understanding of the mathematical foundations of spatial operators. We show that spatial kernel operators are instances of block weighted adjacency matrices for the underlying multibody topology graphs, and that spatial propagation operators are 1-resolvents of these matrices. We explore at an abstract level the properties of such 1-resolvents in order to understand the precise requirements on and the range of applicability of spatial operators to the broad class of dynamics problems.