Graph theoretic foundations of multibody dynamics

Abstract

This is the first part of two papers that use concepts from graph theory to obtain a deeper understanding of the mathematical foundations of multibody dynamics. The key contribution is the development of a unifying framework that shows that key analytical results and computational algorithms in multibody dynamics are a direct consequence of structural properties and require minimal assumptions about the specific nature of the underlying multibody system. This first part focuses on identifying the abstract graph theoretic structural properties of spatial operator techniques in multibody dynamics. The second part paper exploits these structural properties to develop a broad spectrum of analytical results and computational algorithms. Towards this, we begin with the notion of graph adjacency matrices and generalize it to define block-weighted adjacency (BWA) matrices and their 1-resolvents. Previously developed spatial operators are shown to be special cases of such BWA matrices and their 1-resolvents. These properties are shown to hold broadly for serial and tree topology multibody systems. Specializations of the BWA and 1-resolvent matrices are referred to as spatial kernel operators (SKO) and spatial propagation operators (SPO). These operators and their special properties provide the foundation for the analytical and algorithmic techniques developed in the companion paper. We also use the graph theory concepts to study the topology induced sparsity structure of these operators and the system mass matrix. Similarity transformations of these operators are also studied. While the detailed development is done for the case of rigid-link multibody systems, the extension of these techniques to a broader class of systems (e.g. deformable links) are illustrated.

Graph theoretic foundations of multibody dynamics

“…The dynamics formulations derived in [7] for rigid-body, serial-chain and tree-topology, multibody systems are examples of SKO models with and φ as the respective SKO and SPO operators. The existence of an SKO model does not depend on any exceptional or unusual features of tree-topology multibody systems.…”

“…To address this gap, part I of this paper [7] introduced the notions of Spatial Kernel Operators (SKO) and the related Spatial Propagation Operators (SPO) . It showed that key spatial operators are in fact instances of SKO and SPO matrices.…”

“…The first part [7] established the block-weighted adjacency (BWA) matrix structure of spatial operators associated with serial and tree topology multibody system dynamics, and introduced the notions of spatial kernel operators (SKO) and spatial propagation operators (SPO) . This paper builds upon these connections to show that key analytical results and computational algorithms are a direct consequence of these structural properties and require minimal assumptions about the specific nature of the underlying multibody system.…”

Multibody graph transformations and analysis

“…The notion of SKO models have been shown to provide a general way of deriving a broad range of analytical and algorithmic techniques for tree multibody systems, while requiring few assumptions on the underlying low-level properties of the system [7, 8, 10, 11]. This remarkable consistency persists despite significant differences in the component structure of the corresponding spatial operators across the systems.…”

Multibody graph transformations and analysis

“…Recently, graph techniques have been used to identify general structural properties of tree multibody systems that underlie a broad family of analytical techniques and low-order algorithms for the systems [13-17]. It is seen that tree system dynamics can be described using abstract spatial kernel operator ( SKO ) models , and that several operator results including analytical mass matrix inversion follow directly from the SKO model structure—independent of the specific details of the system.…”