Recursive dynamics for flexible multibody systems using spatial operators

Due to their structural flexibility, spacecraft and space manipulators are multibody systems with complex dynamics and possess a large number of degrees of freedom. Here the spatial operator algebra methodology is used to develop a new dynamics formulation and spatially recursive algorithms for such flexible multibody systems. A key feature of the formulation is that the operator description of the flexible system dynamics is identical in form to the corresponding operator description of the dynamics of rigid multibody systems. A significant advantage of this unifying approach is that it allows ideas and techniques for rigid multibody systems to be easily applied to flexible multibody systems. The algorithms use standard finite-element and assumed modes models for the individual body deformation. A Newton-Euler Operator Factorization of the mass matrix of the multibody system is first developed. It forms the basis for recursive algorithms such as for the inverse dynamics, the computation of the mass matrix, and the composite body forward dynamics for the system. Subsequently, an alternative Innovations Operator Factorization of the mass matrix, each of whose factors is invertible, is developed. It leads to an operator expression for the inverse of the mass matrix, and forms the basis for the recursive articulated body forward dynamics algorithm for the flexible multibody system. For simplicity, most of the development here focuses on serial chain multibody systems. However, extensions of the algorithms to general topology flexible multibody systems are described. While the computational cost of the algorithms depends on factors such as the topology and the amount of flexibility in the multibody system, in general, it appears that in contrast to the rigid multibody case, the articulated body forward dynamics algorithm is the more efficient algorithm for flexible multibody systems containing even a small number of flexible bodies. The variety of algorithms described here permits a user to choose the algorithm which is optimal for the multibody system at hand. The availability of a number of algorithms is even more important for real-time applications, where implementation on parallel processors or custom computing hardware is often necessary to maximize speed

A Stable Finite Difference Method for the Elastic Wave Equation on Complex Geometries with Free Surfaces

A stable and explicit second order accurate finite difference method for the elastic wave equation in curvilinear coordinates is presented. The discretization of the spatial operators in the method is shown to be self-adjoint for free-surface, Dirichlet and periodic boundary conditions. The fully discrete version of the method conserves a discrete energy to machine precision

Model Checking Dynamic-Epistemic Spatial Logic

In this paper we focus on Dynamic Spatial Logic, the extension of Hennessy-Milner logic with the parallel operator. We develop a sound complete Hilbert-style axiomatic system for it comprehending the behavior of spatial operators in relation with dynamic/temporal ones. Underpining on a new congruence we define over the class of processes - the structural bisimulation - we prove the finite model property for this logic that provides the decidability for satisfiability, validity and model checking against process semantics. Eventualy we propose algorithms for validity, satisfiability and model checking

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. Mass matrix factorization, inversion, diagonalization, and linearization are among several new insights obtained using such operators. While initially developed for serial rigid body manipulators, the spatial operators and the related mathematical analysis have been shown to extend very broadly including to tree and closed topology systems, to systems with flexible joints, links, etc. This work uses concepts from graph theory to explore the mathematical foundations of spatial operators. The goal is to study and characterize the properties of the spatial operators at an abstract level so that they can be applied to a broader range of dynamics problems. The rich mathematical properties of the kinematics and dynamics of robotic multibody systems has been an area of strong research interest for several decades. These properties are important to understand the inherent physical behavior of systems, for stability and control analysis, for the development of computational algorithms, and for model development of faithful models. Recurring patterns in spatial operators leads one to ask the more abstract question about the properties and characteristics of spatial operators that make them so broadly applicable. The idea is to step back from the specific application systems, and understand more deeply the generic requirements and properties of spatial operators, so that the insights and techniques are readily available across different kinematics and dynamics problems. In this work, techniques from graph theory were used to explore the abstract basis for the spatial operators. The close relationship between the mathematical properties of adjacency matrices for graphs and those of spatial operators and their kernels were established. The connections hold across very basic requirements on the system topology, the nature of the component bodies, the indexing schemes, etc. The relationship of the underlying structure is intimately connected with efficient, recursive computational algorithms. The results provide the foundational groundwork for a much broader look at the key problems in kinematics and dynamics. The properties of general graphs and trees of nodes and edge were examined, as well as the properties of adjacency matrices that are used to describe graph connectivity. The nilpotency property of such matrices for directed trees was reviewed, and the adjacency matrices were generalized to the notion of block weighted adjacency matrices that support block matrix elements. This leads us to the development of the notion of Spatial Kernel Operator SKO kernels. These kernels provide the basis for the development of SKO resolvent operators

Discontinuous Galerkin method for the spherically reduced BSSN system with second-order operators

We present a high-order accurate discontinuous Galerkin method for evolving the spherically-reduced Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system expressed in terms of second-order spatial operators. Our multi-domain method achieves global spectral accuracy and long-time stability on short computational domains. We discuss in detail both our scheme for the BSSN system and its implementation. After a theoretical and computational verification of the proposed scheme, we conclude with a brief discussion of issues likely to arise when one considers the full BSSN system.Comment: 35 pages, 6 figures, 1 table, uses revtex4. Revised in response to referee's repor

Finite Domain Anomalous Spreading Consistent with First and Second Law

After reviewing the problematic behavior of some previously suggested finite interval spatial operators of the symmetric Riesz type, we create a wish list leading toward a new spatial operator suitable to use in the space-time fractional differential equation of anomalous diffusion when the transport of material is strictly restricted to a bounded domain. Based on recent studies of wall effects, we introduce a new definition of the spatial operator and illustrate its favorable characteristics. We provide two numerical methods to solve the modified space-time fractional differential equation and show particular results illustrating compliance to our established list of requirements, most important to the conservation principle and the second law of thermodynamics.Comment: 14 figure

pathGCN: Learning General Graph Spatial Operators from Paths

Graph Convolutional Networks (GCNs), similarly to Convolutional Neural Networks (CNNs), are typically based on two main operations - spatial and point-wise convolutions. In the context of GCNs, differently from CNNs, a pre-determined spatial operator based on the graph Laplacian is often chosen, allowing only the point-wise operations to be learnt. However, learning a meaningful spatial operator is critical for developing more expressive GCNs for improved performance. In this paper we propose pathGCN, a novel approach to learn the spatial operator from random paths on the graph. We analyze the convergence of our method and its difference from existing GCNs. Furthermore, we discuss several options of combining our learnt spatial operator with point-wise convolutions. Our extensive experiments on numerous datasets suggest that by properly learning both the spatial and point-wise convolutions, phenomena like over-smoothing can be inherently avoided, and new state-of-the-art performance is achieved.Comment: ICML 202