Engineering topics

Robert Cardona

Publications and source records attributed to Robert Cardona.

6 records

NASA NTRS2022

A Survey of Mathematical Structures for Lunar Networks

To sustain the current and increasing accessibility of space, a scalable communications infrastructure (i.e. the Solar System Internet, SSI) is necessary. The goal of this paper is to begin the discovery of the fundamental underlying mathematical structure of space networks to help the research community harness these structures for algorithm development and optimization. To ensure the applicability of the research, the approaches are considered through the lens of simulated scenarios inspired by the Artemis Back-to-the-Moon mission set for 2024. We note that any approach to an SSI must fit under the umbrella of Delay Tolerant Networking (DTN), due to celestial mobility, high link latencies, high variance in link latencies, disconnections, lack of end-to-end paths, and so on. These difficulties are exacerbated by the fact that the underlying structure of a space network is a time-evolving network and may experience multiple discontinuities in its topology. In this paper we propose several novel approaches to a mathematical foundation for Delay Tolerant Networking Theory that fall outside the traditional scope of temporal network theory. These techniques include methods from Topological Data Analysis, Dynamic Graph Analysis, Applied Algebraic Geometry, Probability Theory, and Game Theory. Some of these methods include tools adapted to the study of dynamic metric spaces, such as zigzag persistent homology and their higher parameter analogs. We find that several of these methods target desired engineering outcomes such as discovery and automatic sub-netting. While each approach is theoretical, they are also algorithmic in nature and offer immediate practical applications. The paper concludes with comparisons of the various methods along with suggestions for future work.

Delay tolerant networking↗

NASA NTRSDate not supplied

Towards Sheaf Theoretic Analyses for Delay Tolerant Networking

The goal of Delay Tolerant Networking (DTN) is to take a collection of heterogeneous, disparate connections between satellites, space assets, ground stations, and ground infrastructure and bring it together into a cohesive, functioning overlay network. Depending on the systems being considered, one can find links with a one-way light time exceeding minutes (and hours),periodic links which can sometimes be predicted by orbital mechanics, and restrictions based on the variety of capabilities built into these systems. These characteristics preclude traditional network models and routing techniques and have classically led to either rigid routing tables or purely probabilistic models. As the deeper underlying structures remain unknown, development of more DTN-optimized algorithms has lacked the necessary foundation. In a continuation of previous work, the goal of this paper is to identify and study these fundamental structures that exist in delay tolerant networks (DTN), with a focus on space networks. The current routing methodology has been to use contact graph routing (CGR) algorithms. CGR models a series of known contacts as a static graph. For CGR to work, this graph must be globally consistent and must have an accurate picture of the network. Because this is a globally controlled structure, there is little room for flexibility in the event of changes to the network which would naturally occur as the network grows. As a response to the desire for flexibility as the network changes, we introduced the mathematical structure known as sheaves to DTNs last year. The tag-line for sheaves is that they are a mathematically precise way of gluing local data together into unique global data. Thus, sheaves lend extra power to traditional models(and routing algorithms) by taking additional information and merging it, in as consistent a manner as possible, with the representation itself. The clearest example of how Earth-bound networks exhibit behavior that is “sheafy” is link state routers, which build a local-to-global picture of their network by gluing local information together into a global network, exactly as a sheaf would do. For routing within delay tolerant networks to truly exploit this structure, a deeper structure than a graph is required. In this paper, we develop sheaves that can work over directed graphs such as temporal flow networks, we construct a sheaf representation for Dijkstra’s algorithm, and we outline a construction for routing sheaves capable of modeling multicast scenarios. Finally, there is a section of future work suggesting follow-on research.

Robert Short↗

NASA NTRSDate not supplied

Dude Where's My Stars: A Novel Topologically Justified Approach to Star Tracking

In this paper, we consider two novel approaches to celestial navigation for spacecraft. Determining attitude without any prior knowledge using star tracking is known to be a difficult task, particularly given the computational complexity and the many potential sources of misinformation. We consider localization by optimizing matching parameters without explicit star identification in a computationally tractable manner. This is achieved using the mathematical tools of topological data analysis (TDA) and cellular sheaves to study the geometry and distribution of cataloged stars. A framework is gained that enhances the statistical approach to noise handling and false star detection, and heterogeneous sensor fusion. Finally, we discuss confidence bounds and minimum information requirements for successful operation.

Sheaf theory↗

NASA NTRSDate not supplied

Sheaf Theoretic Models for Routing in Delay Tolerant Networks

One key to communications scalability is routing; as such the goal of this paper is to build upon successful efforts towards general routing for space-based networks. With the ever-increasing accessibility of space, the number of assets is increasing, which becomes a critical communications burden in terms of scheduling, spectrum allocation, and resource allocation. In order to mitigate these concerns, a true networking approach is necessary; a standard approach for space systems is Delay Tolerant Networking (DTN). For DTN to be a meaningful answer to the Solar System Internet (SSI) question, DTN must offer meaningful routing solutions that span the heterogeneous collection of links and nodes. This, in turn, depends on the general structure of these disconnected networks -- a structure that remains largely unknown. In ground communications networks, routing decisions are made based on several pathfinding algorithms working in tandem. In previous work, we modeled Dijkstra's pathfinding algorithm using sheaves and provided a more general framework for determining paths using sheaves over graphs. Continuing our sheaf-theoretic approach, we introduce here an expansion of our pathfinding sheaf to handle more general information, and we expand on additional pathfinding algorithms that can be represented using sheaves. Moreover, we demonstrate means of combining multiple algorithms into a single sheaf structure so that changes of scale can be presented in the language of sheaves. In addition, space communications networks rely upon radio transmitter antennas which can establish broadcast and multicast communications options, rather than the primarily unicast options available to wired networks. Last year, we also introduced a multicast routing sheaf for presenting broadcast, unicast, and multicast communications over a graph. Extending that work, we also introduce queuing sheaves so that we can blend these communications options together to simulate a variety of routing options across space networks. In addition, we include examples to illustrate the applicability of this abstract theory to routing in disconnected networks.

Robert Short↗

NASA NTRSDate not supplied

Contact Multigraph Routing: Overview and Implementation

In Delay Tolerant Networking (DTN), the standard routing algorithm used to navigate time-varying networks has been Contact Graph Routing (CGR). In CGR, a globally distributed list of contacts, periods during which two DTN nodes may communicate, is used to construct a contact graph, in which contacts are vertices. A version of Dijkstra’s algorithm can then be used to find paths through this model of the timevarying network. However, since contact graphs may be large compared to the network, potentially growing with the square of the number of network nodes and linearly with the time interval represented, the resulting algorithm does not scale well with the size of the network or time. Any improvement to the routing algorithm will bring significant returns to scale. In a previous paper, we briefly introduced an alternative to the contact graph model for routing. This alternative model is based on a multigraph (a graph in which there may be multiple edges between a pair of vertices) where vertices represent network nodes instead of contacts. A version of Dijkstra’s algorithm in these multigraph models reduces the time needed to perform the same routing computations done in the existing CGR algorithm. Moreover, a modified version of Yen’s algorithm for multigraphs is included. Our variation of CGR, which we call Contact Multigraph Routing (CMR), provides an in-line replacement for the previously used pathfinding algorithms. This paper describes an implementation created based on the CMR approach, and experimental comparisons to traditional CGR are given. In addition, we explore some additional modifications to the routing pipeline traditionally assumed in CGR. These modifications range from the theoretical to the practical in terms of size and scope. We step forward our understanding of sheaftheoretic networking and describe how to model the routing pipeline using sheaves. We detail some enhanced route selection criteria that addresses some of the added complexity of DTNbased systems. We also include a future works section on future improvements and implementations that would be of service to the broader DTN community.

contact graph routing↗

NASA NTRSDate not supplied

Categories of Neural Networks

We introduce new categories of neural networks that we hope will help us expose and explore their underlying structure and relationships. Our categories represent neural networks as objects and present structure for comparing neural networks with the same number of layers. In this paper we discuss the construction, properties, and currently known limitations – all with specific, illustrative examples. There are a number of directions this work can move towards, and we conclude with a section on future work summarizing several of these directions.

Category Theory↗