Voronoi diagrams and Morse theory of the distance function

We consider the (minimal) distance function of a point in the plane to a set P of N points in the plane. The locus of non-dierentiability of this distance function consists (besides of the points of P) exactly of the Voronoi diagram of P. We show that the number of minima (m), maxima (M) and `saddle points' (s) of the distance function satisfy: m - s + M = 1. This is similar to the Morse type of statements for dierentiable functions. The saddle points occur exactly where a Delaunay edge cuts the corresponding Voronoi edge in its interior. The set of those edges form a subgraph of the Delaunay graph, which connects all minima and saddle points. This graph devides the plane into regions. In each of the compact regions, there is exactly one maximum, the non compact regions don't contain a local maximum. At the end we classify all those graphs if P contains of 3 or 4 points

Cooperative Pursuit with Multi-Pursuer and One Faster Free-moving Evader

This paper addresses a multi-pursuer single-evader pursuit-evasion game where the free-moving evader moves faster than the pursuers. Most of the existing works impose constraints on the faster evader such as limited moving area and moving direction. When the faster evader is allowed to move freely without any constraint, the main issues are how to form an encirclement to trap the evader into the capture domain, how to balance between forming an encirclement and approaching the faster evader, and what conditions make the capture possible. In this paper, a distributed pursuit algorithm is proposed to enable pursuers to form an encirclement and approach the faster evader. An algorithm that balances between forming an encirclement and approaching the faster evader is proposed. Moreover, sufficient capture conditions are derived based on the initial spatial distribution and the speed ratios of the pursuers and the evader. Simulation and experimental results on ground robots validate the effectiveness and practicability of the proposed method

Collision-free path planning

Motion planning is an important challenge in robotics research. Efficient generation of collision-free motion is a fundamental capability necessary for autonomous robots;In this dissertation, a fast and practical algorithm for moving a convex polygonal robot among a set of polygonal obstacles with translations and rotations is presented. The running time is O(c((n + k)N + nlogn)), where c is a parameter controlling the precision of the results, n is the total number of obstacle vertices, k is the number of intersections of configuration space obstacles, and N is the number of obstacles, decomposed into convex objects. This dissertation exploits a simple 3D passage-network to incorporate robot rotations as an alternative to complex cell decomposition techniques or building passage networks on approximated 3D C-space obstacles;A common approach in path planning is to compute the Minkowski difference of a polygonal robot model with the polygonal obstacle environment. However such a configuration space is valid only for a single robot orientation. In this research, multiple configuration spaces are computed between the obstacle environment and the robot at successive angular orientations spanning [pi] . Although the obstacles do not intersect, each configuration space may contain intersecting configuration space obstacles (C-space obstacles). For each configuration space, the algorithm finds the contour of the intersected C-space obstacles and the associated passage network by slabbing the collision-free space. The individual configuration spaces are then related to one another by a heuristic called proper links that exploit spatial coherence. Thus, each level is connected to the adjacent levels by proper links to construct a 3D network. Dijkstra\u27s algorithm is used to search for the shortest path in the 3D network. Finally, the path is projected onto the plane to show the final locus of the path

Clustering-Based Robot Navigation and Control

In robotics, it is essential to model and understand the topologies of configuration spaces in order to design provably correct motion planners. The common practice in motion planning for modelling configuration spaces requires either a global, explicit representation of a configuration space in terms of standard geometric and topological models, or an asymptotically dense collection of sample configurations connected by simple paths, capturing the connectivity of the underlying space. This dissertation introduces the use of clustering for closing the gap between these two complementary approaches. Traditionally an unsupervised learning method, clustering offers automated tools to discover hidden intrinsic structures in generally complex-shaped and high-dimensional configuration spaces of robotic systems. We demonstrate some potential applications of such clustering tools to the problem of feedback motion planning and control. The first part of the dissertation presents the use of hierarchical clustering for relaxed, deterministic coordination and control of multiple robots. We reinterpret this classical method for unsupervised learning as an abstract formalism for identifying and representing spatially cohesive and segregated robot groups at different resolutions, by relating the continuous space of configurations to the combinatorial space of trees. Based on this new abstraction and a careful topological characterization of the associated hierarchical structure, a provably correct, computationally efficient hierarchical navigation framework is proposed for collision-free coordinated motion design towards a designated multirobot configuration via a sequence of hierarchy-preserving local controllers. The second part of the dissertation introduces a new, robot-centric application of Voronoi diagrams to identify a collision-free neighborhood of a robot configuration that captures the local geometric structure of a configuration space around the robot’s instantaneous position. Based on robot-centric Voronoi diagrams, a provably correct, collision-free coverage and congestion control algorithm is proposed for distributed mobile sensing applications of heterogeneous disk-shaped robots; and a sensor-based reactive navigation algorithm is proposed for exact navigation of a disk-shaped robot in forest-like cluttered environments. These results strongly suggest that clustering is, indeed, an effective approach for automatically extracting intrinsic structures in configuration spaces and that it might play a key role in the design of computationally efficient, provably correct motion planners in complex, high-dimensional configuration spaces

Information Diffusion on Social Networks

In this thesis we model the diffusion of information on social networks. A game played on a specific type of graph generator, the iterated local transitivity model, is examined. We study how the dynamics of the game change as the graph grows, and the relationship between properties of the game on a graph initially and properties of the game later in the graph’s development. We show that, given certain conditions, for the iterated local transitivity model it is possible to predict the existence of a Nash equilibrium at any point in the graph’s growth. We give sufficient conditions for the existence of Nash Equilibria on star graphs, cliques and trees. We give some results on potential games on the iterated local transitivity model. Chapter 2 provides an introduction to graph properties, and describes various early graph models. Chapter 3 describes some models for online social networks, and introduces the iterated local transitivity model which we use later in the thesis. In Chapter 4 various models for games played on networks are examined. We study a model for competitive information diffusion on star graphs, cliques and trees, and we provide conditions for the existence of Nash Equilibria on these. This model for competitive information diffusion is studied in detail for the iterated local transitivity model in Chapter 5. We discuss potential games in Chapter 6 and their existence on the iterated local transitivity model. We conclude with some suggestions on how to extend and develop upon the work done in this thesis

A Comprehensive Overview of Classical and Modern Route Planning Algorithms for Self-Driving Mobile Robots

Mobile robots are increasingly being applied in a variety of sectors, including agricultural, firefighting, and search and rescue operations. Robotics and autonomous technology research and development have played a major role in making this possible. Before a robot can reliably and effectively navigate a space without human aid, there are still several challenges to be addressed. When planning a path to its destination, the robot should be able to gather information from its surroundings and take the appropriate actions to avoid colliding with obstacles along the way. The following review analyses and compares 200 articles from two databases, Scopus and IEEE Xplore, and selects 60 articles as references from those articles. This evaluation focuses mostly on the accuracy of the different path-planning algorithms. Common collision-free path planning methodologies are examined in this paper, including classical or traditional and modern intelligence techniques, as well as both global and local approaches, in static and dynamic environments. Classical or traditional methods, such as Roadmaps (Visibility Graph and Voronoi Diagram), Potential Fields, and Cell Decomposition, and modern methodologies such as heuristic-based (Dijkstra Method, A* Algorithms, and D* Algorithms), metaheuristics algorithms (such as PSO, Bat Algorithm, ACO, and Genetic Algorithm), and neural systems such as fuzzy neural networks or fuzzy logic (FL) and Artificial Neural Networks (ANN) are described in this report. In this study, we outline the ideas, benefits, and downsides of modeling and path-searching technologies for a mobile robot

Sparse topic modeling via spectral decomposition and thresholding

The probabilistic Latent Semantic Indexing model assumes that the expectation of the corpus matrix is low-rank and can be written as the product of a topic-word matrix and a word-document matrix. In this paper, we study the estimation of the topic-word matrix under the additional assumption that the ordered entries of its columns rapidly decay to zero. This sparsity assumption is motivated by the empirical observation that the word frequencies in a text often adhere to Zipf's law. We introduce a new spectral procedure for estimating the topic-word matrix that thresholds words based on their corpus frequencies, and show that its $\ell_1$-error rate under our sparsity assumption depends on the vocabulary size $p$ only via a logarithmic term. Our error bound is valid for all parameter regimes and in particular for the setting where $p$ is extremely large; this high-dimensional setting is commonly encountered but has not been adequately addressed in prior literature. Furthermore, our procedure also accommodates datasets that violate the separability assumption, which is necessary for most prior approaches in topic modeling. Experiments with synthetic data confirm that our procedure is computationally fast and allows for consistent estimation of the topic-word matrix in a wide variety of parameter regimes. Our procedure also performs well relative to well-established methods when applied to a large corpus of research paper abstracts, as well as the analysis of single-cell and microbiome data where the same statistical model is relevant but the parameter regimes are vastly different