Autonomous Unmanned Aerial Vehicles (UAVs) have gained popularity due to their many potential application fields. Alongside sophisticated sensors, UAVs can be equipped with communication adaptors aimed for inter-UAV communication. Inter-communication of UAVs to form a UAV swarm raises questions on how to manage its communication structure and mobility. In this paper, we consider therefore the problem of establishing an efficient swarm movement model and a network topology between a collection of UAVs, which are specifically deployed for the scenario of high-quality forest-mapping.
When considering three dimensions, the question for optimal node positioning is called the @cite_14 . A related problem in geometry is the @cite_1 , which is the number of non-overlapping unit spheres arranged such that each sphere touches another.
@cite_14 @cite_1
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In this paper the solution to the problem of placing n connected points (or nodes) in r-dimensional Euclidean space is given. The criterion for optimality is minimizing a weighted sum of squared distances between the points subject to quadratic constraints of the form X'X - 1, for each of the r unknown coordinate vectors. It is proved that the problem reduces to the minimization of a sum or r positive semi-definite quadratic forms which, under the quadratic constraints, reduces to the problem of finding r eigenvectors of a special "disconnection" matrix. It is shown, by example, how this can serve as a basis for cluster identification. A finitely presented semigroup RSGn is constructed for n ≥ 2. The centre of RSGn encodes uniquely up to rigid ambient isotopy in 3-space all nonoriented spatial graphs with vertices of degree ≤ n. This encoding is obtained by using three-page embeddings of graphs into the three-page book T × I, where T is the cone on three points, and I is the unit segment. The notion of the three-page complexity for spatial graphs is introduced via three-page embeddings. This complexity satisfies the properties of finiteness and additivity under natural operations. We present the H3 layout technique for drawing large directed graphs as node-link diagrams in 3D hyperbolic space. We can lay out much larger structures than can be handled using traditional techniques for drawing general graphs because we assume a hierarchical nature of the data. We impose a hierarchy on the graph by using domain-specific knowledge to find an appropriate spanning tree. Links which are not part of the spanning tree do not influence the layout but can be selectively drawn by user request. The volume of hyperbolic 3-space increases exponentially, as opposed to the familiar geometric increase of euclidean 3-space. We exploit this exponential amount of room by computing the layout according to the hyperbolic metric. We optimize the cone tree layout algorithm for 3D hyperbolic space by placing children on a hemisphere around the cone mouth instead of on its perimeter. Hyperbolic navigation affords a Focus+Context view of the structure with minimal visual clutter. We have successfully laid out hierarchies of over 20,000 nodes. Our implementation accommodates navigation through graphs too large to be rendered interactively by allowing the user to explicitly prune or expand subtrees. We develop a novel localization theory for planar networks of nodes that measure each other?s relative position, i.e., we assume that nodes do not have the ability to perform measurements expressed in a common reference frame. We begin with some basic definitions of frame localizability and orientation localizability. Based on some key kinematic relationships, we characterize orientation localizability for networks with angle-of-arrival sensing. We then address the orientation localization problem in the presence of noisy measurements. Our first algorithm computes a least-square estimate of the unknown node orientations in a ring network given angle-of-arrival sensing. For arbitrary connected graphs, our second algorithm exploits kinematic relationships among the orientation of node in loops in order to reduce the effect of noise. We establish the convergence of the algorithm, and through some simulations we show that the algorithm reduces the mean-square error due to the noisy measurements.