Hyperbolic navigation

Abstract: 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.

Abstract: The recent Arctic GAkkel Vents Expedition (AGAVE) to the Arctic Ocean's Gakkel Ridge (July-August 2007) aboard the Swedish icebreaker I-B Oden employed autonomous underwater vehicles (AUVs) for water-column and ocean bottom surveys. These surveys were unique among AUV operations to date in requiring georeferenced navigation in proximity to the seafloor beneath permanent and moving ice cover. We report results for long-baseline (LBL) acoustic navigation during autonomous under-ice surveys near the seafloor and adaptation of the LBL concept for several typical operational situations including navigation in proximity to the ship during vehicle recoveries. Fixed seafloor transponders were free-fall deployed from the ship for deep positioning. The ship's helicopter collected acoustic travel times from several locations to georeference the transponders' locations, subject to the availability of openings in the ice. Two shallow beacons suspended from the ship provided near-surface spherical navigation in ship-relative coordinates. During routine recoveries, we used this system to navigate the vehicles into open water near the ship before commanding them to surface. In cases in which a vehicle was impaired, its position was still determined acoustically through some combination of its acoustic modem, the fixed seafloor transponders, the ship-deployed transponders, and an onboard backup relay transponder. The techniques employed included ranging adapted for a moving origin and hyperbolic navigation. © 2008 Wiley Periodicals, Inc.

Abstract: A recently developed procedure [1] for assessing the accuracy of hyperbolic multilateration systems makes it easy to determine basic limitations on accuracy. This paper illustrates how such bounds can be derived. The results include bounds for a variety of geometries that are representative of practical ground-based and satellite-based hyperbolic systems. The results are applicable whenever the ranging errors can be treated as uncorrelated zeromean random variables. In some cases the bounds quantify general knowledge (e. g., the directional dependence of errors). In other cases the bounds represent entirely new limitations (e. g., optimum accuracies for sector-restricted and cone-restricted transmitter/receiver configurations).

Abstract: Spherical positioning systems determine position by measuring acoustic travel times from beacons at known locations. To make this travel time measurement, the receiver must know exactly when each beacon transmitted. Hyperbolic positioning systems determine position by measuring differences in travel time between signals from the beacons. The hyperbolic receiver does not need to know when the beacons transmitted, only that they all transmitted at the same time or with known delays relative to each other. The spherical system, which must know the transmit time exactly, and the hyperbolic system, which does not know transmit time at all can be seen as the two endpoints of a continuum of systems parameterised by the accuracy with which the transmit time is known. This paper demonstrates that the Cramer-Rao bounds for position accuracy approach those of the spherical navigation system as the variance in the transmit time estimate approaches zero, and approach those of the hyperbolic positioning system as the variance in the transmit time estimate becomes infinite. This observation has practical application in that the position accuracy of a hyperbolic navigation system which transmits at regular time intervals may be improved if it estimates not only receiver position, but also the transmit time of the beacons. As the estimate of transmit time improves, the position resolution becomes closer to that of a spherical positioning system. This improvement in position accuracy is shown for a typical underwater acoustic positioning scenario.