Positioning Chapter 10 TexPoint fonts used in EMF. Read the TexPoint manual befo
Norman, Royal, Meteorologist has reference to this Academic Journal, PHwiki organized this Journal Positioning Chapter 10 TexPoint fonts used in EMF. Read the TexPoint manual be as long as e you delete this box.: AAAAAAA Acoustic Detection (Shooter Detection) Sound travels much slower than a radio signal (331 m/s) This allows as long as quite accurate distance estimation (cm) Main challenge is to deal with reflections in addition to multiple events Rating Area maturity Practical importance Theory appeal First steps Text book No apps Mission critical Boooooooring Exciting
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Motivation GPS, Geodetics, etc. Measuring Distances or Angles Rigidity Theory Anchors & Virtual Coordinates Embeddings, Theory in addition to Heuristics Boundary Recognition Overview Motivation Why positioning Sensor nodes without location in as long as mation are often meaningless Avoid having costly positioning hardware Geo-routing Why not GPS (or Galileo) Heavy, large, in addition to expensive Battery drain Not indoors Accuracy Solution: equip small fraction with GPS (anchors) GPS A lot of recent progress, so attaching a GPS receiver to each sensor node becomes an alternative. Example, u-blox footprint size: down to 4x4mm Power supply: 1.8 – 4.8V power consumption: 50 mW power on: < 1 second update rate: 4Hz support as long as Galileo! So GPS is definitely becoming more attractive; however, some of the problems of GPS (indoors, accuracy, etc.) remain. GPS Extensions GPS chips can be extended with other sensors such as gyroscope, direction indications, or tachometer pulses (in cars). With these add-ons, mobile devices get continued coverage indoors. Af as long as dable technology has a distance error of less than 5% per distance travelled, in addition to a direction error of less than 3 degrees per minute. [u-blox] Geodetic Networks Optical Sensors Positions through reference points in addition to object points Very accurate (mm) GNSS Global Navigation Satellite System Allow phase measurements [Geodesy pics by Rainer Mautz] GPS meets Geodesy Use of both GPS frequencies L1 & L2 Phase measurements DGPS (relative measurements) elimination of ionosphere delays mm accuracy More Distance Sensors Geotechnical Sensors one-dimensional, relative measurements Meteorological Sensors correction of other sensors Digital Cameras availability: high timeliness: real time reliability: no failures hybrid systems: to be avoided local installations: none accuracy: mm - cm coverage: global Geodesy vs. Sensor Nodes but: Measurements with reasonably priced hardware Distance estimation Received Signal Strength Indicator (RSSI) The further away, the weaker the received signal. Mainly used as long as RF signals. Time of Arrival (ToA) or Time Difference of Arrival (TDoA) Signal propagation time translates to distance. Routing trip time measurements with specific hardware: accuracy 2-3m Better: Mixing RF, acoustic, infrared or ultrasound. Angle estimation Angle of Arrival (AoA) Determining the direction of propagation of a radio-frequency wave incident on an antenna array. Directional Antenna Special hardware, e.g., laser transmitter in addition to receivers. Example: Measuring distance with two radios Particularly interesting if the signal speed differs substantially, e.g. sound propagation is at about 331 m/s (depending on temperature, humidity, etc.), which is of course much less than the speed of light. If you have line of sight you may achieve about a 1cm accuracy. But there are problems (Ultra)sound does not travel far For good results you really need line of sight You have to deal with reflections s t radio (ultra)sound Ultrasound Lab Example: Multipath Interferometry Interferometry is the technique of superimposing (interfering) two or more waves, to detect differences between them Signals transmitted with a few hundred Hz difference at senders A in addition to B will give different phase offsets at C in addition to D. Using this, one can compute the total distance of the four points A, B, C, D. However, one needs to solve a linear equation system, in addition to one needs very accurate time synchronization (¹s order) Positioning Systems: An Overview Positioning in Networks Task: Given distance or angle measurements or mere connectivity in as long as mation, find the locations of the sensors. Anchor-based Some nodes know their locations, either by a GPS or as pre-specified. Anchor-free Relative location only. Sometimes called virtual coordinates. Theoretically cleaner model (less parameters, such as anchor density) Range-based Use range in as long as mation (distance or angle estimation) Range-free No distance estimation, use connectivity in as long as mation such as hop count. It was shown that bad measurements dont help a lot anyway. Overview with anchors without anchors Positioning (Solution quality depends on anchor density) Distance/Angle based Virtual Coordinates Connectivity based Virtual Coordinates distance/angle in as long as mation connectivity in as long as mation only Trilateration in addition to Triangulation Use geometry, measure the distances/angles to three anchors. Trilateration: use distances Global Positioning System (GPS) Triangulation: use angles Some cell phone systems How to deal with inaccurate measurements Least squares type of approach What about strictly more than 3 (inaccurate) measurements initial anchor becomes anchor in 1st step becomes anchor in 2nd step becomes anchor in 3rd step Cooperative Multihop Multilateration: Iterative Multilateration Problems with Distance Measurements No in as long as mation Pair of nodes simply too far apart Physical obstacles: Line-of-sight needed as long as ultrasound, laser, infrared, Limitations of ranging hardware: Might not be omni-directional. Not enough in as long as mation Too sparse deployment to determine a globally rigid structure Inaccurate in as long as mation Measurement errors will sum up Idea: Reduce depth of iterative multilateration Trying to get an over-constrained system (to decrease errors) Problem well known in mobile robotics Rigidity Theory [Jie Gao] Given a set of rigid bars connected by hinges, rigidity theory studies whether you can move them continuously. Continuous de as long as mations, or flips Rigidity Theory Theory concerned with when we are guaranteed uniqueness of structures in 2D or 3D (or higher D). Applications to structural engineering, molecular structures Definition: An n-point as long as mation P in d-space consists of: Coordinates in d-space as long as points p1, ,pn, in addition to A set of edges between some points (indices). In other words, a graph embedded in d-dimensional space. Definition: An n-point as long as mation P in d-space is globally rigid provided that any other n-point as long as mation Q with the same edges in addition to the same distances on those edges is the same as P, up to translation, rotation, in addition to reflection. Flips are at least locally rigid. Local rigidity makes sense in e.g. structural engineering Continuous de as long as mations are not rigid at all Global Rigidity There are some simple rules that guarantee global rigidity: In 2D, a triangle with distances between all pairs is globally rigid. Starting with a triangle, if we repeatedly add a node in addition to edges (with distances) to at least 3 non-collinear points, it results in a globally rigid point as long as mation. In 3D, a tetrahedron (with all 6 distances) is globally rigid. Starting with a tetrahedron, if we repeatedly add a node in addition to edges (with distances) to at least 4 non-coplanar points, it results in a globally rigid point as long as mation.
Noisy distance estimates Noise can introduce anomalies We might no longer get exact solutions, but may have some error (difference between given in addition to computed distances). Tolerable, unavoidable. Solutions may no longer be unique, even allowing as long as small errors: varying the measurements a tiny amount could yield drastically different best solutions. Errors caused by noisy measurements can become compounded through an iterative coordinate assignment procedure. [Jie Gao] Coping with noisy measurements Idea: Robust quadrilaterals Robust with respect to a bound e on error in distance calculations. 4 nodes in 2D, edges ( in addition to distances) between all pairs. Nodes spaced so, even with errors, only one realization is possible. Method Parameterized by bound e on measurement error. Start with a robust quadrilateral, in addition to then add nodes iteratively. Discussion Avoids certain kinds of ambiguities (flip, discontinuous behavior) But not enough to guarantee global rigidity To guarantee global rigidity, additional constraints are needed, e.g. on angles Intuitively: if angles are too small, then flips are easy, even if noise is bounded Rigidity alternative: Simple hop-based algorithms Algorithm Get hop distance h to anchor(s) Intersect circles around anchors radius = distance to anchor Choose point such that maximum error is minimal Find enclosing circle (ball) of minimal radius Center is calculated location In higher dimensions: 1 < d · h Rule of thumb: Sparse graph bad per as long as mance How about no anchors at all In absence of anchors nodes are clueless about real coordinates. For many applications, real coordinates are not necessary Virtual coordinates are sufficient Geometric Routing requires only virtual coordinates Require no routing tables Resource-frugal in addition to scalable s d Virtual Coordinates Idea: Close-by nodes have similar coordinates Distant nodes have very different coordinates Similar coordinates imply physical proximity! Applications Geo-Routing Locality-sensitive queries Obtaining meta in as long as mation on the network Anycast services (Which of the service nodes is closest to me) Outside the sensor network domain: e.g., Internet mapping Model Unit Disk Graph (UDG) to model wireless multi-hop network Two nodes can communicate iff Euclidean distance is at most 1 Sensor nodes may not be capable of Sensing directions to neighbors Measuring distances to neighbors Goal: Derive topologically correct coordinate in as long as mation from connectivity in as long as mation only. Even the simplest nodes can derive connectivity in as long as mation 1 u v Boundary recognition Open problem One tough open problem of this chapter obviously is the UDG embedding problem: Given the adjacency matrix of a unit disk graph, find positions as long as all nodes in the Euclidean plane such that the ratio between the maximum distance between any two adjacent nodes in addition to the minimum distance between any two non-adjacent nodes is as small as possible. There is a large gap between the best known lower bound, which is a constant, in addition to the polylogarithmic upper bound. It is a challenging task to either come up with a better approximation algorithm or prove a stronger (non-constant) lower bound. Once we underst in addition to this better, we can try networks with anchors, or with (approximate) distance/angle in as long as mation. Generally, beyond GPS this area is in its infancy.
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