5.a Framework for Wireless Sensor Network Localization Using Triangular Flips
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Transcript of 5.a Framework for Wireless Sensor Network Localization Using Triangular Flips
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Proceedings of the National Conference on Mobile and Adhoc Networks, 29th
& 30th
October 2010
Dr. Mahalingam College of Engineering and Technology, Pollachi. 23
A Framework for Wireless Sensor Network Localization using Triangular Flips
E. Rama Krishna, Assoc. Prof, [email protected]
G.Rajesh Kumar, Assoc. Prof, [email protected] VITS, Karimnagar
G. Rajinikar, Assoc. Prof, [email protected] N. Rajendar Reddy, Assistant Professor, [email protected]
VCE, Warangal Abstract: Localization has been regarded as one of the fundamental and supporting technology for many applications of wireless sensor networks. The problem of localization of nodes in Wireless Sensor Networks is approached by the implementation of different methods. After explaining each method, The characteristics of Localization and Positioning is introduced to understand the problem. Classification of every technique, depending on how a solution is approached and also comparisons from each method to another is done. Current Study presents an overview of localization techniques and surveys the currently available algorithms for localization based on Triangulation Methods and Flip based improvements Introduction Schemes for localization in WSN have been developed in the last 20 years, mostly being motivated by military use. Numerous studies have been performed since then for civil uses. Researchers have pointed out the influence of noise on the localization process, and the importance of various system parameters on the accuracy and efficiency of the localization process, but there is no consensus of a single best algorithm for localization in sensor networks. This indeed depends on the environment and the specifications of the used motes. Many applications have a need for localization, be it for locating people or objects. Most of the time, data recorded from a wireless sensor only makes sense if correlated to a position, for example the temperature recorded in a given machine room or cold-store. Similarly, many end-user programs are location-aware, for example people would like to find the closest bus stop or mailbox, and emergency services need to localize persons to be rescued. In the following, we refer to a person, object or computer coupled with a
wireless sensor to be localized as an (unknown) node. In both ubiquitous computing and wireless sensor networks (WSNs), localization has drawn considerable attention. The major difference between these two fields lies in the capabilities of the considered computing devices. Ubiquitous computing usually considers devices such as laptops and PDAs that are rather powerful compared to a wireless sensor. A sensor node has both a very limited memory footprint and CPU power, and energy – provided most of the time by a small battery – is a scarce resource. As such, localization algorithms for wireless sensor networks have to be efficient, both in terms of computation and power consumption. Another difference between ubiquitous computing and wireless sensor networks is that laptops and PDAs have often been considered mobile while most of the existing experiments in wireless sensor networks have concentrated on static networks of sensors. At the moment, few low-cost localization algorithms exist that have been specifically designed with sensor movement in mind. Nowadays, the most simple, off-the-shelf, mechanism to determine the location of a mobile node is to use the Global Positioning System (GPS). GPS offers 3D localization based on direct line-of-sight with at least four satellites, providing an accuracy up to three meters. However, some limitations of GPS ask for alternative localization methods. First, GPS is at the moment barely usable indoors, in cluttered urban areas and under dense foliage. Second, while the cost for GPS equipment has been dropping over the years, it is still not suited for mass-produced cheap sensor boards, phones and even PDAs. Third, GPS equipment requires both hardware space and energy, which are two limiting factors for integration on miniaturized sensor boards. To overcome GPS limitations, researchers have developed fully GPS-free techniques for
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Proceedings of the National Conference on Mobile and Adhoc Networks, 29th
& 30th
October 2010
Dr. Mahalingam College of Engineering and Technology, Pollachi. 24
locating nodes as well as techniques where few nodes, commonly called anchors, use GPS to determine their location and, by broadcasting it, help other nodes in calculating their own position without using GPS. Localization in WSNs Recent advances in hardware and wireless communication technologies have resulted in the development of low cost, low power, multi functional sensor devices called sensor nodes. These tiny nodes with sensing, data processing and communication capabilities collaborate among themselves to establish a multi hop network called Wireless Sensor Networks (WSN). Important WSN applications involve monitoring of the physical world. Location of sensor nodes is crucial information in many sensor network applications like environment monitoring and target tracking. The localization problem in WSN involves… Centralized localization method: Centralized localization algorithms require base station to gather network-wide environment information and with plenty of computational power. Base station determines the location of each node by collected data and transport them back into network. The collection of information performed by message exchange between nodes, hence, with the number of nodes increased in network, centralized localization algorithms become lower energy-efficiency, longer delay and larger network communication traffic. In another hand, it will obtain relative high precise location. Common node has light calculation burden. In general, it was suitable for static small networks. Distributed localization method: In the process of distributed localization, each node independently determined its location with only limited communication with one-hop or multi-hops neighbor nodes. It has the characteristics of small traffic, equal calculation burden of each node, little storage requirements, good scalability. However, due to the lack of global information, location accuracy is susceptible to the number of beacon nodes and the distribution of nodes. Range-based localization method: A Range-based localization method depends on distance or angle between nodes to obtain unknown node’s location. The first step is to know the distance estimates and angle estimates. A number of approaches, such as time of arrival, time difference of arrival, angle of arrival, received signal strength, have been presented. The algorithms based on these
attributes are called anchors or beacons. Range-based localization methods have the advantage of fine resolution. However, extra hardware and additional energy consumption restricted the application of range-based methods. Range-free localization method: Range-free localization methods use the information of topology and connectivity for location estimation. Range-free methods have some advanced characteristics, such as low cost, small communication traffic, no extra hardware and flexible localization precision. Because of these special characteristics of range-free methods, they were been regard as a promising solution for the localization problem in WSN. COMPARISON OF LOCALIZATION ALGORITHMS localization Algorithms evaluation is based on: Scalability: Is a localization algorithm scalable to the number of nodes in network. Self-organized: Because of the lack of localization infrastructure in wireless sensor network, the self-organization of localization based algorithm is necessary. Robustness: Localization algorithm is immune to node failure and distance estimation error. Efficient energy: Message exchange is an indicator of energy consumption. How much communication overhead is required? Distributed calculation: Does localization algorithm only uses localized information? Comparison: different localization algorithms advantages and disadvantages. Finally the aim is to choose a suitable location algorithm for different applications of wireless sensor networks.
Selection of Localization Schemes for Sensor Networks Summarizing the state of the art, a comparison of the present localization techniques is presented. For this three aspects have been considered:
• The accuracy and resolution of the obtained positional data
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Proceedings of the National Conference on Mobile and Adhoc Networks, 29th
& 30th
October 2010
Dr. Mahalingam College of Engineering and Technology, Pollachi. 25
• The spatial range of the localization, and
• The ability to build power-aware nodes, which is from special interest for sensor networks due to the typical need for low power consumption.
In order to give an overview, these considerations are taken on three axes in a coordinate system and the presented localization techniques are classified as
Range-based localization method A relative complete description of adhoc positioning systems is given in, comparing DV-Hop (Distance Vector), DV-Distance and Euclidian propagation methods. The first one computes estimation for the size of one hop, while DV-distance measures the radio signal strength and is propagated in meters. The Euclidian scheme propagates the true distance to the anchor. DV schemes are suitable in most cases, while Euclidian is more accurate, but costs much more communications. DV-Distance Using estimate ranges between neighbours, a node can determine its distance to an anchor. If a node take advantage of this range estimates with a sufficient number of neighbours, the Euclidean distance to a far away anchor can be estimated. This Euclidean method precision depends on the number of nodes. If the number of nodes increases the better accuracy results are obtained.
How the Euclidean method works
DV-Hop Count the number of hops along the shortest path, between any two anchors, to estimate the average length of a single hop by dividing the sum of the distances to other anchors by the sum of the hop counts. Every anchor computes this estimated hop length and spread it into the network, so that other nodes with unknown position could use this information to estimate a multi-hop range and perform the multilateration algorithm.
How DV-hop works Anchors: flood network with known position flood network with avg hop distance Nodes count the number of hops to anchors multiply with avg hop distance Range-free methods A description of ad-hoc localization system is the devices were individually tuned (built-in calibration interface or original long life calibration). In sensor networks, as large number of sensors are used, that cannot be the case. Calamari, an ad-hoc localization system was developed that also integrates a calibration process. Regarding localization, it uses fusion of RF received signal strength information and acoustic time of flight. There is an interesting definition of a distributed algorithm for random WSN. The minimal density of known nodes is presented. The main objective of their algorithm is to broadcast a request (”Do you hear me?”) and compute the estimated localization by the interpretation of the answer of all the known nodes. A related method, called APIT APIT The APIT (Approximate Point In Triangulation) idea is to divide the environment into triangles,
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Proceedings of the National Conference on Mobile and Adhoc Networks, 29th
& 30th
October 2010
Dr. Mahalingam College of Engineering and Technology, Pollachi. 26
given by beaconing nodes. An individual node’s presence or absence in each of those triangles will allow to reduce the possible location area. This goes until all the possible sets are exhausted, or the desired accuracy reached. The APIT algorithm is then run at every node: 1. Receive locations from n anchors. 2. For each possible triangle, test if inside or not. 3. If yes, add it to the Inside Set. 4. Break if accuracy reached. 5. Estimate position as Two triangulations are related by a flip if they are the only two refinements of a polyhedral subdivision that can only be refined by triangulations. We called such a subdivision an almost triangulation the following characterization of “almost-triangulations”: a polyhedral subdivision of a configuration A that is not a triangulation is an almost-triangulation if and only if all its cells are either simplices or have corank one (that is, they have two more elements than their affine dimension), and all the cells which are not simplices share one and the same circuit. Every flip “happens” on a circuit, where a circuit is a minimal affinely dependent subset of points, and splits in a unique way into a pair (Z+,Z_) with the property that conv(Z+)∩conv(Z_)≠0. In the plane there are the following three possibilities, depending on the type of circuit in question. Remember that the type of a circuit (Z+,Z_) is the pair (|Z+|,|Z_|): 1. If the circuit is of type (2;2), that is, it consists of the four vertices of a convex quadrilateral, then the almost triangulation S has a unique non-simplicial cell, consisting of these four points, because a cell strictly containing this circuit has corank greater than one. Hence, the two refinements of S are obtained by inserting one or the other diagonal of this quadrilateral. The flip is normally called a diagonal edge flip, and an example of it is depicted on the top part of Figure below 2. If the circuit is of type (3;1), that is, it consists of a point a in the interior of the triangle with vertices the other three {b;c;d}, then again the almost triangulation S has a unique non-simplicial cell, consisting of these four points. Its two refinements are obtained “forgetting the interior point a” (which produces a non-full triangulation) and inserting a as a
new vertex incident to the three triangles {a,b,c},{a,c,d}, and {a,b,d}. The flip is called an insertion-deletion flip because it inserts or deletes a vertex from the triangulation. Find this situation in Figure. 3. If the circuit is of type (2,1), that is, it consists of three collinear points, then one or two cells of S will contain it, depending whether the collinearity lies in the boundary or the interior of conv(A). In any case, the two refinements are obtained forgetting the central point of the collinearity, or inserting it (which removes one or two triangles of the triangulation and inserts two or four new ones). The bottom part of Figure shows this flip in the case of an interior collinearity.
Strictly speaking there is a forth type of flip that can happen, but we do not need to care much about it. If we consider a configuration A with a repeated point, the two copies of this point form a (1,1)-circuit. The flip on this circuit consists merely in changing our mind as to what copy of this point we want to consider a vertex in our triangulation. All triangulations of a point set in the plane are connected by flips The most natural question to ask about flips is whether any pair of triangulations of a
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Proceedings of the National Conference on Mobile and Adhoc Networks, 29th
& 30th
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Dr. Mahalingam College of Engineering and Technology, Pollachi. 27
configuration A can be connected to one another by a finite sequence of flips. One can also wonder about the diameter of the graph of flips, or its maximum or minimum degree. In this section we discuss these questions. A Delaunay triangulation for a set P of points in the plane is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid skinny triangles For a set of points on the same line there is no Delaunay triangulation (in fact, the notion of triangulation is undefined for this case). For four points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: the two possible triangulations that split the quadrangle into two triangles satisfy the "Delaunay condition", i.e., the requirement that the circumcircles of all triangles have empty interiors. By considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions. Generalizations are possible to metrics other than Euclidean. However in these cases a Delaunay triangulation is not guaranteed to exist or be unique.
A Delaunay triangulation in the plane with circum circles
Many algorithms for computing Delaunay triangulations rely on fast operations for detecting when a point is within a triangle's circumcircle and an efficient data structure for storing triangles and edges. In two dimensions, one way to detect if point D lies in the circum circle of A, B, C is to evaluate the determinant:
Assuming A, B and C to lie counter-clockwise, this is positive if and only if D lies in the circumcircle. Effective enumeration of triangulations There is also the practical problem of computing exactly the number of distinct triangulations for concrete instances of point configurations. A way to list all triangulations is via a depth-first search or breadth-first search traversal of the graph of flips. By the connectivity results we have just seen, we are guaranteed to visit all triangulations. The trouble with this approach is that we may need a large amount of storage during these search traversals (saving all triangulations that have been visited). But there is a memory-efficient method of listing all triangulations of a point set in the plane: The reverse search enumeration of Avis and Fukuda Reverse-search is in fact a very general method of listing all vertices of some directed graphs, under certain special assumptions. The amount of memory used for book-keeping is very small and it is independent of the size of the graph. The actual visit to the nodes is done in a depth-first search order and the running time of the reverse-search enumeration is polynomial on the number of triangulations. Thus the algorithm has a good output-sensitive complexity. For brevity we will not present reverse-search in full generality here, but only outline it for the graph of triangulations of point configurations in the plane. The main point of the algorithm is that, for point configurations in general position and with no co-circular points, the graph of flips can be oriented in such a way that it becomes an acyclic graph with a unique sink, the sink being at the Delaunay triangulation (which is unique under the noncocircularity assumption). From any other triangulation we
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Proceedings of the National Conference on Mobile and Adhoc Networks, 29th
& 30th
October 2010
Dr. Mahalingam College of Engineering and Technology, Pollachi. 28
can reverse back to a parent triangulation by an oriented flip. For each triangulation we have an adjacency oracle that tells us its flip neighbors. We can also define a successor oracle that given a triangulation T assigns a unique successor. This should be another triangulation, closer to becoming a Delaunay triangulation. We saw that a triangulation is the Delaunay triangulation if each edge is locally Delaunay. Order all possible edges of the point configuration in some way, for instance lexicographically. From a triangulation T its successor is T
1 if T
1 is obtained from T
by flipping the first (in our ordering) flippable edge which is not locally Delaunay. Note that such successor T
1 exists unless T is
already the Delaunay triangulation. all triangulations of a generic hexagon with their successors are
Consider a point configuration in the plane A and an A polygon R. Let T be a triangulation of R and l a line that intersects T properly, meaning it cuts through the interior of edges. We define the path of T along l, and denote it pathl(T ) as the unique chain of edges from the triangulation T , such that (a) l properly intersects all edges in the chain, (b) The chain starts and ends at two boundary edges of R such that the segment of l between the two intersection points lies in the interior of R, (c) Alternating vertices on the chain lie in opposite sides of l, and (d) The area bounded by the chain and l has no other points of the configuration.
The main property of paths is that they allow us to “identify” all triangulations by their paths coming from line l and thus we can apply a divide-and-conquer enumeration strategy
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Proceedings of the National Conference on Mobile and Adhoc Networks, 29th
& 30th
October 2010
Dr. Mahalingam College of Engineering and Technology, Pollachi. 29
(a) Given a point configuration in the plane A, a triangulation T of an A-polygon R, and a line l that intersects properly T there exists always a path pathl(T ). (b) Given a point configuration in the plane A, triangulations T 1; T 2 of an A-polygon R, and a line l that intersects properly T 1, any two paths pathl(T 1) and pathl(T 2) with same start and end edges are either identical or properly intersect each other. Proof: We can assume that l is a vertical line, otherwise a rotation can be applied. To construct a path, we proceed inductively, starting with the top boundary edge e1 where l first enters the interior of R. Let T1 be the triangle containing e1 and let e2 be the other edge of T1 crossed by l. Let the vertices of T1 be called p0, p1 and p2 in such a way that e1 = p0p1 and e2 = p0p2. In the general inductive step, assume that we have already constructed a path e1 = p0p1,e2 = p1p2,….., ei = pi-1pi in the desired conditions. If ei is a bottom boundary edge of R we have finished, otherwise let us show how to continue the process. Let p0 be the third vertex of the triangle in T based on ei and below it. Either p0 lies on the opposite side of l from pi, in which case we can continue the path with ei+1 = pip0, or p0 is in the same side as pi in which case we abandon the last edge ei = pi-1pi of our provisional path and make pi-1p0 be the new i-th edge. This does not increase the number of edges of the path, but it makes the path reach lower along l as before, so that the process eventually terminates, once we hit the boundary of R again to exit (the edge by construction has its extremes in opposite sides) To prove the second part of the lemma we proceed by contradiction. If we have two paths pathl(T 1) and pathl(T 2) different paths which do not properly intersect. Thus if they do not have common vertices then path pathl(T 1) lies entirely to the left of pathl(T 2). But since pathl(T 2) is a path, no edge of pathl(T 1) can intersect l. This is a contradiction. Next, suppose the two paths have indeed a common vertex p, but the successor of p is different in each of the paths, say p
1,p
11,
The main point is that now the point p
1 cannot
be placed anywhere without violating part (d) of the definition or forcing a proper intersection between the two paths.
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Proceedings of the National Conference on Mobile and Adhoc Networks, 29th
& 30th
October 2010
Dr. Mahalingam College of Engineering and Technology, Pollachi. 30
Conclusion: The Delaunay triangulation Method addresses the practical problem of computing exactly the number of distinct triangulations for concrete instances of point configurations which helps effective calculation of localization in both rage based and rage free methods and also in distributed and centralized localization methods which follow the triangle approaches References:
[1] Location Management and Routing in Mobile Wireless Networks Amitava Mukherjee Somprakash Bandyopadhyay Debashis Saha
[2] Algorithms for localization range based
Jorge Luis Ariza Alvarez, Jairo Enrique Durango Carrasquilla
[3] http://en.wikipedia.org/wiki/Delaunay_tria
ngulation [4] Algorithms and computation in
mathematics Springer Triangulations: Structures for Algorithms and
Applications Jesus A. De Loera, Jörg Rambau, Francisco Santos
[5] F. Santos and R. Seidel. A better upper
bound on the number of triangulations of a planar point set.
[6] F. Santos. Geometric bistellar flips: the
setting, the context and a construction. In International Congress ofMathematicians
[7] Convex Position Estimation in
Wireless Sensor Networks, Lance Doherty, Kristofer S. J. Pister, Laurent El Ghaoui.
[8] Convex Position Estimation in
wireless sensor network, Lance Doherty/Kristofer S.J.Pister/Laurent El Ghaoui, Dept. of Electrical Engineering and Computer Sciences-University of California, Berkeley
[9] Location System for ubiquitous
computing, Jeffrey Highttower/ Gaetano Borriello, University of Washington