UBI 532 Wireless Sensor Networks Paper Presentation Esra Rüzgar,

David Moore, John Leonard, Daniela Rus, Seth Teller MIT Computer Science and Artificial Intelligence Laboratory ACM SenSys’04, Baltimore, Maryland, USA ROBUST DISTRIBUTED NETWORK LOCALIZATION WITH NOISY RANGE MEASUREMENTS

OUTLINE LOCALIZATION CONCEPTS ABSTRACT INTRODUCTION APPROACH ANALYSIS EXPERIMENTAL RESULTS CONCLUSION

LOCALIZATION WHAT IS LOCALIZATION? A mechanism for discovering spatial relationships among objects WHY DO WE NEED LOCALIZATION? It is useful or even necessary for a node in a wireless sensor network to be aware of its location in the physical world in many applications. For example: Tracking Objects Reporting event origins Evaluating network coverage Assisting with routing Supporting for upper level protocols

PROPERTIES OF LOCALIZATION Physical position versus symbolic location Does the system provide data about the physical position of a node (in some numeric coordinate system) or does a node learn about a symbolic location? For example, “living room”, “office 123 in building 4”? Absolute versus relative coordinates Does the system provides absolute coordinates of nodes or positions with respect to each other but have no relationship to absolute coordinates? Localized versus centralized computation Are any required computations performed locally by nodes or are measurements reported to a central station that computes positions or locations and distributes them back to the nodes?

PROPERTIES OF LOCALIZATION Accuracy and precision Positioning accuracy is the largest distance between the estimated and the true position of an entity. Precision is the ratio with which a given accuracy is reached, averaged over many repeated attempts to determine a position. Scale Two important metrics: Are the area the system can cover per unit of infrastructure? What is the number of locatable objects per unit of infrastructure per time interval? Limitations Measurement noise Obstacles, for example GPS does not work indoors

LOCALIZATION PROBLEM Localization can be formulated as graph realization problem

ABSTRACT OF PAPER Distributed, linear-time algorithm for localizing sensor network nodes is proposed Robust quadrilaterals is introduced Trilateration is used for positioning No absolute position references is needed Mobility of nodes is supported Implemented on physical network Simulated for demonstrating scalability

INTRODUCTION Distributed computation and robustness in the precence of measurement noise are key ingredients for a localization algorithm In this algorithm: Nodes have ability to estimate distance to neighbors Localization problem is formulated as two-dimensional graph realization problem But localization is not easy!

DIFFICULTIES OF LOCALIZATION Insufficient data to compute a unique position assignment for all nodes Noisy distance measurements, compounding effects of insufficient data and creating additional uncertainty Lack of absolute reference points Unscalable algorithms

DIFFICULTIES OF LOCALIZATION To solve this difficulties robust quadrilaterals are introduced Robust quads are able to prevent incorrect realizations of flip ambiguities cope with arbitrary amounts of measurement noise But they are not able to localize well under conditions of high measurement noise and low node connectivity

RELATED WORK Algorithms based on local information Trilateration graphs is constructed and localized in linear time, Eren at al[1] Local clusters is used for localization, Capkun et al[2] Lower error bound for localization is derived, Savvides et al[3] But, none of them consider how measurement noise cause incorrect results!

RELATED WORK Algorithms based on propagation of location information from known reference nodes Received signal strength(RSS) and time of arrival(ToA) is used, Patwari et al[4] Distributed localization is proposed for low power devices based on connectivity, Bulusu et al[5] But, anchor nodes with known positions may not be always available!

CHALLENGES OF LOCALIZATION In graph theory, the problem of finding Euclidean positions for the vertices of a graph is known as the graph realization problem. Saxe showed that finding a realization is strongly NP-hard for the two-dimensional case or higher. However, knowing the length of each graph edge does not guarantee a unique realization, because deformations can exist in the graph structure that preserve edge lengths but change vertex positions.

CHALLENGES OF LOCALIZATION Rigidity theory is used to overcome this problem. Rigidity theory distinguishes between non-rigid and rigid graphs. Non-rigid graphs can be continuously deformed to produce an infinite number of different realizations, while rigid graphs cannot. However, in rigid graphs, there are two types of discontinuous deformations that can prevent a realization from being unique.

CHALLENGES OF LOCALIZATION Flip Ambiguities occur for a graph in a d-dimensional space when the positions of all neighbors of some vertex span a (d-1) dimensional subspace. Neighbors create a mirror through which the vertex can be reflected. Vertex A can be reflected across the line connecting B and C with no change in the distance constraints.

CHALLENGES OF LOCALIZATION Discontinuous flex ambiguities occur when the removal of one edge will allow part of the graph to be flexed to a different configuration and the removed edge reinserted with the same length. If edge AD is removed, then reinserted, the graph can flex in the direction of the arrow, taking on a different configuration but exactly preserving all distance constraints

APPROACH Algorithm can be broken three main phases Phase 1. Cluster Localization: localizes clusters into local coordinate systems. Each node becomes the center of a cluster and estimates the relative location of its neighbors which can be unambiguously localized. Phase 2. Cluster Optimization: refines the localization of the clusters using numerical optimization such as spring relaxation. This phase is optional. Phase 3. Cluster Transformation: computes coordinate transformations between these local coordinate systems by finding common nodes between clusters.

CLUSTER LOCALIZATION Quadrilaterals are relevant to localization because they are the smallest possible subgraph that can be unambiguously localized in isolation. Quadrilaterals are assumed to be globally rigid. Any two globally rigid quadrilaterals sharing three vertices form a 5-vertex subgraph that is also globally rigid. By induction, any number of quadrilaterals chained in this manner form a globally rigid graph. Global rigidity is not sufficient to guarantee a unique graph realization when distance measurements are noisy. Using robust quadrilaterals solves this problem.

CLUSTER LOCALIZATION Algorithm identifies only those triangles with a sufficiently large minimum angle as robust. Those triangles that satisfy following equation are called robust triangles. b is the length of the shortest side, Θ is the smallest angle d min is a threshold based on the measurement noise This equation bounds the worst-case probability of a flip error for each triangle.

CLUSTER LOCALIZATION With noisy measurements, trilateration can have flip ambiguity. An example of a flip ambiguity realized due to measurement noise. Node D is trilaterated from the known positions of nodes A, B, and C.

CLUSTER LOCALIZATION Algorithm uses the robust quadrilateral as a starting point, and localize additional nodes by chaining together connected robust quads. Whenever two quads have three nodes in common and the first quad is fully localized, the second quad can be localized by trilaterating from the three known positions. A natural representation of the relationship between robust quads is the overlap graph.

CLUSTER LOCALIZATION The entire algorithm for Phase I is as follows: 1. Distance measurements from each one-hop neighbor are broadcast to the origin node so that it has knowledge of the between-neighbor distances. 2. The complete set of robust quadrilaterals in the cluster is computed (Algorithm 1) and the overlap graph is generated. 3. Position estimates are computed for as many nodes as possible via a breadth-first search in the overlap graph (Algorithm 2).

CLUSTER LOCALIZATION

COMPUTING INTER-CLUSTER TRANSFORMATIONS In Phase III, the transformations between coordinate systems of connected clusters are computed from the finished cluster localizations. As long as there are at least three non-collinear nodes in common between the two localizations, the transformation can be computed. By testing if these three nodes form a robust triangle, non- collinearity and the same resistance to flip ambiguities are guaranteed as Phase I of the algorithm.

ANALYSIS Proof of Robustness: Flex ambiguities The use of robust quadrilaterals rules out the possibility of flex ambiguities. Discontinuous flex ambiguity occurs only when a rigid graph becomes non-rigid by the removal of a single edge.

ANALYSIS If the graph is such that no single edge removal will make it nonrigid, the graph is redundantly rigid, and no flex ambiguities are possible. The robust quad has six edges. By removing any edge, we are left with a 5-edged graph, which must be rigid according to Laman’s theorem. Robust quad with its missing edge has 4 vertices and 5 edges, satisfying the condition in Laman's Theorem. Since every 3-vertex subgraph has 3 or fewer edges and every 2-vertex subgraph has 1 or fewer edges, the 5-edged quad is rigid. Thus, the 6-edged robust quad is redundantly rigid. Therefore, ex ambiguities are impossible for a graph constructed of robust quads.

ANALYSIS Proof of robustness : Flip Ambiguities

ANALYSIS If measured distance is a random variable X, then the worst-case probability of error is P(X > d + d err ) If measurement noise is zero mean Gaussian with standard deviation σ the worst-case probability of error is

ANALYSIS Computational Complexity Finding set of robust quadrilaterals has worst-case runtime is, m is maximum node degree. Solving position estimates for one cluster is, q is number of robust quadrilaterals. Finding inter-cluster transformations for one cluster has runtime, m is number of transformations. Communication overhead for measuring distances between neighbors is.

EXPERIMENTAL RESULTS The algorithm has been tested on a network constructed of Crickets, a hardware platform developed by MIT. Crickets are hardware-compatible with the Mica2 Motes. This hardware enables the sensor nodes to measure inter-node ranges using the time difference of arrival (TDoA) between Ultrasonic and RF signals. Algorithm has been simulated with 183 nodes in order to show scalability. In simulations, node degree was varied by changing maximum ranging distance. Three different degrees of measurement noise was also considered.

EVALUATION CRITERIA The error that computed localization differs from known ground truth: The mean-square error of the distance measurements: The cluster success rate:

ACCURACY STUDY In this experiment 16 nodes placed randomly. Phase 1 of the algoritm is tested. Results are given below.

ACCURACY STUDY In this experiment 40 nodes placed randomly. Both Phase 1 and Phase 3 are tested. Results are given below.

SCALABILITY STUDY Simulation is done with 183 nodes for testing scalability of algorithm.

SCALABILITY STUDY

ERROR PROPAGATION

LOCATION OF MOBILE NODES

CONCLUSION Algorithm successfully localizes nodes in a sensor network with noisy distance measurements, using no beacons or anchors. Simulations and experiments showed the relationship between measurement noise and ability of a network to localize itself. Even with no noise, each node in the network must have approximately degree 10 or more before 100% node localization can be attained. As noise increases, so will the connectivity requirements. Algorithm adapts to node mobility by filtering the underlying measurements.

REFERENCES [1] Eren, T., Goldenberg, D., Whiteley, W., Yang,Y. R., Morse, A. S., Anderson, B. D. O., and Belhumeur, P. N. Rigidity, computation, and randomization in network localization. In Proc. IEEE INFOCOM (March 2004) [2] Capkun, S., Hamdi, M., and Hubaux, J.-P. GPS-free positioning in mobile ad-hoc networks. In Proceedings of the 34th Hawaii International Conference on System Sciences (2001) [3] Savvides, A., Garber, W., Adlakha, S., Moses, R., and Srivastava, M. B. On the error characteristics of multihop node localization in ad-hoc sensor networks. In Proc. IPSN (Palo Alto, CA, April 2003) [4] Patwari, N., III, A. O. H., Perkins, M., Correal, N. S., and O'Dea, R. J. Relative location estimation in wireless sensor networks. IEEE Trans. Signal Process. 51, 8 (August 2003) [5] Bulusu, N., Heidemann, J., and Estrin, D. GPS-less low cost outdoor localization for very small devices. IEEE Personal Communications Magazine 7, 5 (October 2000) Karl,H., Willig, A. Protocols and Architectures for Wireless Sensor Networks, John Wiley & Sons, Ltd. ISBN: , 2005