CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

1. CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch

2. Lecture 9 • Topic: • Localization • Sources: • Priyantha, Chakraborty, Balakrishnan. Cricket. • Savvides, Han, Srivastava. Dynamic fine-grained localization in ad-hoc networks of sensors. • MIT 6.885 Fall 2008 slides Discrete Algs for Mobile Wireless Sys

3. Localization Problem • Ad hoc setting, with many wireless devices. • Assume sufficiently reliable communication between nearby nodes (achieved somehow by the MAC layer). • Assume, if necessary, that the nodes have synchronized clocks. • Precisely synchronized, or • Approximately synchronized (global or gradient). • Network may (or may not) include anchors: nodes that start out knowing their own locations in 2-dimensional or 3-dimensional space. • Exact or approximate. • Some nodes may have info about their distances to nearby nodes. • Exact or approximate. Discrete Algs for Mobile Wireless Sys

4. Localization Problem • Localization Problem:Each node should determine “reasonable'' coordinates for itself, in 2D or 3D space, in a way that is consistent with the given anchor coordinates and distances. • The first time we are considering geographical information. • Different from Location Problem, in which each node determines the locations of other nodes. Discrete Algs for Mobile Wireless Sys

5. Why Localization? • Location information could be useful for applications in ad hoc networks, e.g.: • For sensor networks that collect data or report events, it's important to know where the data readings or events occurred. • For networks that control robots, cars, or airplanes, it’s important to know where the robots, cars,…are. • Location information can be used in location-aware algorithms, for tasks like: • Routing point-to-point messages • Network-wide broadcasting • Aggregating data on the way to a central location • Establishing spanning trees • Implementing virtual geography-based infrastructure • Allocating nearby resources Discrete Algs for Mobile Wireless Sys

6. Global Positioning System (GPS) • Simplest localization method. • Allow all nodes to have GPS receivers, to obtain position data from GPS satellites. • That's generally a good idea, but: • It doesn't work indoors. • It doesn’t work outdoors with obstacles to line-of-sight from the satellites (e.g., in the woods). • Locations are only approximate---to within about 10-20 feet. • Somewhat expensive---OK for laptops, but may be prohibitive for cheaper devices like sensor nodes (but getting cheaper). • Fairly large, use a lot of power (but getting smaller). • Want a technology that is deployable everywhere, accurate, and fairly low cost. Discrete Algs for Mobile Wireless Sys

7. Localization Solutions Overview • Start (possibly) with some anchors, which know their locations, exactly or approximately. • May learn their locations from GPS, or may be manually configured. • Perform ranging: Determine inter-node distances for some pairs of nodes, usually nodes that can communicate in one hop. • Possibly perform additional phases, determining distances between more pairs of nodes, or refining the existing set of distance measurements. • After obtaining a stable set of distances, compute nodes’ coordinates, using a global optimization strategy or an incremental construction. Discrete Algs for Mobile Wireless Sys

8. Localization Solutions Overview • Computing coordinates of nodes: • If there are “enough” anchor nodes and enough distances, it is possible to assign coordinates uniquely to all the nodes. • If there are too few anchor nodes or distances, it might not be possible to assign coordinates uniquely---different combinations of locations might be consistent with the given information. The algorithm can fail. • Even if there are no anchor nodes, if there are enough distances, it is possible to assign coordinates consistently to all the nodes. It may be possible to obtain a relative rather than an absolute coordinate system. Discrete Algs for Mobile Wireless Sys

9. Localization Solutions Overview • After coordinates have been assigned, can evaluate them: • For each edge having a measured distance (obtained from ranging), compare the measured distance with the distance between the locations of the endpoints that are assigned by the algorithm. • Yields a measure of error for the assigned coordinates. • Error can result from inaccuracies in the initial anchor coordinates or in the distance measurements. • Errors can also be introduced during the computation (roundoff). • If unreliable location info is used to compute other location info, the errors can compound badly. • Adjust the coordinate assignments to reduce the error, using optimization techniques. Discrete Algs for Mobile Wireless Sys

10. Ranging • Some nodes measure distance between themselves and some one-hop neighbors. • Use an algorithm that depends on knowledge of signal propagation characteristics. • Technique 1: Use Received Signal Strength: • Node i receives a radio signal from node j, measures received signal power. • Node i uses knowledge of the sender's signal power to determine the power loss. • Then uses a model for the signal propagation behavior to convert the power loss to a distance. Discrete Algs for Mobile Wireless Sys

11. Ranging • Unfortunately, using received signal strength doesn’t work very well: • Radio signal propagation behavior is highly dependent on the environment (obstacles, metal,...), and highly variable. • The Savvides paper describes experiments, failed in every case except for an empty football field, with nodes all at the same height. • Technique 2: Use signal propagation time: • Radio signals travel at speed of light, essentially instantaneous; not plausible to measure this time. • Devices can be equipped with other kinds of transmitters and receivers, for slower signals. • Popular choice: Ultrasonic (US) signals. Discrete Algs for Mobile Wireless Sys

12. S R Ranging Using Signal Propagation Time • E.g.: Time Difference of Arrival (TDoA) • Sender node S transmits, simultaneously, a radio (RF) and ultrasonic (US) signal. • Receiver node R measures time difference between when it begins to receive the two signals. • From the time difference and the known speed of US signals, node R can estimate distance with high accuracy: around 2 cm for node separations around 3 meters! • Range of US transmitters: Typical ones about 3-4 meters, newer ones 10+ meters. • Cricket system uses this method, with radio and ultrasound signals. Discrete Algs for Mobile Wireless Sys

13. Cricket Overview • Localization system for in-building, mobile, location-dependent applications. Like indoor GPS. • Consists of Beacons fixed on ceilings or walls of rooms, and Listeners, which run on mobile and static nodes. • Listeners learn their locations from Beacons. • System is completely distributed. • Beacons aren't connected with any infrastructure, don't coordinate in any way, don't remember any location information about Listeners. Just help Listeners determine where they are. • Listeners can decide what they want to do with this location info, e.g., integrate themselves into some location-dependent applications: • Maintain an active map, including information about which nodes are nearby. • Finding nearby resources. • Controlling nearby devices. Discrete Algs for Mobile Wireless Sys

14. Cricket Results • Yields coarse-grained location information, not precise coordinates. • Granularity is a space = portion of a room, about 4 feet square (floor area). • Listener learns only which “space” it is in. • Design criteria: • Room-size granularity, low cost, no centralized administration, scalability. • Near 100% accuracy for a Listener to determine which space it is in. • Listener privacy: • Cricket doesn't record presence of Listeners. • Listener chooses whether to announce its location. Discrete Algs for Mobile Wireless Sys

15. Cricket Issues • How Beacons broadcast location information, and how Listeners receive it reliably: • Overcoming bad behavior of indoor wireless channels. • Avoiding interference between different Beacons' broadcasts. • Listener algorithm to decode and correlate the broadcasts it hears. • Using Cricket for applications Discrete Algs for Mobile Wireless Sys

16. Cricket Ranging Technique • Each Beacon is associated with a space, and has an identifier string for that space. • Can have more than one Beacon per space---in which case they all broadcast the same identifier. • String must be set by someone manually. • The only manual configuration required. • Beacon broadcasts its space’s identifier frequently, using radio signals. • Each time Beacon broadcasts an RF signal, it simultaneously broadcasts a US signal. • Each nearby Listener measures time difference between beginning of receipt of the RF signal and beginning of receipt of the UF signal. • Accurate measurement, using local clock. • Uses this to estimate distance from transmitting Beacon. • Usually accurate enough to permit reliable identification of the nearest Beacon, which allows the Listener to identify its space. Discrete Algs for Mobile Wireless Sys

17. Interference Problems • Collisions between transmissions of nearby Beacons: • Could result in loss of data • Not serious, because Beacons broadcast frequently. • More seriously, could cause a Listener to correlate RF signal of one Beacon with US signal of another Beacon. • Multipath effects for ultrasound signals: • Reflections from wall, other objects. • Long time scale, orders of magnitude longer than RF multipath. • Could cause Listener to correlate RF signal of a Beacon with an indirect (late) US signal from the same Beacon, which yields a wrong distance estimate. Discrete Algs for Mobile Wireless Sys

18. Interference Problems • Listener's task (in the face of interference): • Gather various RF and US samples. • Try to correlate them, i.e., figure out which were sent by the same Beacon, as part of the same simultaneous transmission. • Calculate distance from the speed, figure out which Beacon is closest. • Choose the space identifier sent from the closest Beacon. Discrete Algs for Mobile Wireless Sys

19. Avoiding Interference • Beacons use randomization to decide when to broadcast. • Not a full-fledged CSMA/CA-style protocol---simpler, less energy usage. • Beacon chooses successive transmission times randomly, uniformly, from an interval of [R1,R2] milliseconds. • Choice of R1 and R2 based on: • Number of Beacons expected to be within range. • Time it takes for transmitted information to reach Listeners (depends on message size and link bandwidth). • Small R1, R2: More chance of collisions. • Large R1, R2: More time required for localization. Discrete Algs for Mobile Wireless Sys

20. B L Coping with Interference • Attach a Beacon-UID to RF transmission, thus allowing Listeners to identify source of RF signals reliably. • Since RF range is much larger than US range (80’ vs. 30’), Listener who receives US signal is almost certain to also receive the corresponding RF signal. • Slow down RF data transmission rate, to ensure that the interval of receipt of the RF transmission includes the interval of receipt of the UF transmission: • This overlap is enough to allow correct correlation for successive transmissions from the same Beacon. • Multipath: If two US signals arrive, overlapping the same RF signal, then Listener chooses the first one for correlation. Discrete Algs for Mobile Wireless Sys

21. A L B Interference from Different Beacons • Consider Beacon A, interfering Beacon B. • Not-so-bad case: • Listener L gets RF from A, totally including US from B; potential for incorrect correlation. • But then L also receives the corresponding RF from B; since its interval contains that of the US from B, it overlaps with the RF from A. • That should result in a collision, so L won't be fooled into making the incorrect correlation. Discrete Algs for Mobile Wireless Sys

22. A L B Interference from Different Beacons • Consider Beacon A, interfering Beacon B. • A worse case: • L gets RF from A, including reflected, late US from B. • L receives the corresponding RF from B, but it arrives much earlier and doesn't overlap with the RF from A, so doesn't result in a collision. • This can result in an incorrect correlation, and a wrong estimate. Discrete Algs for Mobile Wireless Sys

23. Coping with Wrong Correlations • L calculates distance to Beacon B based on several samples for B, not just one. • Randomization reduces likelihood of repeated wrong estimates. • How Listener L determines estimate from several samples: • Chooses the one that occurs most often---the “mode”. • More precisely, L rounds off calculated distances so estimates fall into discrete “bins”, chooses the bin that appears most often. • Mode works better than mean, because it does a better job of throwing out anomalous readings. • Difference in accuracy is more apparent when more samples are taken---since then there is a higher likelihood of some wrong measurements, which the mean takes into account but the mode doesn't. Discrete Algs for Mobile Wireless Sys

24. More Precise Localization • So far: • Introduced the problem, ranging technique for distance estimation. • Cricket system. • Distance estimation using RF + US signals. • Determines room “space” only, not actual coordinates. • Used to implement space-aware applications. • Next: • Consider determining actual coordinates. • [Savvides], giving basic methods. • [Priyantha et al.], Mobile-assisted methods. • [Moore et al.], Robust methods. • [Aspnes et al.], Rigidity theory, describing when it’s possible to determine coordinates. Discrete Algs for Mobile Wireless Sys

25. Dynamic Fine-Grained Localization in ad hoc Networks of Sensors • [Savvides et al.] • Ad hoc network, some fraction of nodes are anchors (with known coordinates). • 2D • Uses TDoA ranging for distance estimation. • Introduces basic multilateration method: • Iterative method for assigning locations to all nodes in a graph, starting from anchors. • Each step is either atomic (adds one new node) or collaborative (adds several new nodes). • New positions may involve some error; use least-squares estimation to minimize the error. • Claim high success rate for assigning coordinates, high accuracy. • Claim distributed algorithm (but descriptions mostly centralized). Discrete Algs for Mobile Wireless Sys

26. Atomic Multilateration • Assign 2D coordinates (x0,y0) to a node 0 having some anchor neighbors 1,…,k. • Assume: • Coordinates of 1,...,k are known. • Distances di between node 0 and anchor neighbors are known. • For each i, 1 ≤ i ≤ k, calculate: erri = di – sqrt( ( xi - x0 )2 + ( yi - y0 )2 ) ). • Discrepancy between measured and calculated distance. • Function of unknown coordinates (x0,y0). • Want to choose (x0,y0) to minimize the errors, specifically, minimize sum of squares of the erri. • If distance estimates and anchor coordinates are exactly accurate, we can achieve total error = 0. Discrete Algs for Mobile Wireless Sys

27. 0? 1 2 d1 d2 0? Atomic Multilateration • Need 3 anchors. • With only 2, solution may not be unique, even if measurements are exact (see figure) • With 3 non-collinear anchors, the minimization problem has a unique solution. • A trick: If speed s of US signal is unknown, can solve uniquely for s as well as (x0,y0), with four anchor points. Discrete Algs for Mobile Wireless Sys

28. Iterative Multilateration • Repeat atomic multilateration, add one new node at a time. • Heuristic: Choose unknown node with the most anchor neighbors. • Error accumulation: • Really, anchor position and distance estimates aren't exact. • Subject to measurement error, limitations of precision of representation. • Since each new node's position is defined based on previous ones, error can compound. • Authors claim this isn't a problem, because the accuracy of each stage is so high. • Distributed algorithm? • Not described that way---someone must select the next node. Discrete Algs for Mobile Wireless Sys

29. 1 2, U 6 5 4, U 3 Collaborative Multilateration • A variant of atomic multilateration, used by a set of unknown nodes to determine their location estimates ``collaboratively''. • For situations in which none of the individual nodes has enough anchor neighbors, but the entire collection does. • Example: • Anchors = 1, 3, 5, 6 • Unknowns = 2, 4 • Consider an undirected connected graph, where nodes are classified as anchor (A) or unknown (U). • The U nodes may form a collaborative set. • Q: Under what conditions? Discrete Algs for Mobile Wireless Sys

30. Collaborative Multilateration • Set up an error equation for every edge in the graph: • Between an unknown node and an anchor node, or • Between two unknown nodes. • Equations have 2 |U| unknowns, for 2 coordinates of each unknown node. • Need “enough” equations; don’t give characterization. • Suggest that each unknown node should have degree  3 (neighbors may be anchors or unknown). • Clearly we need more than this---need some anchors. Discrete Algs for Mobile Wireless Sys

31. 1 2? 6 5 4? 3 Collaborative Multilateration • Example works: • 5 equations, 4 unknowns. • Distances d21 and d23 yield two possible positions for 2; d45 and d46 yield two possibilities for 4. • Assuming the 4 possible distances are distinct, d24 is enough to disambiguate. Discrete Algs for Mobile Wireless Sys

32. 1 5 4? 2? 3 6 Collaborative Multilateration • Distances might not all be distinct, in which case we don’t get unique solution: • But this happens only for sets of choices of points of “measure 0”. • (Implicitly) generalize the example: • A tree of U nodes, plus some A nodes. • Each U node has degree  3, counting connections to both U nodes and A nodes. • Each “leaf” U node has  2 A neighbors. • (Implicitly) claim this is enough to generate unique positions. • Assuming exact measurements. • Except for a set of point choices of measure 0. • Remains to be proved. • Is this the most general condition? Discrete Algs for Mobile Wireless Sys

33. Forming a Collaborative Set • Also give distributed algorithm for a set of unknown (U) nodes to try to form a collaborative set C. • It produces a rooted tree of U nodes, where: • Each leaf has at least 2 anchor (A) neighbors. • The root has at least 3 neighbors that are either anchors or its children. • Each non-root, non-leaf node has at least 2 neighbors that are either anchors or its children. • Implies the implicit conditions we just discussed. Discrete Algs for Mobile Wireless Sys

34. Algorithm to Form a Collaborative Set • Distinguished U node u0 starts a distributed algorithm to establish a rooted tree of U nodes with itself as the root. • A node in the tree with no children so far can decide it is a leaf if it has  2 Anchor neighbors; then marks itself as in C, sends OK to its parent. • A non-root, non-leaf node can finish its work if the number of children from which it has received OK plus the number of Anchor neighbors is  2; then marks itself as in C, sends OK to its parent. • Root u0 can finish if the number of children from which it has received OK plus the number of Anchor neighbors is  3; then marks itself as in C, and announces the members of C. Discrete Algs for Mobile Wireless Sys

35. 1 2, U 6 5 4, U 3 Forming a Collaborative Set • Q: Race condition: Some node could mark itself after its parent has already collected enough OKs. Gets rejected? • Q: Prove that this works right? • Example: • Node 2 starts the algorithm, sets up a 2-node tree with itself as the root, with the needed properties. • Node 4 declares itself a leaf (2 Anchor neighbors). • Node 2 declares itself done (2 Anchor neighbors + one OK child). Discrete Algs for Mobile Wireless Sys

36. Multilateration • Can also apply collaborative multilateration iteratively, mix with atomic multilateration. • Success of the multilateration algorithms depends on some characteristics of the node and anchor placement, especially: • High network connectivity (density). • High probability that nodes have degree  3. • Prevalence of Anchors. • Distributed coordinate-determination algorithms needed, not given. • Accuracy: • Claim accuracy of iterative multilateration is very good in small networks. • Needs to be improved for larger networks. • Need new strategies to limit error propagation across the network (see Robust Quadrilaterals paper). • Algorithms, especially collaborative strategies, need more theoretical analysis. Discrete Algs for Mobile Wireless Sys