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

Discrete Algs for Mobile Wireless Sys2 Lecture 9  Topic: Localization  Sources: Priyantha, Chakraborty, Balakrishnan. Cricket. Savvides, Han, Srivastava. Dynamic fine-grained localization in ad-hoc networks of sensors. MIT Fall 2008 slides

Discrete Algs for Mobile Wireless Sys3 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 Sys4 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 Sys5 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 Sys6 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 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 Sys7 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 Sys8 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 Sys9 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 Sys10 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 Sys11 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 Sys12 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. SR

Discrete Algs for Mobile Wireless Sys13 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 Sys14 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 Sys15 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 Sys16 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 Sys17 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 Sys18 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 Sys19 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 [R 1,R 2 ] milliseconds.  Choice of R 1 and R 2 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 R 1, R 2 : More chance of collisions. Large R 1, R 2 : More time required for localization.

Discrete Algs for Mobile Wireless Sys20 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. BL

Discrete Algs for Mobile Wireless Sys21 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. AL B

Discrete Algs for Mobile Wireless Sys22 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. AL B

Discrete Algs for Mobile Wireless Sys23 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 Sys24 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 Sys25 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 Sys26 Atomic Multilateration  Assign 2D coordinates (x 0,y 0 ) to a node 0 having some anchor neighbors 1,…,k.  Assume: Coordinates of 1,...,k are known. Distances d i between node 0 and anchor neighbors are known.  For each i, 1 ≤ i ≤ k, calculate: err i = d i – sqrt( ( x i - x 0 ) 2 + ( y i - y 0 ) 2 ) ). Discrepancy between measured and calculated distance. Function of unknown coordinates (x 0,y 0 ).  Want to choose (x 0,y 0 ) to minimize the errors, specifically, minimize sum of squares of the err i.  If distance estimates and anchor coordinates are exactly accurate, we can achieve total error = 0.

Discrete Algs for Mobile Wireless Sys27 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 (x 0,y 0 ), with four anchor points. 0? 1 2 d1d1 d2d2

Discrete Algs for Mobile Wireless Sys28 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 Sys29 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: 1 2, U 6 5 4, U 3 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 Sys30 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 Sys31 Collaborative Multilateration 1 2? 6 5 4? 3  Example works: 5 equations, 4 unknowns. Distances d 21 and d 23 yield two possible positions for 2; d 45 and d 46 yield two possibilities for 4. Assuming the 4 possible distances are distinct, d 24 is enough to disambiguate.

Discrete Algs for Mobile Wireless Sys32  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? ? 4? Collaborative Multilateration

Discrete Algs for Mobile Wireless Sys33 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 Sys34 Algorithm to Form a Collaborative Set  Distinguished U node u 0 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 u 0 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 Sys35 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). 1 2, U 6 5 4, U 3

Discrete Algs for Mobile Wireless Sys36 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.