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

Discrete Algs for Mobile Wireless Sys2 Lecture 10  Topic: More Localization  Sources: Priyantha, Balakrishnan, Demaine, Teller. Mobile-assisted localization in wireless sensor networks. MIT Fall 2008 slides

Discrete Algs for Mobile Wireless Sys3 Mobile-Assisted Localization [Priyantha, Balakrishnan, Demaine, Teller]  Collection of nodes in wireless ad hoc network, 2D or 3D.  Undirected graph, distance measurement on each edge, no anchors.  Want coordinates for all nodes, consistent with measured node distances.  Since no anchors, can’t guarantee uniqueness; instead, find coordinates that are unique up to translations, rotations, and reflections.  Requires “enough” distance estimates. Can be difficult to obtain, e.g., in sparse Cricket deployment.

Discrete Algs for Mobile Wireless Sys4 Mobile-Assisted Localization  Obtaining more distance estimates: Mobile-Assisted Localization (MAL) Introduce (temporarily) some “virtual'' graph nodes at strategic locations, use them to help calculate distances between real nodes. Implement the virtual nodes using mobile devices, which travel around playing the role of the virtual nodes.  Calculating the coordinates: Several possible methods. A new method, Anchor-Free Localization (AFL).

Discrete Algs for Mobile Wireless Sys5 Enough Distance Measurements  Why we might not have enough: Physical obstacles: Line-of-sight needed for ultrasound, laser, infrared; improves radio. Limitations of ranging hardware: Might not be omni-directional. Sparse deployments: E.g., in Cricket, hard to get enough connectivity between nodes in different rooms.  Need enough distinct measurements for uniqueness: To determine a “globally rigid” structure (unique, up to translation, rotation, and reflection). If not, could get seriously wrong coordinates.

Discrete Algs for Mobile Wireless Sys6 Enough Distance Measurements  More distance measurements can reduce error: Reduce “depth” of iterative procedure, allowing some nodes’ coordinates to be determined at earlier iterations. Reduces compounding of measurement errors. Yield more distance equations, leading to an over-constrained system, which tends to decrease errors. Reduce Geometric Dilution of Precision (GDOP)  (computed coordinate error) / (measurement error).

Discrete Algs for Mobile Wireless Sys7 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  An n-point formation P in d-space consists of: Coordinates in d-space for points p 1,…,p n, and A set of edges between some points (indices).  Basically, a graph embedded in d-dimensional space.  An n-point formation P in d-space is globally rigid provided that any other n-point formation Q with the same edges and the same distances on those edges is the same as P, up to translation, rotation, and reflection.  Global rigidity means that the formation is essentially unique: Coordinates determined by the number of points, the pairs connected by edges, and the distances on the edges..

Discrete Algs for Mobile Wireless Sys8 Examples, d = 2 (Figure 1, p. 174)  1(a) is not globally rigid. Can be deformed gradually in 2-space, preserving distances, to yield a different shape.  1(b) is also not globally rigid. It can't be deformed gradually in 2-space. But it can be flipped around to get a different shape.  1(c) is globally rigid

Discrete Algs for Mobile Wireless Sys9 Local Rigidity  The structure cannot be locally deformed while preserving the distance constraints.  Makes sense for structural engineering (for 3-space).  1(a) is not locally rigid (in 2-space). Can be deformed gradually in 2- space.  1(b) is locally rigid (in 2-space), but not globally rigid. Could be deformed in 3-space, but that doesn’t count

Discrete Algs for Mobile Wireless Sys10 Local Rigidity  Rigidity theory has various theorems giving sufficient conditions for local and global rigidity.  Mobily Assisted Localization (MAL) uses a simple conservative strategy, justified by 2 simple theorems.  Triangle (with distances) is globally rigid in 2-space  Tetrahedron (with distances) is globally rigid in 3- space.  Use as starting points for building larger globally rigid structures…

Discrete Algs for Mobile Wireless Sys11 2D Theorem for Global Rigidity  Theorem (2D): Suppose we build a 2D point formation by starting with a triangle with 3 distances, and repeatedly adding a node and edges (with distances) to at least 3 non-collinear points. This results in a globally rigid point formation.  2D theorem is like atomic multilateration result in [Savvides]: distances to 2 points yield two circles, which may intersect in 2 points, 3 rd distance disambiguates.

Discrete Algs for Mobile Wireless Sys12 3D Theorem for Global Rigidity  Theorem (3D): Suppose we build a 3D point formation by starting with a tetrahedron with 6 distances, and repeatedly adding a node and edges (with distances) to at least 4 non-coplanar points. This results in a globally rigid point formation.  For 3D, distances to 3 points yield three spheres, which again intersect in at most 2 points (hard to visualize), fourth distance disambiguates.

Discrete Algs for Mobile Wireless Sys13 How MAL works  MAL (and AFL) are for 3D.  MAL inserts additional edges, with distance estimates, between some of the existing points.  Output of MAL is simply a “denser'' graph, with more edges and distance estimates (but same nodes).  Then AFL, or another algorithm, uses the graph to obtain candidate 3D coordinates for all the nodes.  Key subproblem: Determining distance between two nearby nodes, n 0 and n 1. n0n0 n1n1 ?

Discrete Algs for Mobile Wireless Sys14 Determining Distance Between Two Nearby Nodes  Move a mobile node around, establishing 3 temporary virtual graph nodes m 0, m 1, m 2.  Measure distances (using RF and US), between each n i and each m j, giving 6 new distance measurements.  But the virtual nodes also add new unknowns--- their coordinates, or distances.  They use a trick: Constrain the positions of the virtual nodes to be in a line, and all in one plane containing n 0 and n 1. Also (not said) the m nodes must all be on the same side of the line containing the n nodes. Then we get global rigidity by next theorem… m0m0 m2m2 m1m1 n0n0 n1n1 ?

Discrete Algs for Mobile Wireless Sys15 Determining Distance Between Two Nearby Nodes m0m0 m2m2 m1m1 n0n0 n1n1 ?  Mathematical justification for MAL  Theorem 1: A point formation consisting of 5 coplanar points n 0, n 1, m 0, m 1, and m 2, where m 0, m 1, and m 2 are collinear, and all on the same side of the line between n 0 and n 1, together with edges (n i, m j ) for all i, j, is globally rigid.

Discrete Algs for Mobile Wireless Sys16 Avoiding the Special Assumptions  Try to determine distances among four points n 0, n 1, n 2, n 3 instead of just between two points n 0 and n 1.  Theorem 2: A point formation consisting of 11 coplanar points n 0, n 1, n 2, n 3, m 0, m 1, m 2, m 3, m 4, m 5, m 6, where no four points are coplanar, together with edges (n i, m j ) for all i, j, is globally rigid.  No additional assumptions---the equations arising from these edges are enough.  Proof: LTTR. (?)

Discrete Algs for Mobile Wireless Sys17 Which Distances to Determine?  We have seen how to use virtual nodes to add some edges and distances (between existing nodes) to a given graph.  Q: Which edges and distances should be added?  Enough to support an iterative multilateration strategy for determining coordinates: A structure that can be built from a totally- connected non-planar 4-graph (tetrahedron), by adding one point at a time, each new point connected to 3 previous points.

Discrete Algs for Mobile Wireless Sys18 Building a Structure  Initialization: Mobile node finds initial cluster of 4 nodes that can all be seen from a common mobile location. Moves to 7 positions in range of the 4 nodes, measures distances. Theorem 2 implies that this structure is globally rigid. Use it to compute all distances between the given nodes Mark the four nodes.

Discrete Algs for Mobile Wireless Sys19 Building a Structure (cont’d)  Loop: Mobile node finds 4 real nodes, 3 marked and 1 not. Moves to 7 positions, measures distances. Again, Theorem 2 implies that this structure is globally rigid. Use to compute distances between the new node and the 3 others. Mark the new node.  Claim: Iif any mobile strategy could work, this construction will find it. (Meaning? Proof?) Linear bounds (in the number of given nodes), on number of distance measurements, and on total distance travelled by the mobile node.

Discrete Algs for Mobile Wireless Sys20 Two-Phase Algorithm  Phase 1: MAL adds enough distances to yield a globally- rigid point formation.  Phase 2: Obtain coordinates. Unique only up to translation, rotation, and reflection.  Possible algorithms for Phase 2: Iterative multilateration: Like [Savvides], but in 3D.  Start from a tetrahedron with known distances, assign consistent set of coordinates to these 4 nodes.  Then repeatedly: Determine coordinates for a node with edges to 3 nodes whose coordinates have already been determined. Anchor-Free Localization (AFL): New in this paper

Discrete Algs for Mobile Wireless Sys21 Anchor-Free Localization (AFL)  Assign preliminary coordinates to all nodes, e.g., based on simple “hop counts” in the graph. Use connectivity information only, not distance information. These won't be very good.  Then refine, using a non-linear optimization strategy: Try to obtain slightly perturbed assignment of coordinates that minimizes sum of squares of errors, taken over all edges. For each edge, error = difference between the measured distance, obtained from MAL, and the distance that is computed from the proposed coordinate assignment.  If error = 0, it means that we have achieved an exact distance- respecting embedding of the graph in 3-space.  Since the graph is globally rigid, the resulting coordinates are unique (up to translation, rotation, and reflection).