CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

Spring 2009

Prof. Jennifer Welch

Discrete Algs for Mobile Wireless Sys 2

Lecture 10 Topic:

More Localization Sources:

Priyantha, Balakrishnan, Demaine, Teller. Mobile-assisted localization in wireless sensor networks.

MIT 6.885 Fall 2008 slides


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.


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).


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.


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).


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 p1,…,pn, 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..


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.

1 2

34 3

1 2

4

12

34

1

2

34

21

34

5


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..

1 2

34

12

34


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…


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, 3rd distance disambiguates.


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.


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, n0 and n1.

n0 n1

?


Determining Distance Between Two Nearby Nodes Move a mobile node around, establishing 3

temporary virtual graph nodes m0, m1, m2. Measure distances (using RF and US),

between each ni and each mj, 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 n0 and n1.

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…

m0m2

m1

n0 n1

?


Determining Distance Between Two Nearby Nodes

m0m2

m1

n0 n1

? Mathematical justification for MAL

Theorem 1: A point formation consisting of 5 coplanar points n0, n1, m0, m1, and m2, where m0, m1, and m2 are collinear, and all on the same side of the line between n0 and n1, together with edges (ni, mj) for all i, j, is globally rigid.


Avoiding the Special Assumptions

Try to determine distances among four points n0, n1, n2, n3 instead of just between two points n0 and n1.

Theorem 2: A point formation consisting of 11 coplanar points n0, n1, n2, n3, m0, m1, m2, m3, m4, m5, m6, where no four points are coplanar, together with edges (ni, mj) for all i, j, is globally rigid.

No additional assumptions---the equations arising from these edges are enough.

Proof: LTTR. (?)


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.


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.


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.


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


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).

Documents

CPSC 689: Discrete Algorithms for Mobile and Wireless Systems