Ad-Hoc Localization Using Ranging and Sectoring

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Ad-Hoc Localization Using Ranging and Sectoring

• Krishna Kant Chintalapudi,
• Amit Dhariwal,
• Ramesh Govindan,
• Gaurav Sukhatme
• Embedded Networking Laboratory,
• Robotics Laboratory,
• USC

2
Agenda

• Whats Ad-Hoc Localization?
• Existing Ad-Hoc Localization mechanisms
• Metrics to evaluate Ad-Hoc Localization Systems.
• Performance of existing schemes
• Alternatives?
• Range-bearing
• Range-Sector
• Conclusions

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Whats ad-hoc Localization?

• Sensor Nodes are randomly scattered.
• Where in the world am I?
• Only a small fraction of nodes have GPS.
• Rest have to infer their global positions somehow.

4
The Players

• Sensor Nodes
• have Radio
• may have ranging capability to neighboring nodes
• Anchors
• Sensor nodes with GPS
• Radio Communication Range ? Ranging Radius

Radio
Ranging
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Ranging

• In the robotics community
• Ranging range bearing.
• Localization could be,
• Range-only
• Bearing-only
• Range-Bearing

6
Two requirements in our study

• Anchor fraction
• No control over Node and Anchor Placement

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Why Ad-Hoc Localization?

• GPS is expensive ()
• GPS is power hungry
• GPS does not help in indoor environments/ Foliage

8
Agenda

• Whats Ad-Hoc Localization?
• Existing Ad-Hoc Localization mechanisms
• Metrics to evaluate Ad-Hoc Localization Systems.
• Performance of Existing Schemes
• Alternatives?
• Range-bearing
• Range-Sector
• Conclusions

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Why cant we directly borrow from robotics?

• Low Power requirements

• Low cost ()
• Small form factor
• Emphasis on collaborative localization
• 10-100 robots 1000 sensors (scalability)

• Eg. Laser range finder is prohibitively
• expensive, power hungry, large

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State of the art in a nut-shell

• Schemes are range-only
• Acoustic Ranging is almost in place
• Acoustic based precise bearing measurement is not
  easy to measure at the form factor in question.
• Laser does not suit the cost, power or form
  factor requirements

• Two kinds of localization schemes
• Topological schemes
• Geometric Schemes

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Topological Schemes

• Rely on Topology Information
• DV-hop Niculescu et. al.
• finds average distance/hop davg
• distance davg(no of hops)
• requires no ranging!!!
• DV-dist Niculescu et. al.
• Standard distance vector algorithm
• Metric is measured range
• Constantly over-estimates

d3
d2
d1d2d3
d1
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Geometric Schemes

• Savvides et. al.
• Each anchor defines a coordinate system
• Anchor is origin
• Nodes localize relative to anchor using
  hop-by-hop lateration
• Nodes maintain (Anchor_id, x, y)
• 3 such 3-tuples enable localization
• This is followed by refinement
• Observations
• Local coordinate systems may be rotated,
  translated, flips w.r.t the global coordinate
  system.
• Only distance from anchor is invariant

y
q
x
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Geometric Schemes

• Niculescu et. al.
• Uses a voting scheme instead of lateration.
• Some situations may need only 2 nodes
• A more recent work involves inferring bearings
  and propagating them also.
• Observations
• Degenerate cases may lead to errors
• Some heuristic measures were developed to avoid
  this problem
• Do not localize if you are unsure

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Performance Metrics

• Mean error

True sensor location
Estimated sensor location

• Extent of Localization

Percentage of non-anchor nodes localized within
20 of ranging radius.
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Agenda

• Whats Ad-Hoc Localization?
• Existing Ad-Hoc Localization mechanisms
• Metrics to evaluate Ad-Hoc Localization Systems.
• Performance of existing schemes
• Alternatives?
• Range-bearing
• Range-Sector
• Conclusions

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The Simulations

• The Scene
• No of nodes 1000
• Ranging radius 10mts
• Node locations drawn from a uniform distribution
• Area is square of side
• d is the expected degree of a node (5-15)
• Ranging error is zero mean Gaussian and std.
  increases linearly with range at rate
• Anchor fractions (5, 10, 20)

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Motivation for error model

• We used a off the shelf sonar mounted on the
  Pioneer P3DX8 robot
• Standard deviation increases with range.

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Evaluation of r-only schemes

• 3 schemes
• Savvides et. al (geometric)
• Niculescu et. al (geometric)
• DIV-dist (topological)
• 20 anchors, 1 error
• Localization extent
• Nodes Localized within 2 mts.
• Topological Scheme
• Higher localization extent at low degrees
  compared to Geometric.
• Geometric Schemes
• Low localization extent at degree below 10-11
  nodes.

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Evaluation of r-only schemes

• Localization mean error
• Only localized nodes considered
• age of ranging radius
• Topological Scheme
• High localization error at low degrees compared
  to Geometric.
• Geometric Schemes
• Low localization error but
• remember low localization extent as well.

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In short

• The results are in agreement with Langendoen et.
  al. and serve as another independent
  verification.
• Topological schemes
• High localization extent but high mean error
• Maybe good for application which can tolerate
  high error
• Geometric Schemes
• Low error but low localization extent as well.
• Summary of exiting r-only schemes
• Require a degree of around 10-11 nodes for
  achieving 90 localization with 5 mean accuracy.

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3 Connectedness

• Each node in geometric schemes needs a set of 3
  nodes which
• are in its ranging range
• know their distance from anchor
• every node has a range measurement to at least
  one other node in this set.
• Consequences
• some nodes may not be able to find their
  distances from enough anchors
• high anchor fractions and/or node degrees may be
  required

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Concerns

• What do the results mean?
• 10-11 ranging neighbors on an average at 20
  anchor ratio
• Radio range gt Ranging range (in practice)
• If radio range is 20mts and ranging radius is
  10mts, it translates to having 40-44 neighbors/
  radio range!!!
• From a deployment point of view
• 6 nodes/ radio range gives me 99 radio
  connectivity
• Why should I deploy more than 11 nodes/radio
  range just for localization?
• Maybe I would rather put GPS on most nodes and
  work at lower deployment densities if its
  possible

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Discussion

• Maybe a better r-only scheme exists?
• Indeed this maybe the case,
• but from our experience we believe that the
  problem is fundamental to r-only and new schemes
  will not bring about orders of magnitude
  improvement.
• Just deploy more anchors!!!
• We found that at about 30 and more anchor
  fraction one can work at degree of 8-9.
• However, we seek an order of magnitude
  difference between anchors and nodes (less than
  10)
• How about Deterministic Anchor Deployments
• Tailored anchor deployments may bring about
  improvement
• however probably not by orders of magnitude

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Suppose we buy all that what next?

• Try and use more than just range information
• Bearing or even some vague hints of bearing
• Examine how much improvement we can get more
  bearing information
• Is it justified to start effort in building
  devices which get more that just range?

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Agenda

• Whats Ad-Hoc Localization?
• Existing Ad-Hoc Localization mechanisms
• Metrics to evaluate Ad-Hoc Localization Systems.
• Performance of existing schemes
• Alternatives?
• Range-bearing
• Range-Sector
• Conclusions

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Range-Bearing

• Suppose we could get Range and bearing (laser)
• How much would our situation be improved?
• Is there a good distributed way to solve the
  problem?
• Such a scheme would definitely benefit
  collaborative localization in robotics.
• What information do we have from ranging?
• Range with Gaussian error
• Bearing with Gaussian error
• Heading (from a compass on the device)

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Problem Formulation
Every range measurement gives an equation,
Because of ranging errors,
y
xj
First order approximation of the covariance of
is given by,
xi
x
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The Objective function
Minimize the objective function,
The conditions for global minima can be written
as,
(1)
Here, is the set of nodes in the ranging
neighborhood of node ni
and, exists if ni can range to nj
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In simple words

• Nodes which are neighbors to anchors,
• first estimate of location.

• Nodes announce their current estimate to their
  immediate neighbors

• Nodes use (1) and re-estimate their locations

• It is possible to show that the scheme is
  guaranteed to monotonically converge to a unique
  global minima.

• We call this scheme LMSRB (Least Mean Square with
  Range and Bearing) .

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Is it any good?

• We used the same scenario as for r-only

• Error in was modeled by a zero mean
  Gaussian.

• A std. of is a cone of roughly

• We varied cone angle from 2o to 8o.

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Results

• Localization extent
• 40 cone and 1 ranging error.
• 3 different anchor fractions, 5, 10, 20.
• Even at 5 anchor fraction and node degree 5,
  more than 90 nodes are localized!!

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Results continued

• Localization error
• 40 cone and 1 ranging error.
• 3 different anchor fractions, 5, 10, 20.
• Even at 5 anchor fraction and node degree 5,
  error is less than 1

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Effect of error in bearing

• Localization error
• At 80 cones and node degree 5, error is about
  2.5 of ranging radius.
• We found that error rapidly increases after 120
  cones.
• This may be because of the first order
  approximations no longer hold.

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Discussion

• Clearly having bearing information can
  dramatically improve performance.
• A deployer would find it much more acceptable
• Is this practical?
• For robotics. it definitely is
• For sensor networks maybe (CCD ring?)maybe
  not.
• Can there be a scheme which can get a more
  viable solution?

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Is there a more practical scheme?

• Is there a scheme which can get us the best of
  both worlds?

• It seems more practical that a node can figure
  out an approximate range, say within a 300 - 600
  cone.

• Can we get acceptable error and extent of
  localization at low degrees even for such large
  imprecision in bearing?

• As you must already know the answer there is
  one!!!

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Least-Mean-Square Refinement with range and
sector (LMSRS)
Suppose we only had ranges at our disposal, then
each measurement would yield an equation,
Following the same philosophy as in LMSRB we
could minimize,
Conditions for local minima give us,
(2)
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But wait

• Unlike LMSRB, LMSRS does not minimize a
  quadratic function.
• As such the scheme is susceptible to local
  minima.
• The scheme will converge to the nearest local
  minima.
• A good initial guess for xi is needed.
• Can we use the sector information to get a good
  initial guess?
• If yes? How much sector width can LSMRS handle?

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Using sector information as an initial guess

• Use the the bisector of the sector as the
  bearing and find the initial guess.

F
C

• Can we do better?

B
D
E

• Use random heading distribution and sector
  miss-alignment to our advantage.

E
A
F

• Using information from B, A can narrow down the
  sector from CAD to FAD.

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Results

• Localization extent
• 1 ranging error
• 5 node degree, 10 anchor fraction, 300 sector
  gives more than 90 extent.
• 5 node degree, 20 anchor fraction, 600 sector
  gives more than 90 extent.
• 6 node degree, 10 anchor fraction, 450 sector
  gives more than 90 extent.

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Results continued

• Localization error
• 1 ranging error
• 6 node degree, 10 anchor fraction, 300 sector
  gives more than 4 error.
• 6 node degree, 10 anchor fraction, 450 sector
  gives more than 6 error.
• 6 node degree, 10 anchor fraction, 600 sector
  gives more than 8 error.

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A first cut at implementation

• On pioneer P3-DX8 robots
• Have a laser range finder (mm range accuracy, 10
  bearing accuracy)
• Have a sonar belt with 16 sonars, each spanning
  roughly a 22.50 cone.
• Detection probability falls down dramatically
  after 10 feet.

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Overlap in sonars

• Overlap in the sonars
• depends on distance
• at distances less than 3 feet 2 sonars will
  overlap significantly
• however overlap information can be also be used
  to determine the sector.
• After 4 feet, overlap is very small.

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A fully Distributed implementation

• Pioneer robots are linux boxes with radio so
• Implemented LMSRS on robots
• Identity problem which robot is being
  detected?
• Reflections off of objects in indoor environment
• Physical size of the robot induces error over
  multiple hops
• The algorithm convergence depends on topology
  and number of anchors.

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Some ideas

• Use CCD arrays
• Use multiple directional antennae
• A single directional antenna that can be
  rotated
• Small cheap low resolution LCD cameras
• Anymore ideas?

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In Conclusion

• Range only scheme may have large node degree
  requirements
• Having accurate bearing information can
  significantly improve performance as well as
  reduce node degree requirement.
• Proposed an optimal Range-Bearing algorithm
  LMSRB.
• While LMSRB may be practical for robots, for
  sensors we may need a cruder but cheaper
  solution.
• Proposed LMSRS which uses sector information to
  estimate locations.
• LSMRS can localize even with 45-600 sectors.

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Taxonomy of existing schemes

• 3 sub-problems/stages Langendoen et.al.
• Estimation of distance to 3 or more anchors
• Topological (approximate)
• Geometric (accurate)
• Initial position estimate
• Lateration
• Refinement of estimated positions
• Several schemes

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Steps II and III

• Step II can be a simple lateration from 3 or more
  anchors
• Lateration using 3-4 nodes is sensitive to errors
• Errors may blow up super-linearly with hops
• Iterative Refinement schemes may be required as
  Step III to cope with these errors,
• Iterative multi-lateration
• Kalman filter based
• Bounding box based Heuristic

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Current State

• Schemes are range-only

• 3 steps
• Distance from anchors (multi-hop)
• Topological (very inaccurate)
• Geometric (accurate)
• Initial estimate (lateration)
• Iterative refinement (several schemes)

• Topological schemes rely on Iterative refinement
  (step 3) heavily
• Step 3 does not bring about significant
  improvement in geometric schemes (Langendoen et.
  al. and independently by us)