GPS-based Navigation in Static and Dynamic Environments

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GPS-based Navigation in Static and Dynamic
Environments

• Masters Thesis Presentation
• Shahid Jabbar
• Institut für Informatik
• Universität Freiburg
• Supervisor PD Dr. Stefan Edelkamp
• Co-supervisor Prof. Dr. Th. Ottmann

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The big question .. What are we doing here ?Das
Problemo

• Digital maps available in the market are very
  expensive.
• Most of those maps do not allow updates.
• Not possible to have timed queries.
• The travel time can change drastically during
  different kinds of days like, workdays and
  holidays
• Can even change during different times of a day
  like, from 8 to 9 AM as compared to 10 to 11 PM.

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The big question .. What are we doing here ?The
Solution

• Why not let people make their own maps that they
  can query and update ?
• But how ?
• How to collect the data ?
• How to process that data ?
• Global Positioning System (GPS) Receiver
• Computational Geometry

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What is GPS ?

• A collection of 24 geo-stationary satellites.
• Gives the position of an object in terms of its
  longitude, latitude, and height.

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Data Collection
What about the cost of collecting the data ?
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Data Collection
What about the cost of collecting the data ?
We say .
You only need some Bananas.
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Data Collection
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Data format

• ltlongitudegt, ltlatitudegt, ltdategt, lttimegt
• 48.0070783, 7.8189867, 20030409, 100156
• 48.0071067, 7.8190150, 20030409, 100158
• 48.0071850, 7.8191400, 20030409, 100200
• 48.0071650, 7.8191817, 20030409, 100202
• 48.0071433, 7.8191867, 20030409, 100204
• 48.0071383, 7.8191883, 20030409, 100206
• 48.0071333, 7.8191917, 20030409, 100208
• 48.0071317, 7.8191917, 20030409, 100212

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Not everything that glitters is Gold.Filtering
Rounding

• Kalman Filter
• GPS Information Speed-o-meter reading as the
  inertial information gt removes the outliers
• Douglas-Peuker Line Simplification Algorithm
• Simplifies a polyline by removing the waving
  affect.

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Geometric Rounding Douglas-Peuckers algorithm
resultsusing Hersberger and Snoeyink variant
points T10-7 10-6 10-5 10-4 10-3
1,277 766 558 243 77 22
1,706 1,540 1,162 433 117 25
2,365 2,083 1,394 376 28 7
50,000 48,432 42,218 17,853 4,385 1,185
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Lets sweeeeep Graph Construction

• We have multiple traces.

• We need to convert them into a graph to be able
  to apply different graph algorithms e.g. shortest
  path searching

• Seems very simple, just convert Point ?
  Vertex
• Segment ? Edge

• Bentley - Ottmann Line Segment Intersection
  Algorithm.

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Graph Construction Results of Line sweep
GPS Points Nodes in Graph Time to sweep
1,277 1,473 0.42
1,706 1,777 0.27
2,365 2,481 0.37
50,000 54,267 11.13
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Where am I ?

• I am at building 101 and I want to go to
  CinemaxX.
• Schade!!! I have no existing trace that pass
  through building 101.
• What to do ? Hmmmm interesting problem
• How about going to the nearest place that is in
  my existing traces ?

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Where am I ?

• Voronoi Diagram to the rescue!!!

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Node localizationResults
points queries Construc-tion Time Searching Time Naive Searching Time
1,277 1,277 0.10 0.30 12.60
1,706 1,706 0.24 0.54 24.29
2,365 2,365 0.33 1.14 43.3
50,000 50,000 13.73 14.26 gt10,000
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My floppy is too small how can I carry this
file ?Graph Compression
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My floppy is too small how can I carry this
file ?Graph Compression
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My floppy is too small how can I carry this
file ?Graph Compression (contd)
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My floppy is too small how can I carry this
file ?Graph Compression (contd)Results of
Graph Compression
Nodes Compressed Nodes Time
1,473 199 0.01
1,777 74 0.02
2,481 130 0.03
54,267 4,391 0.59
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I have to reach CinemaxX ASAP .. What to do
?Search

• Dijkstra Single-Source shortest path.
• A - Goal directed Dijkstra
• Number of queries is much more than the updates.
• How about pre-computing some information ?
• How about running All-pairs shortest path
  algorithm and saving all the paths

• Nope O(n²) space

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Accelerating SearchBounding-Box pruning (Wagner,
Willhalm)

• With every edge, save a bounding box that
  contains at least all the nodes that can be
  reached on a shortest path originating from that
  particular edge.

9,24
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Accelerating SearchBounding-Box pruning

• In Dijkstra
• u ? DeleteMin(PQ)
• forall v \in adjacent_edges(u)
• if t \in BB(u,v)
• .....
• .....
• endif
• endfor

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Search Models

• Basic model
• Shortest path
• Time model
• Shortest fastest path
• Absolute-time model
• Timed queries

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Accelerating SearchBounding-Box pruningResults
of 200 queries
Nodes Time Expansions Time Expansions
199 0.34 6,596 0.60 19,595
4,391 8.11 65,726 12.88 217,430
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Schade MeldungDynamics

• Disturbances on road A road accident or a
  traffic jam
• A road not usable at all ? Edge weight inf
• Probably one lane of the road is still opened ?
  Edge weight increases by some delta
• Consequence
• The pre-computed information becomes invalid and
  useless.
• Re-computing bounding boxes is very expensive.
• This disturbance is temporary.

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Types of Disturbances
Disturbances as Geometrical Objects Model
Individual Edge Model
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Affect of disturbances on pre-computed
information

• Which information has become invalid ?
• Everything ?
• Nope, only those bounding boxes that have
  intersections with affected edges are potentially
  affected.

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Affect of disturbances on pre-computed
information
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Graph update off-line approach

• Introduce the affect of disturbances on the graph
• Simple for Individual Edge model just increase
  the edge weight of the affected edge.
• A bit complex for Disturbances as Geometrical
  Object model.
• Problem We need all the edges that are covered
  by a rectangle.
• Solution WindowQuery using Segment trees
• Perform search on the updated graph
• Use pruning information only if it is not affected

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Graph update off-line approachin Disturbances
as Geometrical Objects model
Rectangle Intersection Problem or more
precisely Red Blue Rectangle Intersection
Problem
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Exploration time checking on-line approach

• Observations
• Since the weights are always increased, if the
  shortest path is not affected, it remains to be
  the shortest path.
• It is possible that some of the constraints have
  terminated and no longer be there by the time the
  mobile object will reach that area.

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Exploration time checking on-line approach

• General Strategy
• Before exploring an edge e, check if e is
  affected or not
• if e is affected then check whether the
  constraints would be valid by the time e would be
  traversed.
• if constraints are valid then declare the search
  procedure as invalid and use standard Dijkstra or
  A.
• else continue.
• else continue.

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Exploration time checking on-line approach
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Future Issues

• Handling of large data sets
• The compressed edges should not be considered
  straight gt Curved Edges
• Visualization of route on a topographic map.
• Bridges gt 3D navigation.

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Thesis download

• http//www.informatik.uni-freiburg.de/jabbar/thes
  is.pdf