GPS-based Navigation in Static and Dynamic Environments

GPS-based Navigation 1

GPS-based Navigation in Static and Dynamic Environments

Master’s Thesis Presentation

Shahid Jabbar

Institut für Informatik

Universität Freiburg

Supervisor: PD Dr. Stefan Edelkamp

Co-supervisor: Prof. Dr. Th. Ottmann

GPS-based Navigation Shahid Jabbar 2

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.


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

The big question .. What are we doing here ?

The Solution


What is GPS ?

A collection of 24 geo-stationary satellites. Gives the position of an object in terms of its

longitude, latitude, and height.


Data Collection

What about the cost of collecting the data ?


Data Collection

What about the cost of collecting the data ?

We say ….

You only need some Bananas.


Data Collection


Data format

<longitude>, <latitude>, <date>, <time>

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


Not everything that glitters is Gold.

Filtering + Rounding

Kalman Filter GPS Information + Speed-o-meter reading as the

inertial information => removes the outliers

Douglas-Peuker Line Simplification Algorithm Simplifies a polyline by removing the waving

affect.


Geometric Rounding Douglas-Peucker’s algorithm results

using Hersberger and Snoeyink variant

#points

Θ=10-7 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


Lets sweeeeep … Graph Construction

•Bentley - Ottmann Line Segment Intersection Algorithm.

•We have multiple traces.

Some of them might be intersecting

Road crossings!!!

•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


# 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

Graph Construction Results of Line sweep

we are very much dependent on k


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 ?


Where am I ?

Voronoi Diagram to the rescue!!!


Node localizationResults

# points

# queries

Construc-tion Time

Searching

Time

Naive Searchi

ng 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 >10,000


My floppy is too small … how can I carry this file ?

Graph Compression



Graph Compression



Graph Compression (contd…)



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


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


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

e1

e2

e3

e4

e5

s

t

b1

b2b3

b4

b5


Accelerating SearchBounding-Box pruning

In Dijkstra u DeleteMin(PQ) forall v \in adjacent_edges(u)

if t \in BB(u,v).....

.....

endif endfor


Search Models

Basic model Shortest path

Time model Shortest + fastest path

Absolute-time model Timed queries


Accelerating SearchBounding-Box pruningResults of 200 queries

#NodesTime

+Expansions

+Time

–Expansion

s –

199 0.34 6,596 0.60 19,595

4,391 8.11 65,726 12.88 217,430


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.


Types of Disturbances

Affected Edge

Area affected bya disturbance

Individual Edge ModelDisturbances as Geometrical

Objects Model


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.


Affect of disturbances on pre-computed information

e1

e2

e3

e4

e5

s

t

b1

b2b3

b4

b5


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


Graph update – off-line approachin Disturbances as Geometrical Objects model

Rectangle Intersection Problem or more precisely

Red – Blue Rectangle Intersection Problem

e1

e2

e3

e4

e5

s

t

b1

b2b3

b4

b5


Exploration time checking – on-line approach

Observations: 1. Since the weights are always increased, if the

shortest path is not affected, it remains to be the shortest path.

2. 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.



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.



s

u

Disturbances

e

v


Future Issues

Handling of large data sets The compressed edges should not be considered

straight => Curved Edges Visualization of route on a topographic map. Bridges => 3D navigation.


Thesis download

http://www.informatik.uni-freiburg.de/~jabbar/thesis.pdf

Documents

GPS-based Navigation in Static and Dynamic Environments