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Algorithms behind Global Positioning

Murray Grace

Josh Collins

Slide2Agenda

Global Positioning systems

Overview

How

GPS works

Determining position

Errors & Error correction Algorithms

Routing

Other methods of positioning

Cell Phone Towers & Wi Fi

Features of GPS

Location

Elevation

Velocity

Slide3What is GPS ?

GPS is a space based satellite navigation system that provides location and time information anywhere, anytime and in all weather conditions

First satellite navigation system Transit using 5 satellites tested by U.S. Navy

Timation satellite developed to place accurate clocks in space

Navigation System Using Timing and Ranging or Navstar – GPS program launched

GPS was realized by the U.S. Department of Defense and run with 24 satellites

Icons behind GPS invention

Roger L. Easton

Ivan A. Getting

Bradford Parkinson

Slide4GPS Segments

Space segment

A constellation of 24 Satellites that transmits signals that gives the current GPS position & time

Control segment

User segment

Worldwide control stations to maintain satellite in orbit & adjust satellite clocks

GPS receivers which receive signals from satellite to calculate users position & time

Slide5How does GPS works?

Each satellite sends a unique pseudorandom noise so that they can communicate on the same channel and not interfere with one another (Form of Code Division Multiple Access)Each GPS satellite transmits message that includesTime the message was sentSatellite position at the time of message transmissionA GPS device on Earth decodes ephemeris data from satellites to calculate the timestamp and location of each satellite in the GPS satellite constellation (called the almanac)Range from each satellite is calculated as:Range =Time delay (between message sent & message received )X Speed of light

Slide6Pseudorandom noise & Satellites

Pseudorandom noise is a signal similar to noise ,it lacks any definite pattern but consists of a sequence of pulse which repeats itself after a specific periodModulated with data sent from SatelliteEach GPS satellite has a unique Pseudorandom Noise it generates to distinguish itself from other GPS satellitesData received by GPS receiver is sent to the nearest base station to be decoded

Slide7Determining GPS Position

Suppose distance from satellite to our position is 11,000 miles now the position can be anywhere on the sphere

Satellite A

+

Satellite

B

+

Satellite

C

+

Adding another satellite narrows the position to the intersection

Now taking measurement from a third satellite further narrows down to just 2 points

To accurately determine the true location, we can use measurements from a fourth satellite or eliminate one of the 2 points that gives ridiculous answer(may be too far from earth)

Civilian

GPS fixes under a clear view of the sky are on average accurate to about

5 meters (16

ft.)

horizontally

Slide8Calculating Location

The GPS device uses the distance between itself and at least 3 other satellites along with the distance between each satellite to form a triangle in space

Slide9It’s a Game of Error Correction

Accurate timing is the key to measuring distance to satellitesMany factors, such as ionosphere effects, clock errors (unsynchronized with user device clock), multipath distortion, etc. can cause an error in the timing of the signalClock errors are handle with very accurate atomic clocksEverything else is taken care of with Differential Positioning

Slide10Overview of Error Effect

Sources of User Equivalent Range Errors (UERE)SourceEffect (m)Signal arrival C/A±3Signal arrival P(Y)±0.3Ionospheric effects±5Ephemeris errors±2.5Satellite clock errors±2Multipath distortion±1Tropospheric effects±0.5 C/A±6.7 P(Y)±6.0

Slide11Error correction–Differential Positioning

Uses a Reference receiver

Slide12Differential Positioning (Continued)

Whereρm is mobile user positionRm is the position of the Receiver Stationε m,space/m,user are the space and user segment induced pseudorange errorscδtm is the clock offset

Slide13Carrier Phase based Algorithm

Carrier phase measurements in GPS are well known to enable precision performance at centimeter levelCarrier frequency sent from satellite can vary due to Doppler shiftThe change in this frequency can be used to determine how far away the Satellite is from the receiver

Slide14GPS Navigation

Use of computers and GPS to enable easy navigation of vehicles, precise mapping, bomb targeting, land surveying etc.

Use A* algorithm (Built off of Dijkstra’s)Graph weighted by average speed of road

Fastest Travel Time

Use

A* algorithm (Built off of Dijkstra’s)Graph weighted by road distance

Shortest Distance

Slide15A* Algorithm

Graph search algorithm that solves the single-source shortest path problem for a graph by using an admissible heuristic and by taking the distance it has already traveled into account.

ApproachGreedy (best first search)InputWeighted graph G={E,V}, source vertex v∈V, and goal vertex.OutputLength of shortest path from a given source vertex v∈V to a goal vertex. Time/Space ComplexityConstant heuristic: O(n^2) Optimal heuristic: O(n)

If the

heuristic

h satisfies

h(x) ≤ d(x, y) + h(y) for

every edge (x, y) of the graph (where

d denotes

the length of that edge

), it is optimal.

Slide16A* Example

Slide17A* Example

Slide18Cell Phone Towers & Wi-Fi

Cell phone towers can be used to calculate a relative accurate location of a mobile device such as a cell phone using the ping from at least 3 towersAccuracy is lesser than satellites

Similar to cell phone towers Wi-Fi uses ping from at least 3 routers

Accuracy is lesser than satellites

Slide19Features of GPS

Slide20Conclusion

GPS is mainly used for determining location and for routing and requires at least three satellites to do so

GPS is reliant on accurate timing calculations and thus error correcting

GPS can also calculate velocity and elevation, and use Wi-Fi or cell phone towers for

locating

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