Algorithms behind Global Positioning

Murray Grace

Josh Collins

Agenda

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

What is GPS ? GPS is a space based satellite navigation system that provides

location and time information anywhere, anytime and in all weather conditions

1960’s 1970’s & 1980’s 1990’s & 2000’s

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. EastonIvan A. Getting Bradford Parkinson

GPS 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

How 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 includes

Time the message was sent

Satellite position at the time of message transmission

A 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

Pseudorandom 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 period

Modulated with data sent from Satellite

Each GPS satellite has a unique Pseudorandom Noise it generates to distinguish itself from other GPS satellites

Data received by GPS receiver is sent to the nearest base station to be decoded

Determining 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

Calculating 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

It’s a Game of Error Correction

Accurate timing is the key to measuring distance to satellites

Many factors, such as ionosphere effects, clock errors (unsynchronized with user device clock), multipath distortion, etc. can cause an error in the timing of the signal

Clock errors are handle with very accurate atomic clocks

Everything else is taken care of with Differential Positioning

Overview of Error Effect

Sources of User Equivalent Range Errors (UERE)

Source Effect (m)

Signal arrival C/A ±3

Signal arrival P(Y) ±0.3

Ionospheric effects ±5

Ephemeris errors ±2.5

Satellite clock errors ±2

Multipath distortion ±1

Tropospheric effects ±0.5

 C/A ±6.7

 P(Y) ±6.0

Error correction–Differential Positioning Uses a Reference receiver

Differential Positioning (Continued)

Where

ρm is mobile user position

Rm is the position of the Receiver Station

ε m,space/m,user are the space and user segment induced pseudorange errors

cδtm is the clock offset

Carrier Phase based Algorithm

Carrier phase measurements in GPS are well known to enable precision performance at centimeter level

Carrier frequency sent from satellite can vary due to Doppler shift

The change in this frequency can be used to determine how far away the Satellite is from the receiver

GPS 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

A* 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.

Approach Greedy (best first search)

Input Weighted graph G={E,V}, source vertex v∈V, and goal vertex.

Output Length of shortest path from a given source vertex v∈V to a goal vertex.

Time/Space

Complexity

Constant 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.

A* Example

A* Example

Cell 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 towers

Accuracy is lesser than satellites

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

Accuracy is lesser than satellites

Features of GPS

• GPS devices and Mobile Devices such as cell phones combine data from all sources (Satellites, Cell Phone Towers, Routers) to give the user an accurate position to where they are located on Earth

Location

• GPS device will determine its vertical distance from satellite

• Satellite will communicate with device how far it thinks it is from the center of the earth

• Device takes the difference to determine altitude

Elevation

• GPS device monitors its position over a length of time

• Computes estimate of velocity by Change in distance/Change in time

Velocity

Conclusion

• 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