MATHCHALLENGING ASSIGNMENT

BY :- AMIT GARG

USE OF LINEAR ALGEBRA IN GPS SYSTEM

GPS (GLOBAL POSTIONING SYSTEM……..)

The Global Positioning system is a space-based satellite navigation system that provides location and time information in all weather conditions,

WORKING

• A GPS system consist of transmitter (GPS satellite ) and receiver .• A GPS system uses 4 satellites at a time to find the location of any given

point .• A GPS system uses a concept of TRILATERATION .• 4th satellite is only used to increase the accuracy of the position (so

minimum 3 satellites are required to find the location of any given point )

ReceiverTransmitter

TRILATERATION

Intersection of all three circles .

Satellite A

Satellite B

Satellite C

Required Position of the receiver .

Distance between satellite A and receiver

A Satellite measures two things at a time…..

Time….

Location….• Every Satellite measures the time taken to reach the electromagnetic signal from it

to the receiver .• By multiplying it with the speed of the electromagnetic waves we can calculate the

distance between the satellite and the receiver (Radius of the circle ) .• Satellite also sends its exact location to the receiver so that it can be taken as the

centre of the circle .

Solutions using linear algebra……

• Let us assume that the co-ordinates of the receiver location is (x,y,z) . Now , creating the equations…..• Time will be measured in milli-seconds ……

Any point above sea level satisfies the equation :- x2+y2+z2 = 1

• All four satellites will sent the time(when the data was sent to the receiver) and location .

• Let the data from the four satelltite is given as :-

SATELLITE X,Y,Z- POSITION TIME

1 . (1,2,0) 19.9

2. (2,0,2) 2.4

3 . (1,1,1) 32.6

4 . (2,1,0) 19.9

• Now , the distance between receiver and satellite is measured in terms of earth radii per second (c = 0.047 earth radii per millisecond.)

• D=c(t-tsent) ….where t is the time when the signal is received by the receiver .

• Let (x; y; z) be the position of your GPS receiver on Earth and t be the time the signals arrive.

• For satellite 1…. d=0.047*(t-19.9)…………(1) By using pythagorus theorem in three dimensions… ………(2)• Simillarly , equations can be made for all four satellites ….. now , equation (1) and (2) are equal ,therefore ……….. 0.047*(t-19.9) squaring both sides…..we get… (0.047*(t-19.9))2 =(x-1)2 +(y-2)2 +(z-0)2 =>2x + 4y + 0z − 2(.0472)(19.9)t = 12 + 22 + 02 − .0472(19.9)2+ x2 + y2 + z2 − .0472t2

………….(3)

• Simillarly , for all four satellites equations can be made but all contains same type of squared terms of x,y,z,t.

• Therefore , these variables can be easily removed by subtracting equation 1 from equations 2,3,4 ……

• So we get four linear equations…..which can be solved easily with the help of matrices….

2x − 4y + 4z + 2(.0472)(17.5)t = 8 − 5 + .0472(19.92 − 2.42)0x − 2y + 2z − 2(.0472)(12.7)t = 3 − 5 + .0472(19.92 − 32.62)2x − 2y + 0z + 2(.0472)(0)t = 5 − 5 + .0472(19.92 − 19.92)

Other three equations are……

CONVERSION OF LINEAR EQUATIONS IN MATRIX FORM……..

2 -4 4 .077 3.86

0 -2 2 -.056 -3.47

2 -2 0 0 0

Augmented form

Continuing with the solution of the system, the reduced row echelon form for the augmented matrix is……

1 0 0 .095 5.41

0 1 0 -095 5.41

0 0 1 0.67 3.67

Reduced ech form .

From above method we can easily calculate the position of the receiver……This is how a GPS network work……

THANKING YOU

References :- 1). https://www.youtube.com/watch?v=4O3ZVHVFhes&spfreload=1 2) Research paper of sir kalmen .