How Does GPS work?

Justin Chi

Martin Chi

Elizabeth Tom

What is GPS?

GPS = Global Positioning System

Satellite Navigation System

27 Satellites (24 in use, 3 for emergency)

Applications:

• Military

• Navigation

• Surveying

• Visually Impaired

• Geocaching

• Airplanes

• Time

Types:

• Differential GPS (DGPS)

• Wide Area Augmentation System (WAAS)

• Local Area Augmentation System (LAAS)

• Exploitation of DGPS for Guidance Enhancement (EDGE)

• Carrier Phase Enhancement (CPGPS)

• Wide Area GPS Enhancement (WAGE)

• Relative Kinematic Position (RKP)

Geometry:

Simplified GPS:

• Using 4 satellites

(actual GPS uses 4+ satellites and least squares method)

• xyz-coordinate system

• Earth is the origin

• A position can be expressed as (x, y, z) can be translated into longitude and latitude

• In this system, a point at sea level is: x2 + y2 + z2 = 1• Speed of light is about .047 (units of earth radii per millisecond)

Ship Example:

Satellite 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

Let:

(x, y, z) = ship’s position

t = time when signals arrive

d = 222 )0()2()1( zyx

(x – 1)2 + (y – 2)2 + z2 = .0472(t – 19.9)2

Satellite Position Time

1 (1,2,0) 19.9

Calculating distance for Satellite 1:

d = .047(t – 19.9)

AND

(x– 1)2 + (y – 2)2 + z2 = .0472(t – 19.9)2

2x + 4y – 2(.0472)(19.9)t = 12 + 22 - .0472(19.9)2 + x2 + y2 + z2 - .0472t2

We can do this for the rest of the satellites…

2x + 4y + 0z – 2(.0472)(19.9)t = 12 + 22 + 02 - .0472(19.9)2 + x2 + y2 + z2 - .0472t2

4x + 0y + 4z – 2(.0472)(2.4)t = 22 + 02 + 22 - .0472(2.4)2 + x2 + y2 + z2 - .0472t2

2x + 2y + 2z – 2(.0472)(32.6)t = 12 + 12 + 12 - .0472(32.6)2 + x2 + y2 + z2 - .0472t2

4x + 2y + 0z – 2(.0472)(19.9)t = 22 + 12 + 02 - .0472(19.9)2 + x2 + y2 + z2 - .0472t2

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 - 2.42)

2x – 2y + 0z + 2(.0472)(0)t = 5– 5 + .0472(19.92 – 19.92)

00022

47.3056.220

86.3077.442

00022

47.3056.220

86.3077.442

67.3067.100

41.5095.010

41.5095.001

67.3067.100

41.5095.010

41.5095.001

x = 5.41 - .095t, y = 5.41 - .095t, z = 3.67 - .067t

t is free

(5.41 - .095t – 1)2 + (5.41 - .095t – 2)2 + (3.67 - .067t)2 = .0472(t – 19.9)2

.02t2 - 1.88t + 43.56 = 0

2 answers:

t = 43.1

t = 50.0

(x, y, z) = (1.317, 1.317, .790)

d is about 2

Since the units are Earth radii this means that the point is about 4000 miles above the surface of the Earth

t = 43.1

t = 50.0

(x, y, z) = (.667, .667, .332)

d is about .9997

Since the units are Earth radii this means the point is on the surface of the Earth.

We can convert this into longitude and latitude

Wide Area Augmentation System:

• Improves upon GPS

• Uses a combination of satellites and ground based stations to send correction signals to the receiver

• Can improve accuracy up to 5 times