Background: Global Positioning System (GPS)

The Global Positioning System (GPS) provides three-dimensional position, velocity, and track measurements to users located on the ground, on the water, or in the air. GPS receivers (see Figure 1) are used by surveyors, motorists, boaters, pilots, fishermen, hunters, hikers, and others. Most users purchase GPS receivers as stand alone units, but others use GPS receivers included in automobile navigation systems, watches, and other products.

Figure 1. Garmin GPS Handheld Receiver (from: http://garmin.com/).

The GPS consists of 24 satellites that orbit the earth approximately 20,200 kilometers above the surface of the earth (or 12,000 miles). Each satellite follows one of six elliptical orbits completing a cycle in slightly less than twelve hours (see Figure 2).

Figure 2. Satellite Orbits (from: http://www.eso.org/seaspace/navigation/navgps/navgps-1.html).

Five ground stations (Colorado Springs, Hawaii, Ascension Island in the South Atlantic Ocean, Diego Garcia in the Indian Ocean, and Kwajalein in the North Pacific Ocean) maintain each satellite’s orbital data and precise time. Each satellite continuously transmits data while GPS receivers decode this information, calculate their distances from the satellites, and use a method known as triangulation to calculate users’ positions on the earth. A GPS receiver must receive data from at least three satellites in order to obtain a two-dimensional coordinate location (latitude and longitude), while data from at least four satellites are required for a three-dimensional coordinate location (latitude, longitude, and altitude) (Chew, 1998; Kaplan, 1996; Leick, 1995; Parkinson, 1996).

The GPS was developed during the 1970s in order for military forces to determine their exact coordinate position on the earth. In 1973, a Joint Program Office was formed by the U.S. Department of Defense (DoD) and was directed to improve existing operational satellite-based navigation systems. The present Navigation System Timing and Ranging (NAVSTAR) Global Positioning System (GPS) is the result of this initial directive. This system did not become fully operational until the end of 1994 (Parkinson, 1996).

The signals sent by the satellites broadcast at two frequencies, L₁ (1575.42 MHz) and L₂ (1227.60 MHz). The L₁ signal is modulated with two types of codes, including a P-code (Precise Code) and a C/A code (Coarse Acquisition Code). The C/A code is for civilian use, while the P-code is for military use. Previously, the DoD intentionally degraded the accuracy of the C/A code by introducing errors, known as selective availability (SA), to the satellite signals. Having the SA turned on would prevent military adversaries access to the highly accurate GPS signals and degrade the accuracy of the C/A code to within 100 meters. However, with a Differential GPS (DGPS) module used in conjunction with a receiver, the SA can be eliminated and a user can locate their position to within 1 to 3 meters. In May 2000, the government removed the SA interference from the C/A code, which, without the aid of DGPS, increased the receiver’s accuracy to about 15 meters.  Fortunately, the SA can be turned on when necessary.

Background: GPS Receiver

The Garmin GPS receiver has five primary pages that display information (other receivers have similar pages). These include the Satellite Page, Position Page, Map Page, Navigation Page, and Main Menu Page (see Figures 3 and 4). The figures below provide illustrations and descriptions of these pages.

Figure 3. Page Descriptions from the GPS 12 Personal Navigator Owner's Manual (p. 6).

Figure 4. Page Descriptions from the GPS 12 Personal Navigator Owner's Manual (p. 7).

Part 1: Triangulation

• Discuss why a minimum of three satellite signals is needed to determine a 2-dimensional position (latitude and longitude) on Earth.
  
  
• Imagine that a GPS handheld receiver acquires a signal from Satellite A located 13,000 miles from the receiver. Using Figure 5, sketch the locus of points on the earth that are exactly 13,000 miles from this satellite. Discuss how the locus of points relates to the user’s position on the earth.
  
  
• Suppose the handheld receiver acquires a signal from Satellite B, a second satellite also located 13,000 miles away. Using Figure 5, sketch the locus of points on the earth that are exactly 13,000 miles from Satellite B. Note the possible locations of the handheld receiver if it is located 13,000 miles from Satellite A and 13,000 miles from Satellite B. Discuss why signals from two satellites are not sufficient to determine a 2-dimensional position.

Figure 5. Signals from Satellites A and B.

 

• Assume the handheld receiver acquires a signal from Satellite C, a third satellite located 14,000 miles away, above the plane of Figure 5. Discuss why acquiring a signal from this satellite, in conjunction with the signals from Satellites A and B, is sufficient to determine a 2-dimensional position.

Background: Longitude and Latitude

To interpret coordinate positions, a user must understand lines of latitude and longitude. Lines of latitude circle the earth in an east/west direction while lines of longitude circumnavigate in a north/south direction from pole to pole. Degrees of latitude start at 0° at the equator and increase to either 90° north or 90° south at the poles. An equal distance lies between each line of latitude (approximately 69 miles or 111 km), thus lines of latitude are often referred to as parallels. Degrees of longitude start at 0° at Greenwich, England (Prime Meridian) and move east (positive degrees) and west (negative degrees) to 180°. An equal degree of rotation lies between lines of longitude, which are sometimes referred to as meridians. The distance between lines of longitude varies from 69 miles at the equator to zero miles at the poles. Hence, the number of miles per degree of longitude is a function of latitude.

Figure 6. Latitude, Longitude and Earth’s Grid (from: http://www.hammondmap.com/latlong.html).

Part 2: Determining longitude and latitude coordinates

• Approximate your current coordinate (latitude and longitude) position using the map in Figure 7. Record your approximation.

• Go to an unobstructed outside location and use your GPS receiver to acquire a 3-D Navigation status (lock onto at least four satellites) on the Satellite Page. This process may take several minutes. Interpret the status and strength of the satellites whose numbers are displayed in the circular area. Include in your interpretation the meaning of the bar graph located at the bottom of the page.

• Move to the Position Page and make sure your GPS receiver is set to display readings in decimal form (default). Record the latitude and longitude of your position, at thirty-second intervals over a 3-minute period. Record the data pertaining to these coordinate positions over this time span on the table below (see Table 1).

Table 1.  Time, latitude, and longitude.

• Examine the data in your table and describe the general trend of the data. Compare your readings with your approximations. Comment on your accuracy.
  
  

• Predict the shape of the graphs of time versus latitude, and time versus longitude.
  
  

• Input these values in a spreadsheet or a graphing calculator list. Graph time versus latitude (see Figure 8). How does your prediction compare to the resulting graph? Describe the characteristics of the graph. Do the same for longitude.

Figure 8. Graph of time versus latitude.

• From the data and the graph decide on a more precise longitude and latitude for your position.
  
  

• From the data and the graph, calculate and discuss a “margin of error” of your readings in minutes of a degree.

Part 3: Determining differences between two locations

• Input another group’s 3-minute data in your graphing calculator and simultaneously graph time versus latitude for both groups’ data (see Figure 9). Compare the graphs and discuss the relationship between the graphs. Do the same for longitude.

Figure 9. Graph of time versus latitude for two groups’ data.

• Comment on the statement: “Coordinate values from a GPS receiver fluctuate uniformly.”
  
  

• Use the numerical data and the graph to determine the difference in latitude and the difference in longitude, in minutes of a degree, between these two positions. Discuss how you determined these differences.
  
  

• Convert the difference in latitude between these two locations to a difference in feet.

• Discuss how you would convert the difference in longitude from minutes of a degree to feet.

• Discuss how you would calculate the actual distance between these two coordinate locations, assuming that you had already converted the difference in latitude and the difference in longitude from minutes of a degree to feet.
  
  

• Comment on the appropriateness of your method for calculating the distance between two locations if the locations are as far apart as New York and Los Angeles.