Global Positioning System

Most people have been lost at one time or another, but what if it were possible to know where you are, anywhere on Earth, 24 hours a day? The Global Positioning System (GPS) can give that information, and it is free to anyone with the proper equipment and a basic knowledge of mathematics.

In the 1980s, the U. S. Department of Defense designed GPS to provide the military with accurate, round-the-clock positional information. Twenty-seven satellites orbiting over 10,000 miles above Earth regularly send information back to Earth. A small piece of equipment, called a GPS receiver, uses this information to compute its position to within a few yards. GPS receivers used for surveying can find positions to within less than one centimeter.

The "constellation" of satellites above the Earth is constantly changing; each orbits Earth twice a day. At any given time there are enough satellite

signals to accurately locate oneself in three dimensions: latitude, longitude, and elevation.

GPS is rapidly becoming a common technology, but it is still a mathematical wonder. Ancient sailors looked to the heavens to estimate their position in the vast oceans. Modern sailors also look to the sky for information, but the modern positioning information they receive is so accurate that any errors are less than the width of the pencil they use to mark their map.

Triangulation

The basic concept of GPS is triangulation. Suppose a person is standing in a valley surrounded by several towering mountain peaks. By using a compass to measure the direction to each peak, this person could locate his or her exact location on a map by using triangulation. After writing down the three measurements (remembering that there are 360 degrees in a circle), a line should be drawn from each peak in the opposite direction just measured.

Then 180 degrees is added or subtracted so that the direction the lines are drawn from each peak will fall between 0 and 360 degrees. For example, if one of the measurements is 270 degrees to peak A, the line from peak A back to the person's position would be 90 degrees. The point at which the three lines intersect is the point at which the person is standing.

The GPS satellites are like mountain peaks; they are known points in space from which lines can be drawn in order to specify a location. Each satellite transmits a radio signal that can be received on Earth and recognized by a GPS receiver. Rather than measure direction, however, a GPS receiver uses the time it takes for each satellite's beacon to reach it and calculates a distance.

Because radio waves travel at the speed of light, the receiver divides the time the signal takes to reach the receiver by the speed of light (186,000 miles per second) and determines the distance. These distances can be used to form spheres around the satellites that will intersect at a specific position just as the lines drawn from the mountain peaks will intersect at a specific position.

Understanding GPS Measurements

Assume a GPS receiver is sitting in Nebraska. Once activated it begins to collect signals from GPS satellites 1, 2, 3, 4, 5, and 6. The distance to each satellite can be determined using the distance formula d = rt (distance, d, equals rate, r, multiplied by time, t, or distance equals velocity multiplied by time). Although all the satellites are 10,900 miles from the surface of the Earth, the distances to each one will vary according to its position in orbit. For example, all the street lights in a city may be 15 feet in the air but they are not all 15 feet from a specific point in the city.

The formula for determining these distances may be simple, but the calculations themselves are anything but simple. The satellites must be precisely timed so that each is synchronized with the other satellites in the constellation and with base stations on Earth. Although each satellite will be at a different distance from a particular point, the time it takes to cover those distances at the speed of light does not seem significant. In order to calculate distance, however, this time is significant to the GPS receiver.

Consider that a signal 10,900 miles from a receiver reaches that receiver in 0.058602 seconds. A signal 10,926 miles away, however, reaches the receiver in 0.0587419 seconds. A 26-mile difference translates into less than fourteen one hundred-thousandths (0.00014) of a second. Clearly, the GPS receiver has some very precise mathematics to work with, further complicated by the fact that the satellites are always moving.

After making these complex measurements, the distance to each satellite will be the hypotenuse of a right triangle created by the receiver's position, the satellite's position, and the position on Earth directly under the satellite. Once these distances are known, spheres can be created surrounding each satellite. Each sphere has a radius equal to the computed distance between the satellite and the receiver. The first sphere, around satellite 1, will have an infinite number of points along its surface so the receiver's position could fall anywhere on that sphere, including points in outer space.

Next, the sphere around satellite 2 is introduced, and the two spheres create an intersection that forms a circle. Now the GPS receiver could be anywhere on that circle, even points in space that, of course, it is not occupying. The third sphere, around satellite 3, intersects the first two spheres and limits the receiver's possible position to two points. The receiver is located at one of the two points. The other point is either in the air above the receiver or in the ground directly below the receiver. If the altitude of the receiver is known, then it is possible to determine which one of the two points is correct. The sphere around satellite number 4 will also reduce the two points to only one. It is amazing, yet basically simple, how one receiver and four satellites can reduce an infinite number of possible locations to only one.

Simulating GPS. To simulate the process of the Global Positioning System all that is needed is some string, scissors, tape, several coins, and four stationary points (the corners of a room will work). At any three-dimensional point in the room (on a desk, for example) a coin should be placed. The end of the string should then be taped into the corner of the room, with the other end pulled to the coin, cut, and then placed back in the corner. This process is repeated for the remaining three corners, and the extra coins are placed elsewhere in the room.

During this preparation, a volunteer waits outside the room. The volunteer should then enter the room and be alerted to the availability of the strings. The volunteer can then start pulling the cut ends of the string outward from the corners beginning with any two. By adding the third and fourth strings and finding where they all intersect, the volunteer should be able to eliminate all the extra coins and find the original coin. This is how GPS works in its most basic form.

Advantages and Disadvantages of GPS

Atmospheric inconsistencies can create inaccuracies in the positions computed by a GPS. Additionally, GPS is a "line of sight" system. Although a user cannot actually see the satellites in space, he or she does need an unobstructed view of the sky in order to utilize GPS. This poses serious challenges to those who choose to use GPS in canyons, cities, or other situations where large, solid objects mask out portions of the sky. When working where obstructions exist, careful planning must be done to ensure enough satellites are in "in view" for proper positioning.

Fortunately, GPS has a built-in feature, the almanac, to aid in identifying the location of satellites. Each satellite "knows" the location and direction of every other satellite. Along with the signal used to provide positions, satellites also transmit the almanac to a GPS receiver. Common GPS planning software can use the almanac to plot the entire constellation of satellites so users can plan ahead for their needs.

For example, if one needed to work in a canyon, planning software may indicate the only feasible time would be from noon until 2:00 P.M. Only during that time will the receiver have an unobstructed path to a sufficient number of satellites, all very high above the horizon, from within the canyon.

Global Positioning System

Triangulation

Understanding GPS Measurements

Advantages and Disadvantages of GPS

Internet Resources

APPLICATIONS OF GPS