Industrial Sensors: Global Positioning System

The Global Positioning System (GPS) is a global navigation satellite system (GNSS) developed by the United States Department of Defense and managed by the United States Air Force 50th Space Wing. It is the only fully functional GNSS in the world. It can be used freely by anyone, unless the system is technically restricted. These restrictions can be applied to specific regions by the U.S. Department of Defense. GPS can be used almost anywhere near the earth, and is often used by civilians for navigation purposes. An unobstructed line of sight to four satellites is required for non-degraded performance. GPS horizontal position fixes are typically accurate to about 15 meters (50 ft). GPS uses a constellation of between 24 and 32 medium Earth orbit satellites that transmit precise radiowave signals, which allow GPS receivers to determine their current location, the time, and their velocity. Its official name is NAVSTAR GPS.

Since it became fully operational on April 27, 1995, GPS has become a widely used aid to navigation worldwide, and a useful tool for map-making, land surveying, commerce, scientific uses, tracking and surveillance, and hobbies such as geocaching. Also, the precise time reference is used in many applications including the scientific study of earthquakes and as a required time synchronization method for cellular network protocols such as the IS-95 standard for CDMA.

Basic concept of GPS

A GPS receiver calculates its position by precisely timing the signals sent by the GPS satellites high above the Earth. Each satellite continually transmits messages which include

the time the message was sent
precise orbital information (the ephemeris)
the general system health and rough orbits of all GPS satellites (the almanac).

The receiver measures the transit time of each message and computes the distance to each satellite. Geometric trilateration is used to combine these distances with the satellites' locations to obtain the position of the receiver. This position is then displayed, perhaps with a moving map display or latitude and longitude; elevation information may be included. Many GPS units also show derived information such as direction and speed, calculated from position changes.

Three satellites might seem enough to solve for position, since space has three dimensions. However, even a very small clock error multiplied by the very large speed of light—the speed at which satellite signals propagate—results in a large positional error. Therefore receivers use four or more satellites to solve for x, y, z, and t, which is used to correct the receiver's clock. The very accurately computed time is effectively hidden by most GPS applications, which use only the location. A few specialized GPS applications do however use the time; these include time transfer, traffic signal timing, and synchronization of cell phone base stations.

Although four satellites are required for normal operation, fewer apply in special cases. If one variable is already known, a receiver can determine its position using only three satellites. (For example, a ship or plane may have known elevation.) Some GPS receivers may use additional clues or assumptions (such as reusing the last known altitude, dead reckoning, inertial navigation, or including information from the vehicle computer) to give a degraded position when fewer than four satellites are visible (see , Chapters 7 and 8 of , and ).

Position calculation introduction

To introduce the operation of a GPS receiver, this section ignores measurement errors.

A GPS receiver can use the messages from a minimum of four visible satellites to determine

the times the messages were sent
the satellite positions corresponding to these times.

The x, y, and z components of position, and the time sent, are designated as $\left [x_i, y_i, z_i, t_i\right ]$ where the subscript i is the satellite number and has the value 1, 2, 3, or 4. Knowing the indicated time the message was received $\ tr_i$ , the GPS receiver can compute the indicated transit time, $\left (tr_i-t_i\right )$ . of the message. Assuming the message traveled at the speed of light, c, the distance traveled, $\ p_i$ can be computed as $\left (tr_i-t_i\right )c$ .

A satellite's position and distance from the receiver define a spherical surface, centred on the satellite. The position of the receiver is somewhere on this surface. Thus with four satellites, the indicated position of the GPS receiver is at or near the intersection of the surfaces of four spheres. (In the ideal case of no errors, the GPS receiver would be at a precise intersection of the four surfaces.)

If the surfaces of two spheres intersect at more than one point, they intersect in a circle. The article trilateration shows this mathematically. A figure, Two Sphere Surfaces Intersecting in a Circle, is shown below.

Two sphere surfaces intersecting in a circle

The intersection of a third spherical surface with the first two will be its intersection with that circle; in most cases of practical interest, this means they intersect at two points. Another figure, Surface of Sphere Intersecting a Circle (not disk) at Two Points, illustrates the intersection. The two intersections are marked with dots. Again trilateration clearly shows this mathematically.

Surface of Sphere Intersecting a Circle (not disk) at Two Points

For automobiles and other near-earth-vehicles, the correct position of the GPS receiver is the intersection closest to the earth's surface. For space vehicles, the intersection farthest from Earth may be the correct one.

The correct position for the GPS receiver is also the intersection closest to the surface of the sphere corresponding to the fourth satellite.

Correcting a GPS receiver's clock

The method of calculating position for the case of no errors has been explained. One of the most significant error sources is the GPS receiver's clock. Because of the very large value of the speed of light, c, the estimated distances from the GPS receiver to the satellites, the pseudoranges, are very sensitive to errors in the GPS receiver clock. This suggests that an extremely accurate and expensive clock is required for the GPS receiver to work. On the other hand, manufacturers prefer to build inexpensive GPS receivers for mass markets. The solution for this dilemma is based on the way sphere surfaces intersect in the GPS problem.

It is likely that the surfaces of the three spheres intersect, since the circle of intersection of the first two spheres is normally quite large, and thus the third sphere surface is likely to intersect this large circle. It is very unlikely that the surface of the sphere corresponding to the fourth satellite will intersect either of the two points of intersection of the first three, since any clock error could cause it to miss intersecting a point. However, the distance from the valid estimate of GPS receiver position to the surface of the sphere corresponding to the fourth satellite can be used to compute a clock correction. Let $\ r_4$ denote the distance from the valid estimate of GPS receiver position to the fourth satellite and let $\ p_4\$ denote the pseudorange of the fourth satellite. Let $\ da = r_4 - p_4$ . Note that $\ da$ is the distance from the computed GPS receiver position to the surface of the sphere corresponding to the fourth satellite. Thus the quotient, $\ b = da / c\$ , provides an estimate of