Jeremy Jones, chief of the navigation team for the Cassini Project at NASA's Jet Propulsion Laboratory, offers this explanation.
The accurate navigation of space probes depends on four factors: First is the measurement system for determining the position and speed of a probe. Second is the location from which the measurements are taken. Third is an accurate model of the solar system, and fourth, models of the motion of a probe.
For all U.S. interplanetary probes, the antennas of the Deep Space Network (DSN) act as the measurement system. These antennas transmit radio signals to a probe, which receives these signals and, with a slight frequency shift, returns them to the ground station. By computing the difference between the transmitted and received signals, a probe's distance and speed along the line from the antenna can be determined with great accuracy, thanks to the high frequency of the signals and a very accurate atomic clock by which to measure the small frequency changes. By combining these elements, navigators can measure a probe's instantaneous line-of-sight velocity and range to an accuracy of 0.05 millimeter-per-second and three meters respectively, relative to the antenna.
Many probes also carry cameras that are used to image the destination, whether it be a moon, planet or other body. During the final approach, these images are used when the distance becomes small. For example, the Cassini spacecraft's camera provides an angular measurement with an accuracy of three microradians (three kilometers) at a distance of one million kilometers. The images complement the radio data and provide a direct tie to the target.
Calculation of the trajectory of a space probe requires the use of an inertial coordinate system as well, wherein a grid is laid over the solar system and fixed relative to the star background. For interplanetary missions, an inertial coordinate system with an origin at the center of mass of the solar system is used. Because the measurements provide information on the position of a probe relative to the antenna, knowledge of the antenna's location relative to this inertial coordinate system is used to convert the measurements into elements in the system. Where the antenna is depends not only on its geographic location on Earth's surface, but on Earth's position relative to the solar system center of mass (known as the Earth ephemeris). Measurements of this ephemeris have an accuracy of about 0.5 kilometer and the location of the antenna is known to an accuracy of better than five centimeters.
The third component of interplanetary navigation is an accurate model of the solar system. Gravity is the most important force acting on a spacecraft. Determining these gravitational forces requires accurate knowledge of the locations of all of the major bodies, such as the sun and all the planets, over the course of time. This information is provided by the planetary ephemeris, which has been in continuous development since the beginning of the interplanetary space program. Thanks to this longstanding work, the location of Saturn was known to an accuracy of a few hundreds of kilometers previous to Cassini's final approach when it deployed its camera. After the probe entered Saturn orbit, the moons of the giant planet became important gravitational bodies. Their locations have been determined to an accuracy of a few kilometers relative to Saturn.
The final component of accurate navigation takes all of these other elements and, using models of the forces acting on a probe and orbital dynamics, estimates its location. By taking regular measurements over a period of time, a probe's position and velocity can be determined. For example, Cassini's location is typically determined to a kilometer or less relative to Saturn. Using a probe's known position and velocity, its future positions can be worked out. Navigators compare these positions to the predicted location of the target body--based on the ephemeris--to determine when a probe will reach its target. Then, all that's left to do is to collect the flyby data, take a deep breath, and go on to the next encounter.