Air Force Space Command is interested in improving the accuracy of GPS receiver positioning, navigation, and timing. To this end, it is useful to identify a set of optimal satellite constellations where each correspon...Air Force Space Command is interested in improving the accuracy of GPS receiver positioning, navigation, and timing. To this end, it is useful to identify a set of optimal satellite constellations where each corresponds to a configuration specifying the number of satellites in each orbital plane. These constellations could then be maintained in a library for future use as satellites fail and are launched. We utilize symmetry in the geometry of the GPS satellite orbits to partition the configurations into a much smaller set of equivalence classes where each class has the same overall receiver accuracy performance. We apply a classical algebraic combinatorial result, Polya＇s Theorem, to count and categorize the classes. Incorporating our results into a GPS constellation optimization computer tool will reduce run time by about an order of magnitude. We apply other algebraic and combinatorial techniques in original ways to count the class sizes and the classes that contain a given number of satellites. Finally, we break the equivalence classes into a still smaller set of new ＂structure＂ classes that are useful in applying the GPS computer tool.展开更多
文摘Air Force Space Command is interested in improving the accuracy of GPS receiver positioning, navigation, and timing. To this end, it is useful to identify a set of optimal satellite constellations where each corresponds to a configuration specifying the number of satellites in each orbital plane. These constellations could then be maintained in a library for future use as satellites fail and are launched. We utilize symmetry in the geometry of the GPS satellite orbits to partition the configurations into a much smaller set of equivalence classes where each class has the same overall receiver accuracy performance. We apply a classical algebraic combinatorial result, Polya＇s Theorem, to count and categorize the classes. Incorporating our results into a GPS constellation optimization computer tool will reduce run time by about an order of magnitude. We apply other algebraic and combinatorial techniques in original ways to count the class sizes and the classes that contain a given number of satellites. Finally, we break the equivalence classes into a still smaller set of new ＂structure＂ classes that are useful in applying the GPS computer tool.
