Chinese scientists invent the navigation and positioning system based on commercial communications satellites and develop them successfully into China Area Positioning System (CAPS). In principle, this system is different from the GPS broadcasting satellite navigation class, where the propagation epoch of original navigation signals for pseudo-range measurement is from a ground master control station rather than from satellite transponders. This paper addresses the establishment of the three observation equation models for the navigation and positioning system based on communications satellites, and expresses them identically to operator equations and optimized models. Furthermore, both algorithms of the linear solution for the observable characteristic equation and the least-squares solution for the condition number more than 4 are discussed, with several methods for the exact solution, such as improving the behavior of coefficient matrices, right estimation for the weighted right hand side and selection of iteration forms of solutions, and the influence of the condition number on improving navigation and positioning accuracy is also analyzed carefully. Hopefully, all the works would be contributive to further development of the navigation and positioning system based on communications satellites, and be potentially valuable to other satellite navigation and positioning systems.
CAPS, navigation and positioning, observation equationUnlike GPS and GLONASS satellite navigation system using special navigation satellites, the navigation and positioning system based on communications satellites uses commercial GEO and IGSO communications satellites to solve navigation observation equations for pseudo-range measurement [1,2] , and satellite transponders are known as a spatial reference, where there are no higher precise atomic clock as a time reference and no onboard navigation computer for data processing, and hence, it is called a "transparent bending" transponder. The required atomic clock as a time reference has to be installed on the Master Ground Control Station, where it will generate time service information; navigation message will be transmitted broadcasting to users via satellite transponder. In reception, users receive carrier signals and decode the navigation message, finally, to obtain the observable measured pseudo-ranges from the master ground control station to users via satellites. As the orbital parameters from navigation message for observable pseudo-ranges measurement to satellites are known, the accurate satellite orbit positions can be measured by means of time differential and comparison measurement techniques, and the satellite observation equations will be established and determine finally accurate user's position.The principally geometric model of the navigation and positioning system based on communications satellites is shown in Figure 1. Since ground station references are relatively easy to be known, the satellite orbit positions can be determined by means of the points of spheral inters...