## 1 Introduction  
In the past 10-15, years the problem of GPS (GPS means Global Positioning System) satellite-ground station communications has been replaced by the problem of autonomous navigation and inter-satellite communications (ISC) (links), which has been mentioned yet in 2005 in the monograph [1]. Autonomous navigation means that satellites should have the capability to transmit data between them via inter-satellite cross-link ranging [2] and, thus, to ensure navigation control and data processing without commands from Earth stations in the course of six months. It is important that next generation space missions will attain sub-millimeter precision of measuring distances beyond \(10^{-6}\) m (1 micrometer = 1 \(\upmu\)m = \(10^{-6}\) m) by means of ultrashort femtosecond pulse lasers. The theoretical description of such measurements is inevitably related to general relativity theory (GRT). This means that the relative ranging and relative velocity model should account also for the bending of the transmitting path of the signal, which is significant for such large distances between the satellites due to the action of the gravitational field. The bending of the light (laser) is related to the important and fundamental physical fact that due to the action of gravitational field, the signal travels a greater distance in comparison with the straight path distance in the absence of a gravitational field. In the book [3], autonomous navigation is defined as referring to \"processes in which the spacecraft without the support from the ground-based TT&C (telemetry and tracking control) system for a long time, relying on its onboard devices, obtains all kinds of measurement data, determining the navigation parameters like orbit, time and attitude\". In this monograph, DORIS and PRARE navigation systems are determined as non-autonomous, because \"DORIS system can determine the spacecraft's orbit with high accuracy, but both need to exchange information with the ground stations\". Thus, the main aim of autonomous navigation is to reduce the dependance of spacecrafts on the ground TT&C network and, in this way, to enhance the capability of the system's anti-jam and autonomous survivability. It is evident also that the determination of the satellite's exact location is achieved through the reception, processing and transmitting of ranging signals between the different satellites. Autonomous GPS navigation systems are necessary, in view of the proposal of researchers from the University of Texas, Aerospace Corporation, National Bureau of Standards, International Business Machines Corporation (IBM) and Rockwell Automation Inc., to monitor nuclear explosions, based on the GPS inter-satellite communication link. The development of inter-satellite laser communication systems in space technologies that enable super-high-speed data transfers at rates greater than 1 Gbps is widely applied also in cube-/nano-satellite platforms such as CubeLCT, AeroCube-7B/C, CLICK, LINCS-A/B, SOCRATES and LaserCube [4]. However, in order to establish laser communications, the high performance of the arc-second level pointing system is required. This is a difficulty for the nanosatellite platforms. It can be supposed why there is such a difficulty--since the data-transmission rate of 1 Gbps between the two nanosatellites is at an inter-satellite range of 1000 km, the effects of curving the trajectory of the laser signal may be considerable. In the publication [5], concerning the optical links of small satellites, it has been admitted that the s.c. \"pointing error\" arises not only due to tracking sensors and mechanical vibrations, but also due to the base motion of the satellite. This fact, together with the ranging of signals and the pointing error, illustrates the assumption that it is not only the motion of two satellites, but the effect of curving of the signal trajectory, that is important.