Annotation: The methods are proposed (analytical and numerical based on motion equations integration) to evaluate probability of first approaches to small distances of satellites of cluster uncontrolled in flight in long time intervals. As the number of satellites injected into area of one base orbit grows, the necessity of evaluating such probability constantly increases – already at present their number in some cases exceeds hundred units. In flight, such satellites form in limited area of space rather compact cluster; the satellite density in such cluster exceeds by many orders the density of operating space objects at their functioning altitudes. Due to somewhat different satellite orbiting periods, the distances between them in flight direction continuously change, different precession motion of orbital planes determines their angular spread – approach in flight. It was determined that maximal probability of approach of whatever pair of satellites of cluster to small distances is the case if in some neighborhood of numbers of their flight orbits, simultaneously two events are realized – the satellites approach to minimal distances in flight direction and angular spread of their orb ital planes is close to zero. The conditions are determined of separation of whatever two satellites of cluster (their separation directions and velocities) – that ensure simultaneous realization of the above events in some neighborhood of number of flight orbits. The analytical relations were obtained that allow determining the corresponding numerical values of satellite approach parameters. For particular case – satellite separation at the equator – maximal probability of approach of two satellites of cluster to small distances is the case when their relative separation velocities are equal in flight direction and in perpendicular to this direction. For the option of injecting 12 satellites to the area of one base orbit of ~ 650 km altitude and  98 inclination, the parameters of satellites separation at the equator were determined that realize their uniform dispersion in the first orbits of autonomous flight. For 2 pairs (out of 66 formed for considered injection case) the conditions of maximal probability of their first approaches to small distances are realized. Using two developed methods evaluations of such probability were obtained.

Annotation: High-precision guidance of supersonic flying vehicles maneuvering while descending in the atmosphere with high degree of thermal protection ablation is a well-known problem of space ballistics. The existing methods for calculating the ablation of thermal protection and the subsequent calculation of aerodynamic characteristics lead to scatter of the landing points of a flying vehicle reaching 5 km or more. The functional guidance method, in principle, allows achieving the required guidance accuracy (hundreds of meters), however, it requires a reserve of power of the controls at a level 50% to counter the influence of disturbing factors. The known terminal guidance method, which has recently become widespread, is based on a highly accurate prediction of motion parameters and, in this regard, has little promise. The method has been described in the article that allows 15-20-fold reducing the flight range scatters caused by lack of knowledge (including due to coating ablation) of its current aerodynamic characteristics and ensuring that the accumulated lateral deviation is counteracted in the limit to 1-1.5 km. The method is applicable to the flying vehicles with weight asymmetry (“transverse” displacement of the center of mass), performing maneuvering under conditions of aerodynamic balancing. The method is based on the solution to increase the accuracy of hits by spinning the shells around longitudinal axis. It is proposed that when a flying vehicle moves in the dive mode by means of the onboard CVC, it is regular (at intervals) to calculate its flight path in the (conditionally) autorotation mode. Based on the results of processing single calculations, the corresponding flight ranges of a flying vehicle and the lateral displacement of the touchdown points are determined, the point in time is predicted at which the flight range of the flying vehicle is equal to the specified one and the average lateral deviation is determined. At this moment the angular movement of the flying vehicle is transferred to the autorotation mode. Counteraction of the lateral displacement is introduced by adjusting the half-periods of flying vehicle movement along the angle of the precession. An example of pointing a flying vehicle at a given range, and bringing it to the touchdown point, shifted to the right relative to the original flight path by 1 km. The error of the terminal guidance of a maneuvering while reducing the aircraft using the proposed guidance method is determined.

Annotation: This paper suggests method for analysis of the dynamics of the aircraft with weight asymmetry (transverse displacement of the center of mass) maneuvering in the atmosphere under the impact of the short-time alternating moment of engine thrust, spread out over a period. The engines are installed on the bottom of the aircraft at the maximum distance from its longitudinal axis. Angular motion with nominal and perturbed performances of the aircraft and flight conditions has been consistently considered. Before maneuvering, the aircraft is set at the trimming angle of attack, determined by the magnitude of transverse displacement of the center of mass and aerodynamic characteristics. The direction of the aircraft maneuvering in the atmosphere depends on the acting moments of forces and time diversity of the engine firings to speed up and shutdown the angular motion. In the absence of disturbances, the angular motion of the aircraft shows in part signs of regular precession (almost constant precession velocity and nutation angle) and autorotation (close to zero self-rotation angle). Under the influence of disturbances, the spread of the aircraft angular motion parameters increases, mainly at the angle of precession, which characterizes changes in the direction of maneuvering. Composition of disturbances includes the spread of the aircraft technical characteristics (position of the center of mass, moments of inertia, aerodynamic coefficients, velocity head, etc.), errors associated with the operation of the engines (thrust spread, time of ignition and shutdown, angular alignment of their longitudinal axes). Terminal control was introduced to realize the given final state and to reduce the disturbances impact on the maneuvering parameters based on the registered deviations of the angular motion from the nominal one after the first shutdown of the attitude maneuver engine. Monte Carlo method (1000 variations of random realizations of the acting perturbations) confirmed the effectiveness of the proposed terminal control of the angular motion of the aircraft to provide the specified maneuvering parameters.