__Keywords:__spaceship navigation zenith

S. G. Likhansky

The problem of ballistic data and orbital parameters estimation for group of 4 or more close-flying spaceships by their relative movement only without any external navigation is considered at this paper. Reserve internal navigational method using only presize knowledge of current group configuration is proposed here. This method is expected to duplicate the work of well-known space navigation systems (spaceship orientation star-detectors, positioning systems as GPS, GLONASS etc.) in reference to close groups of N4 spaceships (called in text as N-tandems).
The following main supposition is being accepted at the paper. The 4-tandem is equiped with interior high-presizion laser measuring system which is capable to determine distances between mass centers for any pair of spaceships (error 1 mm) and their derivatives of order 3 by time. Such system is called at the paper as Local Positioning System (LPS).
The second supposition – the main spaceship orientation control must be turned off immediately after the navigation was put out of action. This natural supposition simplifies the task solution.
The third supposition – the generality of 4-tandem initial conditions in Orbital (Siderial, Inertial) Coordinate System (OCS) and current 4-tandem configuration. LPS is being turned-on only in following case: a) the main spaceship orientation control is being turned-off; b) the current «4-tandem pyramid» is «sufficiently non-degenerate» (it’s current volume is strongly non-zero).
The task solution algorithm consists of the following steps (in brief).
1) The mathematic model of 4-tandem spaceships relative movement is being constructed here (in Kepler’s approximation) – both in OCS and in Interior Coordinate System (ICS) of fixed spaceship (equipped with active SAR and called here as main spaceship).
2) By means of Keplerian ballistics and matrix analysis, using presize LPS-data of order 2 and accounting the «free rotation» in OCS of main spaceship, the current ICS-components of main spaceship rotation vector are being calculed.
3) Using the same mathematics, presize LPS-data of order 3 and rotation vector value knowledge in ICS (see step 2), the complete set of current coorinate and velocity ICS-vectors and orbital data (paremeters of Keplerian ellipse and Laplace–Runge–Lenz vectors in ICS) for each spaceship of 4-tandem is being calculed.
4) The whole set of the ICS ballistic and orbital vector data (step 3 exit) is being transformed (in co–ordination) from current non-inertial ICS to current non-inertial Way Orbital Coordinate System (WOCS), associated with main spaceship current OCS velocity vector,
Infortunately, the obtained solution has 2-ambiguity of following type – «zenith–nadir and forward–back along the way». So we must to add the fifth step:
5) To resolve the 2-ambiguity «zenith–nadir» and choose the correct branch of t-curve (t – time parameter) in multidimentional abstract space of ballistic values being estimated, we must rotate the main spaceship, steer the SAR beam and emit the short SAR signal to «hypotetical nadir» direction so as to verify non-zero reflected signal existance. If reflected signal exists – chosen hypothesis is true, otherwise – false.
The spherical symmetry of Keplerian task doesn’t allow to determine the obtained set of ballistic vector values in OCS and Greenwich Coordinate System (GCS) without auxillary SAR data. But, due to knowledge of full ballistic data in WOCS, SAR system of tandem is capable to start work in Stripmap and Moving Target Detection (MTD) modes. Usage of obtained SAR data (Doppler and MTD information, global Earth SAR image formation) in combination with discribed LPS ballistic algorithms and ballistic data extrapolation will allow us to move on (adaptively, step-by-step) to normal navigation and SAR imaging in OCS and GCS.
The main conclusion out of the paper – the expediency of working out LPS of discribed type in reference to close-flying groups of N4 spaceships (as duplicating system of navigation).

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