On renormalization of the approximate solution of the orbital coordinate system equations of orientation

In the quaternion formulation, the problem of mathematical modeling of the spacecraft motion in an elliptical orbit is considered. The control is a modulo-limited acceleration vector from the jet thrust, directed orthogonally to the plane of the spacecraft orbit. The motion of the spacecraft center of mass is described using the quaternion differential equation of the orientation of the orbital coordinate system. An approximate analytical solution of the quaternion differential equation of the orientation of the orbital coordinate system is constructed in the form of a uniformly suitable asymptotic expansion by degrees of eccentricity of the spacecraft orbit (small parameter). To eliminate the secular terms in this expansion, the renormalization method was applied. Taking into account the known solution of the equation of orientation of the orbital coordinate system for the case when the spacecraft orbit is circular, allowed to simplify the form of the above expansion. The nonlinear oscillation frequencies of each component of the desired quaternion were found. Analytical transformations were performed using the SymPy symbolic algebra package. To carry out numerical simulation of the spacecraft motion, a program was written in Python. Calculations based on analytical formulas obtained in the paper (in the absence of secular terms) and previously obtained results in the presence of secular terms are compared. An example of modeling the controlled motion of a spacecraft is given for the case when the initial orientation of the orbital coordinate system corresponds to the orientation of the orbit of one of the satellites of the GLONASS orbital grouping. Graphs were built to show error in the module (and components) of the quaternion describing the orientation of the orbital coordinate system. It is shown that the elimination of secular terms using the renormalization method made it possible to reduce the error in determining this module with an increase in the number of spacecraft revolutions around the Earth. The analysis of the obtained approximate analytical solution is carried out. The features and regularities of the spacecraft motion in an elliptical orbit are established.

spacecraft, orbit, optimal control, quaternion, approximation