Symbolic-numeric Investigations for Stability Analysis of Satellite Systems

An approach for the symbolic-numeric stability analysis of satellite systems correspondingly to structure of gravitational, aerodynamical, gyrostatic and static forces is presented. The satellite system is described by Lagrange differential equations. The equations of motion form a closed system, for which the Jacobi Integral is valid. Stationary solutions of these equations are defined by the multivariate polynomial system. The algebraic polynomial system has been investigated with the help of born the numerical and symbolic analysis. The symbolic investigation was made by means of Resultant and Grobner Basis methods[7]. The stationary motions of a satellite sub ject to gravitational, aerodynamical, gyrostatic and static torques is governed by nine algebraic equations with nine parameters - pro jections of torques vectors onto the frame attached to the body of the satellite. The classes of stationary solutions of these algebraic equations have been found With the help of computer algebra system Maple [S] by applying the Resultant, the Groebner Basis and Factorization methods. Computation of the surfaces of algebraic equations real solutions is performed numerically by the gradient method. The number of grid points was chosen as a function of curvature. As a result of this work 3D boundary surfaces with equal number of equilibrium positions of a satellite are constructed. On the base of this methods the problem of defining the equilibriuYn positions of a satellite in a circular orbit under the influence of external torques was solved [2],[6], [9],[10],[11]. The stability of equilibrium positions are analyzed numerically with Lyapunov’s second method. The Jacobi Integral as Lyapunov’s function is used.