Symbolic-numeric Investigations for Stability Analysis of Lagrange Systems

Sergey A.Gutnik
Institute for Computer Aided Design, Russian Academy of Sciences,
19/18 2-nd Brestskaya str. Moscow, 123056 Russia

Abstract

An approach for the symbolic-numeric stability analysis of the satellites 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 both the numerical and symbolic methods. The symbolic investigation was made by means of Resultant, Grobner Basis and Factorization methods [1] with the help of computer algebra system Maple [2].

On the base of this methods the problem of defining the equilibrium positions of a satellite in a circular orbit under the influense of gravitational, aerodynamical, gyrostatic and static torques was solved [3-6].

The stability of equilibrium positions are analyzed numerically with Lyapunov's second method. The Jacobi Integral as Lyapunov's function is used.

1.

Buchberger B., Theoretical Basis for the Reduction of Polynomials to Canonical Forms. SIGSAM Bulletin (1976), pp. 19-29.

2.

Char B. W., Geddes K. O., Gonnet G. H., Monagan M. B., Watt S. M., Maple Reference Manual. Watcom Publications Limited, Waterloo, Canada, 1988.

3.

Sarychev V. A. and Gutnik S. A., Equilibrium positions of a satellite-gyrostat,Proc. of XXXV-th International Astronautical Congress, IAF, Lausanne, 1984, p. 356.

4.

Gutnik S. A., Application of Computer Algebra to Investigation of the Relative Equilibria of a Satellite, Proceedings of International Symposium on Symbolic and Algebraic Computation Kiev (1993), ISSAC'93, pp. 63-64. ACM Press.

5.

Sarychev V. A. and Gutnik S. A., Equilibria of a Satellite Under the Influence of Gravitational and Static Torques. COSMIC RESEARCH (KOSMICHESKIE ISSLIDOVANIYA), Vol. 32, Nos. 4-5, pp. 386-391, (1995), Plenum Publishing Corporation.

6.

Gutnik S. A., Symbolic-numeric methods for solving satellite equilibrium equations, Proceedings of International Workshop on Symbolic-Numeric Algebra for Polynomials SNAP 96 INRIA Sophia-Antipolis, France (1996), p. 20.