Dynamics of a System of Two Connected Bodies Moving along a Circular Orbit around the Earth

Symbolic–numeric methods are used to investigate the dynamics of a system of two bodies connected by a spherical hinge. The system is assumed to move along a circular orbit under the action of gravitational torque. The equilibrium orientations of the two-body system are determined by the real roots of a system of 12 algebraic equations of the stationary motions. Attention is paid to the study of the conditions of existence of the equilibrium orientations of the system of two bodies refers to special cases when one of the principal axes of inertia of each of the two bodies coincides with either the normal of the orbital plane, the radius vector or the tangent to the orbit. Nine distinct solutions are found within an approach which uses the computer algebra method based on the algorithm for the construction of a Gröbner basis.


Introduction
In this paper, we apply symbolic-numeric methods to investigate the dynamics of a system of two bodies (satellite and stabilizer) connected by a spherical hinge. The system moves in a central Newtonian force field along a circular orbit. The problem is of practical interest for designing composite gravitational orientation systems of satellites that can stay on the orbit for a long time without energy consumption. The dynamics of various composite schemes for satellite-stabilizer gravitational orientation systems was discussed in detail in [1].
We analyze the spatial equilibria (equilibrium orientations) of the satellite-stabilizer system in the orbital coordinate system for certain values of the principal central moments of inertia of the two bodies refers to special cases when one of the principal axes of inertia of each of the two bodies coincides with either the normal of the orbital plane, the radius vector or the tangent to the orbit. Equilibrium orientations of the satellite-stabilizer system are determined by real roots of a system of algebraic equations. To find equilibrium solutions, algorithms for Gröbner basis construction were used [2]. Some classes of equilibrium solutions are obtained explicitly from algebraic equations included in the Gröbner basis. The parameters values that cause the change in the number of equilibrium orientations for the satellite-stabilizer system are found numerically.
The combination of the computer algebra and linear algebra methods for the investigation of equilibrium orientations of the system of two bodies connected by a spherical hinge on a circular e-mail: s.gutnik@inno.mgimo.ru e-mail: vas31@rambler.ru orbit under certain constraints imposed on the parameters of the bodies were successfully used in [3]. It has been shown that, on a circular orbit, the two-body system can have both planar and spatial configurations of the equilibrium orientation.

Equations of motion
Consider a system of two bodies connected by a spherical hinge that moves along a circular orbit [3]. To write the equations of motion of the two bodies, we introduce the following right-handed Cartesian coordinate systems: the absolute coordinate system CX a Y a Z a with the origin at the Earth's center of mass C; the plane CX a Y a coincides with the equatorial plane and the CZ a axis coincides with the Earth axis of rotation, and the orbital coordinate system OXYZ. The OZ axis is directed along the radius vector that connects the Earth center C with the center of mass of the two-body system O, the OX axis is directed along the linear velocity vector of the center of mass O. Then, the OY axis is directed along the normal to the orbital plane. The coordinate system for the ith body (i = 1, 2) is Ox i y i z i , where Ox i , Oy i , and Oz i are the principal central axes of inertia for the ith body. The orientation of the coordinate system Ox i y i z i with respect to the orbital coordinate system is determined using the pitch (α i ), yaw (β i ), and roll (γ i ) angles. The direction cosines a i j of the first body and b i j of the second body in the transformation matrix between the orbital coordinate system OXYZ and Ox i y i z i are expressed in terms of the aircraft angles [1]. Suppose that (a i , b i , c i ) are the coordinates of the spherical hinge P in the body coordinate system ; M i is the mass of the ith body; p i , q i , and r i are the projections of the absolute angular velocity of the ith body onto the axes Ox i , Oy i and Oz i ; and ω 0 is the angular velocity for the center of mass of the two-body system moving along a circular orbit. The kinetic energy of the system writes The force function which determines the effect of the Earth gravitational field on the system of two bodies connected by a hinge [1] is given by The equations of motion for this system can be written as Lagrange equations of the second kind by symbolic differentiation in the Maple system [4] in the case when b and kinematic Euler equations r 1 = (α 1 + 1)a 23 +β 1 cos γ 1 , r 2 = (α 2 + 1)b 23 +β 2 cos γ 2 .

Investigation of equilibria
Assuming the initial condition (α i , β i , γ i ) = (α i0 = const, β i0 = const, γ i0 = const), also A i B i C i , we obtain from (3) and (4)  which allow us to determine the equilibrium orientations of the system of two bodies connected by a spherical hinge in the orbital coordinate system. In (5): Taking into account the orthogonality conditions for the direction cosines, a 21 a 31 + a 22 a 32 + a 23 a 33 = 0, b 21 a 31 + b 22 b 32 + b 23 b 33 = 0, the equations (5) and (6) form a closed algebraic system of equations in the 12 unknown direction cosines that determine the equilibrium orientations of the two body system. For this system the following problem is formulated: for given m 1 , m 2 , n 1 , and n 2 , determine all twelve direction cosines. The other six direction cosines (a 1i and b 1i ) can be obtained from the orthogonality conditions. In [5] planar oscillations of the two-body system were analyzed, all equilibrium orientations were determined, and sufficient conditions for the stability of the equilibrium orientations were obtained using the energy integral as a Lyapunov function. In [3] for this case the system of 12 algebraic equations (5) and (6) was decomposed using linear algebra methods and algorithms for the Gröbner basis construction. Some classes of spatial equilibrium solutions were obtained from the algebraic equations included in the Gröbner basis. Construction of the Gröbner basis for the system (5) and (6) of 12 second-order algebraic equations, the coefficients of which depend on 4 parameters, is a very complicated algorithmic problem. In the general case, the system of algebraic equations (5) and (6) cannot be solved by direct application of the Gröbner basis construction methods.
The system (5) and (6) was solved in the special cases when one of the principal axes of inertia of each of the two bodies coincides with either the normal of the orbital plane, the radius vector or the tangent to the orbit.
Eqs. (7) define equilibrium solutions for the system of two bodies in the orbital plane. In [5] planar oscillations of the two-body system were analyzed and all equilibrium orientations were determined. Case 2: a 2 23 = 1, b 2 23 = 1 (axis Oz 1 of the satellite and axis Oz 2 of the stabilizer coincides with the normal OY to the orbital plane).