A two-dimensional approach to multibody free dynamics in space environment

The equation of motion of a multibody system, described here as a chain of rigid bars and revolute joints orbiting around the Earth, is derived.

For each bar two translational and one rotational equilibrium equations are written. The forces acting on each body are the gravitational forces and the reaction forces (unknown) acting on it's end joints. The complete set of equilibrium equations consists of N_X differential equations, where N_X is the order of the state vector. The total number of unknowns is N_X+N_R where N_R=2N_J and N_J is the number of joints. The N_R additional equations, to make the system determinate, are provided by the nondifferential compatibility equations.

The resulting system is a set of differential algebraic equations (DAE) for which the well-known method of reducing the system to ordinary differential equations (ODE) is applied.

Since the internal forces are associated with the relative displacements between the bodies, which are small fractions of the distance of the multibody spacecraft from the center of the Earth, the task of obtaining these forces from inertial coordinates, from a numerical viewpoint, could be impossible. So the problem is reformulated in such a way that the equation of motion of the system, contains global quantities where no internal forces appear, and local equations where internal forces do appear. In the latter one, only quantities of the same order of the spacecraft dimensions are present. Numerical results complete the work.