Resonant Capture and Separatrix Crossing in Dual-Spin Spacecraft

Abstract

We consider the rotational motion of a spacecraft composed of two bodies which are free to rotate relative to one another about a common shaftS. A motor on one of the bodies provides a small constant internal torque which influences the relative motion of the two bodies, and which may influence the orientation of their common shaft S. Resonant capture refers to the phenomenon that the spacecraft may end up in one of several possible orientations, including a nearly flat spin (transverse to S), in addition to the expected simple rotation aboutS. The method of averaging is used to treat the original equations of motion, and it is shown that the essential mathematical problem involves separatrix crossingin a problem with slowly moving separatrices. Energy changes represented by Melnikov integrals are used to supplement the averaged equations in the neighborhood of the heteroclinic motions. The method is used to predict which initial conditions lead to capture into each of three distinct capture regions. The asymptotic results are compared to those obtained by direct numerical integration of the equations of motion. A dual-spin spacecraft consists of two bodies, called the platform and the rotor, attached to each other by a shaft S about which they can rotate relative to one another, see Figure 1. In addition, the entire assemblage can rotate freely in space. A motor acting along the shaft S may be utilized to apply an equal and opposite torque to both bodies. The motor may be used to control the orientation of the spacecraft in space. In this work we will be concerned with accomplishing a rotational pointing maneuver with a small constant torque supplied by the motor. We model the platform and rotor as rigid bodies. The rotor is assumed to be axisymmetric and balanced, with its symmetry axis coinciding with the shaft S. The platform is assumed to be asymmetric and balanced. Here balanced means that the shaft axisS is a principal axis for the moment of inertia tensor, i.e., no product of inertia terms are present. Also, axisymmetric means that all principal moments of inertia in directions orthogonal to S are equal. We assume that the initial state of the system and the physical parameters are chosen such that the applied torque is able to drive the system towards what turns out to be one of three different classes of motions, a process called resonant capture. Each of these classes of motions will be shown to