Two-Craft Coulomb-Force Formation Dynamics and Stability Analysis with Debye Length Characteristics

Abstract

This paper investigates the linear dynamics and stability of an Earth-orbiting two-craft formation flying. The electrostatic Coulomb forces are used as a means of inter-spacecraft forces to bind two-craft formation. The Clohessy-Wiltshire equations of motion are used for the analysis. This study extends the work by Natarajan and Schaub (2006) by including the Debye length characteristics of the Coulomb forces, which diminish exponentially with the ratio of the relative distance over the Debye length. The analysis incorporates the Debye length effects in the equations of motion as well as in the feedback control law. HIS paper investigates the linear dynamics and stability of an Earth-orbiting two-craft formation flying. The electrostatic Coulomb forces are used as a means of inter-spacecraft forces to bind two satellites in a near-rigid formation. Although the Coulomb forces cannot directly stabilize the attitude of the near-rigid formation, the gravity-gradient torque is exploited to stabilize the Coulomb formation about an orbit nadir direction. The HillClohessy-Wiltshire equations of motion, describing the linearized motion of satellites relative to a circularly orbiting reference point, are used for the analysis. Natarajan and Schaub [1] showed the existence of a static Coulomb two-craft formation in the nadir direction, and investigated the linear dynamics and stability around this equilibrium formation successfully, but by disregarding the Debye length effect. The inter-spacecraft Coulomb forces diminish exponentially with the ratio of the relative distance over the Debye length. Natarajan and Schaub assumed that the two spacecraft move well inside the circular area with a Debye length radius, thereby its effect can be ignored. This paper extends their analysis by including the Debye length characteristics of the Coulomb forces, and the linear analysis and stability of the two-craft formation is investigated. The separation distance and the pitch angle equations of motion are linearized about small variations in separation distance, pitch angle, and in the product charge terms, taking the Debye length terms into account. Following the method by Natarajan and Schaub, but adding the Debye length terms, a feedback law to control the separation distance using the Coulomb forces is established by treating the small charge product variation as a control variable. The inclusion of the Debye length characteristics increases the possible Coulomb force formation regions near to the vicinity of the Debye length radius from the reference point, which was restricted to a very small region within the Debye length radius in the past paper. Furthermore, the possible range for initial deviation of the two-craft position from the reference geometry is identified, in which the formation can be stabilized based on the linear dynamics and control strategy. The inaccessible regions with large initial deviation are also tackled by a step-by-step approach, in which the target interspacecraft length is set step by step controlled to achieve the final target formation. The analysis extends the possibility of two-craft Coulomb formation by incorporating the Debye length effects in the equations of motion as well as in the feedback control law.