Effect of Kinematic Rotation-Translation Coupling on Relative Spacecraft Translational Dynamics

A CCURATEmodeling of the differential translation and rotation between two spacecraft is essential for cooperative distributed space systems, spacecraft formation flying (SFF), rendezvous, and docking. High-fidelity relative motion modeling, as opposed to absolute motion modeling, is particularly important for autonomous missions 1. Point-mass models for relative spacecraft translational motion have been extensively studied over the past 50 years, since Clohessy and Wiltshire (CW) presented a rendezvous model for a circular reference orbit and a spherical Earth 2. Following the work of Clohessy and Wiltshire, variants on the point-mass model were developed, such as generalizations to elliptic reference orbits 3–5 and an oblate Earth 6,7. The growing interest in SFF motivated the research of relative spacecraft motion modeling, yielding more accurate and complete equations and solutions for perturbed relative motion 8–10. However, most of the works focused on point-mass, 3 degrees-offreedom (DOF) spacecraft. Obviously, performing a space mission that consists of several cooperative space vehicles requires modeling the relative rotational motion in addition to the relative translation, that is, 6-DOF models. Models for the relative motion of 6-DOF spacecraft have gained attention in the literature only in recent years. Among the first to suggest treating the spacecraft relative angular velocity in an SFF control problem were Pan and Kapila 11, who addressed the coupled translational and rotational dynamics of two spacecraft. By defining two body-fixed reference frames, one attached to the leader and the other attached to the follower, it was proposed 11 to use a two-part relative motion model: one that accounts for the relative translational dynamics of the body-fixed coordinate frame origins, and another that captures the relative attitude dynamics of the two body-fixed frames. A similar modeling approach was used for relative motion estimation 1. In addition, tensorial equations of motion for a formation consisting ofN spacecraft, each modeled as a rigid body, were derived 12. However, only the absolute equations of motion were developed 12; a relative version of these equations was not given. Moreover, a clear mathematical relationship between the developed models and the traditional nonlinear point-mass relative motion and CW models was not provided. The coupling between the translational and rotational motion in the aforementioned models 1,11 was induced by gravity torques. The kinematic coupling, which is essentially a projection of the rotational motion about the center of mass (c.m.) onto the relative translational configuration space, was neglected. It is this kinematic coupling that the current paper is concerned with. In general, rigid-body dynamics can be represented as translation of the c.m. and rotation about the c.m. 13. Thus, spacecraft relative motion must be composed by combining the relative translational and rotational dynamics of arbitrary points on the spacecraft. Whenever one of these points does not coincide with the spacecraft’s c.m., a kinematic coupling between the rotational and translational dynamics of these points is obtained. The purpose of this paper is to quantify the kinematic coupling effect and to show that this effect is key for high-precision modeling of tight SFF, rendezvous, and docking. This effect is also important in vision-based relative attitude and position control, where arbitrary feature points on a target vehicle are to be tracked. Given two rigidbody spacecraft, the model presented herein is formulated in a general manner that describes the motion between any two arbitrary points on the spacecraft. The relative translational motion is then generated by both the spacecraft orbitalmotion and the rotation about the c.m. In addition, this paper provides a CW-like approximation of the relative motion that includes the kinematic coupling. This new approximation is aimed at alleviating an apparent contradiction in linearized relative motion theories: to obtain linear equations of motion, the spacecraft are assumed to operate in close proximity. However, if the spacecraft are close to each other, then they can no longer be treated as point masses, because the spacecraft shape and size affects the relative translation between off-c.m. points. This effect is accentuated as the distances between spacecraft decrease. The remainder of this paper is organized as follows. First, a background on the relative position and attitude dynamics is given. Then, a new coupled relative spacecraft motion model is presented. The newly developed model is then examined in a simulation.