Attitude dynamics of a rotating chain or rigid bodies in a gravitational field

This paper documents in scalar detail a minimum dimension set of discrete coordinate equations of motion of a spacecraft with a chain of hinge-connected rigid bodies in a gravitational field. The equations are nonlinear in attitude angles in orbital plane and linear in attitude rates. The derivation procedure is based on Newtonian mechanics and employs the "direct path" technique. Unknown constraint forces at hinges are eliminated analytically. Translational motion of all the bodies is expressed in terms of attitude motion by using a kinematical identity. Symmetry of an associated mass matrix and antisymmetry of gyroscopic matrix are proved. Also, symmetry of a stiffness matrix corresponding to linear range of attitude angles is verified. The motion equations are numerically illustrated for a five-body spacecraft. I. . Introduction T HE objective of this paper is to document in scalar detail a minimum dimension set of discrete coordinate equations of motion of a spacecraft with a chain of hingeconnected rigid bodies in a gravitational field. It is believed that these equations are not available elsewhere and that they can be used in a number of instances; for example, 1) for an initial estimate of control requirements in terms of power and energy for a complex spacecraft such as Space Station, and 2) to design optimal, large-angle maneuvers of a multibody spacecraft in the presence of gravitational torques. Indeed, precisely these research topics motivated the present study. The spacecraft studied travels round a planet in a circular orbit and rotates once per orbit about its mass center. In the early part of the paper, three-dimensional attitude dynamics of the spacecraft is considered; subsequently, it is specialized to attitude dynamics in the orbital plane only. Section II establishes notational conventions,and describes idealizations which form the basis of the mathematical model developed herein. In Sec. Ill, gravitational forces and torques acting on the hinged bodies of the spacecraft are summarized. Since attitude dynamics of the spacecraft is influenced by constraint forces acting at the hinges, these forces are explicitly determined in Sec. IV by considering trarislational motion of the spacecraft. Rotational equations are derived in Sec: V. Dynamic analysis in Sees. IV and V is based on Newtonian approach. The equations of motion are nonlinear in attitude angles and linear in attitude rates. Symmetry of the associated mass matrix and antisymmetry of gyroscopic matrix are proved in Sec. V. In addition, the stiffness matrix corresponding to a linear range of attitude angles is shown to be symmetric. In Sec. VI planar attitude dynamics of a five-body spacecraft is illustrated. In order to reveal in explicit detail the structure of the scalar equations of motion derived in generic matrix terms in Sec. V, these matrix equations are expanded for an illustrative two-body spacecraft in Sec. VII. The paper is concluded in Sec. VIII by some closing remarks about the present work. Development of the motion equations in this paper is based on the "direct path" method of Ho1 and Hughes.2 Out symbology is mostly from Ref. 2. Since Keat,3 and Singh and Likins4 have provided comprehensive literature surveys of the multibody spacecraft dynamics field, contributions of the past