Formulation of Equations of Motion for Complex Spacecraft

HE formulation of equations of motion, always an important facet of spacecraft design and performance analysis, is receiving ever greater attention as vehicles become larger and more complex and as the need grows more and more acute to predict spacecraft motions accurately. This article is intended to answer the following question that is clearly central in this context: "What are the relative advantages of various methods available for the formulation of equations of motion of complex spacecraft? " The question just stated can be answered incisively only if the criteria by which a given method is judged are sufficiently well defined, and meaningful criteria can be established only after a clear-cut purpose for the formulation of equations of motion has been identified. We take this purpose to be the production of algorithms to be employed for the simulation of spacecraft motions by means of numerical solutions of initial value problems. The principal criteria by which we judge a method are the simplicity of the equations to which it leads and the amount of labor required for the formulation of the equations. Of course, neither criterion is important when one is dealing with a relatively uncomplicated system. Conversely, these matters can become crucial in connection with truly complex spacecraft, for both computer storage limitations and excessive execution times can then plague one needlessly if equations have not been brought into the simplest possible form, and the labor expended to generate equations which have such a form can easily become prohibitive unless a highly efficient scheme for formulating equations of motion has been employed from the outset. It should be understood, therefore, that we are not concerned with systems so simple that their equations of motion can be produced with essentially equal ease by using almost any method. Since there exist numerous so-called multibpdy programs, as well as analyses intended to be the bases for such programs, and it is quite clear that these are of great value in solving problems of spacecraft dynamics, one may well ask whether or not it is ever really necessary to construct literal equations of motion for complex vehicles. The answer is that the need to do this arises frequently because multibody programs can fail a user in a number of ways. Every such program represents a compromise between two mutually exclusive goals, that of providing a computer code applicable to the solution of the widest possible class of problems, and that of minimizing the effort required by a user to bring his problem into a form compatible with program input requirements. Consequently, a given multibody program can be totally inapplicable to a particular problem, can force a user to make major program additions or other modifications, or can lead to inefficient or inaccurate simulations. Clearly, therefore, algorithms created to meet specific needs are indispensable. Not surprisingly, it is precisely the authors of some of the best multibody programs who, realizing all of this, welcome algorithms developed independently, particularly since these can be used to test the validity of multibody computer codes. Moreover, although it is probably true that the greatest obstacle to be surmounted in producing a multibody program is that of devising a procedure for assembling the equations of motion associated with the various bodies comprising a spacecraft, rather than that of writing equations governing the behavior of a generic body, there can be little doubt that the way in which the latter