Attitude of a satellite spinning with an angular velocity varying in time—an exact solution

Extract

The problem of determining the attitude of a satellite, spinning in space, with its angular velocity being an arbitrarily specified time function, although of significant interest, has long remained analytically intractable in view of the complexity of the non linear coupling in the system of equations governing the Eulerian angles that prescribe the attitude of the satellite. Even the relatively simpler case of a satellite spinning with constant angular velocity poses significant analytical complexity, but has been solved recently. Although, numerical solutions to specific cases are not hard to obtain, the closed form solution, which brings out the inherent properties, is still of interest to the analyst. Accordingly, this study is aimed at an analytical method of solution to the governing equations. It is shown that the three, coupled, non linear, non-autonomous, first order, differential equations can be reduced to a single, equivalent, linear, uncoupled, autonomous equation of second order, through a differential transformation technique. The solution of this equivalent linear equation, is easily obtained and can be mapped onto the original space to derive the Eulerian angles. This backward transformation, using the transformation relations that established the equivalence of the non linear and linear systems of equations, is shown to be feasible as long as certain integrability conditions are satisfied by the input angular velocity time function. The earlier study is easily observed to be a particular case of this generalised analysis.