Concerning a Special Problem of the Optimal Control of Spacecraft's Terminal Orientation

Abstract

The topic of the article is a problem of optimal control of a spacecraft reorientation from an arbitrary initial attitude to a prescribed angular attitude. The case is studied, when the minimized functional combines the time and the integral value of the modulus of the angular momentum for the spacecraft reorientation. Using the necessary conditions of optimality in the form of Pontryagin's maximum principle and the quaternion method for the spacecraft motion control, an analytical solution to this problem was obtained. Solution to the optimal control problem is based on the quaternion equation connecting the angular momentum vector and the orientation quaternion of the body coordinate system. Formal equations were derived and expressions for construction of the optimal control program were given. The dependence of the control variables from the phase coordinates was found (in an explicit form). Using the transversality condition as the necessary condition of optimality, optimum value of key parameter of optimum functions is determined. For a dynamically symmetric solid body, the problem of the spatial reorientation was completely solved - the optimal law of variation of the spacecraft angular momentum (as an explicit function of time) was obtained in an analytical form. The results of the mathematical simulation of the spacecraft motion under the optimal control are presented and demonstrate practical usefulness of the proposed algorithm for the spacecraft attitude control.