Algorithm for the Optimal Turn of a Spacecraft in the Sense of the Minimal Energy Loss under Arbitrary Boundary Conditions

Abstract

The problem of the optimal turn in the sense of minimal energy loss of a spacecraft as a rigid body with an arbitrary distribution of mass without constraint of a control action and under arbitrary boundary conditions is considered in the quaternion statement. In the class of the generalized conical motions a modification was made of the task of optimal rotation, which allowed to obtain analytical solution to the movement equations containing any constants and two any scalar functions (parameters of the generalized conic movement). Concerning these functions and their derivatives the optimizing problem with a square functionality, in which the second derivative of these two functions acts as control, was formulated and solved using Pontryagin maximum principle. Explicit expressions for the optimal angular velocity and the optimal control vectors of a spacecraft are presented. The motion trajectory of a spacecraft is generalized as a conical precession. Algorithm for the optimal turn of a spacecraft is given. The found analytical solution to the modified problem can be considered as an approximate solution to the classical problem of an optimal turn of a spacecraft under arbitrary boundary conditions. Numerical examples are presented showing that the solution of the modified problem well approximates the solution of the classical problem of a spacecraft optimal turn. These examples contain reorientations of the International Space Station and of the Space Shuttle.