Analytical Optimal Solution to the Problem of Energy-Minimal Programmed Control of the Spatial Motion of a Spacecraft in the Class of Helical Generalized Conical Motions

The problem of optimal programmed control of the spatial motion (including maneuvering) of a spacecraft (SC) modeled as a free rigid body of arbitrary dynamic configuration with a quadratic functional of the energy spent on the spacecraft maneuver and a fixed transition time is investigated. In the class of helical generalized conical motions, an optimal analytical solution of the problem is obtained under arbitrary boundary conditions of angular and linear positions and angular and linear velocities of the spacecraft; the solution is presented in algorithmic form. Four–dimensional dual Euler (Rodrigues–Hamilton) parameters, which are components of the biquaternion of the finite Clifford displacement, are used to describe spatial motion. The solution is obtained using the Chasles theorem on the displacement of a free rigid body and the Kotelnikov–Study displacement principle. Numerical examples showing the effectiveness of the proposed solution to the problem are given.