Analytical solution of the optimal slew problem for an axisymmetric spacecraft in the class of conical motions

The traditional problem is discussed of an optimal spacecraft slew in terms of minimum energy costs. The spacecraft is considered as a rigid body with one symmetry axis under arbitrary boundary conditions for the angular position and angular velocity of the spacecraft in the quaternion formulation. Using substitutions of variables, the original problem is simplified (in terms of dynamic Euler equations) to the optimal slew problem for a rigid body with a spherical mass distribution. The simplified problem contains one additional scalar differential equation. A new analytical solution is presented for this problem in the class of conical motions, leading to constraints on the initial and final values of the angular velocity vector. In addition, the optimal slew problem is modified in the class of conical motions to derive an analytical solution under arbitrary boundary conditions for the angular position and angular velocity of the spacecraft. A numerical example is given for the conical motion of the spacecraft, as well as examples showing the closeness of the solutions of the traditional and modified optimal slew problems for an axisymmetric spacecraft.