Analytical algorithm of energy and time quasioptimal turn of a spacecraft under arbitrary boundary conditions
The optimal attitude maneuver control problem without control constraints is studied in the quaternion statement for an axially symmetric spacecraft as a rigid body under arbitrary boundary conditions on angular position and angular velocity of a spacecraft. The performance criterion is given by a functional combining the time and energy used for the attitude maneuver. 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. In the class of generalized conical motions, the traditional optimal slew problem is modified to obtain analytical solutions for motion equations. The solutions contain arbitrary constants and two arbitrary scalar functions (generalized conical motion parameters). The proposed approach fits well with the Poinsot concept that any arbitrary angular motion of a rigid body around a fixed point can be considered as some generalized conical motion of a rigid body. Moreover, for the cases of analytic solvability of the classical problem of optimal reversal spherically symmetric spacecraft, when restrictions are imposed on the edge conditions of the problem (plane Euler turn, conical motion) solutions of the classical and modified tasks are completely the same. An optimization problem is formulated and solved with respect to these functions, the second derivatives of which serve as controls in the optimization problem. The resulting analytical solution of the modified problem can be treated as an approximate (quasioptimal) solution of the traditional optimal slew problem under arbitrary boundary conditions. The quasioptimal algorithm of the optimal turn of a spacecraft is given. Numerical example showing the closeness of the solutions of the traditional and modified optimal slew problems for an axially symmetric spacecraft is given.