A solution of the optimal turn problem of an axially symmetric spacecraft with bounded and pulse control under arbitrary boundary conditions

Abstract

Given a constraint on the modulus of the control action in the quaternion statement, the optimal turn problem of a spacecraft regarded as a rigid body with one symmetry axis is considered under arbitrary boundary conditions with respect to its attitude and angular velocity. The functional combining the time and the integral value of the control vector spent to the spacecraft turn is used as the optimality criterion. Changes of variables allow reducing (in terms of dynamic Euler equations) the initial optimal turn problem of an axially symmetric spacecraft to the optimal turn problem of a rigid body with spherical mass distribution. Two ways of solving the optimal control problem are proposed. In the first case, the Pontryagin maximum principle is used to obtain the expressions of the optimal control and the adjoint system of equations. In the second case, the passage to the limit with the upper value of the control action increasing indefinitely is performed to construct an analytical solution to the pulse optimal turn problem of a spacecraft that implements the double-pulse control scheme. The original procedure of finding the numerical solution to the continuous optimal turn problem of a spacecraft with bounded control is described, and the examples of calculations are given. The numerical approbation of the proposed analytical algorithm of solving the pulse optimal turn problem of a spacecraft is given