Conservative methods of controlling the rotation of a gyrostat

A method for shaping the control of the rotation of a gyrostat consisting of a rigid body, within which there are three rotors rotating about non-coplanar axes rigidly connected to the body, is discussed. The state of the system is defined by the position and angular velocity of rotation of the body, as well as by the angular velocities of the rotors. Control is achieved by torques applied to the rotors. The idea behind the proposed control method is to choose the controlling torques so that the angular velocities of rotation of the rotors are linear functions of the components of the angular velocity vector of the body. The linear dependence thus specified defines a 3 × 3 matrix, that is, a “controlled inertia tensor.” This matrix, which is specified by the parameters of the control selected, does not necessarily have the properties of an inertia tensor. As a result of such a choice of controls, the equations that define the variation of the angular velocity of the body are written in a form similar to Euler's dynamical equations. The system of equations obtained is used to formulate and solve problems of controlling the angular motion of a satellite in a circular orbit. The proposed method for constructing controlling actions enables both the Lagrangian structure of the equations of motion and the fundamental symmetries of the problem to be maintained. Expressions for the torques acting on the rotors and realizing the motion of the required classes are written in explicit form.