On the nonlinear problem of the three-axis reorientation of a three-rotor gyrostat in the game noise model

Abstract

Abstract-The nonlinear game problem of the three axis reorientation of an asymmetric solid body with three flywheels (rotors) has been solved. Acceptable levels of uncontrollable noise depending on given con straints of control moments have been estimated.

Reorientation of three-rotor gyrostat under uncontrolled external disturbances

“…For the sake of convenience let us represent the expressions (5) for moments ui as a sum of two items : ui = ui (1) + ui (2) , where…”

“…Taking into account the equalities (13), the estimate (15) as well as the values (12) for τi, taking into account the phase portrait of the system (6), (11), using Cauchy-Schwarz inequality we get the following inequalities In case αi * − βi * = ε > 0 (where ε is a sufficiently small number), values of |ui (2) | will be arbitrarily small. As a result, based upon conditions (3)…”

“…The controlling moments, applied to flywheels, are offered to be generated by means of a feedback in form of nonlinear functions of phase variables of a considered conflict-controlled system of differential equations, which consists of dynamic Euler equations and kinematic equations based on the quaternion parametrization of attitude kinematics. It follows from [1,2] specific choice of the feedback structure for controlling moments is considered, which enables one to narrow down the nonlinear problem of reorientation to solving a linear game problems. The results found are the further development of the results [2]: 1) the algorithm for solving of the reorientation problem is extended to the case of an arbitrary initial and final angular positions of the solid; 2) the estimates of the accepted level of external disturbances are improved.…”

“…It follows from [1,2] specific choice of the feedback structure for controlling moments is considered, which enables one to narrow down the nonlinear problem of reorientation to solving a linear game problems. The results found are the further development of the results [2]: 1) the algorithm for solving of the reorientation problem is extended to the case of an arbitrary initial and final angular positions of the solid; 2) the estimates of the accepted level of external disturbances are improved.…”

“…Taking into consideration the structure of controlling moments (5), we come to a conclusion, that for their estimates it is enough to have the estimates of Aixi + Jiφi′. This estimates have the form [2] ( )…”