Reorientation of Three-Rotor Gyrostat under Uncontrolled External Disturbances

A nonlinear game problem of the three-axis reorientation of an asymmetric solid using three reaction wheels is considered for case of unconrolled external disturbances. We proposed an approach of reducing the above-mentioned nonlinear problem to a linear game theory problem. Mathematics Subject Classification: 49N90, 49N30, 93C95


Introduction
The article studies the problem of three-axis reorientation of an asymmetric solid.Three reaction wheels are employed to produce necessary torque in the axes of the spacecraft.External uncontrolled disturbances, that have no statistical description, are taken into consideration in the process of reorientation.
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.

Statement of the problem of reorientation
Let us consider a mechanical system: an asymmetric solid and three homogeneous symmetric flywheels whose axes are aligned with the principal central axes of inertia of the solid.The angular motion of the system (gyrostat) under study with respect to the center of mass is described by the equations [1] Here xi are the projections of the instantaneous angular velocity vector of the solid onto the principal central axes of inertia ki of the gyrostat; Аi are the moments of inertia of the system with respect to the ki-axes; Ji, φi are the axial moments of inertia and the relative angular velocities of the flywheels.The controlling moments ui applied to the flywheels.The moments vi describe the moments of external forces and uncontrolled external disturbances.
Here and below i = 1, 2, 3; summation over the index i is from 1 to 3. Let us designate as x, u, v, ϕ′ the vectors consisting of xi, ui, vi and φi′, respectively.
Along with (1), let us consider the kinematic equations, which based on the quaternion parametrization of attitude kinematics We denote as η the vector, involving ηi and η4, respectively.
Let us choose the feedback controlling moments) in the class functions ui = ui(x, η, ϕ′) with measurable realizations ui[t], which obey the constrains Uncontrollable disturbances vi can be realized in the form of any measurable functions vi = vi[t] within the constraints Problem of reorientation.It is required to find for any admissible realizations of vi the controlling moments ui that transfer the solid from the initial state η(t0) = η0 to the given state η(t1) = η1 during a finite time.Both that states are states of rest: x (t0) = x (t1) = 0. Also condition ϕ′(t0) = 0 takes place.The time instant t1 > t0 is not fixed.Without any loss of generality, we assume that η(t1) = (0, 0, 0, 1).

Reorientation of three-rotor gyrostat 8705 3 An auxiliary linear conflict-control system
Let us consider the nonlinear controlling moments (only the expression for u1 is presented, expressions for u2 and u3 are obtained from u1 by the cyclic permutation where ui * are some auxiliary controls. An auxiliary linear conflict-controlled system can be isolated from closed nonlinear system (1), ( 2), (5): * * .
"Auxiliary disturbances" vi * have the form Using the Cauchy-Schwarz inequality and conditions (4), we deduce estimates Let us now solve the problem of the fastest transfer of system (6) to the state for any admissible vi * which satisfy inequalities (7).We interpret this problem as a differential game in which one of the players has at its disposal the auxiliary control ui * and aims at decreasing time τ for taking the system (6) to the required position (8).The second player (opponent), whose objective is to increase τ (or to avoid attaining position (8) at all) has "auxiliary disturbances" vi * at his disposal.
For this problem to be solvable, the admissible levels ui * should be higher than the levels vi * .Let us represent corresponding constraints in the form The outlined game problem for system (6) is reduced [3] to the time-optimal problem for system * (1 ) .
Boundary conditions are the same as for system (6).
determines the minimum guaranteed time τ for attaining position ηi = ηi′ = 0 in the auxiliary linear game-theoretic problem.Let us mention that those subsystems of the system being described that reach the necessary position earlier, than the last of them, will remain in this position.The relevant control ui * in those subsystems will be counteracted by "auxiliary disturbances" vi * .

The algorithm for solving the problem of reorientation
Solving the system related to ηi′ in (2) as a set of algebraic equations for xi, we arrive at the equalities By the basis of (13) we conclude that transferring the system (6) to the position (8) in fact amounts to solving problem of reorientation during the finite time (12) by means of controlling moments (5).
The iteration algorithm of solving includes the following stages.
Let η(t1) ≠ (0, 0, 0, 1).In order to avoid a "singularity", in this case it is sufficient to pass to the control moments obtained from (5) by permutation of indices (or to the combination of such control moments), i.e, along with (5) to apply the constructions of control moments of the form The constructions ( 14) allow one at a certain choice of the functions fi (s) , to select linear auxiliary systems of type (6) from closed system of differential equations ( 1), ( 2), ( 14).In this case, the set of indices i in a system of type (6) will depend upon the index s of variable η in the denominator of expression ( 14).Thus, index s = 4 corresponds to i = 1, 2, 3, index s = 1 corresponds to i = 2, 3, 4, etc.
4 stage.Verifying the constraints (3) imposed on controlling moments ui.To this end we employ equalities (13).As result, this verification is possible along trajectories of linear system (6), (11) and also linear non-homogeneous system of differential equations for φi′.If estimates (3) fail to hold or, vice versa, they are satisfied with a kind of "reserve", the search for the appropriate numbers αi * is to be continued.Otherwise, the reorientation time equals τ.
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] ( )

Estimation of the admissible domain of disturbances
Let us find sufficient conditions for αi, βi, which determine the possibility of solving the problem of reorientation on the basis of the controlling moments (5).
T h e o r e m .Let the admissible domain of the disturbances vi be estimated by the inequalities ( Then the problem of reorientation is solved by controlling moments (5), (11) which comply with given constraints (3).
Presented estimates has a feature of sufficiency, because employed inequalities were overstated.This estimates guarantees the solution of the problem of reorientation, when the value of τ is sufficiently large (although finite).If conditions (16) hold with a "reserve", then the presented iteration algorithm can be used in order to determine the guaranteed time τ.

Proof of the theorem
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 Based upon the analysis of the phase portrait, we come to a conclusion, that for the possible states of the system (6), (11) the following inequalities take place 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) we have ( ) ( ) ( ) Then, using the inequalities βi * < αi * and Cauchy-Schwarz inequality we get the