Control of spacecraft spatial turn with
minimal value of the way functional M.V.Levskii Research Institute of Space systems, In continuation of the former works the problem of
spacecraft transfer into a position with a preset orientation by optimal way is
solved in present article. The integrated index having sense of the generalized
angle of a turn (so-called functional of a way) is used. The way of solution
and formalization of the description of kinematics of spacecraft rotary motion
are based on a quaternion method. In this paper, the most widespread case is
considered when inertial coordinate system is basic. Many publications are devoted to control methods and a
problem of optimal control for reorientation of a solid body in various
statements and formulations. In particular, questions of spacecraft optimal
turn as fast finishing and also with minimum of power expense are examined early.
The analytical solution has been received by Pontryagin principle of maximum
for a case when the area of allowable values of the control moment is limited
to sphere, and spacecraft itself is unwrapped around of a vector of final turn.
Before optimal control was found by method of the combined synthesis on the
basis of algorithm with prognostic model, and the functional of a generalized
work was minimized. Analytical designing by a criterion of generalized work (АDGW) represents enough universal
remedy of the solution of optimal control problems. However, optimization
according to АDGW
does not allow satisfying constraints imposed for control variables. Besides,
the controls synthesized by ADGW method represent continuous (and even smooth)
functions; they provide smooth approach to a required final status (angular
position and velocity). Nevertheless, for fastest ending or criterion of fuel
economy, optimum will be relay controls addition, ADGW method is applicable
only for relatively small initial angular deviations. Unfitness of ADGW for a
case of arbitrary initial angular deviations is its essential drawback. The use
the predicting models at synthesis of controls raises the quality of
regulation, and because of this they are widely applied now by many authors.
But in such algorithms the final result depends mainly on a form of the
predicting model; last completely predetermines type of a turn realized by
received controls. Acceptance some predicting model close to a reality entails
the big computing complexities. In well-known and recent publications on the
mentioned subject the received solution, unfortunately, is not radically new
essentially the control which is realized as a result of the synthesis, leads
to a turn of the spacecraft around the Euler axis, though the principles of
optimization and as such the algorithms of control are different. Given article is devoted to a finding and research of
optimal trajectories of a turn (in sense "functional of a way") and properties
of spacecraft motion during the change of its orientation from one angular
position into another oriented position. In article, the
problem of optimal control of a three-dimensional turn of a spacecraft for
criterion of a minimum of the "way functional" is considered and solved. The
accepted index of optimality most fully characterizes expenses of shares (labor
input) of transferring a solid body from one position into another specified
position. The finding of an optimal regime of space-craft reorientation with
the minimal value of the way functional is sufficiently urgent task. Synthesis of the
optimal program of control was spent using the Pontryagin's maximum principle.
The description of spacecraft's spatial motion around the center of mass is
made in quaternion variables. It considerably simplifies calculation procedures
and essentially reduces computing expenses of algorithm of control, doing its
more attractive for onboard realization. Main attention is given to a
kinematical problem of reorientation. Necessary conditions of optimality are
found and the structure of optimal control is determined. The problem of
optimal control is reduced to the solution of three problems: acceleration of
the spacecraft, spacecraft rotation with calculated velocity of motion and
cancellation of angular velocity. The formalized relationships for
determination of the program of spacecraft motion are received. At the phases
of the boost and braking, the controlling moment is greatest possible, it
delivers a minimum of duration of acceleration and braking. At a stage between
gaining the required velocity and braking, the controlling moment is calculated
under condition that spacecraft motion around the center of mass occurs along
the programmed (appointed) trajectory of rotation determined by the rated
vector of a turn. For a case when times of acceleration (a gaining the required
angular velocity) and braking (cancellation of angular velocities) are
insignificant in comparison with main segment of the turn (or all turn as a
whole), control under which the module of spacecraft's kinetic moment at the
stages of the boost and braking varies under the linear law is constructed.
Such way of control allows to calculate most precisely time parameters of
reorientation maneuver (the duration of the boost and braking), and main thing
is that already during spacecraft turn independently to determine the instant
when braking begins based on the principles of terminal control using the
actual kinematical parameters of the spacecraft motion (under the information
about angular position and the measurements of angular velocity), than high
accuracy of transferring the spacecraft to a specified angular position is
reached. Use the way
functional allows synthesizing control of spacecraft reorientation by the
uniform methodical tool for principle-radically different criteria of
complexity: from a turn with the minimal size of the kinetic moment to a
control which is optimum for fuel expense. Appointing definitely the
coefficients of the way functional, we receive desirable properties of the
spacecraft motion at transition into required final status. For a symmetric
index of complexity the full solution of a problem of the spacecraft
reorientation in the closed form is submitted. Data of mathematical modeling of
dynamics of spacecraft motion during a turn under the described method of
control which confirm a practical reliability of the suggested way of control
for spacecraft's spatial turns are presented. |
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