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Mathematical Modeling of Complex Information Processing Systems 119
OPTIMAL CONTROL IN THE PROBLEM OF UNPACKING A SPACE TELESCOPE
MIRROR
V. V. Alexandrov 1 , D. I. Bugrov 1 , B. A. Khrenov 2 , S. S. Migunov 1 , and L. Gomez Esparza 3
The problem of optimal control in the processes of unpacking a space telescope­detector is considered.
The telescope consists of a number of separate hexagonal segments and is designed to observe the
Earth's atmosphere. At each stage of unpacking, the telescope is regarded as a system of two
absolutely rigid bodies (one is packed and the other is unpacked) linked together by a cylindrical
hinge. The controlling moment in the hinge is caused by an electric motor. It is required to determine
the control (in other words, a level of voltage impressed on the motor) that minimizes the energy
spent for unpacking the telescope.
Let us consider a telescope designed to observe processes in the Earth's atmosphere. Its mirror is directed
toward the Earth [1, 2]. It is assumed that a space station, where the telescope is installed, moves along a
circular orbit. The mirror consists of a number of regular hexagonal segments (Figure 1). Since the mirror is
transported to the orbit, the problem of unpacking the mirror arises. The dotted line in Figure 1 illustrates the
sequence of unpacking the segments.
Fig. 1
In the packed state, all the segments are linked together in such a way that the entire configuration can
be considered as an absolutely rigid body. The process of unpacking progresses sequentially relative to the
initial (zero) segment. The segments are fastened as follows: any intermediate segment is linked only with the
two adjacent ones by cylindrical hinges. Thus, we shall assume that the edge of a segment coincides with the
corresponding edge of another one. Figure 2 demonstrates the unpacking of three segments.
1 Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119899, Russian Federation,
e­mail: valex@moids.math.msu.ru; bugrov@moids.math.msu.ru
2 Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, 119899, Russian Federation,
e­mail: khrenov@eas.npi.msu.ru
3 Facultad de Ciencias Fisico Matematicas, Universidad Autonoma de Puebla, Apdo. Postal 1364, Puebla,
M'exico, e­mail: valex@fcfm.buap.mx

120 Mathematical Modeling of Complex Information Processing Systems
Fig. 2
It is assumed that the unpacked part of the mirror is associated rigidly with the station that moves along
the orbit in such a manner that the unpacked part is directed toward the Earth all the time.
Let us introduce a coordinate system EXY Z with its origin E at the center of the Earth. The axes EX
and EZ are positioned on the plane of the orbit, whereas the axis EY is directed along the vector ! 0 of the
absolute angular velocity of the station. This coordinate system is supposed to be inertial. Let us introduce
another coordinate system Ex s y s z s such that it rotates relative to the system EXY Z with the angular velocity
! 0 = ! 0 e fl in such a way that the axis Ex s passes through the center S of mass of the orbital station (Figure 3).
Fig. 3
The coordinate system Ox 2 y 2 z 2 is obtained by the translation of the system Ex s y s z s at the midpoint O
of the hinge axis along which the unpacking of the next segment takes place (note that Ox 2 k Ex s , Oy 2 k Ey s ,
and Oz 2 k Ez s ). The coordinate system Ox 1 y 1 z 1 is obtained by the rotation of Ox 2 y 2 z 2 about the axis Ox 2 by
an angle ff (this angle is constant for each stage of unpacking). The coordinate system Oxyz associated rigidly
with the segments is obtained by the rotation of the system Ox 1 y 1 z 1 about the axis Oz 1 by an angle '. The
point L is the center of mass for the unpacked part of the mirror (Figure 4).
Assuming that the Earth's gravitational field is central, we obtain the following equation describing the
motion in the coordinate system Ox 1 y 1 z 1 :
C ¨
'=G(') +M mot +M fr
G(') =m! 2
0 l
\Gamma
\Gamma3¸ cos(' + fl) + sin(i cos ff + j sin ff) sin(' + fl) + 0:5 l cos 2 ff sin 2(' + fl)
\Delta
' = '(t); '(0) = 0; '(t 1 ) = ú; —
'(0) = 0; —
'(t 1 ) = 0
Here C is the moment of inertia of the unpacked part of the mirror relative to the hinge axis, m is the mass of
the unpacked part, l is the distance between the center of mass of the unpacked part and the hinge axis, M fr is
the moment of friction in the hinge, M mot is the moment caused in the hinge by the electric motor, ¸, j and i
are the coordinates of the point S in the system Ox 1 y 1 z 1 : OS = ¸e x1 + je y1 + ie z1 .
Further we assume that —
' is proportional to the angular velocity of rotation of the motor's rotor with the
coefficient of proportionality j. As a model of the electric motor, we adopt the equation
L —
I + RI + ''j' = k g U; M mot = aI

Mathematical Modeling of Complex Information Processing Systems 121
Fig. 4
Fig. 5
where I is the current intensity in the coil of the motor moving armature, L is the inductance of the coil, R
is the resistance of the armature, '' is the coefficient of the back electromotive force, j is the gear ratio, k g is
the voltage gain, a is the momentum characteristic of the motor, and U is the external voltage considered as a

122 Mathematical Modeling of Complex Information Processing Systems
control.
The optimization of the process of unpacking the telescope mirror is regarded as finding a law for the
formation of the voltage U such that the functional
J 0 = 1
2
t 1
Z
0
\Gamma
k 1 I 2 + k 2 U 2 \Delta
dÜ ! min
U
reaches its minimal value with a fixed terminal time t 1 and in the absence of any restrictions imposed on U .
The physical meaning of this approach consists in the following. First, we minimize the integral of the
mechanical moment produced by the motor:
t 1
Z
0
k 1 I 2 dÜ = k 1
a 2
t 1
Z
0
M 2
mot dÜ
Second, we minimize the energy spent for the unpacking of the next segment:
t 1
Z
0
k 2 U 2 dÜ
Using Pontrjagin's maximum principle, we reduce this optimization problem to a two­point boundary value
problem in the seven­dimensional space.
The MATLAB program was used to solve the latter problem. Some of numerical results we obtained are
presented in Figure 5.
The question of finding maximal errors arising in the course of unpacking under the programmed control
(given as a time function) is of particular interest. We restrict ourselves to the consideration of two possible
sources of errors: the inaccuracy in initial conditions and the ignoring of quantities of higher order in small­
ness that enter into the equations describing the process of unpacking. Assuming the above errors small and
considering the linear terms only, we obtain the linear second­order error equation with variable coefficients:
¨ x + f 0 —
x + f 1 x = f 2
Here x(t) = '(t) \Gamma ' com (t) is the error, ' com (t) is the computed angle of rotation, f 0 = const, jf 1 (t)j 6 f \Lambda
1 =
const, and jf 2 (t)j 6 f \Lambda
2 = const. Next we solve the problem of finding a maximal value of y(t 1 ) = b T x(t 1 ),
where b T = (1; 0; 0) or b T = (\Gamma1; 0; 0) with jx(0)j 6 x 0 , j —
x(0)j 6 —
x 0 .
The above results can be used to construct space telescopes for the registration of high energy particles.
REFERENCES
1. V.V. Alexandrov et al., Vestnik MSU. Astronomy. Physics, 6: 33, 2000.
2. V.V. Alexandrov et al., Proc. of Workshop on Observing of Extremely High Energy Cosmic Rays from Earth
and Space, AIP Conf. Proc., Woodbury, New York, 2001, in press.