INTRODUCTION
Growing needs of human in communication require larger communication satellites
in size. The Early Bird American communications satellite, Intelsat1, launched
on 6th April 1965 has 39 kg mass and 0.59 m height only. Then, Intelsat4 satellites
launched in 19711975 have 1410 kg launch mass and 5.30 m height. Besides of
batteries, the both types of satellite use solar cells mounted on their bodies
to generate power needs. The increasing power need then requires the bigger
solar cell sizes. They cannot be mounted on the limited dimension of the satellite
body anymore. The solar cells then are covered on the deployed panels. They
form solar wings. Intelsat5 satellites launched in 19801989 have solar wings
in 15.90 m span and 2000 kg launch mass. The following generation satellites,
Intelsat7, launched between 19931996 have 3695 kg launch mass, 21.80 m solar
wing span and 4.20 m height. The Malaysian satellite, MEASAT3, launched in
2006 has 3220 kg mass in the beginning of life in orbit with 26.2 m solar wing
span. Canadian Communication Technology Satellite (CTS) Anik F1 launched 21st
November 2000 has 3,015 kg in orbit mass at the beginning of life with two solar
wings in 40.4 m span. Canadian communication technology satellite (CTS) Anik
F2 launched on 14th July 2004 has 3,805 kg in orbit mass at the beginning of
life with two solar wings in 47.9 m span. From the above examples it can be
concluded that the communication satellites become heavier and heavier, while
their solar wing spans become longer and longer. Based on the limitation of
weight in launching, the materials to construct the satellite must be light
in weight. When the light solar wings have long span, they must be considered
as flexible structures.
The communication satellites are designed to have certain attitude accuracies
in operation. For example, Hwangbo (1992) mentioned
that a satellite required by the Koreasat must have an antenna beam pointing
error less than 0.07° in roll and pitch and less than 0.2° in yaw. To
keep the precise orientation, the satellite needs frequent corrections of its
attitude during its operation in space. Attitude of rigid satellites can be
changed without residual vibration problems after the maneuvers. When the satellite
structures are not rigid, maneuvering the attitude without regard to the system
flexibility will result large amplitude steadystate oscillations, especially
when the system is equipped with constantamplitude onoff jets.
To reduce vibration in a flexible satellite system, which is equipped with
onoff reaction jets, the input shaping methods have been developed by Liu
and Wie (1992), Pao and Singhose (1995) and Rogers
and Seering (1996). Parman and Koguchi (1998, 1999a,
b, 2000) demonstrated an application
of shaped commands to change roll angle of a flexible satellite with a large
number of flexible modes. They studied roll and pitch maneuvers of the flexible
satellite. They showed that the residual vibrations can be reduced drastically
when the satellite is subjected to shaped inputs suppressing the vibration at
two frequencies with largest vibration amplitude.
In this study, computer simulations of attitude maneuvers of the CTSlike spacecraft
equipped with onoff thruster are presented. The spacecraft consists of a rigid
main body and two flexible solar wings. The spacecraft model developed by Parman
and Koguchi (1998) is used and torque inputs resulted by thrusters are at
constant amplitude. The solar panel offset angle is set to 30°. For this
condition of the spacecraft, roll motions couple with the yaw angle motions.
The constantamplitude torque inputs based on Proportional Derivative (PD) control
logic are applied to maneuver the spacecraft attitude.
CTSLIKE SPACECRAFT
Finite element model of the spacecraft: The spacecraft studied in this
study is a CTSlike spacecraft consisted of a rigid main body and two symmetrical
large solar panels. Since, the solar panels are large in size but light in weight,
they are supposed as flexible structures. The finite element method is used
to discrete their elastic deformations. For this purpose, a finite element model
of the spacecraft is developed by using the model studied by Parman
and Koguchi (1998) as shown in Fig. 1. In this model,
each solar panel is divided into 16 rectangular plate elements. By using such
a division, each solar panel has 27 nodal points. The elements are numbered
from 1 through 16 on the righthandside and from 17 through 32 on the lefthandside,
while their nodal points are numbered from 1 through 27 on the righthandside
and from 28 through 54 on the lefthandside. The solar panels are oriented
towards the sun and the declination with respect the X_{b}axis of rigid
main bodyfixed frame is identified by the offset angle δ. Only outofplane
deformations of the solar panels are considered.
The spacecraft attitude with respect to its orbital reference frame is expressed
in Bryant’s angles: roll angle φ, pitch angle θ and yaw angle
Ψ; where the definition of these angles can be seen in Fig.
2. The equations of motion of the spacecraft have been developed by Parman
and Koguchi (1998) by using Lagrange’s formulation. For small attitude
angle displacements, they can be written in the following matrix form:
where, m is the mass of the spacecraft and Q is the coupling matrix for the
translational and rotational displacements of the rigid main body. The coupling
matrix for the translational displacements of rigid main body and the displacements
of flexible solar panels is notated in W, while the inertia matrix of the whole
spacecraft is symbolized in I. The coupling for the rotational displacements
of the rigid main body and the displacements of flexible solar panels is expressed
is A. The matrices M, D and K are the mass matrix, the damping matrix and the
stiffness matrix of the flexible solar panels, respectively. If the angular
velocity of spacecraft orbit is ω_{0}, then
is its skew symmetric matrix. Vector r is the translational displacement of
the rigid main body, while Θ is the rotational displacement of the rigid
main body. Vector Θ is an expression vector of Bryant’s angles. Vector
d is stating the displacements of the flexible solar panels. F_{b} and
T_{b} are external forces and torques vectors acting on the rigid main
body, F_{a} is the vector of external forces and torques acting on the
solar panels, while f is the total number of degrees of freedom of the solar
panels.

Fig. 1: 
The Model of communication technology satellite (CTS)like
spacecraft, Element numbers are in rectangles and nodal point numbers are
in circles, X_{j}Y_{j}Z_{j} (j = 1, 2, …, 32)
are local reference frames for solar panel elements 

Fig. 2: 
The rotations from an observation reference frame F_{o}
(X_{o}Y_{o}Z_{o}) to the main bodyfixed reference
frame F_{b} (X_{b}Y_{b}Z_{b}) 
In this study, the flexible structural subsystems are supposed to have no dissipation
properties, so that D = 0.
Resttorest attitude maneuver of CTSLike spacecraft using “Fast”
constantamplitude feedforward torques: The main body of real spacecraft
contains cameras, control devices, electronics, antennas, etc. They are placed
in suitable positions in the main body structure related to their functions.
For control consideration, the locations of these components are also trying
to follow symmetrical conditions with respect to the main body reference axes.
For simulations, the main body of spacecraft is modeled as 6 lumped masses at
certain positions as shown in Table 1. Following this table,
total mass of the main body is 3,100 kg. The parameters of flexible solar panels
can be seen in Table 2. The offset angle of solar panels is
taken to be 30 degrees. For this configuration, the total mass of the spacecraft
becomes 3,446 kg. The origin of the rigid main body fixed reference frame coincides
with the centre of mass of the whole spacecraft in the undeformed state, I_{xx}
= 42,140 kg m^{2}, I_{yy} = 1,431 kg m^{2}, I_{zz}
= 41,498 kg m^{2}, I_{xy} = I_{yz} = 0 and I_{xz}
= 112 kg m^{2}. The orbital frame moves relative to the inertial frame
with constant angular velocity:
where, j_{i} is the unit vector in Y_{i}axis direction,
ω_{o }= 7.29 x 10^{5} rad sec^{1}, so that F_{o
}performs in F_{i} one rotation per sidereal day.
Using the above parameters, the motion of the spacecraft is expressed in 168
generalized coordinates. The first six coordinates represent translations and
rotations of the main body.

Fig. 3: 
Plot of eigenvector’s components in roll, pitch and yaw
angles vs. Natural frequencies 
Table 1: 
Lumped masses consisting of the rigid main body 

Table 2: 
Parameters of the solar panels of spacecraft 

The system has six natural frequencies in zero values and 162 ones in nonzero
values. The zero values are relating to the rigid body modes: three in translation
and three in rotations. The nonzero values of natural frequencies range from
0.042 Hz until 34.6 Hz. The natural frequencies of the system and components
of related eigenvectors for roll, pitch and yaw angles are plotted in Fig.
3. It can be seen in this figure that the large values of eigenvector’s
components are related to low natural frequencies of the system. The high natural
frequencies are relating to the small values of eigenvector’s components.
So, if the system oscillates, the large vibrations happen at low frequencies.
A system is called to be in a rest condition if the velocity of the system
is zero. For a flexible system, the rest condition doesn’t mean the velocity
to be zero, but it means that the average velocity of the oscillating system
is zero. The maneuver is referred to as resttorest if the conditions of the
system before and after the maneuver process to be in rest. For the system having
translational motion, the resttorest maneuver is to move it from one initial
rest position to another rest position. The resttorest maneuver of the rotational
system is to change its attitude angle, from one rest condition to another rest
condition.
A timeoptimal input is an input given to a certain maneuver in the shortest
time duration. For an input consisted of a series of alternatingsign constantamplitude
pulses, the input to slew the system in resttorest maneuver is a bangbang
in maximum amplitude. The bangbang input will be two sequence pulses in the
alternating sign in the same width.
In this study, the observed spacecraft is supposed to have no control and no
damping properties on the flexible solar panels. The control inputs are only
applied to the rigid main body at the center of mass of spacecraft, as constant
amplitude force or torque pulses resulted by onoff reaction jets. For such
a system, under control or external torques only, remembering Eq.
1, the resultant attitude angle acceleration of spacecraft as a rigid body
motion can be written as:
where I_{xx}, I_{yy}, I_{zz}, I_{xy}, I_{xz},
and I_{yz} are components of the inertia matrix I of the whole spacecraft
and T_{bx}, T_{by} and T_{bz} are components of the
torque input vector T_{b} on the rigid main body. By integrating
Eq. 3 with respect to time we get an expression of desired
attitude angle velocity:
and integrating once more gives a desired roll angle displacement:
The spacecraft has 4° roll angle, 7° pitch angle and 4° yaw angle
at initial condition. These attitude angles must be corrected to 0°.
Torqueses used to maneuver the spacecraft being studied here are resulted by
onoff reaction jets, so the amplitude of torques is constant. The shortest
duration time of constantamplitude feedforward command for resttorest slew
maneuver is a bangbang input. Constraint equations that must be satisfied for
this resttorest slew maneuver are {}^{T}
= {0, 0, 0}^{T} and {φ_{d}, θ_{d}, Ψ_{d}}^{T}
= {0.0698, 0.1222, 0.0698}^{T}. If the amplitude of command, either
T_{bx}, T_{by}, or T_{bz} is determined to be 20 N m,
the profile of bangbang torques needed consist of 24.29 seconds long of T_{bx},
5.91 seconds long of T_{by} and 24.10 sec long of T_{bz} bangbangs.
First, the spacecraft is subjected to the roll torque. Then, the pitch torque
is applied as the roll torque finished. Finally, it is subjected the yaw torque
input.
For simulations done in this paper, the secondorder linear differential equations
system, Eq. 1, is solved using the Newmark Method (Smith
and Griffiths, 2005). As results, under these inputs, all attitude angles
can be changed to 0°. After the torques were removed, the roll and yaw angles
still oscillate with an amplitude dominantly at the period of 23.62 sec which
relates to natural frequency resulted in calculation 0.266 rad sec^{1}.
The dominant pitch angle oscillation has 6.06 sec period, relating to natural
frequency 1.037 rad sec^{1}, with the 0.5° total amplitude. The
total amplitude of residual oscillation is 10.4° for roll angle and 6.1°
for yaw angle as shown in Fig. 4. These oscillations are larger
than the desired attitude angle displacements. Also, all residual attitude angles
oscillations are larger than the permitted attitude errors of such a spacecraft
required by Koreasat.
Shaping constantamplitude torque input based on PD control: The spacecraft
studied here is equipped with onoff reaction jets and it cannot produce variable
amplitude actuation force. The spacecraft must be maneuvered with constant amplitude
torques resulted by thruster pulses. For this purpose, the input is shaped by
following PD control.

Fig. 4: 
Time responses under the bangbang torque inputs: attitude
angle displacement of the main body 
If the desired angular displacement and velocity vectors
of the spacecraft main body are Θ_{d} and Θ_{d}, respectively,
then at time t the vectors of angular displacement and velocity errors are:
and
respectively. In standard PD control, the torque input can be written as:
where, K_{Θ} and K_{Θ} are displacement and velocity
gain constant matrices, respectively. Equation 8 produces
variable values of control command in roll, pitch and yaw directions. For application
of PD control on the torque commands resulted by constantamplitude thrusters,
the following logic is defined to shape T_{b} in Eq. 1:
In Eq. 9, e_{Θ0} and e_{Θ0} are
setting points of attitude angle and angular velocity errors, respectively.
T_{switch} is the setting point of T to switch on the thruster. In this
study, it is selected that:
e_{Θ0} 
= 
{10^{4}, 10^{4}, 10^{4}}^{T}
deg 
e_{Θ0} 
= 
{10^{5}, 10^{5}, 10^{5}}^{T} deg sec^{1} 
T_{max} 
= 
{20, 20, 20}^{T} Nm and 
T_{switch} 
= 
{10, 10, 10}^{T} Nm 
For example, at certain time, when:
e_{Θ0} 
= 
{12x10^{4}, 0.2x10^{4}, 20x10^{4}}^{T}
deg 
e_{Θ0} 
= 
{20x10^{5}, 0.2x10^{5}, 45x10^{5}}^{T}
deg sec^{1} 

= 
{8, 0.2, 11}^{T} Nm 
then
T_{b} 
= 
{0, 0, 20}^{T} Nm 
Although, the roll angle and velocity errors are greater than the error setting
points, but since the magnitude of roll torque following PD control is less
than the thruster switching point, then the roll thruster will be switched off.
Attitude maneuvers of spacecraft under PD based feedback constantamplitude
inputs: The values of gain constant matrices used for simulations presented
in this paper are:
and
A thruster has a certain capability in switching: from switch off to switch
on, from switch on to switch off, or from switch on in one direction to switch
on to the opposite direction. The torques resulted by the spacecraft thrusters
consist of roll, pitch and yaw simultaneously. If the minimum time interval
of sequence thruster switching is a second, the time history of the spacecraft
attitude angles can be seen in Fig. 5a. It can be seen in
this case that the spacecraft has very poor attitude accuracies after the maneuvers.
The roll torque input used is shown in Fig. 5b.
For 0.5 sec minimum switching time interval, the responses of the spacecraft
attitude angles can be seen in Fig. 6. The maneuvers are completed
in all directions after t = 100 sec. Now, after the maneuvers, the oscillations
of attitude angles can be reduced better than the bangbang input case. However,
they are still big enough in amplitude. The pitch angle oscillates in 0.4 deg
resultant amplitude, while the amplitudes of roll and yaw angle oscillations
are still in 0.9 and 0.5 deg, respectively.
When the minimum switching time interval used is reduced to 0.1 sec, the responses
of the spacecraft attitude angles can be seen in Fig. 7. The
spacecraft needs about 92 sec to complete the maneuvers. Here, the attitude
angle oscillations can be reduced slightly. The pitch angle oscillation happens
in 0.03 deg amplitude. However, the amplitudes of roll and yaw angles after
t = 100 sec are still about 0.13 and 0.08 degrees, respectively.
A 0.02 sec minimum time interval of thruster switching results better accuracies
of the spacecraft attitude after the maneuvers. To complete the maneuvers, the
spacecraft consumes about 88 sec. The time histories of the spacecraft attitude
angles for this case are shown in Fig. 8.

Fig. 5(ab): 
Time histories of attitude angles and torques for 1 sec minimum
switching time interval of thrusters 

Fig. 6: 
Time responses for 0.5 sec minimum switching time interval
of thrusters 

Fig. 7: 
Time responses for 0.1 sec minimum switching time interval
of thrusters 

Fig. 8: 
Time responses for 0.02 sec minimum switching time interval
of thrusters 

Fig. 9: 
Time history of torques for 0.02 sec minimum switching time
interval of thrusters 
The figure shows that the oscillation amplitudes of roll, pitch and yaw angles
after t = 100 sec are 0.07, 0.03 and 0.06 degrees, respectively. These attitude
angle oscillations are small enough and all satisfy the attitude accuracy required
by Koreasat as stated by Hwangbo (1992). The roll, pitch
and yaw torque inputs needed for the maneuvers are shown in Fig.
9. In reality, after completing the maneuvers, all thrusters can be switched
off and the attitude of the spacecraft will be maintained by passive stabilization
devices.
ACKNOWLEDGMENTS
This study is supported by MOSTI ScienceFund (Project Number: 040202SF0097).
The authors are thankful to Universiti Teknologi PETRONAS for providing the
research facilities.