1. Introduction
Observation satellites are required high pointing accuracy and
stability to achieve various and complicated missions[1]. For
example, figure 1 shows the overview of Nano-JASMINE
satellite[2]. Nano-JASMINE is developed by Nakasuka
Laboratory at University of Tokyo and National Astronomical
Observatory of Japan. For this satellite, required stability is 740
milli-arcsecond at 8.8 sec. However, reaction wheel assembly
used as an actuator of attitude control system can be a major
source of disturbances[3,4]. Therefore, isolation of disturbance is
one of the most significant issues in designing a precise attitude
control system. On the other hand, reaction wheel is an actuator
of attitude control system. Therefore, disturbance must be
isolated from satellite, and control torque must be transmitted.
The purpose of this study is to propose the new design method of
vibration isolator in consideration of control torque
transmissibility.
2. Numerical model of vibration isolator
A parallel link mechanism is known to achieve a high rigidity
with lightweight construction because it is a truss construction.
Figure 1 Overview of Nano-JASMINE
Reaction wheel
Satellite
Isolator
Figure 2 Concept of parallel link isolator
Therefore, parallel link mechanism is adopted to design a
vibration isolator. Figure 2 shows the concept of parallel link
isolator for reaction wheel assembly.
Numerical model of parallel link isolator is derived by kinematics
or inverse kinematics. In kinematics, position and attitude are
calculated from a length of each leg. However, because of an
indeterminate structure, kinematics is difficult to solve. On the
other hand inverse kinematics, a length of each leg is calculated
from position and attitude of platform. A length of each leg can
be calculated by Jacobian matrix easily.
Therefore in this study, inverse kinematics is adopted to analyze
the isolator.
Figure 3 shows the definition of coordinates and vector.
Coordinates and vectors are described as follows:
{p} Coordinate fixed to the C.M. of platform
{b} Coordinate fixed to the C.M. of base plate
i r Vector from origin of {b} to the bottom of each leg
i p Vector from origin of {p} to the top of each leg
0 X Vector from origin of {b} to origin of {p}
Cb p DCM from {p} to {b}
i e Vector from bottom to top of each leg (unit vector)
Where, i e is written by elevation angle and azimuth angle as
follow,
, , , , , cos cos cos sin sin T
i EL i AZ i EL i AZ i EL i e =?? ??
The position and attitude are described by ( , , , , , )T
x y z w = x y z , and
a length of each leg is described by ( ) 1 2 , , T q = q q ?? . Then, following
relation works out.
T T
i i i i i q=??e e ?e e p?? ??w
The equations of motion are written as follows:
2 {( ) }
1
n
T T
i i i i i RW
i
ms d cs k e e e e p
=
r+ + ?? ? ????w=f
2 {( ) }
1
n
T T
i i i i i i i RW
i
Is d cs k p e e p e e p
=
+ + ???? ??? ????w =
Where, m is a mass of platform, c is a viscosity, k is a spring
constant, I is a inertia moment of platform and s is a Laplace
operator respectively. Force and torque transmitted to the
(x,y,z,x,y,z)
(L1,L2,EEE)
Length
Position & Attitude
Base
Platform
{p}
{b}
ri
pi
X0
ei
Figure 3 Definition of coordinates and vectors
Az
El
Pos Figure 4 Definition of elevation angle and azimuth angle
satellite are calculated by
{( ) }
1
n
T T
iso i i i i i
i
cs k e e e e p
=
f= + ?? ? ?? ??w
{( ) }
1 1
n n
T T
iso i i i i i i i i i
i i
cs k re e re e p
= =
=r~f = + ???? ??? ????w
Then, transfer function from disturbance force, torque and control
torque to transmitted force and torque is written by
Sat RW
Sat RW
G
? ? ? ?
? ?= ? ?
? ? ? ?
f f
3. Design method of vibration isolator
3.1. Break down from mission requirement
This section describes the optimal design condition of vibration
isolator. First, mission requirement is translated into the allowed
disturbance torque for satellites. Attitude variation by
disturbances is assumed as follow:
(t)=asin(t)=asin(2ft)
The amplitude of attitude variation is calculated by mission
requirement as follow
( ) ( )
( )
0
sin s
s
s
f f
a fT
f f
? < < ? ???
? ?
Finally, allowed disturbance is calculated by this amplitude and
property of controller.
( 2 2)2 (2 )2 sat c d c c d =I a ? +
Where, c
is a damping ratio of controller, c
is a natural
frequency of controller respectively.
For example, allowed disturbance for Nano-JASMINE is shown
in figure 5.
10-2 10-1 100 101 102
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
aellowed distugrbanNce
Frequency [Hz]
Amplitude [Nm]
allowed disturbance
Figure 5 Allowed disturbance for Nano-JASMINE
Next, design condition for isolator is derived by allowed
disturbance. Disturbance caused by reaction wheel is modeled
as follow
2
whl = a
Where a is a proportional constant and is a angular velocity
of reaction wheel. Figure 6 shows the allowed disturbance and
disturbance caused by reaction wheel assembly. Finally,
required property for vibration isolator is calculated. Figure 7
shows the design condition for vibration isolator.
Figure 6 Allowed disturbance and reaction wheel disturbance
10-2 10-1 100 101 102
10-1
100
101
102
Frequency [Hz]
gain
Upper Bound
Figure 7 Design condition for vibration isolator
3.2. Optimization
In the previous section, required property is derived from mission
requirement. In this section, parameters of isolator (spring
constant, viscosity, elevation angle, azimuth angle and so on) are
optimized to satisfy the design condition.
Figure 8 shows the overview of optimization loop. First, initial
parameters (spring constant, viscosity and so on) are set and
requirements for attitude stability, control property and robustness
are derived by mission requirement. Then, controller is
designed for augmented system to satisfy these requirements.
Next, required property for isolator is derived and compared with
initial parameters. Finally, evaluate value is calculated and this
loop is iterated to find optimal parameters by sequential quadratic
programming (SQP).
Figure 9 shows the result of optimization. In this figure blue
line shows required isolator property for x and y axis, red line
shows for z axis, and green and light blue line show the result of
optimization for x,y and z axis respectively. Required property
shows the upper bound for isolator. It is confirmed that result of
optimization is lower than upper bound.
controller
Required Isolator property
Iteration
Control property (bandwidth)
Robustness
Attitude stability
Isolator parameter
Isolator property
comparing
Figure 8 Overview of optimization loop
10-2 10-1 100 101 102
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Frequency [Hz]
Amplitude [Nm]
Allowed Torque
RW Disturbance
Figure 9 Result of optimization
4. Validation by numerical analysis
Numerical analysis is conducted to validate proposed
algorithm. Numerical model includes the dynamics of satellite,
reaction wheel assembly and vibration isolator. Satellite is
modeled single rigid body. In terms of isolator, numerical
model written in section 2 is used. Reaction wheel assembly is
modeled by using Herzian contact theorem. Then, two
parameters of isolator (optimal and not optimal) are simulated.
10-1 100 101 102
10-3
10-2
10-1
100
101
102
103
`B
vl
Frequency [Hz]
Transmissibility
Required
Designed
Figure 10 Optimal parameters for validation
10-1 100 101 102
10-3
10-2
10-1
100
101
102
103
`B
vRequilred
Designed
Frequency [Hz]
Transmissibility
Figure 11 Not optimal parameters for validation
299 299.1 299.2 299.3 299.4 299.5 299.6 299.7 299.8 299.9 300
-8
-6
-4
-2
0
2
4
6
8
x 10-4
œKȏꍇ
œKłȂꍇ
vl(+)
vl(-)
Optimal
not optimal
Requirement
Requirement
Time [s]
Attitude [deg]
Figure 12 Result of validation
The result of numerical analysis is shown in figure 12. As a
result, it is confirmed that isolator with optimal parameter satisfy
the requirement. Therefore, the proposed algorithm is validated.
4. Conclusion
Vibration isolator for reaction wheel assembly is needed to
isolate disturbance, and to transmit control torque. Therefore,
there is a trade-off in designing an isolator. In this study, new
design method is proposed in consideration of torque
transmissibility. In proposed method, optimal design condition
of isolator is derived by mission requirement. Numerical
analysis is conducted to verify proposed method. As a result, it
is confirmed that optimal parameters calculated by proposed
method satisfy the trade-off.
References
[1] Norimasa Yoshida, Kiyoshi Ichimoto et al., gSystem
Technology for the Achievement of Ultra-High Pointing
Accuracy of Solar-Bh, 45th Space Sciences and Technology
Conference, 2001
[2] Nobutada Sako, Yoichi Hatsutori, Takashi Tanaka and
Shinichi Nakasuka, gAbout Nano-Satellite for Infrared
Astrometry (Nano-JASMINE) Projecth, 25th ISTS, ISTS
2006-f-09, 2006
[3] R. A. Masterson, D. W. Miller, R. L. Grogan, gDevelopment
and Validation of Reaction Wheel Disturbance Model: Empirical
Modelh, Journal of Sound and Vibration, Vol. 3, No. 249, 2002,
pp. 575-598
[4] Bill Bialke, gHigh Fidelity Mathematical Modeling of
Reaction Wheel Performanceh, Advances in Astronautical
Sciences 98-063, 1998
10-1 100 101 102
10-3
10-2
10-1
100
101
102
103
Frequency [Hz]
Transmissibility
Required property (Tx/Ty)
Designed property (Tx/Ty)
Required property (Tz)
Designed property (Tz)