B. Case 1 Global Star. The periods of the libration and string vibration are 3900sec and 550sec respectively. The shaper for the Case 1 is designed as follows. â¥.
AIAA 2004-5313
AIAA/AAS Astrodynamics Specialist Conference and Exhibit 16 - 19 August 2004, Providence, Rhode Island
An Application of Input Shaping For Electrodynamic Tether System Takeo Watanabe*, Takeshi Makida†, Hironori A. Fujii‡, and Hirohisa Kojima§ Tokyo Metropolitan Institute of Technology, Hino, Tokyo 191-0065 Japan and William Singhose ** Georgia Institute of Technology, Atlanta, GA, 30332,USA
In the present study, Input Shaping method is applied to reduce the initial vibrations of electrodynamic tether system. In the initial phase of the propulsion, the electrodynamic force induces vibrations on the tether. Libration and string vibration are undesirable for any missions of the system. These are caused by the flexibility or orbital motion of the system. The different shapers are applied for the vibrations and the input command is designed by multiplying the both shapers respectively. As examples, two cases of de-orbit missions of space debris are analyzed numerically. The results of this study show significant performances of suppressing by employing the Input Shaping applications for tether system.
I.
Introduction
n this paper Input Shaping method is applied to reduce the initial vibrations of electrodynamic tether system. The concepts of electrodynamic tether propellant system have been proposed for the orbital re-boost and de-orbit. Though electrodynamic tether can create small propel force, it has huge ISP compare with chemical propellant system. Especially, it is effective in the low orbit, and suits for long period missions. For examples, ideas of reboosting of Inter national Space Station (ISS), De-orbit missions of Space debris are proposed. In the initial phase of the propulsion, the electrodynamic force induces vibrations on the tether. Libration and string vibration are undesirable for any missions of the system. These are caused by the interaction of The Lorentz force, and flexibility or orbital motion of the system1-4. Taking the linear density of the tether into consideration, transverse motions of the tether can be treated as vibration of a string, which has a tension gradient along its length. The libration absorbing control and wave absorbing feedback control of tether satellite system by using movable attachment or electro dynamic force have been analyzed numerically5,6,7. Input shaping is a command generation technique that is used to reduce command-induced vibration (as opposed to disturbance-induced vibration). Input shaping is implemented by convolving a sequence of impulses, called an input shaper, with a desired reference command signal8-13. In the present study, two cases of de-orbit missions of space debris, and re-boosting mission of the ISS are analyzed numerically. Given the general qualities of the various input shapers and the needs of a tether system, we utilize a Unity-Magnitude, Zero Vibration (UM-ZV) shaper to reduce the libration (pendulum motion) because it is a long duration oscillation that is easy to estimate. It is a function of only the orbital altitude. This dynamic phenomenon is analogous to crane motion, which has been effectively dealt with the using input shaping. On the other hand, Zero Vibration and Derivative (ZVD) shapers are employed to deal with the string vibration because it is
I
*
Graduate Student, Department of Aerospace Engineering, 6-6 Asahigaoka, Hino, 191-0065, Student Member Graduate Student, Department of Aerospace Engineering, 6-6 Asahigaoka, Hino, 191-0065 ‡ Professor, Department of Aerospace Engineering, 6-6 Asahigaoka, Hino, 191-0065, Associate fellow AIAA § Professor, Department of Aerospace Engineering, 6-6 Asahigaoka, Hino 191-0065 ** Associate Professor, Department of Mechanical Engineering, Atlanta, GA, 30332 †
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Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
a more complex function of tether length, line density, tension, and the motion. The robustness of the shaper can be demonstrated by plotting the residual vibration as a function of the actual system frequency normalized by the modeling frequency. The modeling error is given as the changes of the parameters of the tether system. The robustness of the shapers for the change of parameters is indicated by the sensitivity maps. The results of this study show significant performances of suppressing by employing the Input Shaping applications for tether system.
II.
System Model
A. Equations of motion Figure 1 shows the model used here for a space tether satellite system. Taking the flexibility of tether into consideration, the lamped mass model is employed, and the equation of motion of the space tether system is described by using Kane’s Equations as outlined in Ref 4 and 7. The center of mass of the main body is on the circular orbit, and only motion in orbital plane is considered. The angular velocity of the system is Ω . The vibration of the tether is estimated by the deflection of the tether mid point. B. Electrodynamic Tether propulsion The concepts of electrodynamic tether propellant system have been proposed for the orbital re-boost and de-orbit. Although electrodynamic tether can create small thrusting force, it has huge ISP compared with the chemical propellant system. Especially, it is effective in the low orbit, and suitable for long period missions 2,3. In the present study, a dipole model is employed to be the magnetic field model of the earth (Figure 2). The magnetic field vector is given by the equations:
B=
µm R3
B = −2 B=
µm R3
(u m − 3(u m ⋅ u r )u r )
µm R3
sin(ϖ + f ) sin i
cos(ϖ + f ) sin i
X Y
Electro
Current
Dynamic Force
Magnetic Field
Ω
(1) Figure. 1 System model
(2) (3)
The Lorentz force created by the interference of current in the tether and the magnetic field is given as the following equation:
F = (I i × B i ) l i
(4)
In the initial phase of the propulsion, the electrodynamic force induces vibrations on the tether. Libration and string vibration are undesirable for any missions of the system. These are caused by the flexibility or the orbital motion of the system. In the present study, the inclination of the orbit is assumed 0 degree.
Figure 2 Earth’s Magnetic Field.
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III.
Input Shaping
Input shaping is a command generation technique that is used to reduce command-induced vibration (as opposed to disturbance-induced vibration). Input shaping is implemented by convolving a sequence of impulses, called an input shaper, with a desired reference command signal. To simplify the analysis, we consider the case where only one flexible mode exists. If a system’s natural frequency Figure 3. Input Shaping a Step Input. ω and damping ratio ς are estimated reasonably, the nondimensional residual vibration that results from a sequence of impulses can be described by ambiguities in denominators. Be sure that the symbols in your equation are defined before the equation appears, or immediately following. Where
V (ω , ς ) = e −ςωtn C (ω , ς ) 2 + S (ω , ς ) 2
(5)
n
C (ω , ς ) = ∑ Ai e ςωti cos(ω d t i ) i =1
(6a)
n
S (ω , ς ) = ∑ Ai e ςωti sin(ω d t i ) i =1
(6b)
Ai and ti are the amplitude and time location of the ith impulse, respectively, n is the number of impulses in the impulse sequence, and ω d = ω 1 − ς 2 . By setting (1) to zero and solving it the impulse amplitudes and time locations, we can determine for desired impulse sequence. In order for (5) to be zero, both (6a) and (6b) must equal zero independently. Therefore, the impulses must satisfy n
C (ω , ς ) = ∑ Ai e ςωti cos(ω d t i ) = 0 i =1
(7a)
n
S (ω , ς ) = ∑ Ai e ςωti sin(ω d t i ) = 0 i =1
(7b)
These constraints contains 2n unknown parameters. Without loss of generality, the time location of the first impulse t1 can be set to be zero. The number of unknown parameters is now reduced to 2n-1. In addition, real systems cannot be actuated by a sequence of impulse commands, but by finite duration command signals. Thus, the properties of the impulse command need to be converted to a finite duration command. This conversion can be implemented by convolving the impulse sequence, called the input shaper, with any desired command signal. This process is demonstrated in Fig. 3. If the impulse sequence causes no residual vibration, then the convolved command will induce no vibration. This convolution also ensures that the rigid body motion induced by the input remains the same before and after convolution, provided that the input shaper satisfies: n
∑A i =1
i
=1
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(8)
The number of constraints is now three, as given by (7a), (7b) and (8). Thus, if the number of impulses is two, all the unknown parameters can be determined. For input shapers containing additional impulses, additional constraints are needed and can be used to increase robustness of the input shaping to the modeling errors.
A. Zero Vibration (ZV) Shaper The impulse sequence for the case of n=2 is called Zero Vibration (ZV) control8, and is the simplest input sequence. In this case, by solving the preceding constraints, the amplitude and time location of the input shaping are obtained as follows:
⎡ A1 ⎢A ⎣ 3
A2 ⎤ ⎡1 (1 + K ) K (1 + K )⎤ = A4 ⎥⎦ ⎢⎣ 0 0.5Td ⎥⎦
(9)
where
(
K ≡ exp − ξ π
1−ξ 2
)
Td = 2π ω d
(10)
(11)
B. Zero Vibration and Derivative (ZVD) Shaper In order to increase robustness of the input shaping process with respect to the frequency modeling error, the following constraint can be used: ∂V (ω , ς ) =0 ∂ω
(12)
This condition leads to two more constraints involving the impulse amplitudes and time locations as follows: n
∑ A t eςω i =1
ti
i i
n
∑ A t e ςω i =1
i i
ti
cos(ω d t i ) = 0
sin(ω d t i ) = 0
(13a)
(13b)
In order to satisfy (9), one more impulse with two undetermined parameters (amplitude and time location) is needed. By solving the preceding constraints, the amplitudes and time locations for ZVD control are obtained as follows:
2K K2 ⎤ ⎡A1 A2 A3⎤ ⎡⎢ 1 ⎢t t t ⎥ =⎢1+2K+K2 1+2K+K2 1+2K+K2 ⎥⎥ ⎣ 1 2 3⎦ ⎣ 0 0.5Td Td ⎦
(14)
The improved robustness can be seen by plotting a shaper’s sensitivity curve; the amplitude of vibration vs. modeling error, as shown in Figure 4.
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C. Extra-Insensitive (EI) and Specified-Insensitive (SI) shaper Several other methods have been developed to increase the robustness of input shapers. Many increase the shaper’s robustness in discrete interval, such as in the case of Extra-Insensitive (EI) shaper12. Another method specifies th amount of robustness as a design constraint. Specified Insensitivity (SI) shaper requite the vibration to be below some acceptable threshold13. The insensitivity I, is defined to be nondimensional width of the sensitivity curve that lines below the toleration limit. The sensitivity curve for an SI shaper with I=0.7 is plotted in Figure 4 for the case when the vibration tolerance is 5% of the case when input shaping is not used. D. Unit-Magnitude Zero Vibration (UM-ZV) shaper A fundamental tradeoff in the design of input shapers is vibration reduction vs. rising time. As the input shapers is required to suppress more insensitively, the rise time or the system is increased. This has motivated the development of input shapers that contain negative impulses9. Unit Magnitude Zero Vibration (UM-ZV) Shaper is one of the negative input shaper that is composed unit magnitude impulses. By including negative impulses, the rise time can be significantly reduced.
Figure 4 Sensitivity Curves for Common Shapers. E. Convolved Shaping The design of shapers for multi mode systems can proceed. Shapers for each mode can be calculated separately, and then convolving together10. The convolved shaper is extraordinary robust to modeling error of the higher mode. By using convolved shaper, plural modes of vibrations can be reduced.
UM − ZV
Current
Time
*
ZVD
*
Figure 5 Multimode Shaper.
IV.
Numerical Analysis
Given the general qualities of the various input shapers and the needs of a tether system, we utilize a UnityMagnitude, Zero Vibration (UM-ZV) shaper to reduce the libration (pendulum motion) because it is a long duration oscillation that is easy to estimate. It is a function of only the orbital altitude. This dynamic phenomenon is analogous to crane motion, which has been effectively dealt with the using input shaping. On the other hand, ZVD shapers are employed to deal with the string vibration because it is a more complex function of tether length, line density, tension, and the motion. 5 American Institute of Aeronautics and Astronautics
f string =
1 T 2l ρ
(15)
1 2π
3Ω
(16)
f LIB =
Where, the tension can be estimated simply as T ≅ 3Ω 2l (m + ρl / 2) . A. System Parameters In this study, three cases of electrodynamic tether missions are analyzed. Table 1 shows the system parameters Case 1 and Case 2 are deorbit mission of the satellites after their lives. Case 3 is the re-boost application of electrodynamic tether for the ISS. Note that the current direction of Case 3 is opposite way to Case 1,and 2. Table 1 System Parameters Case 1 Case 2 Globalstar De-orbit Adeos De-orbit
Target Satellite
Case 3 The ISS Re-boost
Orbital Altitude [km]
1414
800
400
Mass [kg]
450
3500
200(subsatellite)
Tether Length [km]
10
10
20
Tether Density [g/m]
10
10
5
Max Current [A]
3
3
(-) 3
B. Case 1 Global Star The periods of the libration and string vibration are 3900sec and 550sec respectively. The shaper for the Case 1 is designed as follows
⎡ Ai ⎤ ⎡0.25 0.5 0.25 − 0.25 − 0.5 − 0.25 0.25 0.5 0.25 ⎤ ⎢t (sec)⎥ = ⎢ 0 275 550 650 925 1200 1300 1575 1850⎥⎦ ⎣i ⎦ ⎣
(17)
Figure 5a show the time response of the deflection of tether mid point Dotted black line shows the unshaped time response. String vibration and libration are induced in the system significantly. Rigid blue line is the result of shaped current profile by using ZVD for String vibration and UM-ZV for the libration. Figure 5 b illustrates shaped current profile. The amplitude of current is switched by the convolved shaper. Figures 5c and 5d show the tension at Main satellite side and Subsatellite side respectively. The results show significant performance of reducing vibrations principal vibration and libration. The shaper is designed to suppress 1st mode of vibration and orbital libration, therefore, higher mode vibrations remain. However , such a high frequency vibrations are reducible naturally.
Figure 5 a, Deflection of tether mid point
Figure 5b Shaped current profile
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Figure 5c Tension at Main Satellite
Figure 5d Tension at Subsatellite
C. Case 2 ADEOS The periods of the libration and the string vibration are 3500sec and 180 sec respectively. The shaper for the Case 2 is designed as follows
⎡ Ai ⎤ ⎡0.25 0.5 0.25 − 0.25 − 0.5 − 0.25 0.25 0.5 0.25 ⎤ ⎢t (sec)⎥ = ⎢ 0 90 180 582 671 762 1165 1255 1345⎥⎦ ⎣i ⎦ ⎣
(18)
Figures 6a-d show the time response of the deflection, shaped current profile, and tensions respectively. In the Case 2, the periods of libration and string vibration are detached, so the shaper acts effectively. The vibrations of tensions are also reduced. Rigid blue line in Figure 6b is the result of shaped current profile by using ZVD for String vibration and UM-ZV for the libration.
Figure 6 a, Deflection of tether mid point
Figure 6b Shaped current profile
Figure 6c Tension at Main Satellite
Figure 6d Tension at Subsatellite
D. Case 3 ISS The periods of the libration and string vibration are 3000 sec and 620 sec respectively. The shaper for the Case 3 is designed as follows
⎡ Ai ⎤ ⎡0.25 0.5 − 0.25 0.25 − 0.5 0.25 − 0.25 0.5 0.25 ⎤ ⎢t (sec)⎥ = ⎢ 0 310 500 620 810 1000 1120 1310 1620⎥⎦ ⎣i ⎦ ⎣
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(19)
Figure 7a is the deflection of tether. Note that the Case 3 is reboost action therefore direction of deflection is headway direction. In this case, the modeling period of string vibration is longer than previous cases, the look of shaped current is different shape (Figure 7b). Shapers for this case is also, ZVD for String vibration and UM-ZV for the libration.
Figure 7c Tension at Main Satellite
Figure 7 a, Deflection of tether mid point
Figure 7d Tension at Subsatellite
Figure 7b Shaped current profile
E. Sensitivity Map The modeling errors are given by the change of subsatellite mass and tether length. The considerable modeling errors can be caused by these two reasons, 1) Trouble of the deployment, 2) Fuel consumption after several reboost missions. In the present study, there are two fundamental factors of vibrations, so the sensitivity plot is illustrated as 3 dimensional maps. Both modeling errors are decrease of parameters. Therefore, the sensitivity plot is illustrated in the area of the decreasing side of parameters. Table 2 shows the configuration of the shapers convolving and characteristics of the target modes. Table 2 Characteristics of Target vibration and Configurations. Shaper for Liration Shaper for String Vibration Period
About 3000 sec
Factors
Orbital Angle veracity
About 620 sec Tension, Tether length, Liner density, Pitch motion, Mass of Subsatellite, etc. Difficult
Estimation
Easy
Figure 8
UM-ZV
ZVD
Figure 9
UM-ZV
EI (5%)
Figure 10
UM-ZV
Offset EI (5%, 720 sec)
Figure 11
UM-ZV
Offset EI (5%, 520 sec)
Figure 12
UM-ZV
UM-ZV
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The vertical axis is the residual vibration. The value of residual vibration is determined the percentage of ratio between shaped response and unshaped response. The modeling error of the tether length is considered from 20000m (nominal) to 10000m (worst case), the modeling error of the subsatellite mass is considered from 200kg (Initial) to 100kg(after fuel consumption) respectively.
Figure 8 Sensitivity Map of UM-ZV (NZV) *ZVD shaper. Figure 8 shows the sensitivity map of the UM-ZV (NZV) *ZVD shaper for the reboost of ISS. The residual vibration is increase in the large modeling errors. The frequency of the system can be described as function of the root of m/l ratio, the residual vibrations in the diagonal zone of the map are suppressed effectively. The shapers are designed for the nominal parameters. Figure 9 shows the sensitivity map of the UM-ZV*EI (5%) shaper. By the employing of the EI shaper, the robustness of the convolved shaper is improved significantly.
Figure 9 Sensitivity Map of UM-ZV*EI shaper. 9 American Institute of Aeronautics and Astronautics
The offset shaper for higher frequency is shown in Figure 10. The decrease of the tether length causes the increasing of frequency. The higher offset application is designed to correspond for the length change particularly. The shaper is sensitive for the change of subsatellite mass. So in the initial phase of mission, this offset application can be helpful.
Figure 10 Sensitivity Map of UM-ZV*EI Higher offset shaper.
Figure 11 Sensitivity Map of UM-ZV*EI lower offset shaper. On the other hand, figure 11 is the offset application for lower frequency. This construction is designed for the significant robustness for change of subsatellite mass. After conform of the tether length, such a shaper is effective.
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Figure 12 Sensitivity Map of UM-ZV*UM-ZV Time Optimal shaper. If the parameters of the system are estimated exactly, we can chose time optimal construction of shapers. Figure 12 shows the sensitivity map of time optimal shaper. The shaper is very sensitive for the modeling error. Figures 13a and 13b compare the rise time of shapers. Time optimal shaper can rise 400sec earlier than the robust shapers.
Figure 13 a Shaped Current profile of Time optimal shaper
V.
Figure 13 a Shaped Current profile of robust shaper
Conclusion
This paper has studied the application of Input Shaping method for the Electrodynamic tether systems. The UMZV shaper and ZVD shaper are applied to design the current input command. The two cases of deorbit missions and the reboost mission of the ISS are employed for numerical simulation. The performances of the shapers are illustrated by the sensitivity maps, and the robustness of the shapers for the modeling errors are analyzed. In addition, the offset applications of the shaper are proposed for the phase of missions. The results have shown significant effect of reducing unwanted vibrations.
Acknowledgments This study is partly supported by the Tokyo Metropolitan Government, and the Academic Frontiers Student Exchange Program of Japanese Ministry of Education, Culture, Sports, Science and Technology.
References 1
Jonsnon, L. and Herrmann, M., ‘International Space Station electrodynamic tether re boost study’, NASA TM1998-208538, Marshall Space Flight Center, Alabama, 1998
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2
R.I.Samanta,D.E. Hastings, and E.Ahedo.Systems analysis of electrodynamic tethers. Journal of Spacecraft and Rockets, Vol.29,No.3,pp415-424,May-June 1992. 3 A.Bade and P. Eichler. The removal of large Space Debris objects with the help of space tethers. Z. Flugwiss. Weltraumforsch, Vol.16,pp.271-282,1992. 4 Williams, P., Blanksby, C., and Trivailo, P.,‘The use of electrodynamic lorenz forces as a tether control actuator’ IAC-02-A.5.04, 52nd International Astronautical Congress, Oct. 10-19, 2002, Huston Texas 5 Fujii, H. A., Watanabe, T., Taira, W., and Kumamoto, T., Analysis of Traveling Wave of Tether Systems, Proceedings of The 22nd International Symposium on Space Technology and Science, MORIOKA, Japan, June 2000, Vol. 1, pp. 665-670. 6 Fujii, H.A., Watanabe, T., Taira, W., and Trivailo, P.2001, ‘An analysis of vibration and wave-absorbing control of tether systems’, AIAA Guidance, Navigation ,and Control Conference, 6-9 Aug., Montreal, Canada. 7 Williams, P., Blanksby, C., Trivailo, P., and Fujii, H.A.,’ Libration Control of Flexible Tethers Using Electromagunetic Forces and Movable Attachment’ AIAA Guidance Navigation and Control Conference and Exibit 11-14 August Austin Texas 8 Neil C. Singer, Warren P. Seering ’Preshaping Command Inputs to Reduce System Vibration’ J. of Dynamic Sys., Measurement, and Control March 1990 vol. 112 9 Singhose, W., Singer, N., and Seering, W., ’Time-Optimal Negative Input Shapers’ ASME J. of Dynamic Systems, Measurement, and Control June 1997 vol. 119 10 Crain, E.A., Singhose, W., and Seering, W., ‘Derivation and Properties of Convolved and Simultanous Twomode Input Shapers’, Proceedings of the 1996 IFAC Congress. 11 Singhose, W., Seering, W., and Singer, N., ‘Shaping inputs to reduce vibration : A vector daiagram approach’, In IEEE int. Conf. On Robotics and Automation, Vol.2, pp.922-927, Cincinnati, OH, 1990 12 Singhose, W. Derezinski, S., and Singer, N., ‘Extra-Insensitive Input Shapers for Controlling Flexible Spacecraft’ AIAA J. of Guidance, Control, and Dynamics 1996 vol. 19 n2 13 Singhose, W., Seering, W., and Singer, N.,’Input Shaping for Vibration Reduction with Specified Insensitivity to Modeling Errors’ Japan-USA Sym. on Flexible Automation 1996 Boston, MA
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