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A construction methodology of the large solar array system of a solar power satellite
Acta Asrronautica Vol. 38, Nos 4-8, pp. 223-229. 1996 Astmnautical Federation. Published by Elsevier Science Ltd Printed in Great Britm SOO94-5765(96)00048-9 0094-5765/96 $lS.oO + 0.00
Copyright 0 1996 ~temational
PII:
A Construction Methodology of the Large Solar Array System of a Solar Power Satellite M. Shigehara* and A. Ohtaka** Dept. of Aerospace Engineering, Tokyo Metropolitan Institute of Technology,
1. INTRODUCTION
ABSTRACT
Japan
(SPS2000)
SPS2000 is a technology demonstration program, as summarized in Table 1, for future Solar Power Satellites (SPS) in Japan. The spacecraft with a triangular configuration will fly on a low Earth equatorial orbit as shown in Figure 1. One face having a transmitting antenna is orienting toward the Earth and the other two faces have amorphous Si flexible solar arrays to generate 18 MW solar power. On the ground, the receiving antennas of wired mesh rectenna are placed to collect power from the satellite. The study on SPS2000 is conducted by the SPS2000 working group and the main purpose of the study is: 1) to design SPS2000 demo model 2) to examine technical feasibility and to develop the technologies for constructing such a large space infrastructure in a simple and cost effective way.
A construction methodology of the large solar array system is investigated, assuming the SPS 2000 model of ISAS, the experimental model for the future Solar Power Satellite in Japan. The structure elements such as trusses, beams and solar arrays are transported from the ground 10 times by using a rocket. First, the basic module is assembled and then the subsequent modules are attached towards N-S direction. The construction sequence how to form this basic module has already been examined under the gravity gradient environment. This paper discusses how to extend the flexible solar array sheets over this module. First, the dynamic model when extending an array is developed, by including the moment of inertia and angular momentum change during the extension. Then, the dynamic behaviors during extension are examined. When an extension is made by one array by one, the momentum by extending array and the moment of inertia change induce larger oscillation around all three axes of the basic module. Thus the method to minimize these changes by extending the pair of the arrays simultaneously is studied for reducing the attitude vibration. In conclusion, a method of extending two arrays simultaneously is judged desirable for a practical application.
l
l
l l
l
Copyright 0 1995 by the International Astronautical Federation. All rights reserved.
l
l
* Professor, ** Graduate student 6-6, Asahigaoka, Hino, Tokyo, 19 1 JAPAN
Technology demonstration program for future Solar Power Satellite (SPS) in Ja$%cept Study, R & D by ISAS SPS2000 WG Orbit : Low(1 IOOkm), equatorial, circular Power 18 MW (on orbit), 1OMW on ground (300kW average) Transmitting Ant. : Phased array, retrodirective (2.45GHz, 150m ) Receiving Ant. : Wired mesh rectenna (2km ) Total mass : 240ton, more than 10 times launches by Ariane 5 Table 1 SPS 2000
223
46rh IAF Congress
224
2. Summary of the Previous Studies
TransmittinghtcMa
/
I\ ’
1 IOOkm
Figure 1 Configuration
of SPS2OOO
Various efforts have being done and are continuing in the various fields for these goals, including an assembling robot, structural members, solar cells, transmitting phased arrays and receiving rectenna. Among them, one of the most critical problems is how we can construct a large structure with solar arrays in space under a passive stabilization such as to use the gravity gradient (G.G.) torque. This is one of the focus of our recent studies. The following topics have already been covered in the previous studies2’.3’. 1) The basic construction scenario has been established for the basic module and the subsequent assembling of the 10 modules for the final configuration. In these analyses, the static stability condition for G. G. was used so as to keep the moment of inertia ratio (MOIR) of the system within a stability region. 2) The dynamic stability has been studied to assure stability during construction for such as configuration change, and construction speed. By these analyses, the effect of an initial rate, main truss extension and robot motion are clarified. 3) The characteristics of disturbing torques such as solar radiation, geomagnetic torque and air drag have been studied. The significant source of the disturbances for SPS2000 is from a solar radiation pressure, which creates the larger torque around pitch. The measure to cancel out this disturbance is proposed by using a geomagnetic torque as control torque. This paper is intended to study the effect of the solar array extension on the surface of the basic triangular module. This may create the angular momentum change and inertia matrix change, and may cause some transient attitude deviation.
2.1 Basic Construction Sequence from Static Stabtlity To be stable under G. G. torque, the parameters relating with MOIR of the system must stay in the stability region as shown in Fig.2, which is derived from Euler equations. By satisfying this static condition, the basic construction sequence was proposed as in Fig. 3. The preferable sequence was Cl*C2+C3, where C 1: frame truss extension along outward local vertical C2: frame truss opening to triangular shape C3: beam truss extension along orbit normal
Figure 2 Expansion Module
(Cl) vedcal
I
((2)
f
Sequence
of Basic
extension of frame
~CUIICopening to triangular shape (0)
beam expansion
* 8
(ST) start PI
cnfg.
F’2 cnfg.
(BU) built up of basic module
Figure 3 G. G. Stability Ratio
Map for Inertia
46th 1AFCongress
2.2 Effect of Configuration Change from Dvnamic Stabilitv Dynamic behaviors by such as an initial rate, transient due to configuration change, robot motion and flexibility of the joints give additional stability restraints. These effects can be studied through a multi-chain bodies model with flexible joints. One of the typical results is shown in Fig. 4, where the dynamic behavior was examined by extending a truss along the local vertical with a speed of 90 ml 100 min. As shown in Fig. 4 a, the moment of inertia (MOI) in pitch and roll increases by more than 10 times after one orbital period. According to the change of this
Af=llMO[kJ] L
= 66027[4m’l
I,
=13887[kgm’]
I, = 2643O(~m’]
n M=lUco[kg]
1,=11147759[4m’] 1,,=1122619[kgm’]
I, =2643O[kgm*]
225
MOI, the attitude of the system deviates backwards from the local vertical. It is due to momentum conservation; since MO1 increases in pitch, the angular rate must decrease to keep the system angular moment constant. After a completion of the extension, G. G. torque restores the deviation and keeps the system stable. Also, the case of decreasing MOI, by taking an example of frame opening to a triangular shape, was examined. Through opening, h401 in pitch reduces to l/3 and this causes a significant increase of angular rate of the system, resulting in the attitude drift forwards from the local vertical. The results show the system becomes unstable, if an opening speed exceeds over a certain limit. This is because G.G. Torque cannot overcome attitude deviation before it reaches 90 degrees. This case is a critical, since a decreasing MO1 reduces G. Restoring torque. It was concluded that a careful attention must be paid for a construction speed when the construction produces decreasing MO1 in pitch. The effect of slew motion of robots and free joints is not significant.
Figure 4a Main Truss Extension along Local Vertical
3. Array Extension 3.1 Definition After a completion of construction of the basic module (hereafter, called as base structure, which does not include the solar array), the solar arrays will be extended on the two surfaces of the triangular base structure. The arrays are assumed to be extended from their containers, #O, 1, 2, 3 attached on the comers of the base structure by using a robot. Physical dimension of the base structure is shown in Fig. 5. Each solar
#0 solar array container
A
)1 \
#3 solar array\
1
51m Figure 4b Attitude Histories under Truss Extension along Local Vertical
Figure 5 Basic .Module with Solar Arrays
46th IAF Congress
226
array weighs 230 kg and the total mass of the module about 8 tons. Attitude requirement for SPS2OOOis assumed within +5”
c.g. of roll parts of # i solar array
in roll/ pitch and within +lO’ in yaw. During extension, the configuration may change in an unsymmetrical fashion and the produced momentum by extending the array with certain speed will create the angular momentum change of the base structure. These changes are supposed to create an additional transient attitude motion of the system and these will be investigated here.
3.2 Mathematical Model The orbit fixed coordinate frame @(x,,y,,b) and the base structure (triangle) fixed frame Q(x,y,z) are defined as in Fig.6, where the origins of both frames coincide. The angles($), w,A) define the attitude deviation of the base structure. The “system frame” is defined as the one of which axes and origin coincide respectively with the principal axes and the origin of the base structure, combined with the solar arrays. Thus, the “system frame” represents the current configuration during extension of the array. The solar array sheet is extended from its container as shown in Fig.7, where 1 is the rolled out length, r radial thickness of the remaining array and b, thickness of the array sheet. From this geometrical con-
c.g. of the base structure Figure 7 Geometry of Solar Array Extension
figuration, the following equations are derived.
&__ 4
--q$p z (2x) &
(I)
dy,= 4 dl;
(2)
a, = tan (3)
North Pole
dr dy; - -i sin( y,,. + 77,)+ sin CY,- 1;COSCX, dt. I drt, _ d/1 q COS( Y,,~ + q,) fi a, 4
sin
(4)
dczi _ dV;
4 Z
South Pole {X.Y.Z}:
inertia frame x,.y,,zo}: orbit fixed frame 1 (x,y,z}: base structure fixed frame Figure 6 Coordinate Frames
dl;
dyi dl,
(6)
where i is the parameters relating with the ith solar array and pi is the rotation angle of the role part of i th solar array. From conservation of momentum and Euler equations, the equations of motion of the base structure are formulated as follows.
46th Iti
221
congress
(7)
(10) creates an angular momentum in pitch. Thus the base structure must rotate in the opposite direction to compensate this momentum change. In the example, the base structure rotates forward. A slight increase of the angular momentum in roll and yaw also causes the base structure rotate in the opposite direction around these two axes. Figure 9 shows variation of the inertia matrix of the system. Since the solar array
where fi,, : position vector of c.g. of the base structure from that of the system. fi : position vector of c.g. of the system from the Earth subscript si, ri : the respective sheet and role part of the i th solar array c;,, : angular velocity of the base structure
*la -tU
bracket [ , ] : outer products of vector, ex.
5.0 101
0.0 loo
&bF=SixFxb [ -1
D :inertia matrix a: initial angular momentum of the system 0 : angular momentum of the system
-5.0 101
-1.0
-2.0
4 Analysis and Discussion
in 0.5 orbit. Figure 8 shows the angular momentum change of the #O panel with respect to the system frame. The array extension velocity
101 0
0. I
0.2
0.3
0.4
05
orbits
Figure 8 Angular Momentum of #O Array during Extension (extension in 0.5 orbit)
8.0 70 F 00 c
6.0
B 2 ._
5.0
b z
4.1 Extension of the #O panel At first, motion of the base structure is examined when a single array is extended
10,
-1.5 1w
given by gravity gradient torque. The equation (7) shows the translational motion of the base structure with respect to the system frame, and Eq.(9) the rotational motion of the base structure. The first bracket of Eq. (9) is the inertia matrix of the system and the first term of the second bracket is the anguler momentum of solar panels around the c.g. of the system. These equations represent the motion of base structure caused by a change of the inertia matrix and motions of the solar panels around the c.g.
hY
2
40 30 20 00
oi
02
03
0.4
05
arbtu
Figure 9 Inertia Matrix of the System during #o Array Extension (extension in 0.5 orbit)
228
46th IAF Congress
--b -
0.0
2.0
pitch’ ....l ... ro,,’ . ..*... . ..*e..
-pitch
roll YIW
4.0
6.0
8.0
10.0
orbits
Figure 10 Attitude Histories of Base Structure when #O Array extended in 0.5 Orbit is extended along the base structure in an unsymmetrical fashion, the products of inertia are created, causing the principal axes of the system frame deviate from that of the base structure. The restoring gravity gradient torque acts around the principal axes and vibrates the system around these axes. The vibration seen from the base structure frame becomes offset from that around the principle axes due to these products of inertia. In Fig. 10 are shown the resulting attitude motion, where the solid lines are the actual attitude variations around the pitch, roll, yaw in the base structure frame. As a total, the base structure will rotate in a positive direction around pitch at the beginning due to a negative angular momentum given by extending array. Then it rotates in a negative direction due to an opposite change of angular momentum to stop extension. After completion of the extension, the system is in almost steady oscillation under the gravity gradient. However, due to the product of inertia, an oscillation has an offset from null axis as shown in the dashed lines. Around the roll and yaw, the change of the angular momentum and inertia is relatively small and creates a small oscillation around roll. In yaw, this small disturbance produces a larger error, since the G.G. Restoring torque is smaller around yaw. 4.2 Symmetrical Extension To keep the deviation as small as possible, the measures to reduce the angular momentum change and the product of inertia are studied, and the following two cases are investigated.
Case. 1 Extending two diagonal arrays (#Oand #Q) at the same time In this case, there are produced no angular momentum change and product of ine&a during extension in pitch and roll. However, a slight change of MO1 in pitch creates a rotational rate change from an initial orbital rate. This is due to conserve the angular momentum, and results in an oscillation as shown in Fig. 11. Because the configuration is not symmetrical around yaw during extension, there are still momentum change and some deviation of the principle axis in yaw. These factors produce the attitude deviation in yaw asinFig.11. Case 2 Extending four arrays at the same time. Figure 12 shows the motion of the base structure when the four arrays are extended at the same time. In this case, due to a symmetrical configuration, there occur no momentum changes in all three axes and no principal axis deviation during extension. Thus, the attitude around roll and yaw do not change, in spite of the change of MO1 around these axes, if there is no initial rate. Attitude around pitch becomes in an OScillation by the same reasons as in case 1. The effect of extension speed can be studied by the following simplified analysis. If there is no gravity gradient torque applied, the motion around pitch axis is expressed from the angular momentum con-
---*a-- pilch’ . ..a.-. roll’ . . . . &... y*w’
orbits
Eigure 11 Attitude Histories of Base Structure when #O and ##2Array extended in 0.5 Orbit)
229
46th IAF Congress
--8-
pitch(O.Sorbit)
-8-
cdch(O.lorbit)
device. Under the G. Stabilization, some sort of the damping device, active or passive, is mandatory and with this damper, the oscillation is sure to be damped out.
5 CONCLUSION
0.0
2.0
4.0
6.0
8.0
10.0
orbits
Figure 12 Attitude Histories of Base Structure #O, #l, #2, #3 Arrays extended in 0.1 and 0.5 Orbit)
(11) where y(t) is pitch angle and R is orbital rate. If the panels are extended in 0.1 orbit, the pitch angle vo,,(T) at the end of extension, t = T = 0.1 orbit is
wo,(T)= (H(T)- T>Q
(12)
The effect of solar array extension along the base structure is investigated, under the gravity gradient stabilization. The momentum change due to an extension rate of the array causes the attitude deviation and make the system in an oscillation. Also, unsymmetrical extension creates the inertia matrix change, which causes the principal axes deviate from that of the base structure. This principal axes travel gives an offset to an oscillation and makes the system oscillate larger in the base-structure frame. For reducing a change of angular momentum and product of inertia, such symmetrical extension is proposed as two diagonal arrays or four corn& arrays extending simultaneously. The latter reduces a change in an ideal fashion, but the former is also acceptable, if a proper damping device is implemented. If we take into consideration of the number of robot required to extend the array, the simultaneous extension of two diagonal arrays is recommended.
where ACKNOWLEDGMENTS If the panels are extended in 0.5 orbit, the pitch angle va J (5T) at the end of extension, t=5T is & 0) = !, (t/5) 0.5
t/&5T)
= R/;r
(14)
01
+” ??
- R(5T)
5(H(T)- T)Q
(15)
Wo.s(5T) = 5Wo.0)
(16)
=
Thus, This means the angle deviation becomes 5 times larger if extension is done in 5 more times. However, in an actual case, the G.G. restoring torque will suppress an increase of the deviation, which is shown also in Fig. 12. The longer the extension speed, the larger deviation is produced, but the influence is not significant. In all above calculation, the system seems to continue to oscillate, because the system is assumed without any damping
Part of this study has been conducted under an academic grant from the Ministry of Education and some under a special research fund from Tokyo Metropolitan Institute of Technology. Thanks are due to the colleagues of SPS 2000 working group. References 1. Nagatomo, M. , “1OMW Satellite Power System; A Space Station Mission Beyond 2000”, Space Power, vol. 6, no. 2, 1986, pp. 299-304. 2. Fukuzawa, S., Modi, V. J., Nagatomo, M., “On the Construction Methodology and Dynamical Formulation for the Proposed Solar Power Satellites SPS2000”, ISTS 94e-06, 19th International Symposium on Space Technology and Science (ISTS), Yokohama, Japan, 1994 3. Shigehara., M., Fukuzawa, S. : A Construction Methodology of the Large Space Structures under Gravity Gradient Stabilization, 3rd Pacific International Congress of Aero and Space Technology (PICAST), Melbourne, Australia, 1995
Copyright 0 1996 ~temational
PII:
A Construction Methodology of the Large Solar Array System of a Solar Power Satellite M. Shigehara* and A. Ohtaka** Dept. of Aerospace Engineering, Tokyo Metropolitan Institute of Technology,
1. INTRODUCTION
ABSTRACT
Japan
(SPS2000)
SPS2000 is a technology demonstration program, as summarized in Table 1, for future Solar Power Satellites (SPS) in Japan. The spacecraft with a triangular configuration will fly on a low Earth equatorial orbit as shown in Figure 1. One face having a transmitting antenna is orienting toward the Earth and the other two faces have amorphous Si flexible solar arrays to generate 18 MW solar power. On the ground, the receiving antennas of wired mesh rectenna are placed to collect power from the satellite. The study on SPS2000 is conducted by the SPS2000 working group and the main purpose of the study is: 1) to design SPS2000 demo model 2) to examine technical feasibility and to develop the technologies for constructing such a large space infrastructure in a simple and cost effective way.
A construction methodology of the large solar array system is investigated, assuming the SPS 2000 model of ISAS, the experimental model for the future Solar Power Satellite in Japan. The structure elements such as trusses, beams and solar arrays are transported from the ground 10 times by using a rocket. First, the basic module is assembled and then the subsequent modules are attached towards N-S direction. The construction sequence how to form this basic module has already been examined under the gravity gradient environment. This paper discusses how to extend the flexible solar array sheets over this module. First, the dynamic model when extending an array is developed, by including the moment of inertia and angular momentum change during the extension. Then, the dynamic behaviors during extension are examined. When an extension is made by one array by one, the momentum by extending array and the moment of inertia change induce larger oscillation around all three axes of the basic module. Thus the method to minimize these changes by extending the pair of the arrays simultaneously is studied for reducing the attitude vibration. In conclusion, a method of extending two arrays simultaneously is judged desirable for a practical application.
l
l
l l
l
Copyright 0 1995 by the International Astronautical Federation. All rights reserved.
l
l
* Professor, ** Graduate student 6-6, Asahigaoka, Hino, Tokyo, 19 1 JAPAN
Technology demonstration program for future Solar Power Satellite (SPS) in Ja$%cept Study, R & D by ISAS SPS2000 WG Orbit : Low(1 IOOkm), equatorial, circular Power 18 MW (on orbit), 1OMW on ground (300kW average) Transmitting Ant. : Phased array, retrodirective (2.45GHz, 150m ) Receiving Ant. : Wired mesh rectenna (2km ) Total mass : 240ton, more than 10 times launches by Ariane 5 Table 1 SPS 2000
223
46rh IAF Congress
224
2. Summary of the Previous Studies
TransmittinghtcMa
/
I\ ’
1 IOOkm
Figure 1 Configuration
of SPS2OOO
Various efforts have being done and are continuing in the various fields for these goals, including an assembling robot, structural members, solar cells, transmitting phased arrays and receiving rectenna. Among them, one of the most critical problems is how we can construct a large structure with solar arrays in space under a passive stabilization such as to use the gravity gradient (G.G.) torque. This is one of the focus of our recent studies. The following topics have already been covered in the previous studies2’.3’. 1) The basic construction scenario has been established for the basic module and the subsequent assembling of the 10 modules for the final configuration. In these analyses, the static stability condition for G. G. was used so as to keep the moment of inertia ratio (MOIR) of the system within a stability region. 2) The dynamic stability has been studied to assure stability during construction for such as configuration change, and construction speed. By these analyses, the effect of an initial rate, main truss extension and robot motion are clarified. 3) The characteristics of disturbing torques such as solar radiation, geomagnetic torque and air drag have been studied. The significant source of the disturbances for SPS2000 is from a solar radiation pressure, which creates the larger torque around pitch. The measure to cancel out this disturbance is proposed by using a geomagnetic torque as control torque. This paper is intended to study the effect of the solar array extension on the surface of the basic triangular module. This may create the angular momentum change and inertia matrix change, and may cause some transient attitude deviation.
2.1 Basic Construction Sequence from Static Stabtlity To be stable under G. G. torque, the parameters relating with MOIR of the system must stay in the stability region as shown in Fig.2, which is derived from Euler equations. By satisfying this static condition, the basic construction sequence was proposed as in Fig. 3. The preferable sequence was Cl*C2+C3, where C 1: frame truss extension along outward local vertical C2: frame truss opening to triangular shape C3: beam truss extension along orbit normal
Figure 2 Expansion Module
(Cl) vedcal
I
((2)
f
Sequence
of Basic
extension of frame
~CUIICopening to triangular shape (0)
beam expansion
* 8
(ST) start PI
cnfg.
F’2 cnfg.
(BU) built up of basic module
Figure 3 G. G. Stability Ratio
Map for Inertia
46th 1AFCongress
2.2 Effect of Configuration Change from Dvnamic Stabilitv Dynamic behaviors by such as an initial rate, transient due to configuration change, robot motion and flexibility of the joints give additional stability restraints. These effects can be studied through a multi-chain bodies model with flexible joints. One of the typical results is shown in Fig. 4, where the dynamic behavior was examined by extending a truss along the local vertical with a speed of 90 ml 100 min. As shown in Fig. 4 a, the moment of inertia (MOI) in pitch and roll increases by more than 10 times after one orbital period. According to the change of this
Af=llMO[kJ] L
= 66027[4m’l
I,
=13887[kgm’]
I, = 2643O(~m’]
n M=lUco[kg]
1,=11147759[4m’] 1,,=1122619[kgm’]
I, =2643O[kgm*]
225
MOI, the attitude of the system deviates backwards from the local vertical. It is due to momentum conservation; since MO1 increases in pitch, the angular rate must decrease to keep the system angular moment constant. After a completion of the extension, G. G. torque restores the deviation and keeps the system stable. Also, the case of decreasing MOI, by taking an example of frame opening to a triangular shape, was examined. Through opening, h401 in pitch reduces to l/3 and this causes a significant increase of angular rate of the system, resulting in the attitude drift forwards from the local vertical. The results show the system becomes unstable, if an opening speed exceeds over a certain limit. This is because G.G. Torque cannot overcome attitude deviation before it reaches 90 degrees. This case is a critical, since a decreasing MO1 reduces G. Restoring torque. It was concluded that a careful attention must be paid for a construction speed when the construction produces decreasing MO1 in pitch. The effect of slew motion of robots and free joints is not significant.
Figure 4a Main Truss Extension along Local Vertical
3. Array Extension 3.1 Definition After a completion of construction of the basic module (hereafter, called as base structure, which does not include the solar array), the solar arrays will be extended on the two surfaces of the triangular base structure. The arrays are assumed to be extended from their containers, #O, 1, 2, 3 attached on the comers of the base structure by using a robot. Physical dimension of the base structure is shown in Fig. 5. Each solar
#0 solar array container
A
)1 \
#3 solar array\
1
51m Figure 4b Attitude Histories under Truss Extension along Local Vertical
Figure 5 Basic .Module with Solar Arrays
46th IAF Congress
226
array weighs 230 kg and the total mass of the module about 8 tons. Attitude requirement for SPS2OOOis assumed within +5”
c.g. of roll parts of # i solar array
in roll/ pitch and within +lO’ in yaw. During extension, the configuration may change in an unsymmetrical fashion and the produced momentum by extending the array with certain speed will create the angular momentum change of the base structure. These changes are supposed to create an additional transient attitude motion of the system and these will be investigated here.
3.2 Mathematical Model The orbit fixed coordinate frame @(x,,y,,b) and the base structure (triangle) fixed frame Q(x,y,z) are defined as in Fig.6, where the origins of both frames coincide. The angles($), w,A) define the attitude deviation of the base structure. The “system frame” is defined as the one of which axes and origin coincide respectively with the principal axes and the origin of the base structure, combined with the solar arrays. Thus, the “system frame” represents the current configuration during extension of the array. The solar array sheet is extended from its container as shown in Fig.7, where 1 is the rolled out length, r radial thickness of the remaining array and b, thickness of the array sheet. From this geometrical con-
c.g. of the base structure Figure 7 Geometry of Solar Array Extension
figuration, the following equations are derived.
&__ 4
--q$p z (2x) &
(I)
dy,= 4 dl;
(2)
a, = tan (3)
North Pole
dr dy; - -i sin( y,,. + 77,)+ sin CY,- 1;COSCX, dt. I drt, _ d/1 q COS( Y,,~ + q,) fi a, 4
sin
(4)
dczi _ dV;
4 Z
South Pole {X.Y.Z}:
inertia frame x,.y,,zo}: orbit fixed frame 1 (x,y,z}: base structure fixed frame Figure 6 Coordinate Frames
dl;
dyi dl,
(6)
where i is the parameters relating with the ith solar array and pi is the rotation angle of the role part of i th solar array. From conservation of momentum and Euler equations, the equations of motion of the base structure are formulated as follows.
46th Iti
221
congress
(7)
(10) creates an angular momentum in pitch. Thus the base structure must rotate in the opposite direction to compensate this momentum change. In the example, the base structure rotates forward. A slight increase of the angular momentum in roll and yaw also causes the base structure rotate in the opposite direction around these two axes. Figure 9 shows variation of the inertia matrix of the system. Since the solar array
where fi,, : position vector of c.g. of the base structure from that of the system. fi : position vector of c.g. of the system from the Earth subscript si, ri : the respective sheet and role part of the i th solar array c;,, : angular velocity of the base structure
*la -tU
bracket [ , ] : outer products of vector, ex.
5.0 101
0.0 loo
&bF=SixFxb [ -1
D :inertia matrix a: initial angular momentum of the system 0 : angular momentum of the system
-5.0 101
-1.0
-2.0
4 Analysis and Discussion
in 0.5 orbit. Figure 8 shows the angular momentum change of the #O panel with respect to the system frame. The array extension velocity
101 0
0. I
0.2
0.3
0.4
05
orbits
Figure 8 Angular Momentum of #O Array during Extension (extension in 0.5 orbit)
8.0 70 F 00 c
6.0
B 2 ._
5.0
b z
4.1 Extension of the #O panel At first, motion of the base structure is examined when a single array is extended
10,
-1.5 1w
given by gravity gradient torque. The equation (7) shows the translational motion of the base structure with respect to the system frame, and Eq.(9) the rotational motion of the base structure. The first bracket of Eq. (9) is the inertia matrix of the system and the first term of the second bracket is the anguler momentum of solar panels around the c.g. of the system. These equations represent the motion of base structure caused by a change of the inertia matrix and motions of the solar panels around the c.g.
hY
2
40 30 20 00
oi
02
03
0.4
05
arbtu
Figure 9 Inertia Matrix of the System during #o Array Extension (extension in 0.5 orbit)
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46th IAF Congress
--b -
0.0
2.0
pitch’ ....l ... ro,,’ . ..*... . ..*e..
-pitch
roll YIW
4.0
6.0
8.0
10.0
orbits
Figure 10 Attitude Histories of Base Structure when #O Array extended in 0.5 Orbit is extended along the base structure in an unsymmetrical fashion, the products of inertia are created, causing the principal axes of the system frame deviate from that of the base structure. The restoring gravity gradient torque acts around the principal axes and vibrates the system around these axes. The vibration seen from the base structure frame becomes offset from that around the principle axes due to these products of inertia. In Fig. 10 are shown the resulting attitude motion, where the solid lines are the actual attitude variations around the pitch, roll, yaw in the base structure frame. As a total, the base structure will rotate in a positive direction around pitch at the beginning due to a negative angular momentum given by extending array. Then it rotates in a negative direction due to an opposite change of angular momentum to stop extension. After completion of the extension, the system is in almost steady oscillation under the gravity gradient. However, due to the product of inertia, an oscillation has an offset from null axis as shown in the dashed lines. Around the roll and yaw, the change of the angular momentum and inertia is relatively small and creates a small oscillation around roll. In yaw, this small disturbance produces a larger error, since the G.G. Restoring torque is smaller around yaw. 4.2 Symmetrical Extension To keep the deviation as small as possible, the measures to reduce the angular momentum change and the product of inertia are studied, and the following two cases are investigated.
Case. 1 Extending two diagonal arrays (#Oand #Q) at the same time In this case, there are produced no angular momentum change and product of ine&a during extension in pitch and roll. However, a slight change of MO1 in pitch creates a rotational rate change from an initial orbital rate. This is due to conserve the angular momentum, and results in an oscillation as shown in Fig. 11. Because the configuration is not symmetrical around yaw during extension, there are still momentum change and some deviation of the principle axis in yaw. These factors produce the attitude deviation in yaw asinFig.11. Case 2 Extending four arrays at the same time. Figure 12 shows the motion of the base structure when the four arrays are extended at the same time. In this case, due to a symmetrical configuration, there occur no momentum changes in all three axes and no principal axis deviation during extension. Thus, the attitude around roll and yaw do not change, in spite of the change of MO1 around these axes, if there is no initial rate. Attitude around pitch becomes in an OScillation by the same reasons as in case 1. The effect of extension speed can be studied by the following simplified analysis. If there is no gravity gradient torque applied, the motion around pitch axis is expressed from the angular momentum con-
---*a-- pilch’ . ..a.-. roll’ . . . . &... y*w’
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Eigure 11 Attitude Histories of Base Structure when #O and ##2Array extended in 0.5 Orbit)
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46th IAF Congress
--8-
pitch(O.Sorbit)
-8-
cdch(O.lorbit)
device. Under the G. Stabilization, some sort of the damping device, active or passive, is mandatory and with this damper, the oscillation is sure to be damped out.
5 CONCLUSION
0.0
2.0
4.0
6.0
8.0
10.0
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Figure 12 Attitude Histories of Base Structure #O, #l, #2, #3 Arrays extended in 0.1 and 0.5 Orbit)
(11) where y(t) is pitch angle and R is orbital rate. If the panels are extended in 0.1 orbit, the pitch angle vo,,(T) at the end of extension, t = T = 0.1 orbit is
wo,(T)= (H(T)- T>Q
(12)
The effect of solar array extension along the base structure is investigated, under the gravity gradient stabilization. The momentum change due to an extension rate of the array causes the attitude deviation and make the system in an oscillation. Also, unsymmetrical extension creates the inertia matrix change, which causes the principal axes deviate from that of the base structure. This principal axes travel gives an offset to an oscillation and makes the system oscillate larger in the base-structure frame. For reducing a change of angular momentum and product of inertia, such symmetrical extension is proposed as two diagonal arrays or four corn& arrays extending simultaneously. The latter reduces a change in an ideal fashion, but the former is also acceptable, if a proper damping device is implemented. If we take into consideration of the number of robot required to extend the array, the simultaneous extension of two diagonal arrays is recommended.
where ACKNOWLEDGMENTS If the panels are extended in 0.5 orbit, the pitch angle va J (5T) at the end of extension, t=5T is & 0) = !, (t/5) 0.5
t/&5T)
= R/;r
(14)
01
+” ??
- R(5T)
5(H(T)- T)Q
(15)
Wo.s(5T) = 5Wo.0)
(16)
=
Thus, This means the angle deviation becomes 5 times larger if extension is done in 5 more times. However, in an actual case, the G.G. restoring torque will suppress an increase of the deviation, which is shown also in Fig. 12. The longer the extension speed, the larger deviation is produced, but the influence is not significant. In all above calculation, the system seems to continue to oscillate, because the system is assumed without any damping
Part of this study has been conducted under an academic grant from the Ministry of Education and some under a special research fund from Tokyo Metropolitan Institute of Technology. Thanks are due to the colleagues of SPS 2000 working group. References 1. Nagatomo, M. , “1OMW Satellite Power System; A Space Station Mission Beyond 2000”, Space Power, vol. 6, no. 2, 1986, pp. 299-304. 2. Fukuzawa, S., Modi, V. J., Nagatomo, M., “On the Construction Methodology and Dynamical Formulation for the Proposed Solar Power Satellites SPS2000”, ISTS 94e-06, 19th International Symposium on Space Technology and Science (ISTS), Yokohama, Japan, 1994 3. Shigehara., M., Fukuzawa, S. : A Construction Methodology of the Large Space Structures under Gravity Gradient Stabilization, 3rd Pacific International Congress of Aero and Space Technology (PICAST), Melbourne, Australia, 1995