Integrated Orbit, Attitude, and Structural Control System Design for Space Solar Power Satellites

11 Pages·2001·0.49 MB·English

by  Wie, Bong

AIAA 2001-4273
INTEGRATED ORBIT, ATTITUDE, AND
STRUCTURAL CONTROL SYSTEM DESIGN
FOR SPACE SOLAR POWER SATELLITES
Bong Wie∗ Carlos Roithmayr†
Arizona State University NASA Langley Research Center
Tempe, Arizona 85287-6106 Hampton, Virginia 23681-2199
Abstract
(1 km diameter). The total mass was estimated to be
The major objective of this study is to develop an in- 50×106 kg. A ground or ocean-based rectenna (recti-
tegrated orbit, attitude, and structural control system fying antenna) measuring 10×13 km would receive the
architecture for very large Space Solar Power Satellites microwave beam on the earth and deliver up to 5 GW
(SSPS) in geosynchronous orbit. This study focuses on of electricity.
the 1.2-GW “Abacus” SSPS concept characterized by a In1995,NASArevisitedtheSpaceSolarPower(SSP)
3.2×3.2kmsolar-arrayplatform,a500-mdiametermi- concept to assess whether SSP-related technologies had
crowavebeamtransmittingantenna,anda500×700m advancedenoughtoaltersignificantlytheoutlookonthe
earth-tracking reflector. For this baseline Abacus SSPS economic and technical feasibility of space solar power.
configuration, we derive and analyze a complete set of The “Fresh Look” study (Ref. 3), conducted by NASA
mathematical models, including external disturbances during 1995-1997, found that in fact a great deal had
such as solar radiation pressure, microwave radiation, changed and that multi-megawatt SSP satellites appear
gravity-gradienttorque,andotherorbitperturbationef- viable, with strong space applications. The study also
fects. The proposed control system architecture utilizes found that ambitious research, technology development
a minimum of 500 1-N electric thrusters to counter, si- and validation over a period of perhaps 15-20 years
multaneously, the cyclic pitch gravity-gradient torque, are required to enable SSP concepts to be considered
the secular roll torque caused by an offset of the center- “ready” for commercial development.
of-mass and center-of-pressure, the cyclic roll/yaw mi- Recent studies by NASA as part of the SSP Ex-
crowave radiation torque, and the solar radiation pres- ploratory Research and Technology (SERT) program
sure force whose average value is about 60 N. have produced a variety of new configurations of Space
Solar Power Satellites (SSPS), including the “Abacus”
configuration, as described in Refs. 4–6. Some of these
1 Introduction
configurations, such as the “Sun Tower” configuration,
are based on the passive gravity-gradient stabilization
A renewed interest in space solar power is spurring
concept. However, most other configurations require
a reexamination of the prospects for generating large
three-axis attitude control to maintain continuous sun
amounts of electricity from large-scale, space-based so-
tracking of the solar arrays in the presence of external
lar power systems. Peter Glaser (Refs. 1–2) first pro-
disturbances including the gravity-gradient torque. A
posed the Satellite Solar Power Station (SSPS) concept
cylindrical configuration, which is not affected by the
in 1968 and received a U.S. patent on a conceptual de-
troublesomepitchgravity-gradienttorque,hasalsobeen
sign for such a satellite in 1973. As a result of a series
considered by NASA (Ref. 6).
of technical and economic feasibility studies by NASA
This study focuses on the 1.2-GW “Abacus” satellite
and Department of Energy in the 1970s, an SSPS ref-
configurationshowninFigure1. ThisAbacussatelliteis
erence system was developed in the late 1970s. The
characterizedbyitssimpleconfigurationconsistingofan
1979 SSPS reference system, as it is called, featured a inertially oriented, 3.2 × 3.2 km solar-array platform, a
very large solar array platform (5.3×10.7 km) and a
500-m diameter microwave beam transmitting antenna
double-gimballedmicrowavebeamtransmittingantenna fixed to the platform, and a 500 × 700 m rotating re-
∗Professor, Dept. of Mechanical & Aerospace Engineering, flector that tracks the earth. Some unique features of
[phone],[email protected],AssociateFellowAIAA. the Abacus satellite relative to the 1979 SSPS reference
†AerospaceEngineer,[email protected],(757)864- system are:
6778, Senior Member AIAA. Copyright (cid:1)c2001 by the American
InstituteofAeronauticsandAstronautics,Inc. Allrightsreserved. • Thetransmittingantennaisnotgimballed;instead,
1
Table 1: Geometric and mass properties of the 1.2-GW
Prismatic
3200 m
Structure Abacus satellite
Solar array mass 21 ×106 kg
Transmitting antenna mass 3 × 106 kg
Roll Reflector mass 0.8 × 106 kg
Total mass m = 25 × 106 kg
1
Platform area A = 3200 m × 3200 m
Area-to-mass ratio A/m = 0.4 m2/kg
Roll inertia J = 2.8 × 1013 kg-m2
1
Pitch inertia J = 1.8 × 1013 kg-m2
Yaw 2
Yaw inertia J = 4.6 × 1013 kg-m2
3200 m 3
3 cm-cp offset 200 m (along pitch axis)
cm-cp offset (uncertainty) ±20 m (along roll axis)
Nadir
Table 2: Solar pressure and microwave radiation distur-
Transmitting bances
Antenna
2
(500 m) Pitch RF Reflector Solar pressure force (4.5E-6)(1.3)(A) = 60 N
Orbit (500 m x 700 m) Solar pressure torque (roll) 60 N × 200 m
Normal
Solar pressure torque (pitch) 60 N × 20 m
Reflector radiation force 7 N (rotating force)
Figure 1: Baseline 1.2-GW “Abacus” satellite configu-
Reflector radiation torque 7 N ×1700 m
ration (Refs. 4–6).
an azimuth roll-ring mounted, rotating reflector isgiveninTable1,togetherwiththetotalmassandarea
provides earth pointing of the microwave beam. of the spacecraft. The mass of the reflector is approx-
• The rotating reflector design thus eliminates mas- imately 3% of the total mass; therefore, the reflector’s
sive rotary joint and slip rings of the 1979 SSPS masscanbeneglectedintheanalysisofattitudemotion,
reference system. simplifyingthetaskintwoimportantrespects. First,the
• Links activated by ball-screw mechanisms tilt the Abacus satellite can be treated as a single body rather
reflector to point to ground stations at various lat- thanamultibodyspacecraft. WhentheAbacussatellite
itudes isregardedasrigid,thespacecraft’smomentsandprod-
ucts of inertia for a set of axes fixed in the solar array
The objectives of this study, in support of the SERT
donotvarywithtime. Second,whentheunsymmetrical
program of NASA, are: (i) to develop preliminary con-
mass distribution of the reflector is left out of account,
cepts for orbit, attitude, and structural control of very
the principal axes of inertia of the spacecraft with re-
large SSPS using a variety of actuators such as control
spect to the spacecraft’s mass center are parallel to the
moment gyros, momentum wheels, and electric propul-
roll, pitch, and yaw axes illustrated in Figure 1. The
sion thrusters; (ii) to develop mathematical models, de-
momentsofinertiafortheseaxes,henceforthconsidered
fineatop-levelcontrolsystemarchitecture,andperform
to be principal moments of inertia, are given in Table
control system design and analysis for a baseline Aba-
1. The center of pressure is located 100 m below the
cus satellite configuration in geosynchronous orbit; and
geometric center of the square platform, the center of
(iii) to determine the required number, size, placement,
mass is located 300 m below the geometric center along
mass, and power for the actuators to control the orbit,
thepitchaxis,and±20%overalluncertaintyinthemass
attitude, and structural motions of the baseline Abacus
properties should be considered in control design.
satellite.
2.2 External Disturbances
2 Mass Properties, Disturbances,
and Control Requirements External disturbances acting on the Abacus satellite in-
clude: solar radiation pressure force, microwave radia-
tion force, gravity-gradient torque, and other orbit per-
2.1 Geometric Properties
turbationforces. Someofthesedisturbanceswith±20%
The three major parts of the Abacus satellite and their overalluncertaintiesaresummarizedinTable2. Distur-
dimensionsareshowninFigure1; themassofeachpart bance torques in units of N-m, due to solar pressure,
2
Table 3: Orbit parameters and control requirements Table 4: A large single-gimbal CMG
Cost $1M
Earth’s gravitational parameter µ = 398,601 km3/s2 Momentum 7,000 N-m-s
Geosynchronous orbit (e,i≈0) a = 42,164 km Max torque 4,000 N-m
Orbit period 23 hr 56 min 4 sec Peak power 500 W
Orbit rate n = 7.292 ×10−5 rad/sec Mass 250 kg
Stationkeeping accuracy ±0.1 deg Momentum/mass 28 N-m-s/kg
Solar array pointing accuracy ±0.5 deg for roll/pitch
Microwave beam pointing accuracy ±5 arcmin
counter the cyclic gravity-gradient torque simply be-
comes
microwave radiation, cm-cp offset, and cm-cp offset un-
3n2
certainty,canbeexpressedalongtheplatform-fixedcon- u =− (J −J )sin2nt (3)
trol axes as: 2 2 3 1
Roll: d ≈12,000−11,900cosnt (1a) with peak values of ±143,000 N-m. If angular mo-
1
mentum exchange devices, such as momentum wheels
Pitch: d ≈1200 (1b)
2 (MWs)orcontrolmomentgyros(CMGs), aretobeem-
Yaw: d3 ≈ −11,900sinnt (1c) ployedforpitchcontrol,thepeakangularmomentumto
be stored can then be estimated as
where n is the orbital rate of the Abacus satellite and t
is time. The constant pitch disturbance torque of 1200 3n
N-m is due to the assumed cm-cp offset of 20 m along Hmax = 2 (J3−J1)=2×109 N-m-s (4)
the roll axis, and ±20% uncertainty in this disturbance
model should also be considered in control design. In This is is about 100,000 times the angular momentum
addition to these disturbances, gravity-gradient distur- storage requirement of the International Space Station
bance torques are also acting on the Abacus satellite. (ISS). The ISS is controlled by four double-gimballed
It is assumed that the electric currents circulate in the CMGs with a total momentum storage capability of
solar array structure in such a way that magnetic fields about 20,000 N-m-s. The double-gimballed CMGs em-
canceloutandtheAbacussatelliteisnotaffectedbythe ployed by the ISS have a momentum density of 17.5 N-
magnetic field of the earth. m-s/kg,andfutureadvancedflywheelsmayhavealarger
momentum density of 150 N-m-s/kg. Basic characteris-
2.3 Orbit Parameters and Control Re- tics of a large single-gimbal CMG are also summarized
quirements in Table 4.
Basedontheprecedingdiscussion,itcanbeconcluded
Basicorbitalcharacteristicsandcontrolrequirementsfor
thatatraditionalmomentummanagementapproachus-
theAbacussatelliteingeosynchronousorbitaresumma-
ing conventional CMGs (or even employing future ad-
rized in Table 3.
vanced flywheels) is not a viable option for controlling
very large Space Solar Power Satellites.
3 Technical Issues To meet the momentum storage requirement of very
largeSSPS,aconceptofconstructinglarge-diametermo-
3.1 Momentum Storage Requirement mentum wheels in space has been studied in the late
1970s (Ref. 7). In an attempt to resolve the angular
Assuming that the gravity gradient torque is the only momentumstorageproblemoflargesun-pointingspace-
external disturbance torque acting along the pitch axis, craft,aquasi-inertialsun-pointing,pitchcontrolconcept
weconsiderthepitchequationofmotionofarigidspace- was also developed by Elrod in 1972 (Ref. 8), and fur-
craft in geosynchronous orbit given by ther investigated by Juang and Wang in 1982 (Ref. 9).
However, such a “free-drift” concept is not a viable op-
3n2
J θ¨ = (J −J )sin2θ +u (2) tion for the Abacus satellite because of the large pitch
2 2 2 3 1 2 2
attitude peak error of 18.8 deg and its inherent sensi-
where J , J , and J are, respectively, the roll, pitch, tivity with respect to initial phasing and other orbital
1 2 3
and yaw principal moments of inertia; θ is the pitch perturbations.
2
angle measured from the LVLH (local vertical and local Because the pitch gravity-gradient torque becomes
horizontal) reference frame; n is the orbit rate; and u naturally zero for cylindrical, spherical or beam-like
2
is the pitch control torque. satellites with J = J , a cylindrical SSPS configura-
1 3
For continuous sun pointing of the Abacus platform tion was also studied by NASA to simply avoid such a
with θ = nt, the pitch control torque required to troublesome pitch gravity-gradient torque problem.
2
3
Solar pressure constant P Formostpracticalcasesofsatelliteswithsmallangles
ofφ,theSRPperturbationforceperunitmassissimply
Surface area A
modeled as
n f(cid:3)=P(1+ρ)(A/m)(cid:3)s (7)
Incoming photons
s where ρ is the overall surface reflectance (0 for a black
φ body and 1 for a mirror) and A/m is the area-to-mass
φ ratio.
Let the SRP perturbation acceleration be expressed
as
f(cid:3)=f (cid:3)e +f (cid:3)e +f (cid:3)e (8)
r r θ θ z z
Specularly where {(cid:3)e ,(cid:3)e ,(cid:3)e } is a set of unit vectors of the so-called
r θ z
reflected photons perifocal reference frame. Ignoring the effects of sea-
sonal variations of the sun vector, we simply obtain
Figure 2: Solar radiation pressure force acting on an f ≈fsinθ and f ≈fcosθ where f =P(1+ρ)(A/m)
r θ
ideal flat surface. and θ is the true anomaly.
From the orbit perturbation analysis (Refs. 11–12),
3.2 Solar Radiation Pressure and Large we have
Area-to-Mass Ratio da 2
= √ {f esinθ+f (1+ecosθ)} (9a)
dt n 1−e2 r θ
Despite the importance of the cyclic pitch gravity- √
de 1−e2
gradienttorque,thisstudyshowsthatthesolarradiation = {f sinθ+f (cosθ+cosE)} (9b)
pressure force is considerably more detrimental to con- dt na r θ
trol of the Abacus satellite (and also other large SSPS) where θ and E are the true and eccentric anomalies,
because of an area-to-mass ratio that is very large com- respectively. For geosynchronous satellites with e ≈ 0,
pared to contemporary, higher-density spacecraft. we obtain
The significant orbit perturbation effect of the solar da 2 2f
= f = sinθ
radiationpressureonlargespacecraftwithlargearea-to- dt n θ n
mass ratios has been investigated by many researchers ⇒∆a=0per day (10)
in the past. A detailed physical description of the solar
and
radiationpressurecanbefoundinarecentbookonsolar
de 1
sailing by McInnes (Ref. 10). = (f sinθ+2f cosθ)
Thesolarradiationpressureforcesareduetophotons dt na r θ
1
impinging on a surface in space, as illustrated in Figure = (fsin2θ+2fcos2θ)
2. Assuming that a fraction, ρ , of the impinging pho- na(cid:2) (cid:3)
s
tons is specularly reflected, a fraction, ρ , is diffusely f 3 1
d = + cos2θ
reflected, and a fraction, ρ , is absorbed by the surface, na 2 2
a
we have 3πf
⇒ ∆e≈ per day (11)
ρs+ρd+ρa =1 (5) n2a
The solar radiation pressure (SRP) force acting on an The solar radiation pressure effect on the longitude
ideal flat surface is then expressed as change can also be found as
(cid:1) (cid:2) (cid:3) (cid:4)
dn 3nda 3n2 3
F(cid:3) =PA((cid:3)n·(cid:3)s) (ρa+ρd)(cid:3)s+ 2ρs((cid:3)n·(cid:3)s)+ 23ρd (cid:3)n λ¨ = dt =−2a dt =−2anfθ =−afθ
3f
(6) =− cosθ (12)
where P = 4.5×10−6 N/m2 is the nominal solar ra- a
diation pressure constant, A is the surface area, (cid:3)n is a For the Abacus satellite, we have
unit vector normal to the surface, and(cid:3)s is a unit vector
Area-to-mass ratioA/m≈0.4m2/kg
pointing from the sun to satellite.
SRP force≈60N
For an ideal case of a perfect mirror with ρ =ρ =0
d a
and ρ =1, we have SRP perturbation acceleration≈2.4×10−6 m/s2
s
3π(4.5×10−6)(1.3)A/m
F(cid:3) =2PAcos2φ(cid:3)n ∆e= ≈1×10−4 per day
n2a
where Acosφ is called the projected area of the surface ⇒Longitude drift ∆λ=2∆e≈0.0115deg/day
under consideration. Also for an ideal case of a black ⇒maximum∆e≈0.018 at the mid year
body with ρs =ρd =0 and ρa =1, we have ⇒maximum∆λ=2∆e≈2deg; maximum∆a≈1.8km
F(cid:3) =PAcosφ(cid:3)s
4
Solar Array
Structure
x 104
4.2168 Space Solar Power Satellite
4.2166
m) dm
a (k4.2164
4.2162 N n3 R ρ b 3 a1
0 5 10 15 20 25 30
x 10-3
3
2 n1 n2 Rc a3 b1
e Earth
1 b2
a
2
00 5 10 15 20 25 30 Orbital Path
0.06 Microwave
Transmitter
0.04
i (deg)0.02 Reflector
0 Figure 4: A large Space Solar Power Satellite (SSPS) in
0 5 10 15 20 25 30
geosynchronous orbit.
Figure3: OrbitsimulationresultsoftheAbacussatellite
with the effects of the earth’s oblateness and triaxiality,
correspond to an initial position and velocity of
luni-solar perturbations, and 60-N solar radiation pres-
sure force (time in units of orbits).
(cid:3)r =XI(cid:3)+YJ(cid:3)+ZK(cid:3) =−11698.237I(cid:3)−40508.869J(cid:3) (km)
(cid:3)v =2.954I(cid:3)−0.853J(cid:3) (km/s)
Consequently, orbit eccentricity control using high-I
sp
ionenginesbecomesnecessary. Theyearlypropellantre- where {I(cid:3),J(cid:3),K(cid:3)} is a set of unit vectors of the Earth-
quirementtocountertheapproximately60-Nsolarpres- Centered-Inertial (ECI) reference frame.
sure force can be estimated as In the next section, we develop an attitude dynam-
(60)(24×3600×365) ics model of sun-pointing spacecraft in geosynchronous
∆m= ≈40,000kg/year orbit for attitude control system architecture design.
5000×9.8
whereanionenginewithI of5000secisassumed. De-
sp 4 Attitude Equations of Motion
tailed discussions of an electric propulsion system pro-
posed for the Abacus satellite will be presented later in of Sun-Pointing Spacecraft
this paper.
Typicalnorth-southandeast-weststationkeepingma- Consider a spacecraft in circular orbit, as illustrated in
neuvers for the Abacus satellite will also require Figure 4. A local vertical and local horizontal (LVLH)
(cid:2) (cid:2) (cid:3)(cid:3) reference frame A with its origin at the center of mass
∆V
∆m=m 1−exp − ≈30,000 kg/year of an orbiting spacecraft has a set of unit vectors
gIsp {(cid:3)a ,(cid:3)a ,(cid:3)a } with (cid:3)a along the orbit direction, (cid:3)a per-
1 2 3 1 2
pendicular to the orbit plane, and(cid:3)a toward the earth,
where m=25×106 kg, ∆V = 50 m/s per year, g =9.8 3
asillustratedinFigure4. TheangularvelocityofAwith
m/s2, and I =5000 sec.
sp respecttoaninertialorNewtonianreferenceframeN is
The results of 30-day simulations of orbital motion
of the Abacus satellite, with the effects of the earth’s ω(cid:3)A/N =−n(cid:3)a (13)
2
oblateness and triaxiality, luni-solar perturbations, and
60-N solar pressure force included, are shown in Figure where n is the constant orbital rate. The angular ve-
3. It is worth noting the extent to which eccentricity locity of the body-fixed reference frame B with basis
and inclination are perturbed. vectors {(cid:3)b ,(cid:3)b ,(cid:3)b } is then given by
1 2 3
The initial values used in the simulations corre-
spond to a circular, equatorial orbit of radius 42164.169 ω(cid:3)B/N =ω(cid:3)B/A+ω(cid:3)A/N =ω(cid:3)B/A−n(cid:3)a (14)
2
km; therefore, the initial orbital elements are: a =
42164.169 km and e = i = Ω = ω = 0. The epoch used where ω(cid:3)B/A is the angular velocity of B relative to A.
to calculate the solar and lunar positions, as well as the Todescribetheorientationofthebody-fixedreference
earth’s orientation in inertial space, is March 21, 2000. frame B with respect to the LVLH reference frame A in
In order to place the spacecraft at an initial terrestrial terms of three Euler angles θ (i = 1,2,3), consider the
i
longitude of 75.07 deg (one of the stable longitudes), a sequence of C (θ )←C (θ )←C (θ ) from the LVLH
1 1 3 3 2 2
true anomaly θ of 253.89 deg is used. These elements reference frame A to a body-fixed reference frame B.
5
For this rotational sequence, we have J ω˙ −(J −J )ω ω =−3n2(J −J )C C
2 2 3 1 3 1 3 1 33 13
     J ω˙ −(J −J )ω ω =−3n2(J −J )C C
(cid:3)b C C C (cid:3)a 3 3 1 2 1 2 1 2 13 23
 (cid:3)b1 = C11 C12 C13  (cid:3)a1  (15) (20)
2 21 22 23 2
(cid:3)b3 C31 C32 C33 (cid:3)a3 where
where C = cosθ cosθ , C = sinθ , C = C =−sinθ cosθ
11 2 3 12 3 13 13 2 3
−sinθ2cosθ3, etc. C23 =cosθ1sinθ2sinθ3+sinθ1cosθ2
Assuming that the gravity gradient torque is the only
C =−sinθ sinθ sinθ +cosθ cosθ
external disturbance torque acting along the pitch axis, 33 1 2 3 1 2
we obtain the rotational equation of motion of a rigid for the sequence of C (θ ) ← C (θ ) ← C (θ ) under
1 1 3 3 2 2
body with an angular momentum H(cid:3) in a circular orbit consideration. For this rotational sequence we have the
as following kinematic differential equations:
(cid:11) (cid:12) (cid:11) (cid:12)
      
dH(cid:3) dH(cid:3) θ˙ cθ −cθ sθ sθ sθ ω 0
dt ≡ dt +ω(cid:3)B/N ×H(cid:3) =M(cid:3) (16)  θ˙1 = 1  03 cθ1 3 −1sθ3  ω1 + n 
N B θ˙23 cθ3 0 sθ1c1θ3 cθ1cθ13 ω23 0
where {d/dt} indicates differentiation with respect to
N (21)
time in reference frame N and {d/dt} indicates differ-
B
entiationwithrespecttotimeinreferenceframeB. The where cθ ≡cosθ and sθ ≡sinθ .
i i i i
gravity-gradient torque M(cid:3) is expressed in vector-dyadic One may linearize Eqs. (20)–(21) “about” an LVLH
form as: orientationwhileadmittingalargepitchangleasfollows.
M(cid:3) =3n2(cid:3)a ×Jˆ·(cid:3)a (17) Assume θ and θ remain small, allow θ to be large,
(cid:13) 3 3 1 3 2
wheren= µ/R3istheorbitalrate,(cid:3)a ≡−R(cid:3) /R ,and assume ω1 and ω3 are small, and ω2 is equal to the sum
c 3 c c of a small quantity and −n. The attitude equations of
Jˆis the inertia dyadic of the spacecraft with respect to
motion that are linear in the small quantities can then
its mass center (Refs. 12–13).
be obtained as (Ref. 12):
SinceH(cid:3) =Jˆ·ω(cid:3)B/N,theattitudedynamicalequations
of motion can be rewritten as J θ¨ +(1+3cos2θ )n2(J −J )θ −n(J −J +J )θ˙
1 1 2 2 3 1 1 2 3 3
Jˆ·ω(cid:3)˙ +ω(cid:3) ×Jˆ·ω(cid:3) =3n2(cid:3)a ×Jˆ·(cid:3)a (18) +3(J2−J3)n2(sinθ2cosθ2)θ3 =u1+d1
3 3 J θ¨ +3n2(J −J )sinθ cosθ =u +d
2 2 1 3 2 2 2 2
where ω(cid:3) ≡ ω(cid:3)B/N and ω(cid:3)˙ = {dω(cid:3)/dt}N ≡ {dω(cid:3)/dt}B. Ex- J θ¨ +(1+3sin2θ )n2(J −J )θ +n(J −J +J )θ˙
pressing ω(cid:3), (cid:3)a , and Jˆ in terms of basis vectors of the 3 3 2 2 1 3 1 2 3 1
3 +3(J −J )n2(sinθ cosθ )θ =u +d
body-fixed reference frame B as 2 1 2 2 1 3 3
(22)
ω(cid:3) =ω (cid:3)b +ω (cid:3)b +ω (cid:3)b (19a)
1 1 2 2 3 3
where u and d are control and disturbance torques,
(cid:3)a =C (cid:3)b +C (cid:3)b +C (cid:3)b (19b) i i
3 13 1 23 2 33 3 respectively.
(cid:14)3 (cid:14)3 However,foraquasi-inertiallystabilized,sun-pointing
Jˆ= J (cid:3)b(cid:3)b (19c)
ij i j SSPS in geosynchronous orbit with small body rates, ω
i
i=1j=1 (i = 1,2,3), and small roll/yaw angles, θ and θ , the
1 3
weobtaintheattitudedynamicalequationsofmotionin kinematic differential equations, (21), can be linearized
matrix form as: in the small quantities, as follows:
   θ˙1 ≈ω1 (23a)
 J11 J12 J13  ω˙1  θ˙2 ≈ω2+n (23b)
J J J ω˙
21 22 23 2 θ˙ ≈ω (23c)
J J J ω˙ 3 3
  31 32 33  3 
0 −ω ω J J J ω The attitude equations of motion of a quasi-inertially
3 2 11 12 13 1
+ ω 0 −ω  J J J  ω  stabilized, sun-pointing spacecraft with small roll and
3 1 21 22 23 2
 −ω2 ω1 0  J31 J32 J33 ω3  yaw angles, θ1 and θ3, can then be found as
0 −C C J J J C J θ¨ =−3n2(J −J )(cos2θ )θ
33 23 11 12 13 13 1 1 2 3 2 1
=3n2 C33 0 −C13  J21 J22 J23  C23  −3(J −J )n2(sinθ cosθ )θ +u +d
2 3 2 2 3 1 1
−C C 0 J J J C
23 13 31 32 33 33 J θ¨ =−3n2(J −J )sinθ cosθ +u +d
2 2 1 3 2 2 2 2
The dynamical equations of motion about the body- J θ¨ =−3n2(J −J )(sin2θ )θ
3 3 2 1 2 3
fixed principal axes become
−3(J −J )n2(sinθ cosθ )θ +u +d
2 1 2 2 1 3 3
J ω˙ −(J −J )ω ω =−3n2(J −J )C C (24)
1 1 2 3 2 3 2 3 23 33
6
Node Number Locations for Normal Modes Results 10-2 10-2
1117 (Nine3 2N0o0dmes S(8h0o warnr aiyns R)ed) 1377 Force at #1 to #11100--64 Force at #1 to #125111000---864
1217
10-8 10-10
10-3 10-2 10-1 100 10-3 10-2 10-1 100
Hz Hz
10-2 10-2
ys)
(16 arra 469 549 729 Force at #1 to #549111000---864 Force at #1 to #1377111000---864
10-10 10-10
10-3 10-2 10-1 100 10-3 10-2 10-1 100
Hz Hz
125
1 325
Figure 6: Bode magnitude plots of reduced-order trans-
Transmitter Reflector (500m x 750m) fer functions from an input force at node #1 to various
(500m diameter)
output locations.
Front View (Abaqus support truss in back)
topic of continuing practical as well as theoretical inter-
Figure5: SelectedFEMnodelocationsforcontrolanaly-
est. However, a significant control-structure interaction
sisanddesign(CourtesyofTimCollinsatNASALaRC).
problem, possible for such a very large Abacus platform
(3.2×3.2km)withtheloweststructuralmodefrequency
ThepitchattitudeanglerelativetotheLVLHframe,θ , ofabout0.002Hz, isavoidedsimplybydesigninganat-
2
is not restricted to be small, but it may be regarded as titude control system with very low bandwidth (< orbit
asum, θ =nt+δθ , whereδθ isasmallpitchattitude frequencyof1×10−5 Hz). Theproposedlow-bandwidth
2 2 2
error. Kinematical and dynamical differential equations attitudecontrolsystem(tobepresentedinthenextsec-
can then be made linear in the small quantities ω , ω , tion), however, utilizes a concept of cyclic-disturbance
1 2
ω , θ , δθ , and θ . For such a case, Eqs. (21) become accommodation control to provide the required ±5 ar-
3 1 2 3
cmin pointing accuracy of the Abacus platform in the
θ˙1 ≈ω1; δθ˙2 ≈ω2; θ˙3 ≈ω3 (25) presence of large, but slowly varying, external distur-
bances and dynamic modeling uncertainties. Conse-
and Eqs. (24) become
quently, the flexible structural control problem is not
J θ¨ +3n2(J −J )[(cos2nt)θ +(1/2)(sin2nt)θ ] further elaborated in this study, while a structural dy-
1 1 2 3 1 3 namicinteractionproblemwiththermaldistortionneeds
=u +d
1 1 to be further investigated in a future study.
J2δθ¨2+3n2(J1−J3)[(cos2nt−sin2nt)δθ2 Variousstructuralconceptsforprovidingtherequired
+(1/2)sin2nt]=u +d stiffness and rigidity of the Abacus platform have been
2 2
J θ¨ +3n2(J −J )[(sin2nt)θ +(1/2)(sin2nt)θ ] presented in Refs 4–5. Selected node locations for con-
3 3 2 1 3 1
trol analysis and design are shown in Figure 5. Typ-
=u +d
3 3 ical pole-zero patterns of reduced-order transfer func-
(26) tions can be seen in Figure 6. Computer simulation re-
sultsofareduced-orderstructuralmodelwiththelowest
where (θ ,δθ ,θ ) are the small roll, pitch, and yaw at-
1 2 3 16modes,confirmthatthecontrol-structureinteraction
titude errors of a sun-pointing spacecraft, respectively.
problem can be simply avoided by a low-bandwidth at-
Equations (24) or (26) are the attitude equations of
titude control system.
motion of the Abacus satellite for control design in the
presenceoftheexternaldisturbances,d ,describedasin
i
Eqs. (1).
6 Control System Architecture
5 Structural Dynamic Models The area-to-mass ratio of the Abacus satellite, A/m =
0.4m2/kg,relativelylargewhencomparedto0.02m2/kg
Dynamics and control problems of large flexible plat- of typical geosynchronous communications satellites, is
formsinspace,suchasthesquareAbacussatellite,have akeyparametercharacterizingtheverylargesizeofthe
been investigated by many researchers in the past. The Abacus satellite. If left uncontrolled, this can cause a
flexible structural dynamics and control problem is a cyclic drift in the longitude of the Abacus satellite of 2
7
deg,eastandwest. Thus,inadditiontostandardnorth-
Table5: Electricpropulsionsystemforthe1.2-GWAba-
south and east-west stationkeeping maneuvers for ±0.1
cus satellite
deg orbit position control, active control of the orbit ec-
Thrust, T ≥ 1 N
centricityusingionthrusterswithhighspecificimpulse,
Specific impulse, I =T/(m˙g) ≥ 5,000 sec
I , becomes mandatory. Furthermore, continuous sun sp
sp Exhaust velocity, V =I g ≥ 49 km/s
tracking of the Abacus satellite requires large control e sp
Total efficiency, η =P /P ≥ 80%
torques to counter various disturbance torques. A con- o i
Power/thrust ratio, P /T ≤ 30 kW/N
trol system architecture developed in this study utilizes i
Mass/power ratio ≤ 5 kg/kW
properlydistributedionthrusterstocounter,simultane-
Total peak thrust 200 N
ously, thecyclicpitchgravity-gradienttorque, andsolar
Total peak power 6 MW
radiation pressure.
Total average thrust 80 N
Total average power 2.4 MW
6.1 Electric Propulsion System Number of 1-N thrusters ≥ 500
Total dry mass ≥ 75,000 kg
Basiccharacteristicsofanelectricpropulsionsystemfor
Propellant consumption 85,000 kg/year
theAbacussatellitearesummarizedinTable5. Approx-
imately85,000kgofpropellantperyearisrequiredforsi- Note: T = m˙Ve, Po = 12m˙Ve2 = 12TVe, Po/T = 12Ve =
multaneous orbit, attitude, and structural control using idealpower/thrustratio, Pi/T = 21ηVe, Isp =T/(m˙g)=
aminimumof5001-Nelectricpropulsionthrusterswith V /g, V = I g where g = 9.8 m/s2, m˙ is the exhaust
e e sp
I =5,000sec. Theyearlypropellantrequirementisre- mass flow rate, P is the input power, and P is the
sp i o
ducedto21,000kgifanI of20,000seccanbeachieved output power.
sp
(as was assumed for the 1979 SSPS reference system).
As I is increased, the propellant mass decreases but
sp
haust gas from an ion thruster consists of large num-
the electric power requirement increases; consequently,
bers of positive and negative ions that form an essen-
the mass of solar arrays and power processing units in-
tiallyneutralplasmabeamextendingforlargedistances
creases. Based on a minimum of 500 1-N thrusters, a
in space. It seems that little is known yet about the
mass/power ratio of 5 kg/kW, and a power/thrust ra-
long-term effect of such an extensive plasma on geosyn-
tio of 30 kW/N, the total dry mass (power processing
chronoussatelliteswithregardtocommunications,solar
units, thrusters, tanks, feed systems, etc.) of an electric
cell degradation, environmental contamination, etc.
propulsion system proposed for the Abacus satellite is
estimated as 75,000 kg.
The capability of present electric thrusters is orders 6.2 Control System Architecture
of magnitude below that required for the Abacus satel-
lite. Ifthexenonfueled,1-kWlevel,off-the-shelfionen- A control system architecture developed in this study is
gines available today are to be employed, the number of showninFigure7. Theproposedcontrolsystemutilizes
thrusterswouldbeincreasedto15,000. Theactualtotal properlydistributedionthrusterstocounter,simultane-
number of ion engines will further increase significantly ously, the cyclic pitch gravity-gradient torque, the secu-
when we consider the ion engine’s lifetime, reliability, lar roll torque caused by an offset of the center-of-mass
duty cycle, and redundancy. andcenter-of-pressure,thecyclicroll/yawmicrowavera-
For example, the 2.3-kW, 30-cm diameter ion engine diation torque, and the solar radiation pressure force
of the Deep Space 1 spacecraft has a maximum thrust whose average value is about 60 N.
level of 92 mN. Throttling down is achieved by feeding A significant control-structure interaction problem,
less electricity and xenon propellant into the propulsion possible for such very large Abacus platform with the
system. Specific impulse ranges from 1,900 sec at the lowest structural mode frequency of 0.002 Hz, is simply
minimum throttle level to 3,200 sec. avoided by designing an attitude control system with
In principle, an electric propulsion system employs very low bandwidth (< orbit frequency). However, the
electrical energy to accelerate ionized particles to ex- proposedlow-bandwidthattitudecontrolsystemutilizes
tremely high velocities, giving a large total impulse for a concept of cyclic-disturbance accommodating control
a small consumption of propellant. In contrast to stan- to provide ±5 arcmin pointing of the Abacus platform
dardpropulsion,inwhichtheproductsofchemicalcom- inthepresenceoflarge,butslowlyvarying,externaldis-
bustionareexpelledfromarocketengine,ionpropulsion turbances and dynamic modeling uncertainties.
is accomplished by giving a gas, such as xenon (which The concept of cyclic-disturbance accommodation
is like neon or helium, but heavier), an electrical charge control has been successfully applied to a variety of
and electrically accelerating the ionized gas to a speed space vehicle control problems (Ref. 12), including the
ofabout30km/s. Whenxenonionsareemittedatsuch Hubble Space Telescope, the International Space Sta-
high speed as exhaust from a spacecraft, they push the tion, flexible space structures, and halo orbit control.
spacecraft in the opposite direction. However, the ex- High-bandwidth, colocated direct velocity feedback, ac-
8
System Uncertainties Solar Pressure
(inertias, c.m.,c.p. etc.) Secular Roll Thrust force direction
Disturbance
Cyclic Disturbance Torque
Rejection Filters #5 #6
-
θ1c= 0 Low-Bandwidth + θ1
Roll Thrusters
+ PID Controller + u
u1c 1 High-Bandwidth Roll/Yaw #11 #1 #9
u3c Feedforward Control Torque Commands Active Dampers CDyonupamledics
+ +
Low-Bandwidth Yaw Thrusters
θ3c= 0 - PID Controller + u3 θ3
Cyclic Disturbance
Rejection Filters Microwave Radiation #4 cp #2
Cyclic Roll/Yaw
Disturbance Torque Roll
cm

Feedforward Control
Torque Command Gravity-Gradient Torque
u2c= 3n2(−J1−-−J 3)(sin 2nt)/2 −3n2(J1−-J 3)(sin 2 θ 2 )/2 #12 #3 #10
θ2c= nt + Low-Bandwidth + Pitch + Pitch
Sun-Pointing - PID Controller + u2 Thrusters + Dynamics θ
Pitch Angle 2
#8 #7
Command Cyclic Disturbance High-Bandwidth
Rejection Filters Active Dampers
Pitch
LVLH Pitch Angle
Figure 7: An integrated orbit, attitude, and structural
Roll: 1/3 Pitch: 2/4 Yaw: 5/6/7/8
control system architecture employing electric propul-
sion thrusters. Orbit Eccentricity, Roll/Pitch Control: 1/3, 2/4

E/W and Yaw Control: 9/10/11/12
tive dampers may need to be properly distributed over
the platform. Detailed technical discussions of practical N/S and Yaw Control: 5/6/7/8
controlsystemsdesignforspacevehiclesinthepresence
structural flexibility as well as persistent external dis- Figure 8: Placement of a minimum of 500 1-N electric
turbances can be found in Ref. 12. Thus, theoretical propulsion thrusters at 12 different locations, with 100
aspects of the control law design problem of the Abacus thrusters each at locations #2 and #4.
satellite are not elaborated upon in this paper.
Placement of approximately 500 1-N electric propul-
sion thrusters at 12 different locations is illustrated in
FigInurceo8n.trast to a typical placement of thrusters at the Error (deg) 105
fe8onumcreincsioymrsnitzeeemrss,,rotehl.lge/.p,pitrecomhppotlshoeryduesdptelafrocrceomtuhepenltin1sg9hs7o9awsSnwSiPenlSlFarisegfutehrree- Roll Attitude -1-0500 1 2 3 4 5 6
emxicniitmatuiomnooff5p0la0tfioornmenoguitn-oesf-polfa1n-eNsttrhurcutsutralelvmeloadrees.reA- Error (deg)105
qtruoilr.edWfohrensimreulilatabnileitoyu,sliaftettiitmued,edauntdysctyactiloe,nkloeweperintghcrounst- Attitude 0
level,andredundancyofionenginesareconsidered,this Pitch -50 1 2 3 4 5 6
number will increase significantly.
6.3 Control Simulation Results Attitude Error (deg)1005
Control system simulation results of an illustrative case w
with 10-deg initial attitude errors are shown in Figure Ya-50 1 2 3 4 5 6
Time (Orbits)
9. In this simulation, various dynamic modeling un-
certainties (i.e., ±20 % uncertainties for the moments Figure 9: Control simulation results for the proposed
and products of inertia, cm-cp offset, external distur- attitude control system architecture.
bances,etc.) areincluded. Theproposedlow-bandwidth
9
compared to contemporary, higher-density spacecraft.
x 104
4.2168 A key parameter that characterizes the Abacus satel-
4.2166 liteisitsarea-to-massratio,A/m,of0.4m2/kg,whichis
m)
a (k4.2164 relativelylargewhencomparedto0.02m2/kgfortypical
geosynchronous communications satellites.
4.21620 5 10 15 20 25 30 A significant control-structure interaction problem,
1.5x 10-4 possible for such very large, flexible Abacus platform
with the lowest structural mode frequency of 0.002 Hz,
1
is simply avoided by designing an attitude control sys-
e
0.5 temwithverylowbandwidth(<orbitfrequency). How-
0 ever, the proposed low-bandwidth attitude control sys-
0 5 10 15 20 25 30
x 10-3 tem effectively utilizes a concept of cyclic-disturbance
2
accommodatingcontroltoprovide±5arcminpointingof
1.5
i (deg) 1 tvhaeryAinbga,cuexstpelrantafolrdmisitnurtbhaenpcreesseanncdeodfylnaarmgei,cbmutosdleolwinlyg
0.5
uncertainties.
0
0 5 10 15 20 25 30 Approximately 85,000 kg of propellant per year is re-
quired for simultaneous orbit, attitude, and structural
Figure10: Orbitcontrolsimulationresultswithcontinu-
control using a minimum of 500 1-N electric propul-
ous (non-impulsive) eccentricity and inclination control
sion thrusters with I = 5,000 sec. The total dry mass
(time in units of orbits). sp
(power processing units, thrusters, tanks, feed systems,
etc.) of an electric propulsion system proposed for the
Abacus satellite is estimated as 75,000 kg.
attitude control system, which effectively utilizes the
concept of cyclic-disturbance accommodation control,
maintains the required ±5 arcmin, steady-state point- 7.2 Recommendations for Future Re-
ing of the Abacus platform in the presence of large, but search
slowlyvarying,externaldisturbancesanddynamicmod-
eling uncertainties. The total thrusting force from the The baseline control system architecture developed for
roll/pitch thrusters #1 through #4 nearly counters the the Abacus satellite requires a minimum of 500 ion en-
60-N solar radiation pressure force; however, any resid- gines of 1-N thrust level. The capability of present elec-
ual ∆V caused by various uncertainties should be cor- tricthrustersisordersofmagnitudebelowthatrequired
rectedduringstandardeast-weststationkeepingmaneu- fortheAbacussatellite. Ifthexenonfueled,1-kWlevel,
vers. off-the-shelf ion engines available today, are to be em-
Orbit control simulation results with the effects of ployed, the number of thrusters would be increased to
earth’s oblateness and triaxiality, luni-solar perturba- 15,000. The actual total number of ion engines will fur-
tions, 60-N solar radiation pressure force, and simul- ther increase significantly when we consider the ion en-
taneous orbit and attitude control thruster firings are gine’s lifetime, reliability, duty cycle, redundancy, etc.
shown in Figure 10. It can be seen that the eccentricity Consequently, a 30-kW, 1-N level electric propulsion
and inclination are all properly maintained. The feasi- thruster with a specific impulse greater than 5,000 sec
bility of using continuous (non-impulsive) firings of ion needs to be developed for the Abacus satellite if an ex-
thrusters for simultaneous attitude and orbit control is cessively large number of thrusters is to be avoided.
demonstrated in this study; however, a further detailed Several high-power electric propulsion systems are
orbit control study is needed. The orbit control prob- currently under development. For example, the NASA
lem of geosynchronous satellites is a topic of continuing T-220 10-kW Hall thruster recently completed a 1,000-
practical interest (Refs. 14–16). hr life test. This high-power (over 5 kW) Hall thruster
provides 500 mN of thrust at a specific impulse of 2,450
sec and 59% total efficiency. Dual-mode Hall thrusters,
7 Summary and Recommenda-
which can operate in either high-thrust mode or high-
tions Isp mode for efficient propellant usage, are also being
developed.
The exhaust gas from an electric propulsion system
7.1 Summary of Study Results
forms an essentially neutral plasma beam extending for
Despite the importance of the cyclic pitch gravity- large distances in space. Because little is known yet
gradient torque, this study has shown that the solar ra- about the long-term effect of an extensive plasma on
diation pressure force is considerably more detrimental geosynchronous satellites with regard to communica-
to control of the Abacus satellite (and also other large tions, solar cell degradation, etc, the use of lightweight,
SSPS)becauseofanarea-to-massratiothatisverylarge space-assembled large-diameter momentum wheels may
10