Thrust Vector Control of Solar Sail Spacecraft

Bong Wie∗

Arizona State University, Tempe, AZ 85287-6106

This paper presents a comprehensive mathematical formulation of the thrust vectorcontrol (TVC) design problem of solar sail spacecraft. A TVC system is part of an attitudeand orbit control system (AOCS) of a sailcraft, and it maintains the proper orientation ofits solar sail to provide the desired thrust vector pointing. The solar pressure thrust vectordirection of a sailcraft is often described by its cone and clock angles with respect to aparticular orbital reference frame. This paper describes various forms of orbital trajectoryequations, which employ two different sets of the cone and clock angles, for the designand simulation of solar sail TVC system. In particular, a quaternion-based TVC/AOCSarchitecture is proposed for solar sailing missions because of its simple interface with theTVC steering commands of the desired cone and clock angles. In practice, any sailcraftTVC system will require frequent updates of both orbital parameters and TVC steeringcommands for frequent trajectory corrections because of the inherent difficulty to pre-cisely model the solar radiation pressure and to accurately point the true thrust vectordirection. The frequency of orbit determination update and TVC command update ismission-dependent; it is determined by many factors such as the trajectory dispersions,the target-body ephemeris uncertainty, and the operational constraints.

I. Introduction

Solar sails have the potential to provide cost effective, propellantless propulsion that enables longermission lifetimes, increased scientific payload mass fraction, and access to previously inaccessible orbits (e.g.,high solar latitude, retrograde heliocentric, and non-Keplerian).1−5 In support of such solar sailing missions,the solar sailing trajectory optimization problem has been extensively studied in the past.6−10 The recentadvances in lightweight deployable booms, ultra-lightweight sail films, and small satellite technologies arespurring a renewed interest in solar sailing and the missions it enables, and near-term solar sailing missionsand trajectories are under current development by NASA/JPL.11−13

For typical solar sailing trajectory optimization problems, trajectory models which are decoupled fromattitude dynamics were used in the past.6−10 However, the effect of attitude motion of large solar sailson the solar sailing trajectory is of current practical concern for solar sail mission designs. Consequently,a six-degree-of-freedom, orbit-attitude coupled dynamical model of solar sail spacecraft is currently underdevelopment for the solar sail trajectory optimization and simulation.14

This paper is intended to provide a comprehensive mathematical formulation of the thrust vector control(TVC) design problem of solar sail spacecraft. As part of an attitude and orbit control system (AOCS) of asailcraft, a TVC system maintains the proper orientation of its solar sail to provide the desired thrust vectorpointing. The solar pressure thrust vector direction of a sailcraft is often described by its cone and clockangles with respect to a particular orbital reference frame such as an osculating orbital reference frame. Thispaper describes several forms of orbital trajectory equations employing two different sets of the cone andclock angles. In particular, a quaternion-feedback attitude control system architecture, developed previouslyin Refs. 15-16, is proposed for the thrust vector control of solar sails because of its simple interface with thetypical trajectory control inputs: the cone and clock angles of the solar thrust vector.

∗Professor, Dept. of Mechanical & Aerospace Engineering, (480) 965-8674, Fax (480) 965-1384, bong.wie@asu.edu, AssociateFellow AIAA.

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Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Y

Z

K

X

Vernal Equinox

I

r

J

y

r

y

f

y

f

Figure 1. Heliocentric ecliptic coordinates (X, Y, Z) and spherical coordinates (r, ψ, φ).

Any solar sail TVC system, with the inherent inability to precisely model the solar radiation pressureand to accurately point the true thrust vector direction, will require frequent updates of both orbital pa-rameters and TVC steering commands for frequent trajectory corrections.17 The frequency of such orbitdetermination and TVC command updates is determined by many factors, such as the trajectory disper-sions, the target-body ephemeris uncertainty, the calibration of a solar radiation pressure model, and theoperational constraints. Although the solar sail navigation problem is of practical importance for future so-lar sail missions,17 this paper will focus on the TVC/AOCS design problem assuming that frequent updatesof both orbital parameters and TVC steering commands will be provided from a solar sail navigation andguidance system.

The remainder of this paper is outlined as follows. In Section II, various forms of orbital trajectoryequations employing two different sets of the cone and clock angles are described. Section III presentsexamples of solar sail trajectory design and simulation. Section IV describes a quaternion-based TVC/AOCSarchitecture and a set of the attitude-orbit coupled equations for TVC design and simulation. A TVC/AOCSdesign example is discussed in Section VI.

II. Solar Sail Orbital Dynamics

In this section we examine various forms of orbital trajectory equations for solar sail trajectory designand simulation. In general, the orbital equation of a sailcraft in a heliocentric orbit is simply described by

r +µ

r3r = F + G (1)

where r is the position vector of the sailcraft from the center of the sun, µ ≈ µ = 132715E6 km3/s2, F

is the solar radiation pressure force vector (per unit mass) acting on the sailcraft, and G is the sum of allthe perturbing gravitational forces (per unit mass) acting on the sailcraft. In this paper, all the perturbinggravitational forces are ignored without loss of generality.

A. Cone and Clock Angles

The orientation of the solar sail thrust vector, ideally normal to the sail plane, is often described in termsof the cone and clock angles. These two angles are the typical trajectory control inputs used in solar sailtrajectory optimization. There are at least two different sets of cone/clock angles used in the literature.In this section, the fundamentals of orbital equations of motion in various coordinates employing such twodifferent sets of cone/clock angles are described for the purposes of trajectory design, TVC design, andAOCS simulation.

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n

y

f

b

ra

y

f

r

2

3

n1 =

Figure 2. Cone angle α, clock angle β, and sailcraft orientation when α = β = 0.

Let I , J , K and r, ψ, φ be respectively a set of right-handed, orthonormal vectors of the heliocentricecliptic rectangular and spherical coordinate reference frames, as illustrated in Fig. 1. These two sets ofbasis vectors are related as

r

ψ

φ

=

cos φ 0 sinφ

0 1 0− sinφ 0 cos φ

cos ψ sinψ 0− sinψ cos ψ 0

0 0 1

I

J

K

(2)

where ψ and φ are called the ecliptic longitude and latitude of the sailcraft position, respectively; 0 ≤ ψ ≤360 deg and −90 deg ≤ φ ≤ +90 deg.

The sailcraft position vector is then expressed as

r = rr

= (r cos φ cos ψ)I + (r cos φ sinψ)J + (r sinφ)K= XI + Y J + ZK (3)

where r = |r| is the distance from the sun to the sailcraft,The orientation of a unit vector normal to the sail plane, n, is described in terms of the cone angle α and

the clock angle β, illustrated in Fig. 2, as follows:

n = (cos α)r + (sinα sinβ)ψ + (sinα cos β)φ (4)

wherecos α = r · n

cos β =r × (n × r)|r × (n × r)| · φ

0 ≤ α ≤ 90 deg

0 ≤ β ≤ 360 deg

As also illustrated in Fig. 2, the sailcraft body-fixed basis vectors 1, 2, 3 are assumed to be aligned withr, ψ, φ when α = β = 0, and the sailcraft roll axis is defined to be perpendicular to the sail surface; i.e.,1 ≡ n. The sailcraft body-fixed basis vectors 1, 2, 3 are then related to r, ψ, φ as follows:

123

=

cos α 0 sinα

0 1 0− sinα 0 cos α

1 0 00 cos β − sinβ

0 sinβ cos β

r

ψ

φ

(5)

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Y

Z h

r

X

Vernal Equinox

W

Line of Nodes

iJ

e

q

i

w

I

K

q

rk

Ecliptic Plane

I '

n

qd

ra

k

Figure 3. Orbital geometry (illustrated for a near-circular orbit).

Figure 4. Cone angle α and clock angle δ (Ref. 2).

Let r, θ, k be a set of basis vectors of an osculating orbital plane, as illustrated in Fig. 3. A different setof the cone and clock angles (α, δ) can then be defined as shown in Fig. 4. A body-fixed rotational sequenceto 1, 2, 3 from I , J , K is then described by successive coordinate transformations of the form18

C2(−α) ← C1(−δ) ← C3(θ) ← C3(ω) ← C1(i) ← C3(Ω)

which becomes

r

θ

k

=

cos(ω + θ) sin(ω + θ) 0− sin(ω + θ) cos(ω + θ) 0

0 0 1

1 0 00 cos i sin i

0 − sin i cos i

cos Ω sin Ω 0− sin Ω cos Ω 0

0 0 1

I

J

K

(6)

123

=

cos α 0 sinα

0 1 0− sinα 0 cos α

1 0 00 cos δ − sin δ

0 sin δ cos δ

r

θ

k

(7)

The orientation of a unit vector normal to the sail plane, n, is then described in terms of α and δ, asillustrated in Fig. 4, as follows:

n = (cos α)r + (sinα sin δ)θ + (sinα cos δ)k (8)

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andcos α = r · n

cos δ =r × (n × r)|r × (n × r)| · k

0 ≤ α ≤ 90 deg

0 ≤ δ ≤ 360 deg

B. Solar Radiation Pressure

An ideal model of the solar radiation pressure (SRP) is used here. The SRP force vector (per unit mass)acting on the sailcraft is described in various coordinates as follows:

F = F0(r · n)2n

= Fr r + Fψψ + Fφφ

= Rr + T θ + Nk

= FX I + FY J + FZK (9)

whereF0 =

(r⊕r

)2

ac (10)

where r⊕ = 1 AU = 149597870.691 km is the distance from the sun to the earth and ac is the so-calledcharacteristic acceleration of the sailcraft at 1 AU.

Furthermore, we have the following relationships:

Fr

Fψ

Fφ

= F0 cos2 α

cos α

sinα sinβ

sinα cos β

(11)

R

T

N

= F0 cos2 α

cos α

sinα sin δ

sinα cos δ

(12)

Fr

Fψ

Fφ

=

cos φ 0 sinφ

0 1 0− sinφ 0 cos φ

cos ψ sinψ 0− sinψ cos ψ 0

0 0 1

FX

FY

FZ

(13)

R

T

N

=

cos(ω + θ) sin(ω + θ) 0− sin(ω + θ) cos(ω + θ) 0

0 0 1

1 0 00 cos i sin i

0 − sin i cos i

cos Ω sin Ω 0− sin Ω cos Ω 0

0 0 1

FX

FY

FZ

(14)

C. Orbital Equations in Rectangular Coordinates

The orbital equation of motion in vector form, Eq. (1), can be expressed in the rectangular coordinates asfollows:

X = −µX

r3+ FX (15a)

Y = −µY

r3+ FY (15b)

Z = −µZ

r3+ FZ (15c)

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where r =√

X2 + Y 2 + Z2 and

FX

FY

FZ

=

cos ψ − sinψ 0sin ψ cos ψ 0

0 0 1

cos φ 0 − sinφ

0 1 0sinφ 0 cos φ

Fr

Fψ

Fφ

(16)

sin ψ =Y√

X2 + Y 2; cos ψ =

X√X2 + Y 2

sinφ =Z

r; cos φ =

√X2 + Y 2

r

When the orientation of n is described in terms of α and δ, as shown in Fig. 4, we use the followingcoordinate transformation:

FX

FY

FZ

=

cos Ω − sin Ω 0sin Ω cos Ω 0

0 0 1

1 0 00 cos i − sin i

0 sin i cos i

cos(ω + θ) − sin(ω + θ) 0sin(ω + θ) cos(ω + θ) 0

0 0 1

R

T

N

For this case of employing (R, T, N), we need to determine (Ω, i, ω, θ) for given (X, Y, Z, X, Y , Z).The six classical orbital elements (a, e, i,Ω, ω, M) for given r = XI + Y J + Z K and v = XI + Y J + Z K

can be determined as follows:18

Leth = r × v (17)

µe = v × h − µ

rr = (v2 − µ

r)r − (r · v)v (18)

Then the eccentricity e and the semimajor axis a are found as

e = |e| (19)

a = − µ

2E (20)

where

E =v2

2− µ

r(21)

The right ascension of the ascending node, Ω, can be determined from a unit vector I ′ towards theascending node, given by

I ′ = cos ΩI + sin ΩJ (22)

which is perpendicular to both h and K; i.e.,

I ′ = K ×h

h(23)

The inclination angle i is obtained from

cos i = K ·h

h(24)

and the argument of the perigee, ω, is also obtained as

cos ω = I ′ · e

e(25)

where a proper quadrant correction must be made for e · K < 0.The true anomaly θ is obtained from

cos θ =r · ere

(26)

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where a proper quadrant correction must be made when r ·v > 0. The eccentric anomaly E is then obtainedfrom

tanE

2=

√1 − e

1 + etan

θ

2(27)

The mean anomaly M is obtained from Kepler’s equation of the form

M = E − e sinE (28)

D. Orbital Equations in Spherical Coordinates

The orbital equations of motion in the spherical coordinates (r, ψ, φ) are given by

r − rφ2 − rψ2 cos2 φ = − µ

r2+ Fr (29a)

rψ cos φ + 2rψ cos φ − 2rψφ sinφ = Fψ (29b)

rφ + 2rφ + rψ2 sinφ cos φ = Fφ (29c)

Let vr = r, vψ = rψ cos φ, and vφ = rφ, then we obtain

r = vr (30a)

ψ =1

r cos φvψ (30b)

φ =1rvφ (30c)

vr =1r(v2

ψ + v2φ) − µ

r2+ F0 cos3 α (30d)

vψ =1r(vψvφ tanφ − vrvψ) + F0 cos2 α sinα sinβ (30e)

vφ = −1r(v2

ψ tanφ + vrvφ) + F0 cos2 α sinα cos β (30f)

whereF0 =

(r⊕r

)2

ac

This set of six trajectory equations is often employed to find the time histories of optimal control inputs(α, β). By numerically integrating this set of trajectory equations for known time histories of (α, β), we mayobtain (X, Y, Z, X, Y , Z) as follows:

X = r cos φ cos ψ (31a)Y = r cos φ sinψ (31b)Z = r sinφ (31c)

X

Y

Z

=

cos φ cos ψ −r cos φ sinψ −r sinφ cos ψ

cos φ sinψ r cos φ cos ψ −r sinφ sinψ

sinφ 0 r cos φ

r

ψ

φ

(32)

E. Logarithmic Spiral Trajectory

An interesting case of applying the orbital equations of motion expressed in spherical coordinates is thelogarithmic spiral trajectory problem.2,19 Although such logarithmic spiral trajectories are not practicallyuseful for an interplanetary transfer between circular orbits, a simple steering law with a fixed sun angle isrequired.

For a simple planar case with φ = 0 and β = 0, we have

r − rψ2 = − µ

r2+ F0 cos3 α (33a)

rψ + 2rψ = F0 cos2 α sinα (33b)

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whereF0 =

(r⊕r

)2

ac

For this planar case, α is often called a pitch angle with −90 deg ≤ α ≤ 90 deg.By defining a sailcraft lightness number λ as

λ =F0

µ/r2=

r2⊕ac

µ=

(149597870E3)2

132715E15ac = 168.6284ac (34)

we rewrite the equations of motion as

r − rψ2 = −(1 − λ)µ

r2cos3 α (35a)

rψ + 2rψ = λµ

r2cos2 α sinα (35b)

For a fixed sail pitch angle α, a particular solution can be assumed as2,19

r = r0 exp(ψ tan γ) (36)

where γ is the flight path angle between the velocity vector and the transverse direction and it is often calledthe spiral angle. The sailcraft velocity can then be obtained as

v =√

r2 + (rψ)2

=√

µ

r

[1 − λ cos2 α(cos α − tan γ sinα)

]1/2(37)

Note that the solar sail velocity is always less than a local circular orbit velocity of√

µ/r. A relationshipbetween the pitch angle α and the spiral angle γ can be found as

sin γ cos γ

2 − sin2 γ=

λ cos2 α sinα

1 − λ cos3 α(38)

The transfer time from r0 to r is then given by

t − t0 =13(r3/2 − r

3/20 )

√2 cot γ

λµ cos2 α sinα(39)

For small spiral angles, we have

tan γ =2λ cos2 α sinα

1 − λ cos3 α(40)

and the transfer time becomes

t − t0 ≈ 13(r3/2 − r

3/20 )

√1 − λ cos3 α

λ2µ cos4 α sin2 α(41)

Finally, an optimal pitch angle α for minimizing the transfer time can be found from the followingequation:2,19

2 − 4 tan2 α

cos α(2 − tan2 α)= λ (42)

F. Gauss’s Form of the Variational Equations (Osculating Orbital Elements)

A set of six first-order differential equations, called Gauss’s form of the variational equations, in terms ofosculating orbital elements, are given by20

a =2a2

h[eR sin θ + T (1 + e cos θ)] ≡ 2a2

h[eR sin θ +

pT

r] (43a)

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e =√

p

µ[R sin θ + T (cos θ + cos E)] ≡ 1

hpR sin θ + [(p + r) cos θ + re]T (43b)

i =r cos(ω + θ)

hN (43c)

Ω =r sin(ω + θ)

h sin iN (43d)

ω = − r sin(ω + θ)h tan i

N +1eh

[−pR cos θ + (p + r)T sin θ] (43e)

θ =h

r2+

1eh

[pR cos θ − (p + r)T sin θ] (43f)

wherep = a(1 − e2)

r =p

1 + e cos θ≡ a(1 − e cos E)

h =√

µp = na2√

1 − e2

n =√

µ/a3

R

T

N

= F0 cos2 α

cos α

sinα sin δ

sinα cos δ

F0 =(r⊕

r

)2

ac

In particular, the inclination equation becomes

i =r cos(ω + θ)

hN =

λ

r

õ

pcos2 α sinα cos δ cos(ω + θ) (44)

We then obtain a simple sail-steering law for maximizing the rate of change of inclination as

α = tan−1(1/√

2) = 35.26 deg

and

δ =

0 deg for cos(ω + θ) ≥ 0

180 deg for cos(ω + θ) < 0

Although this simple steering law indicates that the clock angle has to change ±180 deg instantaneously, anequivalent attitude motion is a ±70-deg single-axis slew maneuver every half orbit (i.e., at ascending anddescending nodes). Furthermore, this simple steering law demonstrates an advantage of employing (α, δ)instead of (α, β).

III. Examples of Solar Sailing Trajectory Design and Simulation

A. Solar Polar Imager (SPI) Mission

Solar sails are envisioned as a propellantless, high-energy propulsion system for future space explorationmissions. NASA’s future missions enabled by solar sail propulsion include the Solar Polar Imager (SPI),L1-Diamond, Particle Acceleration Solar Orbiter (PASO), and Interstellar Probe, which are the Sun-EarthConnections (SEC) solar sail roadmap missions.3−5 In particular, the SPI Vision mission is currently beingfurther studied by NASA/JPL.13 Our current understanding of the Sun is limited by a lack of observationsof its polar regions. The SPI mission utilizes a large solar sail to place a spacecraft in a 0.48-AU heliocentriccircular orbit with an inclination of 75 deg. Viewing of the polar regions of the Sun provides a uniqueopportunity to more fully investigate the structure and dynamics of its interior, the generation of solarmagnetic fields, the origin of the solar cycle, the causes of solar activity, and the structure and dynamics ofthe corona.

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As illustrated in Fig. 5, the SPI mission consists of the initial cruise phase to a 0.48-AU circular orbit,the cranking orbit phase, and the science mission phase. A 160-m, 450-kg solar sail spacecraft is consideredfor such a solar sailing mission. A Delta II launch vehicle is able to inject the 450-kg SPI spacecraft intoan earth escaping orbit with C3 = 0.25 km2/s2, and then the sail is to be deployed. The SPI sailcraft firstspirals inwards from 1 AU to a heliocentric circular orbit at 0.48 AU, then the cranking orbit phase beginsto achieve a 75-deg inclination. The solar sail will be jettisoned after achieving the science mission orbit,and the total sailing time is approximately 6.6 yr. A set of the cone and clock angles, (α, β), chosen by CarlSauer (NASA/JPL) for a baseline SPI mission trajectory design, is also illustrated in Fig. 5.

Figure 6 shows an optimization-based trajectory design by Carl Sauer for achieving a circular orbit at 0.48AU with a 75-deg inclination. The monotonically decreasing semimajor axis and the corresponding variationof eccentricity can be seen in this figure during the initial cruise phase to 0.48 AU. The eccentricity remainsconstant during the orbit cranking phase. The eccentricity is finally nulled after cranking is complete.A somewhat complicated nature of the desired, optimal clock angle command can be noticed in Figure6, although the cone angle is nearly kept constant except for the final orbit correction phase to null theeccentricity.

A possibility of employing simple sail-steering laws (a combination of constant cone/clock angles), basedon Gauss’s form of the variational equations with the cone and clock angles (α, δ), was examined for theSPI mission. Figure 7 shows the result of applying such simple sail-steering laws with (α, δ) to the SPImission. The switch from orbit radius reduction to the cranking phase occurs once the sail reaches the targetsemimajor axis of 0.48 AU. However, an actual transition to the cranking orbit phase was executed at aproper orbital location such that the large eccentricity variation of the initial cruise phase can be removed ascan be seen in Figure 7. The simulation used a nominal characteristic acceleration of 0.3 mm/s2 (at 1 AU),an 8-deg initial inclination, and C3 = 0.25 km2/s2. The sailcraft achieves the desired semimajor axis andinclination but the final orbit is slightly eccentric. The corresponding three-dimensional orbital trajectory isillustrated in Fig. 8.

A comparison of the desired clock angle commands β and δ, shown in Figs. 6 and 7 respectively, suggeststhat the clock angle δ is a better choice for an actual TVC steering command implementation because of itssimplicity.

B. Solar Sailing Kinetic Energy Interceptor (KEI) Mission

A fictional asteroid mitigation problem was created by AIAA for the 2004/2005 AIAA Foundation Un-dergraduate Team Space Design Competition. A similar fictional asteroid mitigation problem, called theDefined Threat (DEFT) scenarios, has been created also for the 2004 Planetary Defense Conference. Oneof the four DEFT scenarios is about mitigating a fictional 200-m Athos asteroid with the predicted impactdate of February 29, 2016.

The fictional asteroid mitigation problem of AIAA is briefly described as follows. On July 4, 2004,NASA/JPL’s Near Earth Asteroid Tracking (NEAT) camera at the Maui Space Surveillance Site discovereda 0.205-km diameter Apollo asteroid designated 2004WR. This asteroid has been assigned a Torino ImpactScale rating of 9.0 on the basis of subsequent observations that indicate there is a 95% probability that2004WR will impact the Earth. The expected impact will occur in the Southern Hemisphere on January 14,2015 causing catastrophic damage throughout the Pacific region. The mission is to design a space systemthat can rendezvous with 2004WR in a timely manner, inspect it, and remove the hazard to Earth bychanging its orbit and/or destroying it. The classical orbital elements of 2004WR are given in the J2000heliocentric ecliptic reference frame as follows:

Epoch = 53200 TDB (July 14, 2004)a = 2.15374076 AUe = 0.649820926i = 11.6660258 degω = 66.2021796 degΩ = 114.4749665 deg

M = 229.8987151 deg

The STK 5.0.4 software package, with a 9th-order Runge-Kutta integrator with variable stepsize and the

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Mission DesignSolar Sail Trajectory Overview

Transfer Flight Path

• General Design Optimizes Thrust VectorPointing

* Cruise trajectory produces 15° heliocentricinclination change

* Thrust vector change rates are minimized

* Solar-vector to Sail-Normal-vector angle isconstrained to ≤ 45°

• 2-phase Approach Optimized for Insertionto OPS Orbit in ~6.8 years

* Cruise trajectory produces 15° heliocentricinclination change

* Cranking orbit effects ~53° inclinationchange

into the OPS orbit 60° heliocentric inclination

* Orbit trim is designed for final orbit shapingand velocity matching

Science OPS Orbit

• Designed for High Latitude Coverage with3:1 Earth Resonance

* Nodal phasing included for control of Earth-Sun-S/C angle

2012 Solar Sail Solar Polar Imager3:1 Resonance, R= 0.48 AU

75 Degrees Heliographic Inclinationac=.35 mm/s2 CP1

Sauer 9-14-04spi340-75-1074

1 2

3

4

5Earth

DATE ∆Days

∆Years

METDays

METYears

Launch C 3=0.25 km2/s

2 05/24/12 0 0 0 0.000

Start of Sail Phase 06/03/12 10 0.027 10 0.027

Start of Cranking Phase 12/10/14 920 2.519 930 2.546

End of Cranking Phase 02/05/19 1518 4.156 2448 6.702

Start of Science OPS Phase 03/02/19 25 0.068 2473 6.771

0

30

60

90

120

150

180

210

240

0 200 400 600 800 1000 1200 1400 1600 1800 2000TIME FROM LAUNCH (days)

Sauer 6-1-01

CONE

CLOCK

SA

IL N

OR

MA

L C

ON

E a

nd

CL

OC

K A

NG

LE

S (

deg

)

CONE & CLOCK ANGLES vs. TIME

ALL CASES:

Nbody Trajectory Integration

10 day Initial Coast

C3 = 0.25 km2/s2

ac =0.315 mm/s2

Trajectory CharacteristicsCone and Clock

SOLARVECTOR

SAIL PLANENORMAL

NORTHECLIPTIC

CLO

CK

CONE

PLANE TOSOLAR VECTOR

PROJECTION OFSAIL NORMAL

PROJECTION OFNORTH ECLIPTIC

• Cone Angles* 35° to 45° majority of mission

* Higher angles for short durations (post-injection,transition to cranking, pre-science orbit trim)

- Max cone angle <62°

• Clock Angles* NO abrupt changes in clock angle - actual maneuver

is a soft (2-days), single-axis maneuver of the sail normalfrom 35° below the solar vector to 35° above the solarvector. Result is to shift the projection of sail normalfrom one hemisphere to the other w.r.t. the North Ecliptic.

Figure 5. SPI mission trajectory design by Carl Sauer at NASA/JPL.13

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0 1 2 3 4 5 6 70.4

0.6

0.8

1

a (A

U)

0 1 2 3 4 5 6 70

0.05

0.1

e

0 1 2 3 4 5 6 70

50

100

Years

i (de

g)

0 1 2 3 4 5 6 70.4

0.6

0.8

1

r (A

U)

0 1 2 3 4 5 6 7

20

40

60

Con

e a

(de

g)

0 1 2 3 4 5 6 70

100

200

Years

Clo

ck b

(de

g)

Figure 6. Optimization-based trajectory design with (α, β) for the SPI mission (Carl Sauer, NASA/JPL).

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0 1 2 3 4 5 6 70.4

0.6

0.8

1

a (A

U)

0 1 2 3 4 5 6 7

0.02

0.04

0.06

0.08

e

0 1 2 3 4 5 6 70

50

100

Years

i (de

g)

0 1 2 3 4 5 6 7

0.6

0.8

1

r (A

U)

0 1 2 3 4 5 6 70

50

100

Con

e a

(deg

)

0 1 2 3 4 5 6 7-100

0

100

200

Years

Clo

ck d

(de

g)

Figure 7. SPI mission trajectory design using simple sail-steering laws with (α, δ).

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Earth

Asteroid 2004 WR at Aphelion(3.5533 AU)

Solar Sail SpacecraftImpact at Perihelion(0.7542 AU)Jan. 1, 2012

Earth Impact or FlybyJan. 14, 2015 Mars

2004WR Epoch 53200 TDBJuly 14, 2004 a = 2.1537 AU e = 0.6498 i = 11.6 deg w = 66.2 deg W = 114.4 deg M = 229.8 degPeriod = 3.16 years

200 m1.1E10 kg

X

Y

-0.5

0

0.5

1-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

-0.4

-0.2

0

0.2

0.4

SPI Trajectory: ac = 0.3 mm/s2, C

3 = 0.25 km2/s2

Y (AU)X (AU)

Z (

AU

)

Figure 8. SPI mission trajectory design result using (α, δ).

Figure 9. Illustration of the solar sailing KEI mission for intercepting/impacting/deflecting a near-Earthasteroid. The final, retrograde heliocentric orbit phase (starting from 0.25 AU) results in a head-on collisionwith the target asteroid at its perihelion (0.75 AU).

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planetary positions from JPL’s DE405, was used by AIAA to create this set of orbital parameters of 2004WR.It is further assumed that 2004WR is an S-class (stony-silicate) asteroid with a density of 2,720 kg/m3

and that its estimated mass is 1.1×1010 kg. If 2004WR is an M-class (nickel-iron) asteroid, then its estimatedmass would be 2.2 × 1010 kg.

A solar sailing kinetic energy interceptor (KEI) mission concept applied to the fictional asteroid mitigationproblem of AIAA is illustrated in Fig. 9. The proposed mission requires at least ten 160-m, 300-kg solar sailspacecraft with a characteristic acceleration of 0.5 mm/s2, as proposed in Ref. 21 as a viable near-term optionfor mitigating the threat posed by near-Earth asteroids (NEAs). The solar sailing phase of the proposedKEI mission, which is very similar to that of the SPI mission, is comprised of the initial cruise phase from1 AU to 0.25 AU, the cranking orbit phase (for a 168-deg inclination change), and the final retrograde orbitphase prior to intercepting the target asteroid at its perihelion.

Simple sail-steering laws based on Gauss’s form of the variational equations with the cone and clockangles (α, δ) were examined for the preliminary KEI mission design. Figure 10 shows the result of applyingthe simple sail-steering laws to the KEI mission. A solar sail trajectory optimization result for intercepting,impacting, and deflecting NEAs is also provided in Fig. 11. A detailed discussion of such optimal trajectorydesign for the KEI mission can be found in Ref. 22. In practice, the KEI mission will require precisionnavigation and guidance, and thus it will require frequent updates of the TVC steering commands.

IV. Solar Sail Attitude Dynamics and Control

As described in the preceding sections, various forms of orbital trajectory equations employing two setsof the cone and clock angles are available for the solar sail trajectory design and simulation. It was shownthat a preliminary trajectory design can be performed by employing simple sail-steering laws as applied toa set of orbital equations, called Gauss’s form of the variational equations with (α, δ).

In this section a quaternion-feedback attitude control system (ACS), previously developed in Refs. 15-16,is proposed for solar sail TVC applications because of its simple interface with the typical trajectory controlinputs: the desired cone and clock angles of the solar thrust vector. The practicality and simplicity of theproposed TVC approach will be demonstrated in this section. Furthermore, it will be shown that frequentupdates of the orbital parameters as well as the TVC steering commands will be required for any solar sailTVC system.

A. Solar Sail TVC/AOCS Architecture

One method of properly controlling the attitude of a large sailcraft is to employ small reflective control vanesmounted at the spar tips. Another method is to change the center-of-mass location relative to its center-of-pressure location. This can be achieved by articulating a control boom with a tip-mounted payload/bus.Various dynamical models and attitude control systems of solar sails, utilizing passive stabilization, spinstabilization, control vanes, a 2-axis gimbaled control boom, or translating/tilting sail panels, can be foundin Ref. 23.

A baseline TVC/AOCS architecture illustrated in Fig. 12, consisting of a propellantless primary ACSand a microthruster-based backup ACS, is proposed for a 160-m sailcraft of the SPI mission as well as theKEI mission. As illustrated in Fig. 13, the primary ACS employs trim control masses (TCMs) running alongmast lanyards for pitch/yaw control together with roll stabilizer bars (RSBs) at the mast tips for quadranttilt control. The robustness of such a propellantless primary ACS would be further enhanced by a backupACS utilizing tip-mounted, lightweight pulsed plasma thrusters (PPTs). Such a microPPT-based ACS canbe employed for attitude recovery maneuvers from off-nominal conditions as well as for a spin-stabilized safemode. It can also be employed as a backup to the conventional ACS of the spacecraft prior to sail deploymentand during pre-flight sail checkout operation, if necessary. A conventional bus ACS is required for the SPImission as the sail is jettisoned at the start of the science mission phase. As an alternative to conventionalapproaches, the microPPT-based ACS option promises lower mass, lower cost, and greater redundancy.

An attitude determination system (ADS), included in Fig. 12, is a critical subsystem of most spacecraftAOCS. An ADS of particular interest for solar sail applications is the Inertial Stellar Compass (ISC) recentlydeveloped by Draper Laboratory for an NMP ST6 flight validation experiment. The ISC is a miniature,low-power ADS developed for use with low-cost microsatellites. It is suitable for a wide range of future solarsail missions because of its low-mass, low-power, and low-volume design and its self-initializing, autonomousoperational capability. The ISC is composed of a wide field-of-view active-pixel star camera and microgyros,

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0 1 2 3 4 5 60.2

0.4

0.6

0.8

1

a (A

U)

0 1 2 3 4 5 60

0.1

0.2

e

0 1 2 3 4 5 60

100

200

Years

i (de

g)

0 1 2 3 4 5 60.2

0.4

0.6

0.8

1

r (A

U)

0 1 2 3 4 5 60

50

100

Con

e a

(deg

)

0 1 2 3 4 5 6-100

0

100

200

Years

Clo

ck d

(de

g)

Figure 10. KEI trajectory design using simple sail-steering laws (Ref. 21).

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0 1 2 3 4 5 60

0.5

1

1.5

a (A

U)

0 1 2 3 4 5 60

0.2

0.4

e

0 1 2 3 4 5 60

100

200

Years

i (de

g)

0 1 2 3 4 5 60

0.5

1

1.5

r (A

U)

0 1 2 3 4 5 620

40

60

Con

e a

(deg

)

0 1 2 3 4 5 6-200

0

200

400

Years

Clo

ck d

(de

g)

Figure 11. Optimization-based trajectory design for the KEI mission (Ref. 22).

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Active-PixelStar Camera

MicroGyros

AttitudeDetermination Algorithm(Quaternions)

3-axis AttitudeStabilization & Thrust VectorControl Logic

Sail Attitude Control System (SACS)

Pulsed Plasma Thrusters (total 20 kg, 100 W)

Inertial Stellar Compass (ISC) Attitude Determination System 2.5 kg, 3.5 W, 0.1 deg (1ssss)))),,,, 5 HHHH z

Propellantless 3-axis Trim and Control Mechanisms (total 10 kg, 10 W)

Secondary ACS for Backup Mode

Primary ACS for Normal Flight Mode

5-kg Trim Ballast

Roll Stabilizer Bar

Spacecraft Bus ADCS attitude sensors, reaction wheels thrusters (total 30 kg, 100 W)

pitch

yaw

Sailcraft attitude stabilization prior to sail deployment, during post-deployment checkout, for pre-flight standby mode,and during science mission phase

160-m, 450-kgSailcraft

Figure 12. An integrated TVC/AOCS architecture proposed for the SPI mission.

with associated data processing and power electronics. It has a total mass of 2.5 kg, a power requirement of3.5 W, and an accuracy of 0.1 deg (1σ). It is planned to be flight validated within few years. Some recentadvances in microsatellite technologies, including the ISC, need to be exploited to complete an integratedlow-cost, low-risk, low-mass, low-power, and low-volume AOCS for sailcraft.

Detailed analysis and design of a similar TVC/AOCS architecture for a fight validation mission of a40-m solar sail in a dawn-dusk sunsynchronous orbit can be found in Refs. 24-27. The proposed baselineTVC/AOCS architecture will be applicable with minimal modifications to a wide range of future solarsail flight missions with varying requirements and mission complexity, including a solar sailing mission forintercepting, impacting, and deflecting near-Earth asteroids.21−22 Detailed control design study results ofthe proposed TVC/AOCS architecture for the SPI mission can be found in Refs. 28-29.

A navigation and guidance system, which is not shown in Fig. 12, will also be required for most solar sailmissions. Because of the inherent inability to precisely model the solar radiation pressure and to accuratelypoint the true thrust vector direction, any solar sail TVC system will require frequent updates of bothorbital parameters and TVC steering commands for frequent trajectory corrections. The navigation andguidance system provides such required updates, and the update frequency is determined by many factors,such as the trajectory dispersions, the target-body ephemeris uncertainty, the calibration of a solar radiationpressure model, and the operational constraints. In Ref. 17 the interplanetary navigation problem of asolar sail spacecraft was examined and found to be analogous to that of solar electric propulsion spacecraft.The study result in Ref. 17 also indicates that solar sail spacecraft can be navigated without significantdifficulties by applying techniques previously developed for solar electric propulsion missions. Although thedevelopment of a solar sail navigation system is of practical importance for future solar sail missions, thispaper focuses on the TVC/AOCS design problem assuming that the navigation and guidance system can

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Figure 13. ATK’s solar sail mast with a trim ballast mass running along a lanyard tape and tip-mounted rollstabilizer bars attached to sail panels (Courtesy of ATK Space Systems).

provide the required, frequent updates of both orbital parameters and TVC steering commands.

B. Quaternion-Feedback Attitude Control

Euler’s attitude dynamical equations of motion of a rigid sailcraft are given by

I1ω1 − (I2 − I3)ω2ω3 = u1 + d1 (45a)I2ω2 − (I3 − I1)ω3ω1 = u2 + d2 (45b)I3ω3 − (I1 − I2)ω1ω2 = u3 + d3 (45c)

where (ω1, ω2, ω3) are the angular velocity components, (I1, I2, I3) the principal moments of inertia, (u1, u2, u3)the attitude control torques, and (d1, d2, d3) the disturbance torques.

The kinematic differential equations in terms of attitude quaternions are given by

q1

q2

q3

q4

=

12

0 ω3 −ω2 ω1

−ω3 0 ω1 ω2

ω2 −ω1 0 ω3

−ω1 −ω2 −ω3 0

q1

q2

q3

q4

(46)

where (q1, q2, q3, q4) are the inertial attitude quaternions.The quaternion-feedback attitude control logic, proposed for solar sailing applications, is simply a PID

control logic of the form15−17

u1 = −k1

(e1 +

1τ

∫e1dt

)− c1ω1 (47a)

u2 = −k2

(e2 +

1τ

∫e2dt

)− c2ω2 (47b)

u3 = −k3

(e3 +

1τ

∫e3dt

)− c3ω3 (47c)

where (e1, e2, e3) are the roll, pitch, and yaw components of attitude-error quaternions (e1, e2, e3, e4) and(ki, τ, ci) are control gains to be properly determined. The attitude-error quaternions are computed us-ing the desired or commanded attitude quaternions (q1c, q2c, q3c, q4c) and the actual attitude quaternions(q1, q2, q3, q4), as follows:

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e1

e2

e3

e4

=

q4c q3c −q2c −q1c

−q3c q4c q1c −q2c

q2c −q1c q4c −q3c

q1c q2c q3c q4c

q1

q2

q3

q4

(48)

A saturation control logic of accommodating the actuator torque and slew rate constraints is given by

ui = − satUmax

ki sat

Li

[ei +1τ

∫ei] + ciωi

; i = 1, 2, 3 (49)

and the variable limiter Li is self-adjusted as

Li =ci

kimin

√2ai|ei|, ωmax

(50)

where ωmax is the maximum slew rate (if required) and ai is the maximum angular acceleration. Details ofthis nonlinear PID control logic can be found in Refs. 16 and 17.

C. Trajectory Control Inputs (α, β)

Consider the body-fixed rotational sequence of the form: C2(−α) ← C1(−β) ← C2(−φ) ← C3(ψ). For thiscase of employing (α, β) as the trajectory control inputs or the TVC steering commands, we have

r

ψ

φ

=

cos φ 0 sinφ

0 1 0− sinφ 0 cos φ

cos ψ sinψ 0− sinψ cos ψ 0

0 0 1

I

J

K

(51)

123

=

cos α 0 sinα

0 1 0− sinα 0 cos α

1 0 00 cos β − sinβ

0 sinβ cos β

r

ψ

φ

(52)

The equivalent coordinate transformation in terms of quaternions is expressed as18

0− sin(α/2)

0cos(α/2)

←

− sin(β/2)00

cos(β/2)

←

0− sin(φ/2)

0cos(φ/2)

←

00

sin(ψ/2)cos(ψ/2)

(53)

which results in

q1c

q2c

q3c

q4c

=

cos(α/2) 0 sin(α/2) 00 cos(α/2) 0 − sin(α/2)

− sin(α/2) 0 cos(α/2) 00 sin(α/2) 0 cos(α/2)

cos(β/2) 0 0 − sin(β/2)0 cos(β/2) − sin(β/2) 00 sin(β/2) cos(β/2) 0

sin(β/2) 0 0 cos(β/2)

×

cos(φ/2) 0 sin(φ/2) 00 cos(φ/2) 0 − sin(φ/2)

− sin(φ/2) 0 cos(φ/2) 00 sin(φ/2) 0 cos(φ/2)

00

sin(ψ/2)cos(ψ/2)

(54)

where (q1c, q2c, q3c, q4c) are the desired attitude quaternions of a sailcraft whose orientation provides thedesired thrust vector direction as commanded by (α, β). The preceding result indicates that a TVC systemwill require frequent updates of both the orbital information (φ, ψ) and TVC steering commands (α, β) dueto the inherent difficulty to precisely point the true thrust vector of a large solar sail.

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Given the actual attitude quaternions, (q1, q2, q3, q4), the actual cone and clock angles, (α, β), can bedetermined from the following relationship:

1 − 2(q22 + q2

3) 2(q1q2 + q3q4) 2(q1q3 − q2q4)2(q2q1 − q3q4) 1 − 2(q2

1 + q23) 2(q2q3 + q1q4)

2(q3q1 + q2q4) 2(q3q2 − q1q4) 1 − 2(q21 + q2

2)

=

C11 C12 C13

C21 C22 C23

C31 C32 C33

=

cos α 0 sinα

0 1 0− sinα 0 cos α

1 0 00 cos β − sinβ

0 sin β cos β

cos φ 0 sinφ

0 1 0− sinφ 0 cos φ

cos ψ sinψ 0− sinψ cos ψ 0

0 0 1

Given a direction cosine matrix, quaternions can also be determined as

q4 =12

√1 + C11 + C22 + C33

q1

q2

q3

=

14q4

C23 − C32

C31 − C13

C12 − C21

; when q4 = 0

However, this approach has a singularity problem when q4 = 0.

D. Trajectory Control Inputs (α, δ)

For a case of employing (α, δ) as the trajectory control inputs or the TVC steering commands, consider abody-fixed rotational sequence of the form: C2(−α) ← C1(−δ) ← C3(θ) ← C3(ω) ← C1(i) ← C3(Ω). Forthis case, we have

r

θ

k

=

cos(ω + θ) sin(ω + θ) 0− sin(ω + θ) cos(ω + θ) 0

0 0 1

1 0 00 cos i sin i

0 − sin i cos i

cos Ω sin Ω 0− sin Ω cos Ω 0

0 0 1

I

J

K

(55)

123

=

cos α 0 sinα

0 1 0− sinα 0 cos α

1 0 00 cos δ − sin δ

0 sin δ cos δ

r

θ

k

(56)

0− sin(α/2)

0cos(α/2)

←

− sin(δ/2)00

cos(δ/2)

←

00

sin(ω+θ2 )

cos(ω+θ2 )

←

sin(i/2)00

cos(i/2)

←

00

sin(Ω/2)cos(Ω/2)

(57)

q1c

q2c

q3c

q4c

=

cos(α/2) 0 sin(α/2) 00 cos(α/2) 0 − sin(α/2)

− sin(α/2) 0 cos(α/2) 00 sin(α/2) 0 cos(α/2)

cos(δ/2) 0 0 − sin(δ/2)0 cos(δ/2) − sin(δ/2) 00 sin(δ/2) cos(δ/2) 0

sin(δ/2) 0 0 cos(δ/2)

×

cos(ω+θ2 ) sin(ω+θ

2 ) 0 0− sin(ω+θ

2 ) cos(ω+θ2 ) 0 0

0 0 cos(ω+θ2 ) sin(ω+θ

2 )0 0 − sin(ω+θ

2 ) cos(ω+θ2 )

cos(i/2) 0 0 sin(i/2)0 cos(i/2) sin(i/2) 00 − sin(i/2) cos(i/2) 0

− sin(i/2) 0 0 cos(i/2)

×

00

sin(Ω/2)cos(Ω/2)

(58)

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The preceding result indicates that a TVC system employing (α, δ) will also require frequent updates of boththe orbital parameters (Ω, i, ω, θ) and TVC steering commands (α, δ) due to the inherent inability to pointthe true thrust vector direction of a large solar sail.

Given the actual attitude quaternions, (q1, q2, q3, q4), the actual cone and clock angles, (α, δ), can bedetermined from the following relationship:

1 − 2(q22 + q2

3) 2(q1q2 + q3q4) 2(q1q3 − q2q4)2(q2q1 − q3q4) 1 − 2(q2

1 + q23) 2(q2q3 + q1q4)

2(q3q1 + q2q4) 2(q3q2 − q1q4) 1 − 2(q21 + q2

2)

=

cos α 0 sinα

0 1 0− sinα 0 cos α

1 0 00 cos δ − sin δ

0 sin δ cos δ

×

cos(ω + θ) sin(ω + θ) 0− sin(ω + θ) cos(ω + θ) 0

0 0 1

1 0 00 cos i sin i

0 − sin i cos i

cos Ω sin Ω 0− sin Ω cos Ω 0

0 0 1

(59)

E. Relationship of β and δ

The two different clock angles, β and δ, are related as

1 0 00 cos β − sinβ

0 sinβ cos β

cos φ 0 sinφ

0 1 0− sinφ 0 cos φ

cos ψ sinψ 0− sinψ cos ψ 0

0 0 1

=

1 0 00 cos δ − sin δ

0 sin δ cos δ

cos(ω + θ) sin(ω + θ) 0− sin(ω + θ) cos(ω + θ) 0

0 0 1

1 0 00 cos i sin i

0 − sin i cos i

cos Ω sin Ω 0− sin Ω cos Ω 0

0 0 1

V. TVC Design Example with (α, δ)

Although various forms of orbital trajectory equations employing two different sets of the cone andclock angles are available for the solar sail trajectory design and simulation, we choose here a set of orbitalequations, called Gauss’s form of the variational equations with (α, δ), for the preliminary TVC design andsimulation.

A complete set of 13 first-order differential equations of attitude-orbit coupled dynamical model of asailcraft can be summarized as

I1ω1 − (I2 − I3)ω2ω3 = u1 + d1 (60a)I2ω2 − (I3 − I1)ω3ω1 = u2 + d2 (60b)I3ω3 − (I1 − I2)ω1ω2 = u3 + d3 (60c)

q1

q2

q3

q4

=

12

0 ω3 −ω2 ω1

−ω3 0 ω1 ω2

ω2 −ω1 0 ω3

−ω1 −ω2 −ω3 0

q1

q2

q3

q4

(61)

a =2a2

h[eR sin θ + T (1 + e cos θ)] ≡ 2a2

h[eR sin θ +

pT

r] (62a)

e =√

p

µ[R sin θ + T (cos θ + cos E)] ≡ 1

hpR sin θ + [(p + r) cos θ + re]T (62b)

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i =r cos(ω + θ)

hN (62c)

Ω =r sin(ω + θ)

h sin iN (62d)

ω = − r sin(ω + θ)h tan i

N +1eh

[−pR cos θ + (p + r)T sin θ] (62e)

θ =h

r2+

1eh

[pR cos θ − (p + r)T sin θ] (62f)

where

R

T

N

= F0 cos2 α

cos α

sinα sin δ

sinα cos δ

=

(r⊕r

)2

ac cos2 α

cos α

sinα sin δ

sinα cos δ

(63)

ui = − satUmax

ki sat

Li

[ei +1τ

∫ei] + ciωi

; i = 1, 2, 3 (64)

e1

e2

e3

e4

=

q4c q3c −q2c −q1c

−q3c q4c q1c −q2c

q2c −q1c q4c −q3c

q1c q2c q3c q4c

q1

q2

q3

q4

(65)

q1c

q2c

q3c

q4c

=

cos(α/2) 0 sin(α/2) 00 cos(α/2) 0 − sin(α/2)

− sin(α/2) 0 cos(α/2) 00 sin(α/2) 0 cos(α/2)

cos(δ/2) 0 0 − sin(δ/2)0 cos(δ/2) − sin(δ/2) 00 sin(δ/2) cos(δ/2) 0

sin(δ/2) 0 0 cos(δ/2)

×

cos(ω+θ2 ) sin(ω+θ

2 ) 0 0− sin(ω+θ

2 ) cos(ω+θ2 ) 0 0

0 0 cos(ω+θ2 ) sin(ω+θ

2 )0 0 − sin(ω+θ

2 ) cos(ω+θ2 )

cos(i/2) 0 0 sin(i/2)0 cos(i/2) sin(i/2) 00 − sin(i/2) cos(i/2) 0

− sin(i/2) 0 0 cos(i/2)

×

00

sin(Ω/2)cos(Ω/2)

(66)

(67)

Detailed control design results using the preceding set of attitude-orbit coupled equations for a 160-m SPIsailcraft controlled by trim control masses, roll stabilizer bars, and tip-mounted microPPTs can be found inRefs. 28-29.

VI. Conclusion

This paper has described various forms of orbital trajectory equations, which employ two different setsof the cone and clock angles, for solar sail TVC design and simulation. It was shown that a preliminarytrajectory design can be performed by employing simple sail-steering laws applied to a set of orbital equations,called Gauss’s form of the variational equations with the cone and clock angles (α, δ). The simplicity of aquaternion-based TVC/AOCS architecture was emphasized. However, any solar sail TVC system will requirefrequent updates of both the TVC steering commands (α, δ) and orbital parameters because of its inabilityto precisely point the true thrust vector direction of a large flexible solar sail.

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Acknowledgments

The work described in this paper was funded by the In-Space Propulsion Technology Program, whichis managed by NASA’s Science Mission Directorate in Washington, D.C., and implemented by the In-Space Propulsion Technology Office at Marshall Space Flight Center in Huntsville, Alabama. The programobjective is to develop in-space propulsion technologies that can enable or benefit near and mid-term NASAspace science missions by significantly reducing cost, mass or travel times. The author would like to thankE. Montgomery, G. Garbe, J. Presson, A. Heaton, and M. Whorton at NASA Marshall Space Flight Centerfor their financial and technical support.

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Sustained Thrust,” American Journal of Physics, Vol. 27, 1959, pp. 12-18.20Battin, R. H., An Introduction to the Mathematics and Methods of Astrodynamics, AIAA Education Series,

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22Dachwald, B. and Wie, B., “Solar Sail Trajectory Optimization for Intercepting, Impacting, and DeflectingNear-Earth Asteroids,” AIAA-2005-6176, AIAA Guidance, Navigation, and Control Conference, San Francisco, CA,August 15-18, 2005.

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24Murphy, D. and Wie, B., “Robust Thrust Control Authority for a Scalable Sailcraft,” AAS-04-285, 14thAAS/AIAA Space Flight Mechanics Conference, Maui, Hawaii, Feb. 8-12, 2004.

25Wie, B., Murphy, D., Thomas, S., and Paluszek, M., “Robust Attitude Control Systems Design for Solar SailSpacecraft (Part One): Propellantless Primary ACS,” AIAA-2004-5010, AIAA Guidance, Navigation, and ControlConference, Providence, RI, August 16-19, 2004.

26Wie, B., Murphy, D., Thomas, S., and Paluszek, M., “Robust Attitude Control Systems Design for Solar SailSpacecraft (Part Two): microPPT-based Backup ACS,” AIAA-2004-5011, AIAA Guidance, Navigation, and ControlConference, Providence, RI, August 16-19, 2004.

27Thomas, S.. Paluszek, M., Wie, B., and Murphy, D., “Design and Simulation of Sailcraft Attitude ControlSystems Using Solar Sail Control Toolbox,” AIAA-2004-4890, AIAA Guidance, Navigation, and Control Conference,Providence, RI, August 16-19, 2004.

28Wie, B., Thomas, S., Paluszek, M., and Murphy, D., “Propellantless AOCS Design for a 160-m, 450-kg Solar SailSpacecraft of the Solar Polar Imager Mission,” AIAA-2005-3928, 41st AIAA Joint Propulsion Conference, Tucson,AZ, July 10-13, 2005.

29Wie, B. and Murphy, D., “MicroPPT-Based Secondary/Backup ACS for a 160-m, 450-kg Solar Sail Spacecraft,”AIAA-2005-3724, 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Tucson, AZ, July 10-13,2005.

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