AIAA 2002-4572

Dynamic Modeling and Attitude Control ofSolar Sail Spacecraft: Part I

Bong Wie∗

Arizona State UniversityTempe, AZ 85287-6106

Abstract

Study results for developing dynamic models and at-titude control systems of solar sail spacecraft are pre-sented. The overall study objective was to advance sail-craft attitude control technology so that a sail space-flight experiment for validating sail attitude stability& controllability and thrust vector pointing & steer-ing performance can be conducted within the next 5years. In particular, this paper (Part I) presents anoverview of sailcraft attitude control issues, with spe-cial emphasis on a simple spin stabilization concept forcountering the significant effect of a solar pressure dis-turbance torque, caused by an uncertain center-of-massand center-of-pressure offset, on sailcraft stability andpointing. A sailcraft controlled by means of translat-ing and/or tilting sail panels is also described in thispaper. Detailed dynamic modeling and attitude controldesign for a sailcraft employing a two-axis gimbaled con-trol boom and/or control vanes are presented in Part II.

1 Introduction

A renewed interest in solar sailing due to its potentialfor propellantless space propulsion is spurring recent de-velopments of near-term sail missions and the associ-ated sailcraft technologies (Refs. 1-4). For example, theCosmos-1 solar sail mission of The Planetary Society,with an anticipated launch date of 2002, may becomethe first experimental flight mission for solar sailing inan earth-centered orbit. The sailcraft development forthe Cosmos-1 mission is being conducted in Russia.

Near-term applications of solar sailing technology in-clude high-performance science missions to Mercury,Venus, and the inner solar system. Non-Keplerian or-bits, high-velocity missions to the outer planets, andhigh-velocity interstellar precursor missions (all based onsolar sailing technology) are also envisioned by NASA.

∗Professor, Dept. of Mechanical & Aerospace Engineering,Associate Fellow AIAA; (480) 965-8674, Fax (480) 965-1384,bong.wie@asu.edu. Copyright c©2002 by the American Instituteof Aeronautics and Astronautics, Inc. All rights reserved.

Figure 1: A large solar sail spacecraft controlled by fourcontrol vanes mounted at the spar tips; it was designedby JPL in 1977 for a comet Halley rendezvous mission(Ref. 6).

A rendezvous mission for Halley’s comet, studied atJet Propulsion Laboratory in 1977, required an 800 ×800 m solar sail for an 850-kg payload/bus (Figure 1).However, future solar sail missions will most likely re-quire relatively smaller (< 100 m) sails due to recentadvances in ultra-lightweight sail films, lightweight de-ployable booms, and the miniaturization of spacecraftavionics hardware. Detailed historical as well as techni-cal discussions of solar sailing and the associated tech-nologies can be found in Refs. 5-8.

Following Garwin’s publication on solar sailing in 1958(Ref. 9), attitude stabilization of a space vehicle bymeans of solar radiation pressure was first proposed bySohn in 1959 (Ref. 10). Since then, the concept of usingsolar radiation pressure for attitude stabilization as wellas stationkeeping control of various satellites has beenstudied extensively by many researchers during the lastthree decades (Refs. 11-15).

In fact, such a solar pressure attitude control concepthas been successfully implemented on a certain type ofcommercial communications satellites in geosynchronousorbit as well as on several interplanetary spacecraft. Forexample, the large solar radiation disturbance torque

1

AIAA Guidance, Navigation, and Control Conference and Exhibit5-8 August 2002, Monterey, California

AIAA 2002-4572

Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

N

S

δS

δNδN

δS

δN = 0

δS = 0

-

Pitch Axis

(a) (b) (c)

Figure 2: Illustration of a solar pressure attitude controlconcept successfully implemented on geosynchronouscommunications satellites, such as OTS, TELECOM 1,and INMARSAT 2 (Refs. 16-19).

caused by an asymmetrical solar array configuration ofINSAT with only one solar array wing on the south sideis countered by a solar sail on a long boom mounted onthe north side. The roll/yaw control systems of OTS,TELECOM 1, and INMARSAT 2 successfully utilizedthe solar pressure attitude control concept. An asym-metrical offsetting of the solar array wings from theirnominal sun-pointing orientation generates the so-calledwindmill torque, as illustrated in Figure 2(a). If thenorth and south solar array wings are not rotated asym-metrically (i.e., δN = −δS), then an additional roll/yawtorque, perpendicular to the windmill torque, is also gen-erated as illustrated in Figure 2(c). Additional flapsmounted to the outermost solar panels substantially in-crease the roll/yaw control torques. Note that such flapsproduce the windmill torque even when δN = δS = 0.The maximum offsetting of each array is constrained bythe acceptable electrical power loss (nominally 1%). De-tailed descriptions of this solar pressure attitude controltechnique can be found in Refs. 16-19.

The idea of asymmetrically twisting solar panels tocreate the windmill torque about the roll axis was alsosuccessfully applied to the Mariner 10 spacecraft duringits flight to Mercury. However, its roll attitude controlwas done by commands from the mission controllers be-cause there was no onboard controller for that task.

Even though the effectiveness of the solar pressure at-titude control has been in-flight validated as discussedpreviously in this section, the solar radiation pressureis often considered as an external disturbance for mostsatellites. Recently, the solar pressure effects on for-mation flying of small satellites were investigated inRefs. 20-21. The significant effects of solar radiationpressure on attitude/orbit control of very large Space

Solar Power Satellites (SSPS) in geosynchronous orbitwere also recently studied in Ref. 22.

During the last three decades a variety of advanced dy-namic modeling and spacecraft control techniques havebeen developed. Detailed descriptions of such advancedtechnologies associated with spacecraft dynamics andcontrol problems can be found in Refs. 19, 23-26.

However, there exist various practical implementationissues to be resolved in applying these advanced controltechniques to active three-axis attitude control of near-term sailcraft as well as to future advanced sailcraft. Allpractical spacecraft control designs are often subjectedto the physical limits of actuators, sensors, spacecraftstructural rigidity, and other mission constraints. In par-ticular, when a gimbaled control boom and/or sail con-trol vanes (instead of conventional thrusters, reactionwheels, and magnetic torquers) are to be employed asprimary actuators for active three-axis attitude controlof solar sail spacecraft, there exist a variety of practicalissues to be resolved. Therefore, solar sail attitude andflight control technology needs to be rapidly advanced sothat a sail spaceflight experiment for validating sail atti-tude stability & controllability and thrust vector point-ing & steering performance can be conducted within thenext 5 years.

The remainder of this paper (Part I) is outlined asfollows. In Section 2, three different sail configurationsand the associated attitude control issues are briefly dis-cussed. Section 3 describes solar radiation pressure forcemodeling of a non-perfect sail. Section 4 describes a sim-ple spin stabilization concept for countering the signifi-cant effect of a solar pressure disturbance torque (causedby an uncertain cm/cp offset) on sailcraft stability andpointing. In Section 5, a sailcraft controlled by meansof translating and/or tilting sail panels is investigated.

Detailed dynamic modeling and attitude control de-sign for a sailcraft employing a two-axis gimbaled controlboom and/or control vanes are presented in Part II [27].

2 Sailcraft Attitude Control Is-sues

Three basic types of near-term sailcraft are shown inFigure 3. These configurations do have their own ad-vantages and disadvantages in terms of control author-ity, controllability, packaging, deployment, and othersystem-level tradeoff issues (i.e., mass, cost, etc.). Se-lecting a particular sail configuration for a specific mis-sion is a complex problem, requiring detailed system-level tradeoffs. This study focuses on a square sail con-figuration which is most likely to be chosen for variousnear-term sail missions (Refs. 1-4).

As discussed in Refs. 10-11, an interplanetary space-craft is often said to be statically stable when its center ofmass lies between the sun and its center of pressure. Al-though any point along the resultant solar pressure force

2

Square Sail

Spinning Disk Sail

Heliogyro

Figure 3: Three basic types of near-term sailcraft(Ref. 1).

direction can be considered as the center of pressure, thelocation along a spacecraft reference line through whichthe resultant force is acting is often defined as the cen-ter of pressure. Whenever a statically stable sailcraftrotates away from its neutral orientation, a restoring(stabilizing) torque is generated. The dynamical be-havior such a statically stable sailcraft is analogous tothat of a gravity-gradient stabilized satellite. That is, ifdisturbed, the sailcraft will oscillate indefinitely. If thecenter of pressure lies between the sun and the center ofmass, a destabilizing torque is generated whenever thesailcraft rotates away from its null or trim orientation.

A 76 × 76 m square sailcraft is currently under de-velopment by Team Encounter (Ref. 2). It is scheduledto be launched as a secondary payload on an Ariane 5in 2006. A combination of passive and active attitudecontrol techniques are employed for this 76 × 76 m sail-craft. The Team Encounter sailcraft of a total mass of18 kg is required to achieve solar system escape within3 or 5 years. Its sun-pointing orientation is passivelystabilized by means of “trim tabs.” A constant pitch an-gle of 25 deg with respect to the sun is required duringthe first 300 days after separated from a carrier space-craft. The 25-deg pitch trim angle is passively main-tained by an intentional cm/cp offset caused by a 3-kgpayload tied to the side with a burnwire. The rotationalmotion about the sun vector is actively controlled. Anonboard star camera measures the sailcraft orientationwith respect to a fixed star field and the two circular con-trol vanes provide the necessary windmill control torque.After 300 days, an onboard timer will power the burn-wire to release the payload restrained by a suspension

wire. Consequently, the center of mass will move to thesailcraft center and the sailcraft will be passively stabi-lized for a zero pitch trim angle. Detailed preliminarymission/systems design results of the Team Encountersailcraft can be found in Ref. 2.

A spin-stabilized, 76× 76 m square sailcraft has beenproposed for the New Millennium Program Space Tech-nology 5 (ST5) Geostorm warning mission which wouldprovide real-time monitoring of solar activity (Ref. 4). Itwould operate inside the L1 point of the sun-earth sys-tem toward the sun and increase the warning time forgeomagnetic storms compared to a vantage point closerto the earth. For such a spin-stabilized square sailcraftwith the moments of inertia of (44000, 22000, 24000) kg-m2 and an uncertain cm/cp offset of 1 m, a spin rateof 0.45 deg/s was selected to keep the angular momen-tum vector within 1 deg of the sunline. Thrusters areused for precession/nutation control as well as spin ratecontrol of this sailcraft.

Although a passive or spin stabilization technique canbe very cost effective for certain missions, such as theTeam Encounter mission or the Geostorm warning mis-sion, an active three-axis attitude control will be nec-essary for most sailcraft requiring frequent, large-anglethrust vector steering maneuvers.

One method of actively controlling the attitude of athree-axis stabilized sailcraft is to change its center-of-mass location relative to its center-of-pressure location.This can be achieved by articulating a control boom witha tip-mounted payload/bus. Such a concept of articulat-ing a 2-axis gimbaled control boom was investigated for a40 × 40 m square sailcraft proposed for the New Millen-nium Program Space Technology 7 (ST7) sail flight vali-dation experiment. The sail surface is comprised of fourtriangular panels supported by four deployable booms.Euler column buckling limits this type of design to about150 × 150 m sailcraft, but four or more of these sails canbe linked together as structural modules to construct alarger sailcraft.

Another method is to employ small reflective controlvanes mounted at the spar tips, as can be seen in Fig-ure 1, or to tilt and/or translate sail panels. Similarto the problem inherent to the various sail configura-tions, these different attitude control methods do alsohave their own advantages and disadvantages in termsof control authority, controllability, and other system-level tradeoff issues.

Although the essential idea behind all these “center-of-mass/center-of-pressure methods” appears simple,there are challenging hardware implementation prob-lems to be solved. Some technical issues inherent tothe development of an attitude control subsystem for asquare sailcraft are briefly discussed below.

Attitude control and stabilization of a sailcraft may bepossible by use of a typical attitude determination andcontrol subsystem (ADCS) which is often a necessarypart of sailcraft bus system. However, small reaction

3

S

n

t

Incoming Photons

SpecularlyReflected Photons

α

α

βF

γ = α − β

S

wheels and/or a propulsion subsystem with a limitedamount of propellant to be employed for a typical 100-kgbus may be inadequate, impractical, or even undesirablefor a fully deployed sailcraft because of its large momentsof inertia, its large solar pressure disturbance torque,and its extended sailing voyages. For example, a 40 × 40m, 160-kg sailcraft with a nominal solar pressure force of0.01 N and a cm/cp offset of ±0.1 m has a solar pressuredisturbance torque of ±0.001 N-m, which is about 100times larger than that of typical geosynchronous com-munications satellites. A conventional three-axis atti-tude control system will require large reaction wheelsand also a prohibitively large amount of propellant tocounter such a major disturbance torque.

Consequently, the use of a gimbaled control boom,control vanes, sail panel translation/rotation, control-mass translation, or possibly reflectivity modulation isnecessary for three-axis attitude control of sailcraft.In addition to these propellantless sail control actua-tors, three-axis attitude information is crucial for ac-tive three-axis attitude control and thrust vector point-ing/steering. It is assumed that three-axis inertial atti-tude information in terms of quaternions or Euler an-gles will be available from an attitude determinationsubsystem, consisting of sun sensors, earth sensor, starcamera/tracker, and also possibly inertial measurementunits (IMUs). The IMUs consist of rate gyros andaccelerometers, and they are critical components of aspacecraft attitude determination subsystem.

One of the critical parameters of sailcraft is the sun an-gle, often denoted by α, between the sail surface normaland the sun. Its significant effects on the overall perfor-mance, stability, and control of a sailcraft are similar tothe effects of the aircraft angle of attack, α, on aircraftperformance, stability, and control. Similar to the so-called high-α control problem of high-performance air-craft, a high-performance sailcraft may also have a sim-ilar high-α control problem because of its thrust vectorpointing requirement of typically α ≈ 35 deg.

The basic principle behind various aerodynamic con-trol surfaces of aircraft, such as ailerons, elevator, rud-der, flaps, trim tabs, and spoilers, should be exploitedin developing a sail attitude and flight control subsys-tem. Furthermore, uncertainties inherent to solar radia-tion pressure modeling of non-perfect and non-flat sailsshould be taken into account in designing a sail attitudecontrol subsystem.

3 Solar Radiation Pressure Mod-els

The solar radiation pressure (SRP) forces are due tophotons impinging on a surface in space. If a fraction,ρa, of the impinging photons is absorbed, a fraction, ρs,is specularly reflected, and a fraction, ρd, is diffusely

reflected by a surface, then we have

ρa + ρs + ρd = 1 (1)

and the SRP force acting on a flat surface located at 1astronomical unit from the sun can be modeled as

F = PA[ρa(S · n)S + 2ρs(S · n)2n + ρd(S · n)(S +23n)]

= PA(S · n)

(ρa + ρd)S +[2ρs(S · n) +

23ρd

]n

= PA(S · n)

(1 − ρs)S +[2ρs(S · n) +

23ρd

]n

(2)

where P = 4.563 × 10−6 N/m2 is the nominal solar ra-diation pressure constant at 1 astronomical unit fromthe sun, A is the surface area, n is a unit vector normalto the surface, and S is a unit vector pointing from thesun to the surface. The solar radiation pressure variesinversely with the square of the distance from the sun.(Note that solar pressure force equations given in Ref. 28need to be properly changed to the forms of Eq. (2).)

Figure 4: Solar radiation pressure force model of a non-perfect flat surface.

Since S = cos αn + sinαt where α is the sun anglebetween the surface normal and the sunline and t is thetransverse unit vector, as shown in Figure 4, the SRPforce can also be expressed as

F = Fnn + Ftt (3)

where

Fn = PA

(1 + ρs)cos2α +

23ρd cos α

Ft = PA(1 − ρs) cos α sinα

For a case with ρd ≈ 0, we may also express the SRPforce as

F = PA cos α

(1 − ρs)S + 2ρs cos αn

= PA cos α

(1 − ρs)S + 2ρs cos α(cos αS + sinαS⊥)

= PA cos α(1 − ρs + 2ρscos2α)S + 2ρs cos α sinαS⊥= PA cos α(1 + ρs cos 2α)S + ρs sin 2αS⊥= Fs

S + F⊥S⊥ (4)

4

where S⊥ is a unit vector perpendicular to S and is inthe same plane as S and n.

For sails in sun-centered orbits, the components alongthe sunline and perpendicular to the sunline are some-times called the “drag” and “lift” components, respec-tively. The transverse component, F⊥, provides an effec-tive thrust control for orbital maneuvering of sailcraft insun-centered orbits. For example, the sun angle whichmaximizes the transverse component can be found as

d

dαF⊥ = 0 =⇒ d

dαcos2α sinα = 0 (5)

which gives α = tan−1(1/√

2) = 35.26 deg. This opti-mal sun angle of 35.26 deg for maximizing the transversecomponent is often selected as the desired pitch orien-tation of an interplanetary sailcraft in a sun-centeredorbit.

The normal and transverse components of the SRPforce acting on a flat sail surface with its more detailedoptical and thermal properties are also described by(Refs. 6-7)

Fn

PA= (1 + rs) cos2 α + Bfr(1 − s) cos α

+efBf − ebBb

ef + eb(1 − r) cos α (6)

Ft

PA= (1 − rs) cos α sin α (7)

F =√

F 2n + F 2

t (8)

tanβ =Ft

Fn(9)

Bf , Bb = non-Lambertian coefficients for frontand back surfaces

ef , eb = front and back surface emission coefficientsr = reflectivity of front surfaces = specular reflection coefficientβ = angle of SRP force vector from surface normal

For a square sailcraft similar to the one illustrated in Fig-ure 1, we have the following optical properties (Refs. 6-7):

Bf = 0.79; Bb = 0.55ef = 0.05; eb = 0.55r = 0.88; s = 0.94

and the resulting normal and transverse components ofthe SRP force become

Fn

PA= 1.8272 cos2 α + 0.0417 cos α − 0.0526 cos α

= 1.8272 cos2 α − 0.0109 cos α (10a)Ft

PA= 0.1728 cos α sinα (10b)

In practice, the pressure distribution is not uniformacross the surface of a sail due to curvature (billow). Anumerical integration of the Fn and Ft equations acrossthe curved surface of the sail is needed to determine theresulting pressure distribution. This requires an itera-tive process since the pressure distribution is a functionof the sail shape, and vice versa the shape is a functionof the pressure distribution. For a sailcraft shown inFigure 1, such an iterative process was used by JPL tofind a parameterized model of the following form:

F = ηPA(0.349 + 0.662 cos 2γ − 0.011 cos 4γ) (11)

where η = 1.816 and γ = α − β.The SRP force acting on a sail surface with an area A

is also often approximated as

F ≈ ηPAcos2α (12)

where η is called the overall sail thrust coefficient, typi-cally around 1.8 for a real sailcraft with sail wrinkles andbillowing, with an ideal maximum value of ηmax = 2.

4 Spin Stabilization of Sailcraft

A simple solution to the problem of maintaining a de-sired orientation of a sailcraft in the presence of a cm/cpoffset is to spin the sailcraft in the fashion of spinningrocket about its longitudinal (roll) axis. A thrust vectormisalignment with the roll axis caused by a cm/cp offsetwill cause the sailcraft to tumble in the absence of spin-ning or active three-axis control. However, a spinningsailcraft possesses a gyroscopic stiffness to external dis-turbances, and its motion under the influence of externaldisturbances is characterized by the precession and nu-tation of the roll axis about the desired direction of theroll axis (e.g., Ref. 24, pp. 352-355). The orientationof a spinning sailcraft can be changed by precession ofthe sailcraft using thrusters. Tilting and/or translatingsail panels can also provide an effective precession con-trol torque to a spinning sailcraft with a large angularmomentum.

A spin-stabilized, 76× 76 m square sailcraft has beenproposed for the NMP ST5 Geostorm warning missionin Ref. 4. A spin rate of 0.45 deg/s was selected tokeep the angular momentum vector within 1 deg ofthe sunline for the sailcraft with moments of inertia of(44000, 22000, 24000) kg-m2 and an uncertain cm/cp off-set of 1 m. Precession control as well as spin rate controltorques are obtained by using thrusters.

4.1 Spinning Sailcraft with a Center-of-Mass/Center-of-Pressure Offset

For the purpose of studying the effect of a cm/cp off-set on a spinning sailcraft, consider a sailcraft possess-ing a body-fixed reference frame B with basis vectors

5

b1,b2,b3 and with its origin at the center of mass.The reference frame B coincides with principal axes. Itis assumed that the 1st axis is the roll (spin) axis per-pendicular to the sail surface and the 2nd and 3rd axesare the pitch/yaw (transverse) axes. The solar pressureforce vector is nominally aligned along b1 through thecenter of pressure of the sailcraft.

Euler’s rotational equations of motion of a rigid sail-craft are given by

J1ω1 − (J2 − J3)ω2ω3 = T1 (13a)J2ω2 − (J3 − J1)ω3ω1 = T2 (13b)J3ω3 − (J1 − J2)ω1ω2 = T3 (13c)

where ωi ≡ bi · ω are the body-axis components of theangular velocity of the sailcraft and Ti are the externaltorque vector components along the body axes.

For a square (or an axisymmetric circular) sailcraftwith J2 = J3 = J , the rotational equations of motionbecome

J1ω1 = 0 (14a)Jω2 − (J − J1)ω3ω1 = T2 (14b)Jω3 − (J1 − J)ω1ω2 = T3 (14c)

where T2 and T3 are the solar pressure torque vectorcomponents caused by a cm/cp offset. The windmilltorque about the spin (roll) axis is ignored here; i.e., itis assumed that T1 ≈ 0.

From Eq. (14a), we have

ω1 = constant = Ω (15)

where the constant Ω is called the spin rate of the sail-craft about its roll axis b1. For simplicity, it is furtherassumed that the pitch/yaw transverse axes are chosensuch that T2 = 0 and T3 = εF , where ε is a cm/cp offsetdistance and F is the solar pressure force. Equations(14b) and (14c) then become

ω2 = − λω3 (16a)ω3 = λω2 + a (16b)

where λ = Ω(J1 − J)/J and a ≡ εF/J denotes the dis-turbance acceleration resulting from a cm/cp offest. Al-though the solar pressure force F is a function of thespin-axis orientation, it is assumed to be nearly con-stant.

To describe the rotational motion of the spinning sail-craft as seen from an inertial reference frame, we con-sider the body-fixed rotational sequence of C1(θ1) ←C2(θ2) ← C3(θ3). For this rotational sequence, we havethe following kinematic differential equations:

θ1 = ω1 + (ω2 sin θ1 + ω3 cos θ1) tan θ2 (17a)θ2 = ω2 cos θ1 − ω3 sin θ1 (17b)θ3 = (ω2 sin θ1 + ω3 cos θ1)/ cos θ2 (17c)

For small θ2, these kinematic differential equations be-come

θ1 = ω1 + θ3θ2 (18a)θ2 = ω2 cos θ1 − ω3 sin θ1 (18b)θ3 = ω2 sin θ1 + ω3 cos θ1 (18c)

Assuming θ2θ3 << ω1, we can further approximate θ1

asθ1 ≈ ω1 = Ω = constant (19)

and θ1 ≈ Ωt.Finally, we obtain a set of linearized equations of mo-

tion as follows:

ω2 = − λω3 (20a)ω3 = λω2 + a (20b)θ2 = ω2 cos Ωt − ω3 sin Ωt (20c)θ3 = ω2 sin Ωt + ω3 cos Ωt (20d)

The solutions of Eqs. (20a) and (20b) for a constantdisturbance acceleration, a, can be found as

ω2(t) = ω2(0) sinλt − ω3(0) cos λt − a

λ(1 − cos λt)

ω3(t) = − ω2(0) cos λt − ω3(0) sinλt +a

λsinλt

For a case with ω2(0) = ω3(0) = 0, Eqs. (20c) and (20d)become

θ2 =a

λ

cos

J1

JΩt − cos Ωt

(21a)

θ3 =a

λ

sin

J1

JΩt − sin Ωt

(21b)

Integrating these equations with respect to time for theinitial conditions of θ2(0) = θ3(0) = 0, we obtain

θ2 = Ap sinωpt − An sinωnt (22a)θ3 = Ap(1 − cos ωpt) − An(1 − cos ωnt) (22b)

where Ap =a

λΩJ

J1= precessional amplitude

An =a

λΩ= nutational amplitude

ωp =J1

JΩ = precessional frequency

ωn = Ω = nutational frequency

4.2 Simulation Results

For a 40 × 40 m sailcraft with zero initial conditions andthe following parameter values:

(J1, J2, J3) = (6000, 3000, 3000) kg-m2

J1

J= 2; Ω = 0.5 deg/s

λ =J1 − J

JΩ = 0.5 deg/s

F = 0.01 N; ε = 0.1 m

a =εF

J= 3.3 × 10−7 rad/s2

6

40 m

Spreader

Bar (4)

30 m Spar (4 )

Sail Panel Radial Position Actu ator (4)

Sail Panel Radial Position Tether (4)

Sail Panel Co nstant ForceT ensioning Sp ring (8 )

Sail Hub

Boom

Rotati onAxis

y

x

z

Pitch

YawRoll

Spreader Bar

Sail Film

Flat Panel

Tilted Panel

Boom Tip

Spreader BarSail Film

Boom TipNominal Plane of Sail

-4 -3 -2 -1 0

x 10-3

-2

-1

0

1

2x 10

-3

ω2 (deg/s)

ω3 (

deg/

s)

-0.2 0 0.2-0.5

-0.4

-0.3

-0.2

-0.1

0

Pitch θ2 (deg)

Yaw

θ3 (

deg)

0 500 1000 1500 2000-0.4

-0.2

0

0.2

0.4

time (sec)

Pitc

h θ 2 (

deg)

0 500 1000 1500 2000-0.6

-0.4

-0.2

0

0.2

time (sec)

Yaw

θ3 (

deg)

Figure 5: Simulation results for a cm/cp offset of 0.1 m.

we have Ap = 0.1254 deg, An = 0.2508 deg, ωp = 1.0deg/s, and ωn = 0.5 deg/s. A spin rate of 0.3-0.5 deg/skeeps the thrust vector pointing error within ±1 deg forthis sailcraft with a 0.1-m cm/cp offset and a 0.01-Nsolar pressure force. The linear analysis results agreevery well with the nonlinear simulation results shown inFigure 5. The plot of θ3 vs. θ2 shows the path of the tipof the roll axis in space.

5 Attitude Control by Sail PanelTranslation/Rotation

5.1 Translating/Tilting Sail Panels

As illustrated in Figure 6, a sailcraft configuration con-sidered here has four triangular sail panels supportedbetween the four booms. A significant feature of thisconfiguration is the outer two corners of each sail panelattached by constant-force springs to spreader bars atthe end of the booms. The inner corner of the sail is at-tached to a tether that feeds from a spool with an actua-tor. The sail panel can be radially translated inboard oroutboard by reeling in or paying out the inboard tetherfrom the spool. This moves the center of pressure in orout for pitch/yaw control. Additionally, each boom canbe rotated at the base about its long axis. This twiststhe spreader bars at the end of the booms out of theplane for the sail. This, in turn, lifts or lowers the outercorners of the sail panels out of plane and can be usedfor roll control. If each boom is rotated clockwise at thebase (as viewed from the sail hub) the sail panels willform a pinwheel and provide a roll-axis windmill torqueto the sail.

Because of uncertainties associated with manufactur-ing, mechanization, and deployment of such translat-ing/tilting sail panels, an uncertain cm/cp offset of±0.25 m, instead of ±0.1 m, is assumed here for the

Figure 6: A 40 × 40 m sailcraft and its attitude con-trol mechanism, proposed by JPL for the NMP ST6 saildeployment and control validation experiment.

40 × 40 m sailcraft shown in Figure 6.The study objective for this type of sailcraft is to de-

velop an innovative propellantless sail attitude controlsubsystem, develop attitude control algorithms, and val-idate the concept of an airplane-like control of a sailcraftusing control surfaces such as ailerons, elevators, andrudder, as illustrated in Figure 7.

Mass properties and basic characteristics of this 40 ×40 m sailcraft are summarized in Tables 1 through 3.

5.2 Sail Flight Experiment Design

To validate the concept of propellantless space propul-sion using solar sails, it is proposed that propellantlessorbit-raising maneuvers be flight tested. The proposedpropellantless attitude control system will provideproper sail attitude control and thrust vector steeringto accomplish such a primary objective. A summarydescription of the proposed flight validation mission andthe proposed sail attitude control subsystem (SACS) isprovided below.

• Performance measures for the SACS of a 40 × 40 msailcraft in geosynchronous orbit:

- thrust modulation: 0 < F < 11 mN- thrust vector pointing range: 0 < α < αmax

- thrust vector pointing accuracy: ±1 deg (3σ)- orbit-raising performance: ∆a ≈ 50 km/day

7

y

zYaw

Pitch

b/2

b/2

δ1

δ2

aileron

aileron

elevator

rudder

L L

LL

δ4

δ3

backuprudder

backupelevator

backupaileron b/2

d

b = 40 m L = 30 m

m = 156 kgF = 11.6 mN

-x

d y

-z

δe

δ r δa

δa

cp

solar radiationforce

cp

cp

cp

f

ff

f

Table 1: Sail characteristics

Sail film mf = 6.1 kgBooms (4x) mb = 4 × 3.575 = 14.3 kg

EI > 100 N-m2

L = 30 mρ = 3.575/30 = 0.1191 kg/m2

Hub platform mh = 19.6 kgSolar sail ms = mf + mb + mh = 40 kgPayload/bus mp = 116 kgTotal mass m = ms + mp = 156 kgSail area A = 1400 m2

Solar pressure P = 4.563 e-6 N/m2

Thrust coefficient η = 1.816 (ideal ηmax = 2)Maximum thrust Fmax = ηPA = 0.0116 NArea-to-mass ratio A/m = 8.97 m2/kgAreal density σ = m/A = 0.111 kg/m2

Acceleration ac = Fmax/m = ηPA/m= ηP/σ = 73.7 e-6 m/s2

Table 2: Attitude control characteristics

Roll inertia Ix = 5917 kg-m2

Pitch inertia Iy = 2957 kg-m2

Yaw inertia Iz = 2957 kg-m2

Thrust modulation 0 < F < 11 mNThrust pointing range αmax > α > 0Thrust pointing accuracy ±1 deg (3 σ)Maximum turn rate 0.01 deg/scm/cp uncertainty ±0.25 m (max)Maximum sun angle αmax = 60 degAileron deflection δa = ±1 deg (max)Elevator deflection δe = ±0.5 m (max)Rudder deflection δr = ±0.5 m (max)Maximum thrust Fmax = 11.6 mNControl force fmax = Fmax/4 = 2.9 mNRoll control torque Tx = ±1.34 e-3 N-m (max)Pith control torque Ty = ±1.45 e-3 N-mYaw control torque Tz = ±1.45 e-3 N-mRoll acceleration ax = ±13.0 e-6 deg/s2

Pitch acceleration ay = ±28.1 e-6 deg/s2

Yaw acceleration az = ±28.1 e-6 deg/s2

Control bandwidth 0.0003 HzStructural mode ≈ 0.08 Hz >> 0.0003 HzActuator bandwidth > 0.03 HzActuator nonlinearity stiction/friction (TBD)

Figure 7: Airplane-like attitude and flight control bymeans of tilting and/or translating four sail panels.

• Basic characteristics of the proposed SACS:- pitch turning rate requirement/constraint (0.01

deg/s)- cm/cp offset uncertainty (±0.25 m)- control saturation (±1 deg, ±0.5 m), slew rate limit,

etc.- control bandwidth ≈ 30 × orbital rate = 0.0003 Hz⇒ attitude settling time < 2 hours

- first anti-symmetric bending mode frequency≈ 0.08 Hz >> control bandwidth

- sun/earth acquisition mode, transition mode, on-orbit normal mode, emergency safe mode, and/orground-based backup mode need to be further de-fined.

5.3 Solar Radiation Pressure Model

Solar radiation pressure (SRP) force modeling of thesailcraft is summarized as follows.

An ideal solar radiation pressure force model:

PA = (4.563 e-6) × 1400 = 0.0063882 NFmax = ηPA = 1.816PA = 0.0116 N = 11.6 mN

f = Fmax/4 = 0.0029 N = 2.9 mN

8

Table 3: Orbit control characteristics

Gravitational parameter µ = 398,601 km3/s2

Geosynchronous orbit a = 42,164 kmOrbit period 23 hr 56 min 4 secOrbit rate n = 7.292e-5 rad/sLongitude location TBDStationkeeping accuracy TBDThrust pointing accuracy ±1 degThrust modulation 0 < F < 11 mNMaximum pitch turn rate 0.01 deg/sOrbit-raising maneuver* ∆a ≈ 55 km/day

∗ ∆a =4ac

µa3 ; ideal on-off pitch steering

∆a =16ac

3µa3 ; orbit rate steering

∆a =5.52ac

µa3 ; optimal steering (Ref. 7)

where ac = 73.7 e-6 m/s2

————————————————————

Tx = (f cos2 α)d(cos2 δ1 sin δ1 + cos2 δ4 sin δ4)= (f cos2 α)2d cos2 δa sin δa when δ1 = δ4 = δa

≈ 0.0029 ∗ 2 ∗ 13.3 δa for small α, δa

Ty = (f cos2 α) δe ≈ 0.0029δe

Tz = (f cos2 α) δr + (f cos2 α)d(cos3 δ1 − cos3 δ4)≈ 0.0029 δr when δ1 = δ4

Tx = Ty = Tz = 0 when α = 90(control singularity)

A non-perfect non-flat sail thrust model:

Bf = 0.79; Bb = 0.55ef = 0.05; eb = 0.55r = 0.88; s = 0.94

Ft

PA= (1 − rs) cos α sinα = 0.1728 cos α sinα

Fn

PA= (1 + rs) cos2 α + Bfr(1 − s) cos α

+efBf − ebBb

ef + eb(1 − r) cos α

= 1.8272 cos2 α + 0.0417 cos α − 0.0526 cos α

F =√

F 2n + F 2

t

= ηPA(0.349 + 0.662 cos 2γ − 0.011 cos 4γ)= 0.0116(0.349 + 0.662 cos 2γ − 0.011 cos 4γ)

γ = α − β

β = tan−1(Ft/Fn)

5.4 Dynamical Models

A set of simplified dynamic models of the sailcraft isprovided here for the purpose of preliminary controldesign and computer simulation.

Attitude dynamical equations: The equations ofmotion of a sun-pointing spacecraft with small roll andyaw angles in an earth-centered elliptic orbit are givenby

Jxθx + (θ2 +3µ

r3cos2θy)(Jy − Jz)θx

+3µ

r3(Jy − Jz)(sin θy cos θy)θz = Tx (23a)

Jy θy +3µ

r3(Jx − Jz) sin θy cos θy = Jy θ + Ty (23b)

Jz θz + (θ2 +3µ

r3sin2θy)(Jy − Jx)θz

+3µ

r3(Jy − Jx)(sin θy cos θy)θx = Tz (23c)

where θ is the true anomaly and r is the orbital radiusfrom the center of the earth.

The pitch angle relative to the LVLH frame, θy, canbe expressed as

θy = θ − π

2+ α (24)

where α is the sun angle between the surface normaland the sunline. The pitch equation of motion of a sun-pointing sailcraft then becomes

Jyα − 3µ

r3(Jx − Jz) sin(α − θ) cos(α − θ) = Ty (25)

Orbital equations in ECI coordinates:

X = − µX

r3− F cos2 i

mcos γ (26a)

Y = − µY

r3+

F cos2 im

sin γ (26b)

Z = − µZ

r3+

F sin2 im

(26c)

r =√

X2 + Y 2 + Z2 (26d)

F = 0.0116(0.349 + 0.662 cos 2γ − 0.011 cos 4γ)α = sun angle between the surface normal and the sunγ = α − β = thrust angle (see Figure 3)β = thrust offset due to non-flat sail surface(θx, α, θz) = sailcraft roll, pitch (sun), yaw angles(ωx, ωy, ωz) = sailcraft roll, pitch, yaw rates(Tx, Ty, Tz) = sailcraft roll, pitch, yaw control torquesi = inclination angle from the ecliptic plane

(−23.45 < i < 23.45)

9

5.5 Attitude Control Logic

Nonlinear PID (proportional-integral-derivative) controllogic (Refs. 25 and 26) is considered as follows:

Tx = − satU

K sat

L[(θx − θxc) +

1τ

∫(θx − θxc)] + Cωx

Ty = − satU

K sat

L[(α − αc) +

1τ

∫(α − αc)] + Cωy

Tz = − satU

K sat

L[(θz − θzc) +

1τ

∫(θz − θzc)] + Cωz

where U is the maximum control torque available in eachaxis. A variable limiter in the attitude-error feedbackloop has the self-adjusting saturation limit L as

L =C

Kmin

√2a|e|, |ω|max

where e is the attitude error, |ω|max is the specified max-imum slew rate, and a = U/J is the maximum controlacceleration for each axis. Detailed discussions of thisnonlinear PID control logic can be found in Refs. 25 and26.

In terms of the standard notation for PID controllergains, KP , KI , and KD, we have

KP = K, KI = K/τ, KD = C

The PID controller gains are determined as

KP = J(ω2n + 2ζωn/τ)

KI = J(ω2n/τ)

KD = J(2ζωn + 1/τ)

and the time constant τ of integral control is selected asτ ≈ 10/(ζωn).

Instead of the control logic of using attitude error an-gles, quaternion-feedback control logic in Refs. 25 and26 can also be employed if an attitude determinationsystem provides attitude-error information in terms ofquaternions.

5.6 Attitude and Orbit Control Simula-tion Results

For a solar sail demonstration mission in an earth-centered orbit, it is important to be in a flight regime(> 2000 km altitude) where the solar pressure force ismuch greater than the aerodynamic drag. To meet costconstraints, the main launch options may be a secondarypayload on a commercial or government launch, or fromthe Space Shuttle with an additional chemical propul-sion system to get into a higher orbit. A sail validationmission may include spiraling out to increase the orbitalradius, changing orbit inclination, or demonstrating alevitated orbit (Ref. 1).

In order to achieve good ground station visibility witha limited number of stations, to avoid eclipses, and to

provide the most likely mission design consistent with aride as secondary payload, a geosynchronous orbit wasconsidered in Ref. 1. More specifically, the geostation-ary disposal orbit was selected, which is GEO + 300 km.This is to avoid interference with geostationary satel-lites.

As illustrated in Figure 8, a simple “on-off switching”orbit-raising flight control experiment can be conductedfor such a sail flight validation mission in a geosyn-chronous orbit. The sail is oriented to face the sunwhen the sail is moving away from the sun, but it isoriented edgewise toward the sun when the sailcraft ismoving sunward. Such a simple steering profile requirestwo rapid 90-deg pitch maneuvers twice per orbit. Notethat in an earth-centered orbit, almost no orbital energycan be gained when a sailcraft is flying sunward. Moresophisticated orbit-raising, sail steering profiles can befound in Ref. 7; however, the performance improvementover the simple steering profile is quite marginal.

Attitude and orbit control simulation results for sucha simple orbit-raising sail steering profile are shown inFigures 9 through 10 for the following conditions:• initial roll/pitch/yaw attitude errors: 10 deg• initial roll/pitch/yaw rates: 0.01 deg/s• pitch turning rate constraint: 0.02 deg/s• cm/cp offset uncertainty: 0.25 m• control saturation: δa = ±1 deg, δe, δr = ±0.5 m

Simulation results indicate that the proposed sail atti-tude control system provides proper sail steering in thepresence of various constraints. Because of a near-zerocontrol torque when α > 60 deg, two 50-deg pitch ma-neuvers are performed twice per orbit. Furthermore,the propellantless orbit-raising performance of ∆a ≈ 50km/day is demonstrated in Figure 10.

6 Conclusions

An overview of sailcraft attitude control issues has beenpresented, with particular emphasis on a simple spinstabilization concept for countering the significant effectof a solar pressure disturbance torque (caused by anuncertain center-of-mass and center-of-pressure offset)on sailcraft stability and pointing. A propellantlesssail attiude control concept by means of translatingand/or tilting sail panels was also described in thispaper. A 40 × 40 m, 160-kg sailcraft with a nominalsolar pressure force of 0.01 N and the moments ofinertia of (6000, 3000, 3000) kg-m2 was chosen as abaseline sailcraft to illustrate the various concepts andprinciples involved in dynamic modeling and attitudecontrol design. The study results will provide the sailmission designer with options and approaches to meetthe various requirements of near-term sails as well asfuture advanced solar sails.

10

X

Y

t = 0

x

z

x

z

x

z

α

Fβ

γ

x

z

90 degPitch TurnManeuver

Roll

Yaw

Apogee

Perigee

SRP

Figure 8: An ideal orbit-raising sail steering profile thatrequires two rapid 90-deg pitch maneuvers. Because ofa near-zero control torque when α > 60 deg, two 50-deg pitch maneuvers (instead of 90 deg) at perigee andapogee are to be performed.

Acknowledgment

This research has been supported by the NASA SolarSail Technology Program (JPL Contract No. 1228156).It was also in part supported by New MillenniumProgram Space Technology 7 (ST7) Solar Sail StudyProject. The author would like to thank Hoppy Priceat the Jet Propulsion Laboratory, Randy Baggett andLes Johnson at NASA Marshall Space Flight Center,and Chris Moore at NASA Langley Research Center fortheir supporting this research.

References

[1] Price, H., et al.,“Design for a Solar Sail Demonstra-tion Mission,” presented at the Space Technology andApplications International Forum (STAIF 2000), Albu-querque, New Mexico, February 12-14, 2001.[2] Rogan, J., Gloyer, P., Pedlikin, J., Veal, G., andDerbes, B., “Encounter 2001: Sailing to the Stars,” pre-sented at the 15th Annual AIAA/USU Conference onSmall Satellites, Logan, Utah, August 13-16, 2001.[3] Salama, M., McInnes, C., and Mulligan, P., “Gos-samer Sailcraft Technology,” in Gossamer Spacecraft:Membrane/Inflatable Structures Technology for SpaceApplications, AIAA Progress in Astronautics and Aero-nautics Series (C. H. Jenkins, ed.), 2001.[4] West, J. L. and Derbes, B., “Solar Sail Vehicle SystemDesign for the Geostorm Warning Mission,” AIAA 2000-5326, presented at the AIAA Space 2000 Conference,Long Beach, CA, Sept. 20-21, 2000.

0 10 20 30 40 50-10

0

10

20

θ x (de

g)

0 10 20 30 40 50-50

0

50

100

α (

deg)

0 10 20 30 40 50-10

0

10

20

time (hours)

θ z (d

eg)

0 10 20 30 40 50-2

-1

0

1

δ a (de

g)

0 10 20 30 40 50-1

0

1

δ e (m

)

0 10 20 30 40 50-1

-0.5

0

0.5

time (hours)

δ r (m

)

Figure 9: Attitude & orbit control simulation results.

[5] Friedman, L., Star Sailing: Solar Sails and Interstel-lar Travel, John Wiley & Sons, New York, 1988.[6] Wright, J. L. Space Sailing, Gordon and Breach Sci-ence Publishers, 1992.[7] McInnes, C. R., Solar Sailing: Technology, Dynamicsand Mission Applications, Springer Praxis Publishing,1999.[8] Garner C. E. and Leipold, M., “Developments andActivities in Solar Sail Propulsion,” AIAA Paper 2000-3858, presented at the 36th AIAA Joint Propulsion Con-ference, Huntsville, AL, July 16-20, 2000.[9] Garwin, R. L., “Solar Sailing–A Practical Method ofPropulsion Within the Solar System,” Jet Propulsion,Vol. 28, No. 3, March 1958, pp. 188-190.[10] Sohn, R. L., “Attitude Stabilization by Means ofSolar Radiation Pressure,” ARS Journal, Vol. 29, May1959, pp. 371-373.[11] Acord, J. D. and Nicklas, J. C., “Theoretical andPractical Aspects of Solar Pressure Attitude Control for

11

0 10 20 30 40 500

50

100

γ, α

(de

g)

0 10 20 30 40 500

5

10

15

β (d

eg)

0 10 20 30 40 500

5

10

15

Thr

ust (

mN

)

time (hours)

0 10 20 30 40 504.215

4.22

4.225

4.23x 10

4

a (k

m)

0 10 20 30 40 500

2

4x 10

-3

ecce

ntric

ity e

0 10 20 30 40 50-1

0

1

time (hours)

Z (

km)

Figure 10: Attitude & orbit control simulation results.Propellantless orbit-raising of ∆a ≈ 50 km/day can beseen in this figure.

Interplanetary Spacecraft,” in Progress in Astronauticsand Aeronautics, Vol. 13, Guidance and Control II, Aca-demic Press, New York, 1964, pp. 73-101.[12] Modi, V. J. and Kumar, K. “Attitude Control ofSatellites Using the Solar Radiation Pressure,” Journalof Spacecraft and Rockets, Vol. 9, No. 9, 1972, pp. 711-713.[13] Joshi, V. K. and Kumar, K. “New Solar AttitudeControl Approach for Satellites in Elliptic Orbits,” Jour-nal of Guidance and Control, Vol. 3, No. 1, 1980, pp. 42-47.[14] Stuck, B. W., “Solar Pressure Three-Axis Atti-tude Control,” Journal of Guidance and Control, Vol. 3,No. 2, 1980, pp. 132-139.[15] Angrilli, F. and Bortolami, S., “Attitude andOrbital Modelling of Solar-Sail Spacecraft,” EuropeanSpace Agency Journal, Vol. 14, No. 4, 1990, pp. 431-446.

[16] Renner, U., “Attitude Control by Solar Sailing: APromising Experiment with OTS-2,” European SpaceAgency Journal, Vol. 3, No. 1, 1979, pp. 35-40.[17] Lievre, J., “Solar Sailing Attitude Control of LargeGeostationary Satellite,” IFAC Automatic Control inSpace, Pergammon, Oxford, UK, 1985, pp. 29-33.[18] Azor, R., “Solar Attitude Control Including ActiveNutation Damping in a Fixed-Momentum Wheel Satel-lite,” Proceedings of AIAA Guidance, Navigation, andControl Conference, Hilton Head Islands, SC, August10-12, 1992, pp. 226-235.[19] Sidi, M. J., Spacecraft Dynamics and Control: APractical Engineering Approach, Cambridge UniversityPress, 1997, pp. 229-236.[20] Burns, R., et al., “Solar Radiation Pressure Effectson Formation Flying of Satellites with Different Area-to-Mass Ratios,” AIAA Paper 2000-4132, AIAA/AASAstrodynamics Specialist Conference, Denver, CO, Au-gust 14-17, 2000.[21] Williams, T. and Wang, Z.-S., “Potential Non-Propulsive Stationkeeping Techniques for PicosatelliteFormation Flight,” AIAA Paper 2000-4134, AIAA/AASAstrodynamics Specialist Conference, Denver, CO, Au-gust 14-17, 2000.[22] Wie, B. and Roithmayr, C., “Integrated Orbit, At-titude, and Structural Control Systems Design for SpaceSolar Power Satellites (SSPS),” AIAA Paper 2001-4273,AIAA Guidance, Navigation, and Control Conference,Montreal, Canada, August 6-9, 2001 (also NASA/TM-2001-210854, June 2001).[23] Bryson, A. E., Jr., Control of Spacecraft and Air-craft, Princeton University Press, Princeton, NJ, 1994.[24] Wie, B., Space Vehicle Dynamics and Control,AIAA Education Series, AIAA, Washington, DC, 1998.[25] Wie, B. and Lu, J., “Feedback Control Logic forSpacecraft Eigenaxis Rotations Under Slew Rate andControl Constraints,” Journal of Guidance, Control,and Dynamics, Vol. 18, No. 6, 1995, pp. 1372-1379.[26] Wie, B., Heiberg, C., Bailey, D., “Rapid Multi-Target Acquisition and Pointing Control of Agile Space-craft,” Journal of Guidance, Control, and Dynamics,Vol. 25, No. 1, 2002.[27] Wie, B., “Dynamic Modeling and Attitude Controlof Solar Sail Spacecraft: Part II,” Paper No. 2002-4573,AIAA Guidance, Navigation, and Control Conference,Monterey, CA, August 5-8, 2002.[28] Agrawal, B. N., Design of Geosynchronous Space-craft, Prentice-Hall, Englewood Cliffs, NJ, 1986, pp. 133-135.

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