MODELING AND SIMULATION

OF A POWER SAIL

Matthew P. Wilkins�

Texas A&M University, College Station, TX 77843

Kamesh SubbaraoyThe MathWorks Inc. 3 Apple Hill Dr. Natick, MA 01760

Kyle T. AlfriendzTexas A&M University, College Station, TX 77843

and Srinivas R. VadalixTexas A&M University, College Station, TX 77843

The objective of this work is to model and simulate the behavior of a new type of

satellite system called the Power Sail. The Power Sail concept includes a large solar array,

measuring 400m2, that is own separately from the host satellite but connected via rigid

links. The concept calls for two rigid links, which also act as power conduits, connected

by a joint. The joint is modeled as being locked with some desired angle between the

links. The attitude of the Power Sail system is controlled via thrusters located on the

Power Sail solar panel. The natural motion of this system is found to be unstable in the

desired nominal con�guration. A control law is developed to maintain a desired reference

attitude. To minimize reaction forces on the host satellite, a constant reference attitude

with the center of mass located on the host orbit is found to be desirable.

Nomenclature

m1 Point mass located at the host, kgm2 Point mass located at the joint, kgm3 Point mass located at the Power Sail, kgmt Total mass, kgL Length of the rigid links, m� Gravitational parameter, m3=s2

�1 Spherical coordinate for rhj , rad�2 Spherical coordinate for rjp, rad�1 Spherical coordinate for rhj , rad�2 Spherical coordinate for rjp, rad�12 �2 - �1, radx ECI position, x-coordinate, my ECI position, y-coordinate, mz ECI position, z-coordinate, mu UVW position, radial direction, mv UVW position, along-track direction, mw UVW position, out-of-plane direction, mC Transformation matrix from UVW to ECIq State vectorx State vectorr Position Vector!h Angular velocity vector of host, rad=s! Argument of the perigee, radf True Anomaly, rad Right Ascension of the Ascending Node, radi Inclination, rad� Argument of the Latitude, ! + f , radn orbit frequency, rad=s

�Graduate Research AssistantyDeveloper, Control System ID ToolboxeszProfessor, Aerospace Engineering Dept.xProfessor, Aerospace Engineering Dept.

F Force, N

Subscripts

h Host satellitep Power Sailj Jointhj Vector from host to jointjp Vector from joint to Power Sailcm Center of Masspert Perturbation

Superscripts

(_) 1st Time Derivative(�) 2nd Time Derivative(~) Skew-symmetric matrixO LVLH Reference FrameI ECI Reference FrameB Body Reference FrameT Transpose

Introduction

THE Power Sail concept is a novel concept whichattempts to address a major issue in satellite de-

sign - available power.1 In this concept, the hostsatellite is connected by two rigid links to a separate,large, solar array or \sail" (see section on Geometry).This sail, with a planform area of 400 m2, dwarfs anormal solar array on a satellite. The attachment be-tween the links and the host satellite is bu�ered bya long stroke mechanical isolator which is designed tominimize transfer of perturbations to the host. The ul-timate goal of this endeavor is to allow for the sail tobe sun-pointing while minimizing reaction forces im-parted to the host.

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Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

In the Power Sail Phase I Final Report ,2 a modelwas constructed using Hill's Equations to study fuel re-quirements necessary to treat the Power Sail system asa formation ying system and maintain the formationat a �xed distance relative to the host satellite. Thehost and sail were treated as point masses connectedby a exible umbilical cord (a previous design). It wasfound that the sail should be �xed in an o�set posi-tion from the local horizontal plane of the host in orderto counter a portion of the solar radiation pressure ef-fects which, in turn, minimizes fuel requirements. Thiso�set allowed for the countering a portion of the ef-fects of solar radiation pressure on the sail withoutapplying thrust. This �xed position is o� to the sideopposite the sun and either leading or following thehost. The Phase I Final Report also showed that so-lar radiation pressure force must be almost continuallycountered to maintain the desired attitude. This paperwill also show that the solar radiation pressure must becontinually countered to avoid undesirable behavior.However, we will not treat the rigid body torque equi-librium attitude condition to eÆciently counter SRPperturbations at this time.This paper will model the Power Sail system as three

separate point masses (the host satellite, the joint con-necting the two links, and the Power Sail). The massof the two links and associated power cabling is as-sumed to be evenly lumped into the aforementionedpoint masses. In order to simulate the presence of therigid links, these three free- ying point masses mustbe constrained such that the magnitude of the host-joint vector and the Power Sail-joint vector remainsconstant. This was done by incorporating the motionof each individual point mass into a single equationof motion for the center of mass of the system. Us-ing Lagrange's Equations, the rotational equations ofmotion for the system were derived. This will be dis-cussed further in the Equations of Motion section.In considering the types of perturbations this sys-

tem will encounter, we looked to a similar problemof formation ying. In this case, di�erential gravita-tional accelerations lead to so-called gravity gradienttorques. These gravity gradient torques become thedominate perturbation in formation ying. However,the Power Sail has a large planform area which is inclose proximity to the host satellite. This makes thesolar radiation pressure force and aerodynamic forcesthe dominant perturbations. Additionally, we need toworry about reaction forces being transmitted to thehost. Through a simple analysis, one can show that re-action forces on the host can be minimized by ensuringthat the center of gravity of the arm-sail sub-systemlies on the orbit of the host.

Geometry

A simpli�ed representation of the geometry of thePower Sail system can be seen in Figure 1. Although

Rigid Link/Electrical Conduit

HostSatellite

ConstrainedJoint

IsolatorPower Sail

rjp

rhj

Fig. 1 Nominal geometry of the Power Sail sys-tem.

θ1

rhj

φ1

u

v

w

Joint

Host

Fig. 2 De�nition of �1 and �1.

the actual design of the Power Sail system has a num-ber of additional links and actuators to make the sys-tem physically realizable, we are assuming that we canaccurately model this system by lumping certain seg-ments together. Because we are assuming that thehost and Power Sail are both perfectly controlled, wedo not need any angles describing their orientation.Thus, we have only to worry about the orientation ofthe two links. Since the length of each link is speci�ed,we only need two angles to adequately de�ne their ori-entation. Thus there are four degrees of freedom inthis system. In reality, the joint connecting the twolinks together will have an actuator tightly controllingthe angle between the two links. This leads us to treatthe joint as being �xed and rigid which further reducesthe number of DOF to three. This constraint will bediscussed further at the end of this section and in theEquations of Motion section.

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θ2

rjpφ

2

u

v w

Joint

Sail

Fig. 3 De�nition of �2 and �2.

The link connecting the host to the joint is called rhjwhile the link connecting the joint to the Power Sailis called rjp. rcm is the Earth Centered Inertial (ECI)vector describing the center of mass of the system. Theinertial positions of each point mass can be written asfollows:

rIh = [xh yh zh]T (1)

rIj = rIh + CrOhj (2)

rIp = rIj + CrOjp: (3)

Using the geometry described above, we can constructthe vectors rOhj and r

Ojp as the following (see Figure 2):

rOhj = [L cos �1 cos�1 L sin �1 cos�1 L sin�1]T (4)

rOjp = [L cos �2 cos�2 L sin �2 cos�2 L sin�2]T (5)

where L is the length of the rigid links.We have assumed that the transformation matrix,

C, from LVLH to ECI is approximately the same forboth rhj and rjp. This assumption is valid due to thefact that the distance between the host and the jointis negligibly small compared to the distance betweenthe host and the center of the Earth.We now de�ne the state vector q as:

q = [�1 �2 �1 �2]T (6)

where �1 and �1 are as de�ned in Figure 2. �2 and �2are de�ned identically from a coordinate system whoseorigin is placed at the joint (see Figure 3). Again, thetransformation between these two coordinate systemsis assumed to be negligibly small.

As mentioned previously, we need to lock the jointor, more to the point, constrain the motion of the linksrelative to each other. For small �i, we can constrain�2 to be a function of �1 as follows:

�2 = �1 + � � � (7)

where � is the prescribed interior angle between thetwo links. Except where noted, we assume this suf-�ciently enforces the joint to be \locked." Also notethat, due to this constraint, we must enforce the fol-lowing conditions on the derivatives of �2:

_�2 = _�1 ��2 = ��1 (8)

Equations of Motion

As mentioned in the introduction, in order to in-corporate the physical constraints on the system (i.e.rigid links between the masses), the equation of motionfor the center of mass was constructed using Newton's�rst principles. This equation was then rearrangedsuch that we could solve for the acceleration of thehost. The host equation of motion is as follows:

mt�rIh = �(m2 +m3)[ �Cr

Ohj + 2 _C _rOhj + C�rOhj ]

�m3[ �CrOjp + 2 _C _rOjp + C�rOjp]

� �

"m1r

Ih

jrIhj3+m2r

Ij

jrIj j3+m3r

Ip

jrIpj3

#+Fpert (9)

where C is the transformation matrix from the LVLHto ECI reference frame and

mt = m1 +m2 +m3: (10)

Note that m1 is the host mass plus one half the massof the link from the host to the joint, m2 is the massof the joint plus two one-halves of the link masses,and m3 is the mass of the solar panels plus one halfthe mass of the link from the joint to the solar pan-els. Fpert contains all the perturbation forces acting onthe system such as solar radiation pressure and Earthoblateness e�ects. Also note that we are assuming thatthe attitude of the host and the Power Sail are bothperfectly controlled. Based upon this assumption, weassume that the entire solar panel is facing the sun atall times for purposes of computing the solar radiationpressure acting upon the system.By integrating this host equation of motion, and

knowing the link angles at any time, we can solve forthe position of the other two masses using Eq. 2 andEq. 3. Thus, to complete these equations, the instan-taneous values of the state vector q and its derivativesmust be provided.The equations describing the motion of the link an-

gles can be derived using Lagrange's equations:

d

dt

�@L

@ _q

��@L

@q= Q+Bu (11)

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with

L = T � V (12)

where T is the kinetic energy of the system,

T =1

2

3Xi=1

mi _ri � _ri; (13)

V is the potential energy,

V = �

3Xi=1

�mi

jrij; (14)

Q is a generalized torque and u is a generalized control.

By deriving the kinetic and potential energy terms,along with their partial derivatives, we can rearrangeLagrange's equations to get the following form:

M �q = F�G+Q+Bu (15)

where M = M(q;mi; L) is a symmetric 4 � 4 matrixwith values of

M11 = (m2 +m3)L2 cos2 �1

M12 = m3L2 cos�1 cos�2 cos �12

M13 = 0

M14 = �m3L2 cos�1 sin�2 sin �12

M21 =M12

M22 = m3L2 cos2 �2

M23 = m3L2 sin�1 cos�2 sin �12

M24 = 0 (16)

M31 =M13

M32 =M23

M33 = (m2 +m3)L2

M34 = m3L2(sin�1 sin�2 cos �12 + cos�1 cos�2)

M41 =M14

M42 =M24

M43 =M34

M44 = m3L2;

G = G(q; _q;mi; L) is a 4 � 1 column vector with

values of

G1 = �L2(m2 +m3) _�1 _�1 sin 2�1

� L2m3( _�2

2+ _�22) cos�1 cos�2 sin �12

� 2L2m3_�2 _�2 cos�1 sin�2 cos �12

G2 = �2m3L2 _�1 _�1 sin�1 cos�2 cos �12

�m3L2 _�2 _�2 sin 2�2

+m3L2( _�21 +

_�21) cos�1 cos�2 sin �12 (17)

G3 = �m3L2 _�2

2cos�1 sin�2

� 2m3L2 _�2 _�2 sin�1 sin�2 sin �12

+m3L2( _�2

2+ _�2

2) cos�2 sin�1 cos �12

+ (m2 +m3)L2 _�21 sin�1 cos�1

G4 = �m3L2 _�21 sin�1 cos�2

+m3L2( _�2

1+ _�2

1) cos�1 sin�2 cos �12

+ 2m3L2 _�1 _�1 sin�1 sin�2 sin �12

+m3L2 _�2

2cos�2 sin�2;

and F = F(q; _q;mi; L) is a 4 � 1 column vector de-scribed by the following equation:

F =�m2C[@

@ _q_rOhj ] � (�r

Ih + 2 _C _rOhj +

�CrOhj)

�m3C

�[@

@ _q_rOhj ] + [

@

@ _q_rOjp]

��h

�rIh + 2 _C(_rOhj + _rOjp) + �C(rOhj + rOjp)i

� �m2

�C[

@

@qrOhj ] � r

Ij

�(rIj � r

Ij )�3=2

� �m3

�C

�[@

@qrOhj ] + [

@

@qrOjp]

�� rIp

�(rIp � r

Ip)�3=2;

(18)

Q = Fpert �@r

@q; (19)

u = Fthrust �@r

@q; (20)

where r is a vector to the point of application of theforce, and, �nally, the control matrix B is a 4 � 4identity matrix.Note that the 1st and 2nd time derivatives of the

transformation matrix C must be known. These canbe easily obtained using the identity:

_C = �[~!h]C (21)

which leads to an equation for the 2nd time derivative:

�C = �[ _~!h]C + [~!h][~!h]C (22)

where [~!h] is the skew symmetric cross-product matrixconstructed using the angular velocity vector for the

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host satellite. !h is de�ned as:

!h = [ _ sin i sin � + _i cos � _ sin i cos � � _i sin �

_� + _ cos i]T

(23)

Recognize that perturbations on the host will translateinto changes in the node, inclination, and argument ofthe latitude which directly impact the computation ofthe transformation matrix. Thus we can compute _,_i, and _� along with �, �i, and �� as follows:

_ =jrhj sin �

h sin i(Fh;pert)3 (24)

_i = _ sin i cot � (25)

_� =h

jrhj2� _ cos i (26)

where (Fh;pert)3 is the 3rd-component of the pertur-bation vector.

� =

"_jrhj sin � + jrhj _� cos �

h sin i

�jrhj _h sin �

h2 sin i�jrhj_i cos i sin �

h sin2 i

#(Fh;pert)3 (27)

�i =

"_jrhj cos � � jrhj _� sin �

h�jrhj _h cos �

h2

#(Fh;pert)3

(28)

�� =_h

jrhj2�

2h _jrhj

jrhj3� � cos i+ __i sin i (29)

Fh;pert contains only the perturbations acting uponthe host. One such perturbation that must be in-cluded, Fhj , is an internal reaction force along the linkconnecting the host satellite to the joint. This internalforce can be calculated as

Fhj = m1�rh +�m1rh

jrhj3(30)

In the above equations, we assumed that _Fhj is negli-gibly small.

Reduction of Order

As discussed earlier, the angle �2 is not a free an-gle due to our constraints. Therefore, the equation ofmotion for �2 is not independent and we must incor-porate this fact. One way to accomplish this, is to usea Lagrange multiplier. Looking back at Equation 15in simpli�ed form:

M �q = F (q; _q;u)

We also have a constraint equation of the form:

Cq�D = 0

C _q = 0 (32)

C�q = 0

Modifying our equations with a Lagrange multiplier,�,

�M CT

C 0

���q�

�=

�F0

�(33)

we can now solve for q and �.

However, a more compact way is to de�ne a matrixG which is orthogonal to CT such that the diagonalentries of G are 1's for the variables that will be treatedas independent. If z is a new state vector of indepen-dent coordinates, then we can write:

q = GT z+ constant (34)

Using Equation 34, we can write our reduced orderequations of motion as:

�z = (GMGT )�1GF (35)

where, in our case,

G =

241 1 0 00 0 1 00 0 0 1

35 (36)

This reduces the number of independent equationsfrom 4 to 3. Our control technique will be appliedto this reduced order system.

Control

There are two primary objectives in the control ofthe Power Sail system. First, we would like to ensurethat the solar panel can track the sun without fear ofthe links impinging upon its range of motion. Second,we want to control the system such that a minimum offorces and moments are imparted to the host. The onlyavailable thrusters for control are on the Sail. Thesethrusters are intended to orient the Sail with respectto the sun, control the orbit of the system, and achieveany desired attitude.

This paper will look primarily into the attitude con-trol of the system. We assume that the host and thesail are controlled appropriately to achieve their mis-sion objectives. Now, notice that there really is no\optimal" attitude for this system. There are an in-�nity of possibilities which would satisfy our controlobjectives. Thus, we can prescribe a desired attitudefor the system and then attempt to control to that ref-erence attitude. To accomplish this task, we use thelinear quadratic regulator (LQR) control technique.

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LQR Controller

The standard LQR controller is of the followingform:

_x = Ax+Bu; t � t0 (37)

J =1

2xT (T )S(T )x(T ) +

1

2

Z T

t0

(xTQx+ uTRu)dt

(38)

S(T ) � 0; Q � 0; R > 0 (39)

� _S = ATS + SA� SBR�1BTS +Q; (40)

for t � T and given S(T )

K = R�1BTS (41)

u = �Kx (42)

where x is the state, u is the control input, S is an in-termediate variable that must be computed o�-line, Qand R are weighting matrices on the state and control,A is the plant matrix, B is the control matrix, and Kis the optimal gain matrix.

However, because we have a non-linear system, weare forced to linearize the motion of the Power Sailabout some desired reference state. Our system model,in �rst order form, looks like the following:

_x = f(x; t;d) +

�0

M�1B

�u (43)

where

x =�q _q

�T; (44)

and d is a disturbance vector.

Using a standard Taylor series expansion, the lin-earized equations take the following form:

_xR+Æ _x = f(xR; t;d)+@f

@x

????xR

Æx+H:O:T:+

�0

M�1B

�u

(45)where xR is the desired reference trajectory whichmust be speci�ed.Rearranging, we have:

Æ _x �@f

@x

????xR

Æx+ f(xR; t;d)� _xR+

�0

M�1B

�u (46)

If we set�0B�

�u� = f(xR; t;d)� _xR +

�0

M�1B

�u (47)

and

A =@f

@x

????xR

(48)

Then we obtain the desired form

Æ _x = AÆx+

�0B�

�u� (49)

where one can easily see that the optimal solution tou�, which controls the departure from the desired tra-jectory, can be written as

u� = �KÆx: (50)

Rearranging,�0

M�1B

�u =

��

�0B�

�KÆx� f(xR; t;d) + _xR

�(51)

The actual control u can be found by solving the lowerhalf of Eq. 51. In the next section, we will apply thiscontrol and examine results.

Results

Unless otherwise speci�ed, we set up the followingorbital parameters in order that results can be com-pared between each scenario:

a = 7378 km

e = 0

i = 90Æ

= 90Æ (52)

! = 0Æ

f = 0Æ

The epoch date is set to March 20, 1998, 19:55:00.00(Vernal Equinox). This particular set of parameters,beginning at this epoch, allows us to get purely out-of-plane solar radiation pressure force. The solar ra-diation pressure model used can be found in Vallado'sFundamentals of Astrodynamics and Applications, 2nd

Edition.3 The sail is assumed to be perfectly sun-pointing and has a 400 m2 e�ective area exposed tothe sun.The mass of each component is as follows:

m1 = 3018:66 kg

m2 = 38:32 kg (53)

m3 = 318:66 kg

These masses are approximate values based upon pre-liminary design data for the host satellite, link massper unit length, and mass of the solar panels.Nominally, we would like the Power Sail system to

maintain the following attitude:

q0 = [55Æ 125Æ 0Æ 0Æ]T (54)

_q0 = [0 0 0 0]T Æ=s:

Each of the cases presented will have initial conditionscorresponding to this nominal state. We desire thatthe host and Power Sail maintain a 12m separation.This makes the length of each link 7:32m. Figure 4illustrates the nominal con�guration of the Power Sail.

All integration was performed using a Runge-KuttaOrder 4 method with a 1 second integration time step.

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Z

YEquator Host

Sail

1000 km orbit

U

V

Earth

Joint

Fig. 4 Nominal con�guration of the Power Sailshowing the inertial and relative coordinate sys-tems at initial epoch. The sun is located directlybehind the Earth. The X and W axes are comingout of the page.

Natural Motion

First, we would like to examine the motion of thePower Sail system in an unperturbed and uncontrolledenvironment. Figure 5 shows the results of placing

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−100

−50

0

50

θ 1 (de

g)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−50

0

50

100

θ 2 (de

g)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−2

−1

0x 10

−17

φ 1 (de

g)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

x 10−17

φ 2 (de

g)

Orbits

Fig. 5 Natural motion of the simpli�ed Power SailSystem (uncontrolled and unperturbed).

the Power Sail system in the nominal con�guration andobserving its behavior for two orbits. Clearly, the sys-tem is weakly unstable. The system will stay near itsinitial attitude until gravity gradient torques take overand causes it to ip. Like a pendulum, the system ipsback and forth in a stable oscillation before returningto its initial state in a period of two orbits. Notice,that since there are no out of plane perturbations, theangles �1 and �2 remain zero.

One of our objectives is to minimize the forces ap-plied to the host satellite. Figure 6 shows the internalreaction force along the link connecting the host tothe rest of the system. Notice that there are uctua-

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2

0

2

4x 10

−5

Fhj

u (N

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2

−1

0

1

2x 10

−5

Fhj

v (N

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−5

0

5

10x 10

−26

Fhj

w (

N)

Orbits

Fig. 6 Internal reaction force along link betweenhost and joint in LVLH coordinates (uncontrolledand unperturbed).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2

0

2x 10

−16

diff x (

m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2

0

2

diff y (

m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2

0

2

diff z (

m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

100

200

Rcm

Ang

le (

deg)

Orbits

Fig. 7 ECI Position di�erence between the systemcenter of mass and circular reference orbit. RcmAngle is the angle between the center of mass andthe host orbit (rcm � rh).

tions at very speci�c times - about 0.5 and 1.5 orbits.Looking at Figure 7 gives insight into this phenomena.This �gure shows the movement of the system centerof mass relative to the host's circular orbit. Addition-ally, a plot of the angle between the host orbit and thevector from the host to the center of mass is provided.Clearly, by examining Figures 6 and 7, we can see that,as the center of mass is accelerated, internal stressesare generated along the link. This analysis shows thatwe need to minimize the motion of the center of massin order to minimize forces imparted to the host whichsuggests that we should attempt to control the PowerSail system to a constant state.

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Perturbed but Uncontrolled Motion

In this section, we would like to see what e�ectadding in solar radiation pressure will have on the un-controlled motion of the Power Sail System. Recallthat we are assuming that the solar panels are per-fectly sun pointing. Also recall that due to the orbitselected, the force will be purely out of plane. We haveassumed that the joint is small enough that the SRPforce is negligible. Figure 8 shows the results of thisscenario.

0 0.5 1 1.5 2 2.5 3 3.5 4

−150−100

−500

50

θ 1 − θ

1r (

deg)

0 0.5 1 1.5 2 2.5 3 3.5 4−100

0

100

θ 2 − θ

2r (

deg)

0 0.5 1 1.5 2 2.5 3 3.5 4−200

−100

0

φ 1 − φ

1r (

deg)

0 0.5 1 1.5 2 2.5 3 3.5 4−200

−100

0

φ 2 − φ

2r (

deg)

Orbits

Fig. 8 Natural motion of the simpli�ed Power SailSystem under the in uence of solar radiation pres-sure (uncontrolled).

As expected, the out-of-plane angles �1 and �2 areno longer constant or zero. In the unperturbed case,we saw bounded, near sinusoidal motion for the in-plane angles. However, with the addition of solarradiation pressure force to the model, we see that the�i are no longer in a stable oscillation although thecharacter of the motion is similar. The �i also seem tohave a secular growth component which translates intoan undesirable tumbling motion. Notice that responseof �1 to the SRP perturbation lags behind �2. This isdue to the fact that the SRP force is applied only tothe Sail. We examined a number of randomly chosenorbits and similar results were produced for all thesecases. Because of the large out-of-plane excursions,our \locked" joint constraint is violated; however, thisis acceptable for this demonstration simulation.

Controlled Motion

We can now turn to control to attempt to correct thede�ciencies pointed out in previous sections. First, areference attitude must be selected which we wouldlike to achieve. From our analysis of the uncontrolledmotion, we saw that to minimize the reaction forces inthe links, we should try to keep the center of mass frommoving. Thus, common sense dictates that we specifya constant reference attitude for our controller. In thiscase, we will control the Power Sail system such thatit maintains its desired initial attitude.

There are essentially two aspects to the control ofthe Power Sail system. First, the controller mustovercome initial condition errors. Second, the con-troller must overcome external perturbations such asgravity gradient torques and solar radiation pressure.Third, the controller must counteract any residualforce model di�erences between the actual and desiredreference attitude.Several parameters must be selected to tune the

LQR controller used in this paper. The Q and Rmatrices can be selected by examining the LQR costfunction (Equation 38) in non-dimensional space. Thisleads to values of:

Q = diag(�1 1 1 n�2 n�2 n�2

�) (55)

R = n�4 � I3 (56)

where I3 is a 3 � 3 identity matrix and n is the or-bit frequency. Next we solve the steady-state Riccattiequation using our A and B matrices along with theabove Q and R values to �nd a steady-state value forS(T ). The numerical solution for the S(T ) matrix isnot presented for brevity. S(T ) can be selected inde-pendently from Q and R; however, the integration ofthe control during the simulation can become unstableif S(T ) is not selected very carefully.

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

θ 1 − θ

1r (

deg)

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

θ 2 − θ

2r (

deg)

0 0.5 1 1.5 2 2.5 3 3.5 4−1

0

1

2

φ 1 − φ

1r (

deg)

0 0.5 1 1.5 2 2.5 3 3.5 4

−1

0

1

φ 2 − φ

2r (

deg)

Orbits

Fig. 9 Link angle tracking errors for initial errorof N(0Æ, 5Æ).

In Figure 9, a Gaussian random error was appliedto the initial angles with a N(0Æ, 5Æ) distribution. Forthis case, k = 1, and we achieve our goal withinroughly 2 orbits. Notice that the tracking error of thein-plane angles, �i, is damped out much more rapidlythan that of the out-of-plane angles, �i. By multi-plying the Q matrix by a factor of k = 1; 2; 3; : : :,we can further tune the performance to achieve thedesired reference state within a speci�ed number oforbits. Figure 10 illustrates the generalized torquesrequired to achieve the reference attitude. Notice thatalthough the initial errors are damped out, the mag-nitude of the control does not go to zero. This is due

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0 0.5 1 1.5 2 2.5 3 3.5 4

−0.04

−0.02

0

u θ1 (

N m

)

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.01

0.02

0.03

u φ1 (

N m

)

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.01

0.02

0.03

u φ2 (

N m

)

0 0.5 1 1.5 2 2.5 3 3.5 4

10−2

|u| (

N m

)

Orbits

Fig. 10 Generalized control requirement for cor-recting initial condition errors.

to the fact that we must continually counteract thee�ects of SRP.

0 0.5 1 1.5 2 2.5 3 3.5 4

0

2

4

6

8x 10

−3

u u (N

)

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

x 10−4

u v (N

)

0 0.5 1 1.5 2 2.5 3 3.5 4

2

4

6

x 10−3

u w (

N)

0 0.5 1 1.5 2 2.5 3 3.5 4

10−2

|u| (

N)

Orbits

Fig. 11 Actual control requirement for correctinginitial condition errors.

The exact location of thrusters on the Power Sailsolar panels will be determined during the design pro-cess; however, for this paper, we assume that a thrustforce will be applied at the center of mass of the PowerSail. Thus, we can use Eq. 19 to back-solve for theactual thrust requirement using a least-squares tech-nique for over-determined systems.4 Figure 11 showsthe actual thrust requirement in the relative coordi-nate frame to remove initial condition errors. We seehere that the predominant control e�ort is applied inthe radial (u) and out-of-plane (w) directions. Theradial control counteracts the pitching motion seen inFigure 5 while the out-of-plane control counteracts thee�ects of the SRP force.Note however, that because we were solving an over-

determined system of equations, we will invariablyhave an error in the solution. This error is shown inFigure 12 and illustrates that we need an additional

0 0.5 1 1.5 2 2.5 3 3.5 4

0

5

10

x 10−5

err θ (

N m

)

0 0.5 1 1.5 2 2.5 3 3.5 4−2.5

−2

−1.5

−1

−0.5

x 10−4

err φ1

(N

m)

0 0.5 1 1.5 2 2.5 3 3.5 4

0.5

1

1.5

2

2.5

x 10−3

err φ2

(N

m)

Orbits

Fig. 12 Error in the control solution for correctinginitial condition errors.

control beyond the three-component thrust force tocontrol the system precisely. This can be most eas-ily accomplished by applying a torque at the joint tocompensate for these real world controller errors.

0 0.5 1 1.5 2 2.5 3 3.5

4.1

4.2

4.3

4.4

4.5x 10

−7F

hju (

N)

0 0.5 1 1.5 2 2.5 3 3.5−2

−1

0

1

2x 10

−8

Fhj

v (N

)

0 0.5 1 1.5 2 2.5 3 3.5

−1.6131

−1.6131

−1.6131

−1.6131

−1.6131x 10

−6

Fhj

w (

N)

Orbits

Fig. 13 Internal reaction force along the link be-tween the host and the joint in LVLH coordinatesfor the nominal con�guration with no initial error.

Looking to our requirement of minimizing the re-action forces on the host, we examine Figure 13. Toavoid ambiguity, the initial angle error was removed.Here we �nd that, for this case, the reaction force ispredominantly in the radial and out-of-plane directionsas might be expected. These reaction forces are unde-sirable, and we would like to eliminate them if possible.

Although diÆcult to tell in Figure 7, the center ofmass starts at 2:27Æ away from the host orbit. Usinga simple analysis, one can see that, if two objects areconnected in orbit and if they do not follow preciselythe same orbit, there will be a reaction force betweenthem. This problem can be easily solved by moving

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0 0.5 1 1.5 2 2.5 3 3.5−2

−1

0

1

2x 10

−8

Fhj

u (N

)

0 0.5 1 1.5 2 2.5 3 3.5−2

−1

0

1

2x 10

−8

Fhj

v (N

)

0 0.5 1 1.5 2 2.5 3 3.5

−1.6131

−1.6131

−1.6131

−1.6131

−1.6131x 10

−6

Fhj

w (

N)

Orbits

Fig. 14 Internal reaction force along link betweenhost and joint in LVLH coordinates with the centerof mass controlled.

the initial condition angles such that the center of massof the arm-Power Sail subsystem will lie in the orbitof the host. Using a steepest descent algorithm,5 thecenter of mass equation was back-solved to �nd thefollowing angles:

q0 = [57:275Æ 127:275Æ 0Æ 0Æ]T (57)

With these values used for the initial state, the sim-ulation was run again. Again, the initial error wasremoved. The reaction forces for this case can be seenin Figure 14. Comparing to Figure 13, where the cen-ter of mass has not been moved, we can see that thereis a marked reduction in force by an order of magni-tude in the radial direction. The reaction force in theout-of-plane direction remains the same in both casesdue to the SRP force. This indicates that the center ofmass of the link - Power Sail subsystem should lie inthe same orbit as the host satellite to aid in minimizingthe reaction forces imparted on the host.

Concluding Remarks

This paper demonstrates the natural motion of thePower Sail system. We found that this system is easilydestablized by SRP perturbations. A control systemwas developed to maintain a desired reference attitude.Our two primary goals were to control the attitude ofthe system in such a way that the links do not impingeupon the range of motion of the solar panels and tominimize the reaction forces on the host. This wasaccomplished by establishing a constant reference atti-tude which satis�es any interference issues that mightarise and then ensuring that the center of mass of thesystem lies on the same orbit as the host. As the PowerSail system orbits the Earth, the solar panels and/orthe links may need to completely reorient such thatsun-pointing can be achieved. This re-orientation canbe accomplished easily by giving the controller a new

reference attitude. We also found that internal reac-tion forces along the links will be all but unavoidableduring maneuvers or by maintaining attitudes wherethe center of mass of the arm-Power Sail subsystemdoes not lie on the orbit of the host. Because of somediÆculties in doing analysis with the choice of angles,future work on this project will include a new set ofangles. Additionally, with this new angle set, a torqueequilibrium attitude analysis will be completed as away to reduce control e�ort.

References1Meing, T., Reinhardt, K., Luu, K., Blankinship, R., Huy-

brechts, S., and Das, A., \Power Sail - A High Power Solution,"

AIAA Paper 00-581, 2000.2Alfriend, K. T., Vadali, S. R., Junkins, J. L., and Subbarao,

K., \Power Sail Phase I Final Report," Texas A&M University,

November 20, 2000.3Vallado, D. A., Fundamentals of Astrodynamics and Appli-

cations, McGraw-Hill, New York, 2nd ed., 2001.4Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel,

J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling,

S., McKenney, A., and Sorensen, D., LAPACK Users' Guide,

Society for Industrial and Applied Mathematics, Philadelphia,

PA, 3rd ed., 1999.5Plybon, B. F., An Introduction to Applied Numerical Anal-

ysis, PWS-Kent Publishing Company, Boston, MA, 1992.

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