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Attitude Maneuvers of a Rigid Spacecraft in a Circular
OrbitTaeyoung Lee, N. Harris McClamroch
Department of Aerospace EngineeringUniversity of Michigan, Ann
Arbor, MI 48109
{tylee, nhm}@umich.edu
Melvin LeokDepartment of Mathematics
University of Michigan, Ann Arbor, MI [email protected]
Abstract A global model is presented that can be used tostudy
attitude maneuvers of a rigid spacecraft in a circularorbit about a
large central body. The model includes gravitygradient effects that
arise from the non-uniform gravity fieldand characterizes the
spacecraft attitude with respect to theuniformly rotating local
vertical local horizontal coordinateframe. An accurate
computational approach for solving anonlinear boundary value
problem is proposed, assuming thatcontrol torque impulses can be
applied at initiation and attermination of the maneuver. If the
terminal attitude conditionis relaxed, then an accurate
computational approach for solvingthe minimal impulse optimal
control problem is presented.Since the attitude is represented by a
rotation matrix, this ap-proach avoids any singularity or ambiguity
arising from otherattitude representations such as Euler angles or
quaternions.
I. INTRODUCTIONThe attitude dynamics of an uncontrolled rigid
spacecraft
in a circular orbit about a large central body, includinggravity
gradient effects, have been extensively studied; see[1], [2]. There
are 24 distinct relative equilibria for which theprincipal axes are
exactly aligned with the local vertical localhorizontal (LVLH)
axes, and the spacecraft angular velocityis identical to the
orbital angular velocity of the LVLHcoordinate frame. Linear
rotational equations of motion thatdescribe small perturbations
from any relative equilibriumsolutions are well known. Linear
attitude control of a rigidspacecraft in a circular orbit,
including linear gravity gradienteffects, has also been addressed
in [2]. However, linearcontrollers have the limitation that they
are only applicableto small attitude change maneuvers.
The emphasis in this paper is on large angle attitude ma-neuvers
of a rigid spacecraft in a circular orbit about a largecentral
body, including gravity gradient effects. A nonlinear,globally
defined model is introduced. This model describesthe attitude of
the spacecraft, relative to the uniformlyrotating LVLH coordinate
frame, by a rotation matrix. Themodel includes gravity gradient
terms that reflect the rotationof the LVLH frame, and terms that
reflect control inputtorques. In the problems studied in this
paper, independentimpulsive control torques can be applied about
each principalaxis. Gravity gradient moments are significant in
Earth orbitsfor orbit altitudes between 400 km and 40, 000 km and
forattitude maneuver times that are not small compared withthe
orbital period.
This research has been supported in part by NSF under grant
DMS-0504747, and by a grant from the Rackham Graduate School,
University ofMichigan.
This research has been supported in part by NSF under grant
ECS-0244977.
E3
E1
E2
x
e3
e1
e2
LVLH frame
Inertial frame
Fig. 1. Coordinate frames
We study open loop attitude maneuvers that can be ac-complished
by using two impulsive torque controls, oneoccurring at the initial
time and one occurring at the terminaltime of the maneuver. The
attitude motion in between theinitial time and the final time is
uncontrolled. Two classesof attitude maneuver problems are studied.
In Section III, aterminal attitude is specified so that there is a
unique impulsesequence satisfying the boundary conditions. This
problemcan be solved computationally use a root finding
algorithm.In Section IV, the specified terminal attitude condition
isrelaxed such that there are many impulse sequences thatsatisfy
the boundary conditions. In this case we seek theminimum total
impulse sequence that satisfies the boundaryconditions. This
problem can be solved computationallyusing a constrained
minimization algorithm.
For both classes of attitude maneuver problems, ourcontribution
is to demonstrate how sensitivity derivatives,used in the
computational algorithms, can be determinedeffectively and
accurately such that they satisfy the globalgeometry of the
problem. Since the attitude is represented bya rotation matrix in
the special orthogonal group SO(3), andthe sensitivity derivatives
are expressed in terms of the Liealgebra so(3), this approach
completely avoids singularitiesor ambiguities that arise from other
representations of therotation group, such as Euler angles and
quaternions.
II. RIGID SPACECRAFT MODELS IN A CIRCULAR ORBIT
We assume that a rigid spacecraft is on a circular orbit witha
constant orbital angular velocity 0 R. In this section,
thecontinuous equations of motion and a geometric
numericalintegrator, referred to as Lie group variational
integrator, forthe attitude maneuver of a spacecraft in a circular
orbit aregiven following the results of [3]. We identify TSO(3)
Proceedings of the 2006 American Control ConferenceMinneapolis,
Minnesota, USA, June 14-16, 2006
WeC10.3
1-4244-0210-7/06/$20.00 2006 IEEE 1742
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SO(3)so(3) by left translation, and we identify so(3) R3by an
isomorphism S() : R3 so(3).
Continuous equations of motion: We define three rotationmatrices
in SO(3);
Rbi : from the body fixed frame to the inertial frame,Rli : from
the LVLH frame to the inertial frame,Rbl : from the body fixed
frame to the LVLH frame,
where the inertial frame and the LVLH frame are illustratedin
Fig. 1. Thus, Rbl = RliT Rbi.
The on-orbit spacecraft equations of motion are given by
+ = Mg, (1)Rbi = RbiS(), (2)
Rli = RliS(0e2), (3)Rbl = RblS( 0RblT e2), (4)
where , R3 are the angular momentum and the angularvelocity of
the spacecraft expressed in the body fixed frame,respectively, and
Mg R3 is the gravity gradient moment.The isomorphism between R3 and
so(3) is defined suchthat S(x)y = x y for any x, y R3. Since the
orbitalangular velocity 0 is constant, the solution of (3) is
givenby Rli(t) = Rli(0)eS(0e2)t.
Gravity gradient moment: The gravity gradient momentis derived
in [2]. We present an alternative way to obtain thegravity gradient
moment directly using the gravity potential;
U = B
GM
x + Rbi dm,
where x R3 is the position of the spacecraft in the
inertialframe, and R3 is a vector from the center of mass of
thespacecraft to a mass element in the body fixed frame. G isthe
gravitational constant and M is the mass of the Earth.
From [3], the gravity gradient moment Mg can be deter-mined by
using the following relationship;
S(Mg) =U
Rbi
T
Rbi RbiT URbi
. (5)We derive a closed form for Mg from (5), by assuming
thatthe spacecraft is on a circular orbit so that the norm of x
isconstant, and the size of the spacecraft is much smaller thanthe
size of the orbit.
As shown in Fig. 1, the coordinate of the spacecraftposition in
the LVLH frame is r0e3, where r0 R is theradius of the circular
orbit, and e3 = [0, 0, 1]T . Therefore,the position of the
spacecraft in the inertial frame is givenby x = r0Rlie3. Using this
expression,
U
Rbi=B
GM r0Rli e3
T
r0e3 + Rbl 3dm,
=GM
r0
B
(Rlie3
T)
r0[1 + 2
(eT3 R
bl)
r0+
2
r20
] 32
dm,
where = R3 is the unit vector along the directionof . Assume
that the size of the spacecraft is significantly
smaller than the size of the orbit, i.e. r0 1. Using a Tay-lor
series expansion, we obtain the 2nd order approximation.
U
Rbi=
GM
r0
BRlie3
T
{r0
3eT3 Rbl2r20
}dm.
Since the body fixed frame is located at the mass centerof the
spacecraft,
B dm = 0. Therefore, the first term
in the above equation vanishes. Because eT3 Rbl is a
scalarquantity, we can rewrite the above equation as
U
Rbi= 320Rlie3eT3 Rbl
(12
tr[J ] I33 J)
, (6)
where 0 =
GMr30
R is the orbital angular velocity, andJ R33 is the moment of
inertia matrix of the spacecraft.Substituting (6) into (5), and
using the property S(x y) =yxT xyT for x, y R3, we obtain an
expression for thegravity gradient moment as follows.
Mg = 320RblT e3 JRblT e3. (7)
Discrete equations of motion: In the continuous equationsof
motion, the structure of (2), (3), and (4) ensures thatRbi, Rli,
and Rbl evolve on the special orthogonal group,SO(3). However,
general numerical integration methods,including the popular
Runge-Kutta methods, do not preservethe orthogonality property of
this group. For example, if weintegrate (4) by a typical
Runge-Kutta scheme, the quantityRbl
T
Rbl inevitably drifts from the identity matrix as thesimulation
time increases.
The rotation matrix is commonly parameterized by Eulerangles or
quaternions. These attitude kinematics equationscan be numerically
integrated and are used to recomputethe rotation matrix. However,
Euler angles are not globalexpressions of the attitude since they
have associated singu-larities. The analytical expressions for
sensitivities are hardto develop since many trigonometric terms are
encountered.Quaternions have no singularity, but quaternions must
lie onthe three sphere S3. General numerical integration methodsdo
not preserve the unit length of a quaternion. There-fore,
quaternions have the same numerical drift problemas rotation
matrices. Furthermore, quaternions, which arediffeomorphic to
SU(2), double covers SO(3). So there areinevitable ambiguities in
expressing the attitude.
These cause significant inaccuracies in numerical simula-tions
based on quaternions and Euler angles. In particular,the gravity
gradient moment, as given in (7), depends onRbl directly, and
consequently, errors in computing Rbl causeerrors in the gravity
gradient moment. These effects are morepronounced when the
simulation time is large.
Lie group variational integrators preserve the
orthonormalstructure of SO(3) without any reprojection or
parame-trization. They also conserve the momentum map, and
thesymplectic property of rigid body dynamics. In addition,the
total energy is well-behaved, as it only oscillates in abounded
fashion about its true value. So, Lie group varia-tional
integrators are geometrically exact. Using the results
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given in [3], a Lie group variational integrator for the
attitudedynamics of a spacecraft in a circular orbit are given
by
k+1 = FTk k +h
2FTk M
gk +
h
2Mgk+1, (8)
hS(k +h
2Mgk ) = FkJd JdFTk , (9)
Rbik+1 = Rbik Fk, (10)
Rlik+1 = Rlik e
S(0e2)h, (11)Rblk+1 = e
S(0e2)hRblk Fk, (12)where the subscript k denotes variables at
the kth time step,and h R is the integration step size. The matrix
Jd R33is a nonstandard moment of inertia matrix defined by Jd =12
tr[J ] I33 J .
The matrix Fk = RbiT
k Rbik+1 SO(3) is the relative
attitude between integration steps, and it is obtained bysolving
(9). Since Fk and eS(0e2)h are in SO(3), and SO(3)is closed under
matrix multiplication, Rbik , Rlik , and Rblkevolve in SO(3)
automatically for all k according to (10),(11), and (12). The
actual computation of Fk is done in theLie algebra so(3) of
dimension 3, and the rotation matricesare updated by multiplication
with the exponential of a skew-symmetric matrix. So, Lie group
variational integrators arenumerically efficient, and there is no
excessive computationalburden in updating the 9 elements of the
rotation matrix. Theproperties of these discrete equations of
motion are discussedmore explicitly in [3] and [4].
Since 0 is constant, the solution of (11) is given byRlik =
R
li0 e
S(0e2)kh. (13)III. SPACECRAFT ATTITUDE MANEUVERS
A. Problem formulationA two point boundary value problem is
formulated for
spacecraft rest-to-rest maneuver between two given attitudesfor
a fixed maneuver time. This is a Lambert type boundaryvalue problem
on SO(3). The initial attitude and the desiredterminal attitude are
expressed as rotation matrices withrespect to the LVLH frame,
namely Rbl0 , RblNd SO(3). Twoimpulsive control torques are applied
at the initial time andthe terminal time. We assume that the
control torques arepurely impulsive, which means that each impulse
changesthe angular velocity of the spacecraft instantaneously, but
itdoes not have any effect on the attitude of the spacecraft atthat
instant. The rotational motion of the spacecraft betweenthe initial
time and the terminal time is uncontrolled. Thisassumption is
reasonable when the operating time of thespacecraft control moment
actuator is much smaller than thefixed maneuver time.
We define this boundary value problem directly on SO(3)instead
of using parameterizations such as Euler angles andquaternions.
This approach allows us to define the attitude ofthe spacecraft
globally without singularities and ambiguities.We use the discrete
equations of motion given in (8)(12)for the problem formulation and
for the following analysis.
We transform this two point boundary value problem intoa
nonlinear root finding problem. For a given initial angular
momentum 0 and an initial attitude of the spacecraft Rbl0 ,the
terminal angular momentum N and the terminal attitudeRblN are
determined by the discrete equations of motion. Bychoosing the
initial angular momentum so that the terminalattitude of the
spacecraft is equal to the desired attitude, i.e.RblN = R
blNd
, we obtain the initial impulse and the terminalimpulse. Thus,
the nonlinear boundary value problem for thespacecraft attitude
maneuver is formulated as
given : Rbl0 , RblNd , Nfind : 0
such that RblN = RblNd subject to (8)(12),where N N is the
number of integration steps determinedby N = Th for the fixed
maneuver time T and the fixedintegration step size h.
B. Computational approachWe solve a sequence of linear boundary
value problems
whose solutions converge to the solution of the
nonlinearboundary value problem.
Linearization: The equations of motion are linearizedabout a
given trajectory, and they are expressed in termsof the Lie algebra
so(3). Consider small perturbations froma given trajectory denoted
by k, Rbi,k , Rbl,k , F k :
k = k + k, (14)Rbi,k = R
bik + R
bik +O(2), (15)
Rbl,k = Rblk + R
blk +O(2), (16)
F k = Fk + Fk +O(2), (17)where R. Since the orbital angular
velocity is constant,Rlik = 0.
The infinitesimal variation of the angular momentum kcan be
expressed in R3. The variation of the rotation matrixRbi,k SO(3)
can be expressed as
Rbi,k = Rbik e
S(k),
where k R3 and S(k) so(3) is a skew-symmetricmatrix. Since the
map S() is an isomorphism between so(3)and R3, k is well defined.
Then, the infinitesimal variationRbik is given by
Rbik =d
d
=0
Rbi,k = Rbik S(k). (18)
Fk is obtained from definition (10), and (18) as
Fk =d
d
=0
Rbi,T
k Rbi,k+1,
= S(k)Fk + FkS(k+1). (19)Since Rlik = 0, Rblk is given by
Rblk = RliT
k Rbik = R
blk S(k). (20)
In summary, equations (18), (19) and (20) describe thevariations
of rotation matrices in SO(3).
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Substituting (14), (16), and (17) into (8), and (9), andignoring
higher-order terms, the linearized discrete equationsof motion are
given by
k+1 = FTk k + FTk k
+h
2FTk M
gk +
h
2FTk M
gk +
h
2Mgk+1,
(21)
hS(k +h
2Mgk ) = FkJd JdFTk , (22)
where
Mgk = 320
[Rbl
T
k e3 JRblT
k e3 + RblT
k e3 JRblT
k e3
].
(23)These equations are not in standard form since (22) is
an
implicit equation in Fk. By using (18), (19) and (20), wewill
obtain an explicit solution of (22), and rewrite the aboveequations
in standard form.
Substituting (20) into (23), and using the propertyS(x)y = S(y)x
for all x, y R3, Mgk can be writtenas
Mgk = 320
[ S(JRblTk e3)S(Rbl
T
k e3)
+ S(RblT
k e3)JS(RblT
k e3)]k,
=Mkk, (24)where Mk R33.
Using (19) and the property S(Rx) = RS(x)RT for x R
3, R SO(3), the right hand side of (22) is written asFkJd JdFTk=
{S(k)FkJd + JdFTk S(k)}
+{S(Fkk+1)FkJd + JdFTk S(Fkk+1)
}. (25)
Substituting (24) and (25) into (22), and using the
propertyS(x)A + ATS(x) = S({tr[A] I33 A}x) for x R3,A R33, (22) can
be transformed into an equivalent vectorform;
hk +h2
2Mkk = {tr[FkJd] I33 FkJd} k
+ {tr[FkJd] I33 FkJd}Fkk+1.Multiplying both sides by FTk
{tr[FkJd] I33 FkJd}1 andrearranging, we obtain
k+1 = FTk
[h2
2{tr[FkJd] I33 FkJd}1Mk + I33
]k
+ hFTk {tr[FkJd] I33 FkJd}1 k, Akk + Bkk, (26)
where Ak,Bk R33. Substituting (26) into (19), Fk isobtained
as
Fk = kFk + Fkk+1,= S(k)Fk + FkS(k+1),= S(k)Fk + FkS(Akk + Bkk).
(27)
Equation (27) is an explicit solution of (22).
Now we rewrite (21) in the standard form of linear
discreteequations of motion. Substituting (24), (27) into (21),
andrearranging, we obtain
k+1 =[FTk S(k +
h
2Mgk ) {I33 + FkAk}
+h
2FTk Mk +
h
2Mk+1Ak
]k
+[S(FTk
{k +
h
2Mgk
})Bk + FTk +
h
2Mk+1Bk
]k,
Ckk +Dkk, (28)where Ck,Dk R33.
In summary, (26) and (28) are linear discrete
equationsequivalent to (21), (22) and (23), and they can be written
as[
k+1k+1
]=[Ak BkCk Dk
] [kk
],
Ak[
kk
], (29)
where Ak R66. Equation (29) is the linear discreteequation for
perturbations of the attitude dynamics of aspacecraft in a circular
orbit, expressed in terms of R3 so(3). The important feature of
(29) is that it is linearizedin such a way that it respects the
geometry of the specialorthogonal group SO(3).
Linear boundary value problem: The solution of (29) isgiven by
[
NN
]=
(N1k=0
Ak
)[00
],
[11 1221 22
] [00
], (30)
where ij R33 for i, j = 1, 2.For the given boundary value
problem, 0 = 0 since the
initial attitude of the spacecraft is given and fixed, and Nis
free since the terminal angular momentum is compensatedby the
terminal impulse. Then, we obtain
N = 120.
This equation provides 12, the sensitivity derivative ofthe
terminal attitude with respect to a change in the initialangular
momentum. It states that for a given trajectory, if weupdate the
initial angular velocity by 0, then the terminalattitude is changed
from RblN to RblNeS(N ) = RblNeS(120).
We choose the change of the initial angular momentumso that the
updated terminal attitude is equal to the desiredterminal attitude;
RblNeS(120) = RblNd , or equivalently,
0 = 112 S1(logm
(Rbl
T
N RblNd
)), (31)
where S1() : so(3) R3 is the inverse mapping ofS(), and logm
denotes the matrix logarithm. Equation (31)provides a solution of
the linear boundary value problemfor the attitude dynamics of a
spacecraft in a circular orbit,assuming that 12 is invertible.
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Nonlinear boundary value problem: The linear boundaryvalue
problem is solved successively so that its solutionconverges to the
solution of the nonlinear boundary valueproblem. A numerical
algorithm is summarized as follows.
1: Set Error = 2S .2: Guess an initial condition (0)0 .3: Set i
= 0.4: while Error > S .5: Find (i)k , R
bl(i)k using
(i)0 and (8), (9), (12).
6: Compute the error;(i)N = S
1(logm
(R
bl(i)T
N RblNd
)), Error =
(i)N .7: Update the initial condition; (i+1)0 =
(i)0 +c
112
(i)N .
8: Set i = i + 1.9: end while
Here the superscript (i) denotes the ith iteration, and S , c R
are a stopping criterion and a scaling factor, respectively.
This computational approach utilizes an exact and
efficientmethod to compute sensitivity derivatives in the
specialorthogonal group SO(3). The sensitivity derivatives are
thenused to solve the two point boundary value problem.
C. Numerical exampleThree spacecraft rest-to-rest maneuvers
between relative
equilibrium attitudes are considered. The resulting motionsare
highly nonlinear, large angle maneuvers.
The mass, length and time dimensions are normalized.The moment
of inertia of the spacecraft and simulationparameters are chosen as
J = diag [1, 2.8, 2], s = 1014,c = 0.1, h = 0.001. Each maneuver is
completed in aquarter of the orbit, T = 2 . The boundary conditions
and thecorresponding computed impulsive control are as follows.
(i) Rotational maneuver about the LVLH axis e1:Rbl0 = I33, R
blNd
= diag [1,1,1] ,0 = [2.116, 1.531, 1.782]T ,N = [2.116, 1.531,
1.782]T .
(ii) Rotational maneuver about the LVLH axes e1 and e2:
Rbl0 = diag [1,1,1] , RblNd =1 0 00 0 1
0 1 0
,
0 = [1.323, 1.798, 0.932]T ,N = [0.397, 1.586, 1.310]T .
(iii) Rotational maneuver about the LVLH axes e2 and e3:
Rbl0 = diag [1,1,1] , RblNd = 0 1 01 0 0
0 0 1
,
0 = [1.047, 0.437, 2.800]T,
N = [1.416, 1.761, 1.159]T .Fig. 2 shows the attitude maneuver
of the spacecraft,
and the angular velocity response for each case.
(Simpleanimations which show these spacecraft maneuvers can befound
at http://www.umich.edu/tylee.)
IV. OPTIMAL SPACECRAFT ATTITUDE MANEUVERSA. Problem
formulation
An optimization problem is formulated as a rest-to-restmaneuver
of an axially-symmetric spacecraft from a giveninitial attitude to
a given terminal reduced attitude for a fixedmaneuver time. The
initial attitude is expressed by a rotationmatrix with respect to
the LVLH frame, namely Rbl0 . Theterminal desired attitude is given
by the reduced attitude,Nd = R
blNe3 S2. This reduced attitude represents the
direction of the spacecraft axis of symmetry e3 in the
LVLHframe. Two impulsive control moments are applied at theinitial
time and the terminal time, and the maneuver of thespacecraft
between the initial time and the terminal time isuncontrolled.
In the problem studied in section III, the initial parameter0 is
exactly prescribed by the constraint RblNd = R
blN , and
the discrete dynamics. In this section, we relax the
terminalconstraint by only specifying it up to a rotation about
theaxis of symmetry of the spacecraft. Then, we can formulatean
optimal attitude maneuver problem.
The performance index is the sum of the magnitudes of theinitial
impulse and the terminal impulse. Equivalently, onecan minimize the
change in the initial angular momentumand the change in the
terminal angular momentum. Sincethe initial attitude and the
terminal time are fixed, N andN can be considered as functions of 0
through the discreteequations of motion. The optimization problem
is equivalentto
given : Rbl0 ,Nd , N,min0
J = 0 0RT0 Je2+ 0RTNJe2 N ,= H0+ HN ,such that C = N Nd2 =
0,
subject to (8)(12).B. Computational approach
This problem is optimized by the Sequential QuadraticProgramming
(SQP) method using analytical expressions forthe sensitivity
derivatives of the performance index and ofthe constraint
equation.
The variation of the performance index is
J = HT0
H00 +HTNHN
{0S(N )RTNJe2 N} .Since the initial attitude is given and fixed,
the perturbationof the initial attitude 0 is zero. Therefore N =
220and N = 120 from (30). Then, J is given by
J =[
HT0H0 +
HTNHN
{0S(RTNJe2)12 22
}]0.
(32)Since Nd is fixed and N S2, the variation of the
constraint can be written as
C = 2TNdN = 2TNdRblNS(e3)N ,
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where N = 120 from (30). Thus, C isC = [2Td RblNS(e3)12] 0.
(33)
Equations (32) and (33) are analytical expressions for
thesensitivity derivatives of the performance index and
theconstraint.
C. Numerical exampleThe mass, length and time dimensions are
normalized.
The moment of inertia of the spacecraft is chosen as J =diag [3,
3, 2], so that e3 is the axis of symmetry of thespacecraft.
The desired maneuver is to rotate the axis of symmetryfrom the
radial direction to the normal to the orbital planeduring a quarter
orbit. The boundary conditions are given by
Rbl0 = diag [1, 1, 1] , Nd = [0, 1, 0]T .We use MATLABs fmincon
function as an optimization
tool. The sensitivity derivatives of the performance indexand
the constraint are provided by (32) and (33). The initialguess of
the initial angular momentum is chosen as (0)0 =J [1, 1, 0]T . The
optimized performance index and the vio-lation of constraints are J
= 6.771, C = 4.80 1014. Thecorresponding angular momenta and the
terminal attitude are
0 = [2.915, 2.347, 2.734]T ,N = [0.343, 2.686, 2.734]T ,
RblN =
0.633 0.733 0.0000.000 0.000 1.0000.773 0.633 0.0000
,
so that N = RblNe3 = [0, 0, 1]T = Nd . Fig. 3 shows theoptimal
maneuver of the spacecraft.
V. CONCLUSIONA global model for a rigid spacecraft in a circular
orbit
about a large central body is presented. This model
includesgravity gradient effects that arise from the
non-uniformgravity field.
The sensitivity derivatives for attitude dynamics of a rigidbody
are derived while satisfying the global geometry of theproblem.
Accurate computational approaches for solving anonlinear boundary
value problem and the minimal impulseoptimal control problem for
spacecraft attitude maneuversare studied using sensitivity
derivatives.
The attitude dynamics are represented by a rotation matrixin the
Lie group SO(3), and it is updated by Lie groupvariational
integrators that preserve the structure of SO(3)as well as other
geometric invariants of motion. The sensi-tivity derivatives are
expressed in terms of the Lie algebraso(3). This approach
completely avoids the singularities andambiguities associated with
Euler angles or quaternions, andit leads to a geometrically exact
and numerically efficientmethod for rigid body attitude dynamics
problems.
Although the development in this paper includes a
gravitygradient moment and the rotation of the LVLH frame, the
0 0.05 0.1 0.15 0.2 0.251
2
3
0 0.05 0.1 0.15 0.2 0.251
0
1
:
0 0.05 0.1 0.15 0.2 0.251.2
1
0.8
t/T
(a) Rotation about the LVLH axis e1
0 0.05 0.1 0.15 0.2 0.252
1
0
0 0.05 0.1 0.15 0.2 0.250.5
0.6
0.7
:
0 0.05 0.1 0.15 0.2 0.250
1
2
t/T
(b) Rotation about the LVLH axes e1 and e2
0 0.05 0.1 0.15 0.2 0.250.5
1
1.5
0 0.05 0.1 0.15 0.2 0.250
0.5
1
:
0 0.05 0.1 0.15 0.2 0.252
0
2
t/T
(c) Rotation about the LVLH axes e2 and e3Fig. 2. Spacecraft
attitude maneuvers
0 0.05 0.1 0.15 0.2 0.251
0
1
0 0.05 0.1 0.15 0.2 0.251
0.8
0.6
:
0 0.05 0.1 0.15 0.2 0.252
1
0
t/T
Fig. 3. Optimal spacecraft attitude maneuver
results presented reduce to the case of a free rigid bodyif 0 =
0. That is, the computational approach suggestedapplies directly to
attitude maneuvers of the free rigid body.
REFERENCES[1] P. C. Hughes, Spacecraft attitude dynamcis. John
Wiley & Sons, 1986.[2] B. Wie, Space Vehicle Dynamics and
Control. AIAA, 1998.[3] T. Lee, M. Leok, and N. H. McClamroch, A
Lie group variational
integrator for the attitude dynamics of a rigid body with
application tothe 3D pendulum, in Proceedings of the IEEE
Conference on ControlApplication, Toronto, Canada, Aug 2005, pp.
962967.
[4] , Lie group variational integrators for the Full Body
problem,Computer Methods in Applied Mechanics and Engineering,
submitted,Available: http://arxiv.org/abs/math.NA/0508365.
1747
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