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AIM-90-3317-CB
AN INNOVATIVE APPROACH TO THE MOMENTUM MANAGEMENT CONTROL
FOR SPACE STATION FREEDOM
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317
Jalal Mapar t
G- u' man Space Station Engineering and Integration Contractor
1760 Business Drive Center
Reston, Virginia 22090
Abstract
In all the previous work, the space station
Equations Of Motion (EOM) have been linearized
about the Local-Vertical Local-horizontal (LVLH).
It is assumed that the products of inertia remain
small, i.e., principal axes nearly aligned with the
LVLH, allowing the pitch axis to be decoupled
while the roll/yaw axes remain coupled. These
assumptions are used by Wie et al. [31 in the
development of a continuous Momentum Management
System (MMS) that uses the Linear Quadratic
Regulator (LQR) technique for generating the
feedback gains. The continuous MMS as proposed by
Wie et aJ. [41 is shown in simple block diagram form
in Figure 1. The weighting matrices associated with
the LQR are obtained by trial and error. This process
is often time consuming and does not always yield
desirable closed-loop poles.
More recently, Wie et nJ. I41 have presented the
EOM for the case of large pitch TEA, which may be
encountered during the assembly flights of the space
station, but, in order to simplify the equations, they
lhave neglected the cross products of inertia. Since
the large TEASare chiefly due to large cross products
of inertia, the applicability of the equations to
early stages of the station is debatable. However,
when applied to the assembly complete vehicle, the
controller, which also includes disturbance rejection
filters to minimize the steady state effects of the
aerodynamic torques, is shown to stabilize the
system by achieving the TEA. Sunkel and Shieh 151
have applied the regional pole placement and
optimal control techniques to the linearized model of
the space station and have solved directly for both
the feedback gains and the weighting matrices.
They use the matrix sign function algorithm for the
solution of the Riccati equations and show drastic
time savings associated with the manual assignment
of the weighting matrices as used by Wie ef al. [3,41.
Again, the momentum management algorithm is
applied to the assembly complete vehicle and
stability is obtained as before.
A new approach to the Control Moment Gyro
(CMG) momentum management and attitude control
of the Space Station Freedom is presented. First, the
nonlinear equations of motion are developed in terms
of body attitude and attitude rate with respect to
the Local Horizontal Local Vertical (LVLH); then,
they are linearized about any arbitrary stable point
via the use of perturbations techniques. It is shown
that for some assembly flights, linearization of
equations of motion about the LVLH may not be
valid and that a better choice would be to to
linearize about a Torque Equilibrium Attitude (TEA).
Next, a three-axis-coupled control law is used and
the controller gains are determined via a
combination of the optimal control and regional pole
placement techniques. Finally, It is shown that the
proposed linearization process, together with the
coupled control laws, can stabilize a previously
uncontrollable space station assembly flight.
Introducb'on
The Space Station Freedom will require several
space shuttle flights to complete. When activated
after a few assembly flights, it will use Control
Moment Gyros (CMGs) as the primary attitude
control devices during the normal coasting flight
operations. Since the CMGs are momentum exchange
devices, external torques must be applied to prevent
momentum saturation.
Several momentum
management methods, both discrete and continuous,
have been developed [l-61. These methods are based
on the use of the environmental torques to drive the
vehicle to a Torque Equilibrium Attitude (TEA) thus
minimizing the CMG momentum usage.
Manager, flight mechanics branch, member AIAA.
Copyright @ 1990 American Institute of Aeronautics and
Astronautics, Inc. All rights reserved,
23
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317
While the simplifying assumptions used in the
previous papers [l-61 are valid for the assembly
complete type space station configurations, they may
not be applicable for the build-up stages. Holmes [71
has applied the momentum management algorithm
developed by Wie et al. [3,4] to one of the station
assembly flights and has shown that the large cross
products of inertia produce a large TEA; thus,
invalidating the assumptions used in the previous
papers [1-61.
This paper presents a new approach to modeling
the rigid body dynamics by developing the EOM and
linearizing them about any arbitrary point along the
attitude trajectory. A suitable stable point is
proposed for linearization of the EOM. It is shown
that a three-axis-coupled control law, whose gains
are determined by regional pole placement [5], can
successfully stabilize an assembly flight which was
not controllable previously. The proposed momentum
management method is given in its generic form so
that i t can be easily applied to the assembly
complete type configurations as a special case.
Finally, a comparison is made between the proposed
method and the previous algorithms by presenting
simulation results for a space station assembly
vehicle.
and the ( * ) represents the time derivative. The
non-control torque, T,,, is the sum of the gravity
gradient and aerodynamic torques which will be
given later, and u is the control torque which is
related to the CMG momentum, H , by
The body rate can be expressed as
where WB/L, is the body rate vector with respect to
LVLH and w~ is the LVLH rate vector in the bodyaxes. For a pitch-yaw-roll (2-3-1) Euler rotation
sequence, WB/L and q are given by [81
Mathematical Models
In this section the rotational EOM are presented
in matrix form and linearized about any arbitrary
point. Next, a suitable stable point is proposed for
linearization and the EOM are simplified. For
simplicity, the station is assumed to be a single rigid
body in a circular orbit. The nonlinear rotational
EOM in terms of components along the fixed bodyaxes are given by the well-known Euler's equations.
I c 3 + z I o = T,,+ u
(1)
where C=cos and S=sin; #/ 0, and ware the body
attitude with respect to the LVLH and n is the
orbital rate. Eqs. ( 3 a ) a n d (3b) can be used to
express the total body rate as
U=F&+G
where
F=
1
0
0
sv
0
C4Cw
S#
-S#Cy
C#
Here, the (7 represents the skew symmetric matrix
and
and 0 represents the roll, pitch, and yaw attitude
angles with respect to LVLH.
where ( wn / 4 , o, ) are the body-axis components
of the angular rate with respect to the inertial
frame; (I, I f, , I, 1 are the moments inertia, and
(Ixy ,I,, I I y z ) are the negative products of inertia;
hl
The aerodynamic torque is modeled as bias plus
cyclic terms [3,4,51 in the body axes as
Note that the total body rate is now expressed in
terms of F and G which are functions of the attitude
and attitude rate with respect to LVLH and orbit
rate. Eq. (4) can be differentiated to give the body
angular acceleration as
Cj = F& + F i
where
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317
0
+
rn
A ksin
(k nt + tpk)
(9)
k= 1
+c
W W
0
T,,,,=Bias
where, usually, m=2. The cyclic components at once
and twice the orbital rate are due to the diurnal
bulge effect and the rotating solar panels,
respectively. Substituting Eqs. (7)and (9) into Eq..
( 6 ) and rewriting the EOM in a more compact
functional form yields
1
0
&= f(@,i,u)
Because the expansion of EOM will result in
rather lengthy expressions, we will proceed with
the matrix notation until the final result is obtained.
Equation (10) represents the most general form of the
nonlinear EOM in terms of the body attitude and rate
with respect to LVLH. Linearization of the EOM is
performed by applying small perturbations to the
nominal solution of the Eq. (10). Let %(r) be such a
nominal solution. Then, for small perturbations from
a point on the nominal trajectory, the solution of Eq.
(lo), to first order, is given by
G=n
Substituting Eqs. (4) and ( 5 ) into Eq. (1) and
simplifying yields
-
& = (fF)-’(TnC- (F& + G ) f (Fcb + G ) )
- F - * ( F &+G) + ( f F ) - ’ u
Eq. ( 6 ) represents the nonlinear rotational dynamics
of a body with respect to the rotating LVLH
coordinate system. By integrating Eq. (6), one can
obtain both the body attitude and attitude rate with
respect to LVLH directly. Note that in Eq. ( 6 ) no
assumption on the cross products of inertia has been
introduced.
The gravity gradient torque, Tgg,
is given by
where 6 q t ) represents any small perturbation from
the nominal trajectory. Applying Eqs. (11) to Eq.
(IO), and noting that &D(t) would give rise to a
perturbation in u , will yield the desired linearized
EOM.
u
T,,
= 3n2Rf R
(7)
where R is the unit vector from the center of earth to
the center of mass of the vehicle. For a (2-3-1)
rotation sequence, R can be expressed as
where the subscript N means “at the nominal point.”
Equation (12) can be written in first order form as
-c#ce + s#se s~
24
that minimizes the performance index in (21) is
given by
u = -Kx = R ' 1B TP X
(21)
3
3
where K is the feedback gain and P, an nxn nonnegative definite symmetric matrix, is the solution
of the Riccati equation
*
K,66+KAy6Hy+Kn 6 H y + K i 2 a z +
3
3
K~~ h2 +K&
HY
3
&z +K&
&r + K $ ~ Y +
(19c)
1 T
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317
P B R - B P - P A - A ~ -P Q = o
(22)
The regional pole placement used to generate the
feedback gains in Eqs. (19)is adapted from the
algorithm by Sunkel and Shieh [51. They solve for Q,
R and K so that the closed loop system (A-BK)has
eigenvalues on or within a specified region, as shown
in Figure 2, without explicitly using the eigenvalues
of the open-loop system. The process involves the
use of matrix sign function techniques for the solution
of the modified Riccati equation of (22). It is shown
in [SI that the use of matrix sign function greatly
reduces the computational time required for the
solution of the Riccati equations. The design
procedure is given in detail in [SI and will not be
repeated here.
The MMS presented in this paper is shown in
simple block diagram form in Figure 3. The
algorithm depends only on an estimate of the inertia
properties of the vehicle. The estimated TEA angles
are extracted from the transformation matrix from
principal to body axes. The matrix is obtained by
simply calculating the eigenvector matrix associated
with the inertia matrix. Next, these angles are used
to compute A and B . These matrices are then used to
generate the gains for the control laws, as given in
Eqs. (191, via the LQR and regional pole placement
[SI. Note that, in contrast to the method shown in
Figure 1, the above procedure does not require any
trial and error iterations on the weighting matrices
and can be automated easily.
where the gain superscripts (1, 2, 3) refer to the roll,
pitch, and yaw axes. ln order to avoid CMG
momentum buildup, we have included the integral of
the CMG momentum vector H I i.e.,
Note that, in order to minimize the steady state
oscillations of roll, pitch, yaw attitude and CMG
momentum, we have used the cyclic disturbance
rejection filters as proposed by Wie et al. 131. The
filter equations are given below for momentum
rejection in roll and attitude rejection in pitch and
yaw at frequencies n and 2n.
Also, in the control laws we have chosen the
attitude rate with respect to LVLH in all three axes.
To compute the gains associated with the control
laws, we use the LQR with the regional pole
placement technique. Let the quadratic performance
index for the system in (13)be
Resulb
In this section we present simulation results for a
space station assembly flight and compare the
proposed MMS with that of Reference [31. The
vehicle under consideration, assembly flight #5
(MB-51, is shown in Figure 4. This vehicle was also
used by Holmes [7]to demonstrate the limitations of
the linearization techniques for vehicles with large
cross products of inertia. Although the solar
dynamic collectors will not be used on the station the
vehicle was used as a test model to check the MMS
algorithm. The vehicle properties together with a
where the weighting matrices Q and R are nxn nonnegative and rnxm positive definite symmetric
matrices, respectively. The feedback control law
25
representative aerodynamic torque are given in
Table 1. For MB-5, the principal-to-body angles
were computed to be
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317
#, = -0.4'
e, = -17.1'
y-, = 2.4'
For gain calculation, the parameter h (see Figure 2)
was set at 0.375n. The gains for the three-axiscoupled controller are given in Table 2. In order to
perform a consistent comparison with the method in
[3], we applied the same pole placement method
with the same parameters to the A and B matrices
of [3]. However, since the control laws of 131
destabilized the vehicle in less than an orbit, we
used the three-axis-coupled control laws presented
in this paper. The resulting controller gains are also
given in Table 2.
The results, obtained from nonlinear simulations,
are presented in Figures 4-9. For all the cases, the
vehicle was aligned with the LVLH, i.e., zero
attitude and attitude rates, and the initial filter
states were set equal to zero. The roll, pitch, and
yaw attitude histories, Figures 5a thru 7a, show
that the proposed MMS converges to a TEA of
approximately -330 in pitch and the CMG
momentum, Figures 5b thru 7b, remains bounded after
about 3 orbits. Note that, even though the
aerodynamic torque has shifted the estimated pitch
TEA ( principal-to-body pitch angle ) by about 14O,
the controller has stabilized the system. The
stability can be attributed to the fact that the
linearized EOM about an estimated TEA provide a
better representation of the total system dynamics.
In contrast, the MMS based on [31 is not stable and
diverges from the beginning. The instability is
mainly due to the large pitch TEA (produced by
large cross products of inertia and the aerodynamic
torque) being well outside the linear-angle range. In
fact, the instability suggests that the A and B
matrices of [31, obtained from the EOM linearized
about the LVLH, do not adequately describe the
dynamics of the nonIinear system. As a result,
omission of the products of inertia, and the use of
small angle approximation, do not appear to be
valid assumptions for MB-5 and other stages that
follow the same trend on the inertia properties.
The MMS algorithm presented in this paper has
been developed such that the only inputs to it are
the characteristics of the vehicle to be controlled,
i.e, the inertia matrix, and the parameter h for the
gain calculation. Since linearization about any
arbitrary point is included in the EOM, the MMS
algorithm can compute the A and B matrices and the
controller gains automatically. The total process is
computationally fast and can be automated so that
the MMS can adapt to any configuration by simply
acquiring the minimum amount of information about
the system to be controlled.
A new MMS based on linearizing the EOM about
an estimated TEA has been presented. A set of threeaxis-coupled attitude control laws was also given. It
was shown that, for space station vehicles that
have strong inertia coupling, the use of small angle
approximation and the omission of the moss products
of inertia in the linearized EOM may not provide a
good representation of the nonlinear model.
Although a set of gains can be computed based on the
linearized models of the space station, their use in
nonlinear simulations has been shown to destabilize
the vehicle.
The proposed MMS is computationally fast and
converges quickly. Since the algorithm depends only
on the inertia properties and the desired region for
the closed loop system poles, it can adapt to the
system if the aforementioned inputs are provided. In
fact, depending on the frequency of the gain
calculation, the MMS can be used for control of the
space station during the payload moving maneuvers.
For such maneuvers, the inertias can change
drastically. Instead of scheduling the controller
gains at discrete intervals, the proposed MMS can be
executed at the proper frequency so that the
controller gains could be thought of as time varying
adaptive gains. For specific changes in the inertias,
a new estimated TEA and set of gains can be
computed. It is anticipated that these time varying
gains would stabilize the system because they will
be based on the new linearized EOM presented in
this paper.
In this section the A and B matrices representing
the linearized EOM about any arbitrary point are
presented. Applying small perturbations to Eq. (1)
will yield
-
Z 6
; +( WZ - I @ ) 60 = 6Tnc+ 6~
(A1)
Equations (4) and (5) are used to compute expressions
for 6w and 6h.
i W =aF
-6@,
*
aF
aF
6F=-&+-c%b
(A4)
Note that the coefficients of 6@and Sh, are
evaluated at the desired reference trajectory. These
coefficients form the components of the A and B
matrices which are then complemented with CMG
momentum, Eq. (141, and filter equations, Eqs. (1618), in order to compute the system matrices for gain
computation.
sG=-6Q,,
aG
s b = " 6 @ + - S iab
(AS)
Reference
6h =( 6F)6 + F6&
+ (66) + F6& + SC ( A 3 )
where
a@
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3317
a@
a@
a@
ai
ai
1. Hattis, P. D., " Predictive Momentum of
Guidance, Control, and Dynamics, Vol. 9, No. 4, JulyAug 1986, pp. 454461.
It is easily shown that
acab
-- a@ a i
The term
2. Bishop, L. R., et al., " Proposed CMG Momentum
Management Scheme for Space Station, " Paper No.
87-2528, AIAA Guidance, Navigation, and Control
Conference, August 1987.
ST, is also provided below.
N
-
GTnc= 3n2(RI - I R ) 6R
3. Wie, B., Byun, K. W., Warren, W., Geller, D.,
Long, D., and Sunkel, J., " A New Momentum
Management Controller for the Space Station, "
Journal of Guidance, Control, and Dynamics, Vol. 12,
NO. 5, Sep-Oct 1989, pp. 714-722.
where
Equations (A2) a n d (A31 can be further
simplified by noting that 6 and 6 vanish on
thereference trajectory implying that
F=O,G=O,
4. Warren, W., Wie, B., Geller, D., " PeriodicDisturbance Accommodating Control of the Space
Station for Asymptotic Momentum Management, "
Paper No. 89-3476, AIAA Guidance, Navigation, and
Control Conference, August 1989.
ad = O
a@
and
5 . Sunkel, J. and Shieh, L., " An Optimal
Momentum Management Controller for the Space
Station, " Paper No. 89-3474, AIAA Guidance,
Navigation, and Control Conference, August 1989.
Equation (A101 is then solved to obtain the
linearized EOM about any arbitrary point.
6. Woo, H. H., Morgan, H. D., and Falangas, E. T.,
"Momentum Management and Attitude Control
Design for a Space Station, " Journal of Guidance,
Control, and Dynamics, Vol. 11, No. 1, JanuaryFebruary 1988, pp. 19-25.
Finally, Eqs. (A9) and (A10) are used in Eq. ( A l l ) to
any
On the
an@ar rates*
- -z)@ - ( G I
a#
7. ~
~ E.,
l The Limitations
~
~
of ~ the
Linearization Techniques on Nonlinear Space
Station Equations of Motion, " Grumman Space
Station Engineering Integration Contractor, Report
NO. PSH-341-RP89-OO2, August 1989.
+ (IF)-'6u
8. Junkins, J. L. ,Turner, J. D., Optimal Spacecraft
Rotational Maneuvers, " Studies in Astronautics 3,
Elsevier Scientific Publishing Company, New York,
NY, 1985.
'1
3n2(RI
I'
26
,
Table 1 Vehicle parameters
Aerodynamic torque (ft-lbs)
Inertia ( slugs-ft2
I,
fy
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I,
3.9676E7
2.5588E6
4.0544E7
I,,
I,,
I,,
-1.5544E6
2.6162E5
-2.6641E5
Roll
Pitch
Yaw
0.2 + 0.1 sin(nf)+ 0.01 sin(2nt)
1.2 + 0.4 sin(nt) + 0.1 sinQnt)
-0.02 + 0.02 sin(nt) + 0.05 sin(2nt)
Table2 Controller gains for the propsed MMS and MMS of [31
Gain
Gains for the promed MMS
Roll
Pitch
Yaw
Gains for the MMS of I31
Roll
Pitch
Yaw
-4.2531E+3
-8.8315E+1
-5.9569E+3
-5.8531E+3
-9.4582E+2
-6.3198E+3
-6.0154E+6
2.1047E+5
-1.9527E+6
-6.3867E+6
8.8480E+l
-2.5776E+6
-1.3474E-1
2.4400E-3
-4.8447E-2
-1.4449E-1
-2.7309E-3
-6.315OE-2
1.7773E-5
-1.7318E-6
-1.9954E-5
1.7274E-5
-4.7783E-6
-1.6951E-5
-5.1888E-8
3.6896E-9
3.5004E-8
-4.8944E-8
8.9199E-9
2.5976E-8
-4.2272E-5
3.8643E-7
-1.2175E-5
-5.2958E-5
-3.6524E-6
-1.9888E-5
1.8615E-9
9.3212E-10
8.4531E-9
2.4228E-9
-5.0495E-9
-9.3830E-9
-5.3360E-5
1.6374E-6
-4.4354E-5
-4.3641E-5
-4.6196E-6
-5.3872E-5
-3.4 605E+ 1
-2.6634E+2
-5.8056E+O
1.5639E+2
-2.2254E+2
2.4804E+2
-2.2558E+4
-1.7873E+5
-2.3200E+2
1.8017E+5
-1.7853E+5
1.4912E+5
-3.1247E-3
-5.6951E-2
2.4451E-3
-2.1484E-2
-5.8201E-2
1.5664E-2
-5.3356E-7
-1.1399E-5
5.1890E-7
-3.2993E-6
-1.1469E-5
2.9243E-6
1.0415E-5
-3.7762E-5
9.3871E-6
4.2359E-5
-2.3379E-5
2.9997E-5
1.9171E-3
-2.3821E-1
7.5065E-3
8.8315E-2
-2.2327E-1
3.0972E-2
2.5716E-5
-5.6787E-4
2.6766E-5
9.2073E-5
-5.1726E-4
2.1017E-5
-2.9007E-3
-2.6711E-1
-4.5491E-3
3.5036E-2
-2.6814E-1
2.8538E-2
7.2451E+3
-3.2708E+2
-1.2731E+3
7.2557E+3
-2.9491E+2
-8.6966E+2
4.1 226E+6
-3.4183E+5
-2.2137E+6
2.4071E+6
-5.3573E+5
-1.5240E+6
1.0292E-1
-8.9361E-3
-3.9890E-2
6.1373E-2
-1.3294E-2
-2.2266E-2
1.8243E-5
-2.1006E-8
6.1781E-6
2.2800E-5
-1.1865E-7
6.3630E-6
4.0537E-3
-1.7697E-4
9.8062E-4
2.6247E-3
-3.0929E-4
1.3267E-3
2.1789E+O
-6.1987E-3
-2.5900E+0
4.1 198E+O
-4.0102E-1
-2.1667E+O
6.6404E-3
-2.5999E-4
-5.2998E-3
5.3139E-3
- 1.4189E-3
-5.7521E-3
4.5404E-1
1.1749E-2
-4.3803E+O
I .8650E+0
-1.0042E-1
-4.0506E+0
Compute theA and B matrice from
the EOM linearized about the LVLH.
7
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x = Ax
:g
-a
!i
rq
i
+ Bu
Find u to minimize
I=
TQx + uTR u) dr
Controller gains
-
7space Station Dynamics
Solution:
Fig. 1 Simple block diagram of the MMS. After Wie et al. [4].
P
Im
Re
x
openlooppole
before design
0
Fig. 2 Pole assignment sector.
27
Closedl~poles
after design
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I A andB
I
1
Define h for the Regional pole placement algorithm [51
Controllergains
Fig. 3 Simple block diagram of the proposed MMS.
Fig. 4 A representative assembly flight #5.
lo
5
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-20
-I5
-25
c
1' !
- ............. L ..._........_.______........
;...............
2
0
4oo00
_____________
I
4
6
T i (orb&)
8
I
10
2
0
Fig. 5a Roll attitude comparison.
4
6
Time (orbits)
8
10
Fig. 5b Roll momentum comparison.
20
15ooo
0
a
3
1
P
g
-20
loo00
i4
B
-60
ii;
-80
-100
-120
0
2
4
6
T i (orbits)
8
10
Fig. 6a Pitch attitude comparison.
0
2
4
6
Time (orbin)
8
2
0
4
6
T i w (orbits)
8
10
Fig. 6b Pitch momentum comparison.
10
0
Fig. 7a Yaw attitude comparison.
2
4
6
T i( d i t s )
8
10
Fig. 7b Yaw momentum comparison.
New method
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