AIAA Infotech@Aerospace Conference <br>and<br>AIAA Unmanned...Unlimited Conference
6 - 9 April 2009, Seattle, Washington
AIAA 2009-1836
Optimization in Choosing Gimbal Axis Orientations of
a CMG Attitude Control System
Frederick A. Leve∗∗
University of Florida, Gainesville, FL, 32611-6250, US
George A. Boyarko†
Naval Postgraduate School, Monterey, CA, 93943-5100, US
Norman G. Fitz-Coy‡
University of Florida, Gainesville, FL, 32611-6250, US
Control momentum gyros (CMGs) are often chosen for satellites where high attitude
precision and torque are needed while using minimal input power. Control of these types
of systems is complicated and is directly dependent on the number of actuators and their
gimbal axis orientations with respect to the satellite body frame. This paper discusses the
potential benefits of optimizing these gimbal axis configurations and compares these results
to existing configurations such as the box, rooftop, and pyramid. A static optimization is
performed to find the correct gimbal axis configuration in terms of Euler angles for an
attitude control system (ACS) consisting of four CMGs. A four CMG configuration is
chosen for minimal redundancy in avoiding singularities.
The paper also proposed a method of reconfiguring the CMG gimbal axis orientations
online. Reconfiguring the CMGs online can be beneficial to larger systems with deployables
and/or systems with on-orbit assembly which can afford the mass and volume of extra
mechanisms for onboard reconfiguration.
Nomenclature
A
A+
ASR
δ̇
δ
θ̇
θ
φ̇
φ
h
ḣ
τ
ω
ω̃
J
Jacobian matrix
Moore-Penrose pseudo-inverse
Singularity Robustness pseudo-inverse
Column matrix of CMG gimbal rates
Column matrix of gimbal angles
Column matrix of CMG inclination rates
Column matrix of CMG inclination angles
Column matrix of CMG spacing rates
Column matrix of CMG spacing angles
CMG angular momentum vector
CMG output torque vector
CMG control torque vector
Spacecraft angular velocity vector
Skew symmetric matrix of the spacecraft angular velocity vector
Spacecraft centroidal inertia tensor
∗ Graduate Student, Mechanical and Aerospace Engineering, 231 MAE-A, P.O. Box 116250 Gainesville FL 32611-6250, AIAA
Student Member.
† Graduate Student, Mechanical and Astronautical Engineering, Watkins Hall 700 Dyer Rd, Monterey, CA, 93943-5100,
AIAA Student Member.
‡ Associate Professor, Mechanical and Aerospace Engineering, 231 MAE-A, P.O. Box 116250 Gainesville FL 32611-6250,
AIAA Associate Member.
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Aeronautics
Copyright © 2009 by the American Institute of Aeronautics and
Astronautics,
Inc. All of
rights
reserved. and Astronautics
m
Singularity measure
γ
Singularity parameter
M
Optimization cost functional
Projection matrix for mapping the torque null space
P
[ ]0 Variables at initial time t = t0
θ∗ , φ∗ Optimal solutions for Euler angles
k, c Controller gains
ess Steady state error in degrees
I.
Introduction
n unconstrained optimization scheme is employed to determine the two Euler angles defining the spacing
A
φ and inclination θ of each control moment gyroscope (CMG) gimbal axis with respect to the body
frame F . Figure 1 shows the Euler angles for a four pyramidal CMG cluster as an example. Use of the
i
i
B
Euler angles confines this problem to a parameter optimization for a set of eight constants. The optimization
assumes that a given control logic, spacecraft inertias, and initial conditions of the gimbal states, attitude,
and angular velocity are already known. The results of this paper will determine the optimal configuration
of the gimbal axes and hence the CMG configurations for a given slew or set of slews. The process of
determining the optimal Euler angles for the gimbal axes orientations is not an online process and it is
suggested that this process would be beneficial when performed before the satellite design phase assuming a
given set of slews is already understood.
!Gi
!B
Figure 1. CMG Euler angle representations
A.
Dynamics
C onsidering the satellite and the CMG as a rigid body system, the equations governing the satellite’s
motion and its attitude (error) are given by Eqs.(1)-(3). The rotational equations of motion for the satellite
in Eqs.(1)-(3) represent the states angular velocity ω , quaternion error vector [ee4 ]T , and CMG angular
momentum h.
ω̇ = J −1 [τ − ω̃J ω]
" #
"
1 −ω̃
ė
=
2 −ω T
e˙4
ω
0
#" #
e
e4
ḣ = −τ − ω̃h
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(1)
(2)
(3)
A 3-2-3 Euler sequence is used to transform representations in the body frame FB to each gimbal frame
FGi through the (DCM) in Eq.(4). The direction cosine matrix (DCM) is a function of three Euler angles,
two of which are the optimized constants inclination angle θi , spacing angle φi , and the third is the gimbal
angle δi .
CG
iB
= C 3 (δi )C 2 (θi )C 3 (φi )
(4)
The gimbal frame FG of each CMG is transformed to the body frame through the direction cosine
matrix (DCM) in Eq.(4). This DCM is a function of three Euler angles, two which are optimized constants
inclination angle θi , spacing angle φi , and the gimbal angle δi which completes the rotation sequence. The
angular momentum of each CMG in the system is transformed to the body frame using this DCM and
therefore the resultant angular momentum of the CMG system is a instantaneous function of only the
gimbal angles for single-gimbal CMGs.
The representation of each CMGs angular momentum is transformed to the satellites body frame using
Eq.(4) showing the dependence of the systems angular momentum on the Euler angles. A time derivative of
this representation yields Eq.(5) which shows the dependence on the gimbal angle, δ, the only time dependent
Euler angle. It is customary to represent the variation of the angular momentum with respect to the gimbal
angle as the Jacobian matrix A.
ḣ =
∂h ∂δ
= A δ̇
∂δ ∂t
(5)
A CMG system containing more than three CMG gimbals theoretically has redundancy and therefore a
null space of the Jacobian matrix exists. With this null space, there are infinite minimum error solutions to
Eq.(6). The basis for this null space is the a projection matrix given by [1 − A+ A] where any null vector
d will provide null motion. Furthermore, the Jacobian matrix A may become rank deficient and therefore
singular. When this happens the torque needed in the singular direction cannot be produced with finite
gimbal rates.
δ̇ =
B.
1 +
A ḣ + γ[1 − A+ A]d
h0
(6)
Steering Logics
Steering logics are methods which are used to steer away (avoid) or escape internal singularities of CMG
systems. There are two general classes of steering logics, singularity avoidance algorithms which use null
motion to steer the gimbals away from singularity while in maneuver and singularity escape algorithms which
escape singularity with regulation of the Jacobian matrix’s eigenvalues at the expense of torque error
1.
Null Motion Algorithms
Null motion is used in avoiding some internal singularities and therefore the choice of the null vector d is
nontrivial. The null vector’s magnitude is regulated by the singularity parameter γ defined in Eq(7) with
constants γ0 and µ which are design parameters that control amplitude and rate of null motion decay. The
singular parameter decays with the singularity index m, in Eq.(8) which is a measure of how far the set of
gimbal angles are from a singular configuration.
Situations occur where singularity is unavoidable through use of the Moore-Penrose pseudo-inverse itself.
Online singularity avoidance methods using null motion are known as local gradient methods. Here the null
vector d is the gradient of a function f which maximizes the distance away from singularity. An often used
function is shown in Eq.(9) as a function of the singularity index m in Eq.(8).1–3 Offline methods exist that
attempt to calculate offline intermediate gimbal angles to steer to using null motion before performing a
maneuver.4–6
Null motion will not be considered in the optimization for selecting the Euler angles because the null
vector d is generally a very complicated nonlinear function of the CMG gimbal angles. Additionally null
motion is unable to avoid internal elliptic singularities.7–10
γ = γ0 exp(−µm)
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(7)
m=
q
det(AAT )
f = −m2
2.
(8)
(9)
Pseudo-Inverse Solutions
Pseudo-Inverse solutions do not avoid singularity but rather provide a mechanism for escaping the singularity
by ensuring a solution to Eq.(6) exists at the singularity. This is done through the addition of minimal torque
error. One of these methods is known as the Singularity Robustness (SR) inverse shown in Eq.(10) which
uses the singularity parameter in in Eq.(7) to scale the amount of torque error added when near singularity.
ASR = AT (AAT + γ 1)−1
(10)
In this paper SR inverse is the steering logic utilized in the optimization due to its low computational
burden and its effectiveness in escaping both internal hyperbolic and elliptic internal singularities. It is worth
noting other singularity escape methods exist which are variations of the SR inverse.11–14
C.
Control Logic
In this paper is a static parameter optimization is used and therefore the control logic is assumed known
beforehand. A nonlinear PD control logic is chosen for the optimization with gain constants k and c in
Eq.(11). This eigen-axis rest-to-rest control logic is globally asymptotically stable when exact model knowledge is assumed.15 It should be noted that the results of the optimization are dependent on the control
logic and thus will differ with the choice of selected control logic. However, the insights gained from the
proposed process may be helpful in the selection of the CMG configuration and the associated control logic
for a specific spacecraft mission.
τ = −2kJ e − cJ + ω̃J ω
II.
(11)
Optimization
The cost function for this optimization is continuous and shown in Eq.(12). The first term of the cost
function is an explicit function of the Euler angles. The second term of the cost function comes from the
torque error in Eq.(13) which is added to the cost to limit the torque error coming from the use of the SR
inverse’s escape of singularity.
Z tf
M=
−m2 + τ e T τ e dt
(12)
t0
τ e = ḣ − A δ̇ act
(13)
A discretized approach is taken in the optimization for Eq.(12) where the cost function for the ith slew
assuming a rest-to-rest maneuver P
is given in Eq.(14). These costs are summed for the multiple set of slews
n
going from rest-to-rest (i.e., M = i=1 Mi ).
Z ti+1
Mi =
−m2 + τ e T τ e dt
(14)
ti
To consider the gimbal rates in the minimization, an energy term is added to the cost function in Eq.(14)
giving Eq.(15).
Z ti+1
T
Mi =
−m2 + τ e T τ e + δ̇ δ̇ dt
(15)
ti
The equations for the costate derivatives pertaining to the CMG dynamics are complicated and thus
application of an indirect method such as the shooting method is impractical. For this reason the optimization is performed numerically using a fourth order fixed-timestep discrete Runga-Kutta integration of
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the dynamics and incorporates them into the summation of the cost. A block diagram of the optimization
process shown in Figure 2 where the dynamics incorporating each choice of θ and φ are integrated and then
summed in the cost. After the cost is computed the unconstrained optimization function in Matlab f minunc
is used to choose the next iteration of θ and φ. The initial guess for the optimized Euler angles are random
numbers between [0, 360]deg. The solutions of the optimization will be verified by cost comparison to the
common four CMG configurations of pyramid and box.
!"#$%&'()*+,-./)0&1(2)
*2$2.)!"#$%&'()
3#2.41$5+#))
!
ẋ = f (x, θj , φj )
!
"
x= ω h δ
t0
7.82)32.1$5+#)
[θj+1 , φj+1 ] ⇐= min(M )
tf
!
M=
xdt
6+(2)
tf
t0
T
−m2 + τ e T τ e + δ̇ δ̇ dt
Figure 2. Optimization process block diagram
A.
Results
Results were obtained for the cost functions in Eqs.(14) and (15). All simulations in this section have an
integration step size of ∆t = 0.02 seconds. The results are for a rest-to-rest maneuver with initial quaternion
error e0 and termination steady state error of ess in Eq.(16).
ess = min[2sin−1 (||e||2 ), 2π − 2sin−1 (||e||2 )]
(16)
The cost of the system with the solved Euler angles is compared to the that of the box and pyramid
configurations. Comparisons of these costs will show that the solved Euler angles are more optimal in terms
of the cost than the standard configurations of box and pyramid.
It should be mentioned that without knowledge of the initial positions of the gimbals, this optimization
will not be valid. This is due to the fact that the chance of entering singularity is directly dependent on the
initial gimbal angles. The first simulation uses initial gimbal angles δ0 = [0 0 0 0]deg that are consistent with
a zero-momentum state for a four-CMG pyramid configuration. The choice for the first simulation is based
on the fact that the zero-momentum state for a four-CMG pyramid configuration is away from singularity.
Therefore this set of initial gimbal angles is chosen to observe whether or not the solution of the optimization
results in a CMG configuration which diverges away from the pyramid configuration during the given slew.
However, the gimbal angles that place the four-pyramid configuration in a zero-momentum state, place the
box configuration in a state of singularity. For this reason, results for the four-CMG box configuration are
not included in the first simulation.
The second simulation uses a set of gimbal angles at δ0 = [105 105 105 105]deg which is near an internal
elliptic singularity for the four-CMG pyramid configuration. For this simulation, it is expected that the
solution should diverge away from the pyramid configuration which is near an internal singularity. The
initial optimized Euler angles were chosen randomly in Matlab. Even with a single slew, these simulations
give physical insight into what capabilities would be appropriate for optimizing over multiple slews.
The two simulations were conducted using both cost functions Eqs.(14) and (15) which gives a total of
four simulations.
1.
Results using Eq.(14)
The results from this section are based on Eq.(14). The first simulation has the parameters shown in Table
1.
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Table 1. Simulation Model Parameters
Variable
J
δ0
e0
ω0
h0
k
c
∆t
ess
Value

100 −2.0 1.5


−2.0 900 −60 
1.5
−60 1000
[0 0 0 0]T
[0.04355 − 0.08710 0.04355 0.99430]T
[0 0 0]T
128
0.05
0.15
0.02
0.0001
Units

kgm2
deg
−−
deg/s
N ms
Nm
N ms
s
deg
Simulation plots in Figure 3 (a) and (b) show divergence of the optimized configuration away from the
pyramid configuration and a loss of symmetry with solutions of the Euler angles θ∗ = [170.2 13.6 85.5 168.0]T deg
and φ∗ = [17.7 167.0 304.3 92.5]T deg. The gimbal rates of the optimized configuration in Figure 4 (a) are on
the same order as the pyramid configuration (b) but have a smoother trajectory. Smoothness of the gimbal
rates is inherent to the lower amount of torque error used by the optimized case in Figure 5. Torque error
is increased as a function of the singularity index m for the SR inverse, therefore larger amounts of torque
error, and hence discontinuous gimbal rates are due to the pyramid configurations singularity encounter in
Figure 6 (b). Additionally the singularity encounter of the pyramid configuration gave a better cost for the
optimized case in Figure 7 (a). Recall that the zero-momentum configuration was chosen as a control in that
it was initially far away from singularity for the pyramid configuration. The results of Figure 6 (b) show
that despite the distance away from singularity for the pyramid configuration initially, the maneuver still
encountered singularity. If a gimbal axis configuration for any general slew is to be optimized this method
will not be as effective.
(a) External singular surface
(b) Internal singular surface
Figure 3. CMG singular surfaces for the optimized configuration at θ∗ = [170.2 13.6 85.5 168.0]T deg and φ∗ =
[17.7 167.0 304.3 92.5]T deg
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500
300
dδ1/dt
400
dδ1/dt
dδ /dt
200
300
dδ3/dt
100
dδ3/dt
200
dδ4/dt
0
dδ4/dt
dδ /dt
2
dδ/dt(deg/s)
dδ/dt(deg/s)
2
100
−100
0
−200
−100
−300
−200
0
20
40
Times(s)
60
−400
0
80
20
40
Times(s)
60
80
(a) CMG gimbal rates for optimized (b) CMG gimbal rates for pyramid conconfiguration
figuration
Figure 4. CMG gimbal rates for the optimized and pyramid configurations at δ0 = [0 0 0 0]T deg
0.15
1
0.5
0.1
τe
τe
0
0.05
−0.5
0
−0.05
0
−1
20
40
Times(s)
60
−1.5
0
80
20
40
Times(s)
60
80
(a) CMG torque error for optimized (b) CMG torque error for pyramid conconfiguration
figuration
Figure 5. CMG torque error for the optimized and pyramid configurations at δ0 = [0 0 0 0]T deg
0.65
0.8
0.64
0.6
m
m
0.63
0.62
0.4
0.61
0.2
0.6
0.59
0
20
40
Times(s)
60
80
0
0
20
40
Times(s)
60
80
(a) Singularity index for optimized con- (b) Singularity index for pyramid configuration
figuration
Figure 6. CMG singularity index for the optimized and pyramid configurations at δ0 = [0 0 0 0]T deg
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−3
10
x 10
2
8
1.5
6
1
M
Mp
4
2
0.5
0
0
−2
−4
0
20
40
Times(s)
60
80
(a) Cost for optimized configuration
−0.5
0
20
40
Times(s)
60
80
(b) Cost for pyramid configuration
Figure 7. Optimization cost for the optimized and pyramid configurations at δ0 = [0 0 0 0]T deg
The second simulation used the parameters in Table 2. The simulation is also of a single rest-to-rest slew
maneuver which has initial gimbal angles near an internal elliptic singularity for a four-CMG pyramid at
δ 0 = [105 105 105 105]T deg.
Table 2. Simulation Model Parameters
Variable
J
δ0
e0
ω0
h0
k
c
∆t
ess
Value

100 −2.0 1.5


−2.0 900 −60 
1.5
−60 1000
[105 105 105 105]T
[0.04355 − 0.08710 0.04355 0.99430]T
[0 0 0]T
128
0.05
0.15
0.02
0.0001
Units

kgm2
deg
−−
deg/s
N ms
Nm
N ms
s
deg
The singular surfaces in Figure 8 for the optimized choice of Euler angles of θ∗ = [233.1 279.3 40.6 341.0]T deg
and φ∗ = [−87.1 143.4 321.0 106.1]T deg diverge away from the previous result in Figure 3. The gimbal rates
in Figure 9 for the three configurations of the optimized (a), pyramid (b), and box (c) are smooth and are
all around the same magnitude with that in (b) slightly larger. The magnitude difference of torque error
among the configurations in Figure 10 is not that significant when comparing the optimized (a) to the pyramid configuration (b) but is approximately half the magnitude of the torque error for the box configuration
(c). The singularity index in Figure 11 shows that the optimized case in (a) diverges rapidly away from
singularity where the pyramid (b) and box (c) configurations oscillate toward and away from singularity
(i.e., pyramid configuration in Figure 11 comes within proximity of internal singularity two times). Figure
12 for the optimized (a), pyramid (b), and box (c) configurations show that the optimal solution to the Euler
angles maintains a lower cost.
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(a) External singular surface
(b) Internal singular surface
Figure 8. CMG singular surfaces for the optimized configuration at θ∗ = [233.1 279.3 40.6 341.0]T deg and φ∗ =
[−87.1 143.4 321.0 106.1]T deg
dδ3/dt
dδ4/dt
200
0
dδ/dt(deg/s)
dδ2/dt
400
dδ/dt(deg/s)
600
dδ1/dt
200
dδ2/dt
200
dδ3/dt
100
dδ3/dt
dδ4/dt
0
−200
−400
−400
0
−600
0
40
Times(s)
60
80
dδ1/dt
dδ2/dt
−200
20
300
dδ1/dt
400
dδ/dt(deg/s)
600
dδ4/dt
0
−100
−200
20
40
Times(s)
60
80
−300
0
20
40
Times(s)
60
80
(a) CMG gimbal rates for optimized (b) CMG gimbal rates for pyramid con- (c) CMG gimbal rates for box configuconfiguration
figuration
ration
Figure 9. CMG gimbal rates for the optimized, pyramid, and box configurations at δ = [105 105 105 105]T deg
1.2
2
4
1
3
1
0.8
2
τe
τe
1
τe
0.6
0
0.4
0
0.2
−1
−1
0
−0.2
0
−2
20
40
Times(s)
60
80
−2
0
20
40
Times(s)
60
80
−3
0
20
40
Times(s)
60
80
(a) CMG torque error for optimized (b) CMG torque error for pyramid con- (c) CMG torque error for box configuconfiguration
figuration
ration
Figure 10. CMG torque error for the optimized, pyramid, and box configurations at δ = [105 105 105 105]T deg
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0.7
0.45
0.14
0.65
0.12
0.6
0.4
0.1
0.5
m
0.08
m
m
0.55
0.35
0.06
0.45
0.4
0.04
0.3
0.02
0.35
0
20
40
Times(s)
60
80
0.25
0
20
40
Times(s)
60
0
0
80
20
40
Times(s)
60
80
(a) Singularity index for optimized con- (b) Singularity index for pyramid con- (c) Singularity index for box configurafiguration
figuration
tion
Figure 11. CMG singularity index for the optimized, pyramid, and box configurations at δ = [105 105 105 105]T deg
2.5
0.8
2
0.6
1.5
M
0.4
1
0.2
0.5
0
0
−0.2
0
20
40
Times(s)
60
80
(a) Cost for optimized configuration
−0.5
0
20
15
Mb
3
1
Mp
1.2
10
5
20
40
Times(s)
60
80
(b) Cost for pyramid configuration
0
0
20
40
Times(s)
60
80
(c) Cost for box configuration
Figure 12. Optimization cost for the optimized, pyramid, and box configurations at δ = [105 105 105 105]T deg
2.
Results using Eq.(15)
The results shown here are for the second cost function in Eq.(15). Here an additional term containing
the kinetic energy from the gimbal rates is included within the cost to optimize for a configuration that
will minimize the gimbal rates. This optimization is without constraints on the gimbal rates, therefore the
gimbal rates in all of the simulations may not be implementable physically but this optimization is for proof
of concept not for application.
The external and internal singular surfaces for the Euler angle of θ∗ = [110.6 71.6 61.5 100.2]T deg and
∗
φ = [−23.9 110.2 146.0 288.5]T deg in Figures 13 (a) and (b) for this result are different than the previous
result in Figure 3 using the initial cost function Eq.(14). This is expected because there are different variables
present in the cost function. When compared to the pyramid configuration in Figure 4 (b) for the same
simulation, the plots of in Figure 14 (a) show that the the optimized configuration has gimbal rates of the
same magnitude. The torque error in Figure 14 (b) is smoother with less oscillations than that of Figure 5
(b) for the pyramid configuration. Also there was no singularity encounter of the optimized configuration for
this simulation as shown in Figure 14 (c). The pyramid has two singularity encounters in Figure 6 (b). This
simulation proves that the optimized configuration produces a lower and smoother cost than the pyramid
configuration shown in Figure 15 (a) and (b).
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(a) External Singular Surface
(b) Internal Singular Surface
Figure 13. CMG singular surfaces for the optimized configuration at θ∗ = [110.6 71.6 61.5 100.2]T deg and φ∗ =
[−23.9 110.2 146.0 288.5]T deg
200
100
0.1
dδ2/dt
1.5
0.08
dδ3/dt
1
dδ4/dt
0.06
m
0
τe
dδ/dt(deg/s)
2
dδ1/dt
0.5
−100
0.04
0
−200
−300
0
0.02
−0.5
20
40
Times(s)
60
−1
0
80
20
(a) CMG gimbal rates
40
Times(s)
60
0
0
80
(b) Torque Error
20
40
Times(s)
60
80
(c) Singularity measure
Figure 14. Simulation results for the optimized configuration at θ∗ = [110.6 71.6 61.5 100.2]T deg and φ∗ =
[−23.9 110.2 146.0 288.5]T deg
50
60
50
40
40
30
M
Mp
30
20
20
10
10
0
0
0
20
40
Times(s)
60
80
−10
0
(a) Cost
20
40
Times(s)
60
80
(b) Cost
Figure 15. Optimization cost for the optimized and pyramid configurations at δ0 = [0 0 0 0]T deg
The second simulation of this set has the initial gimbals again set near elliptic singularity of a fourCMG pyramid configuration. The results in Figure 16 with the solution to the Euler angles of θ∗ =
[192.3 256.0 24.5 223.2]T deg and φ∗ = [−64.0 192.3 355.0 211.2]T deg shown an obvious improvement over
the cost to that of the pyramid and box configurations in Figures 17.
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300
200
dδ/dt(deg/s)
2.5
0.14
dδ /dt
2
0.12
dδ3/dt
1.5
0.1
dδ4/dt
1
0.08
dδ1/dt
2
m
τe
100
0.5
0.06
0
0.04
−0.5
0.02
0
−100
−200
0
20
40
Times(s)
60
−1
0
80
20
(a) CMG gimbal rates
40
Times(s)
60
0
0
80
20
40
Times(s)
60
(b) Torque Error
(c) Singularity measure
(e) External Singular Surface
(f) Internal Singular Surface
80
40
M
30
20
10
0
0
20
40
Times(s)
60
80
(d) Cost
Figure 16. Simulation results for optimized choice of inclination and spacing angles starting with initial gimbal
angles near elliptic singularity with θ∗ = [192.3 256.0 24.5 223.2]T deg and φ∗ = [−64.0 192.3 355.0 211.2]T deg
200
50
40
150
Mb
Mp
30
100
20
50
0
0
10
20
40
Times(s)
60
80
(a) Cost
0
0
20
40
Times(s)
60
80
(b) Cost
Figure 17. Simulation results for pyramid (a) and box (b) configurations from the zero momentum configuration
III.
On-orbit Gimbal Axis Reconfiguration
In some cases it may be beneficial to reconfigure an attitude control system of CMGs on-orbit. This
on-orbit reconfiguration would allow the user to adjust the momentum envelope when needed for situations
where the system’s inertia properties change and/or the given configuration of CMGs are saturated.
Such a situation has occured on the International Space Station (ISS).16 Here the Russian attitude
control thrusters were offline and the CMGs onboard the ISS became saturated and could not keep up with
the current angular momentum of the station. If the system were reconfigurable, the angular momentum
envelope could have possibly been reshaped as an ellipse lengthwise about the axis of the ISS angular
momentum and therefore redistribute the angular momentum to a different direction until the Russian
attitude control thrusters were back online. To do this, mass and volume of extra mechanisms would need
to be added. This is not the subject of this paper and therefore will not be discussed here.
The equations for the dynamics while reconfiguring on orbit are shown below in Eqs. (17) and(18)
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including Eq.(6).
∂h
∂h
∂h
δ̇ +
θ̇ +
φ̇ = A δ̇ + P
ḣ =
∂δ
∂θ
∂φ
θ̇
φ̇
!
(17)
If it is assumed that the Euler rate trajectories for inclination and spacing are known, the only degree of
freedom would lie in altering the gimbal rates to negate torque on the spacecraft while reconfiguring. This
is preferrable because the speed of reconfiguration can be easily constrained. The homogeneous solution of
the CMG state equation is added in Eq. (18) to aid in avoiding singularity when configuring but also keep
/ null(A+ P Ẋ). The null vector d requires more computation in on-orbit reconfiguration because it is now
δ̇ ∈
changing with all three Euler angles instead of solely the gimbal angles. If it is possible to use the null space
to move the ACS into a configuration which is far away from singularity (i.e., zero momentum), this should
be done before and after reconfiguration. It should also be noted that as there are many different rotation
sequences which transform a frame, there are also many ways of reconfiguring an ACS containing CMGs to
its final Euler angles. There are different Euler angles which will give the same angular momentum envelope
for a given configuration (i.e., box configuration is the same as a rooftop configuration at an angle θ = 45o ).
!
θ̇
+
+ γ[1 − A+ A]d = A+ P Ẋ + γ[1 − A+ A]d
δ̇ = A P
(18)
φ̇
A simulation was carried out that used a null vector d as the gradient function f in Eq.(9) and a proportional controller in Eq.(19) with k = 0.1 for the inclination and spacing rates. The simulation reconfigured
from a four-CMG pyramid configuration to a four-CMG rooftop or specifically a box configuration. Plots of
the gimbal angles, inclination and spacing angles, and Euler angle error are in Figures 18 (a), (b), and (c).
" #
"
#
k θ − θf
θ̇
(19)
=
2 φ − φf
φ̇
δ1
300
θ1
δ
250
θ
200
θ3
δ3
δ4
0
2
θ4
150
φ1
100
φ2
50
−5
0
100
200
300
0
0
100
time(sec)
(a) Gimbal Angles
200
error(deg)
2
Euler Angles (deg)
Gimbal Angles (deg)
5
90
θe
80
θe
70
θe
60
θe
50
φ
1
2
3
4
e
1
40
φe
30
φ3
φe
20
φ4
φ
10
300
0
0
time(sec)
(b) Euler Angles θ, φ
2
3
e
4
50
100
150
time(sec)
200
250
300
(c) Euler Angle Error
Figure 18. Simulation results for for on-orbit reconfiguration of four CMGs from pyramid to box
The results from Figures 18 show a smooth transition in reconfiguration of the CMG gimbal axis orientations. The proportional controller used could have been replaced by one with more desirable properties
because the settling time of the Euler angle error is directly proportional to the time of the reconfiguration
and hence the choice of the controller. Also the configuration returned to the initial gimbal angles after
completion.
IV.
Conclusion
With the correct choice of CMG configurations, the attitude control performance of any spacecraft can
be enhanced. This paper proposed a method of finding this configuration beforehand and explored another
method to apply the optimization on-orbit. The results of this paper will prove to be a valuable asset to
future satellite mission planning.
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