# Singularity Robust Control Law for Periodic ```計測自動制御学会 産業論文集
Vol.10，No.24，198/208 (2011)

Singularity Robust Control Law for Periodic Maneuvering Satellites
with CMGs
Takuya Ohmura*, Takehiro Higuchi**, and Seiya Ueno***
Control moment gyro (CMG) is a device used as an actuator for attitude control of spacecrafts and provides high agility.
This study focuses on control law that is especially efficient for periodic maneuvering satellites with single-gimbal CMGs
(SGCMGs) arranged like a pyramid. The periodic maneuvers of satellites can be utilized for observation satellites to widen
the swath and observe multiple targets. Minimum-energy CMG control law for periodic maneuvering satellites is
formulated. Simulation results show its performance of stabilizing gimbal angles at desired optimal gimbal angles during
periodic maneuvers. When the state approaches a singularity, the control law cannot steer gimbal angles toward desired
optimal gimbal angles. Using Discrete Lyapunov Stability, this paper analyzes the singularity problem of proposed control
law. Lyapunov analysis explains why the singularity problem occurs, and control law is modified inserting diagonal matrix
to avoid this problem. The modified control law makes the system asymptotically stable at desired optimal gimbal angles.
Simulation results demonstrate the useful merits of the modified control law to stabilize at desired optimal gimbal angles
even if a singularity exists. The singularity robust control law will enable spacecrafts, especially for next-generation
observation satellites, to perform periodic maneuvers much more effectively.
Key Words : Optimal control, Control moment gyros, Periodic control, Stability, Singularity problem
1. INTRODUCTION
Control moment gyro (CMG) is a device used as an
actuator for attitude control of spacecrafts. CMG contains a
spinning rotor with large constant angular momentum. The
spinning rotor can be gimballed to produce gyroscopic
torque, which is orthogonal to both spin and gimbal axes.
The major benefit of utilizing CMG for the satellite control
is torque amplification property that can provide satellites
with high torque capability. High agility with CMG is very
important particularly for earth observation satellites such as
abilities using CMGs. For example, rapid multiple target
imaging as shown in Fig. 1, global observation in short
period, high resolution disaster monitoring, and so on.
CMG is capable of expanding the ability of observation
performance for an attitude controller.
Considerable studies have been conducted on control
problems of CMG systems, including singularity avoidance.
Kurokawa2) conducted the survey of singularity avoidance
steering laws and grouped as exact methods, offline
planning, and SR inverse family. As example of singularity
* Department of ocean-space system engineering, Yokohama
National University, Hodogaya-ku, Yokohama-shi, Kanagawa,
Japan(E-mail: d09GB202@ynu.ac.jp)
** Interdisciplinary Research Center, Yokohama National
University
*** Research Institute of Environment and Information Sciences,
Yokohama National University
avoidance laws, Bedrossian et al.3) applied singularity robust
inverse steering law to control CMGs to avoid singularities
with allowing some torque error. Vadali et al.4) proposed
the method based on back integration of the gyro torque
equation from desired terminal condition to determine
preferred initial gimbal angles that can avoid singularities.
Wie5) developed CMG steering logic that uses an additional
weighting matrix to avoid explicitly singularity encounters.
These results assume that the satellites with CMGs perform
the one-way maneuver from the initial attitude to the
specified terminal condition. This paper focuses on periodic
maneuvering satellites, as shown in Fig. 1, with singlegimbal CMGs arranged like a pyramid. Periodic maneuvers
of satellites can be utilized to widen the swath and observe
multiple targets.
Satellite
Fig. 1 Schematic image of periodic control of satellite
for multiple targets
TRIA 024/11/1024 &copy; 2010 SICE
Ueno6) 7) has proposed optimal control law for adaptive
structure with periodic constraint. It was considered in
these studies that the periodic control enables the
manipulators to track and capture the rotating structures,
especially the space debris that is out of control. Control
law in this study is well related to the approach proposed by
Ueno. This study discusses the effectiveness of applying
periodic optimal control strategy to satellite attitude control
problems with CMGs. The purposes of this study are (1) to
design minimum energy CMG control law for periodic
maneuvering satellites, (2) to analyze singularity problem
using the proposed control law numerically and propose
singularity robust control law, and (3) to demonstrate useful
merits of the singularity robust control law from simulation
results.
The rotational equation of motion of a rigid satellite with
CMGs is described by,
  ω H  0.
(1)
H
H=[Hx, Hy, Hz]T is the total angular momentum vector of
the system that is sum of the satellite main body angular
momentum I and the CMG angular momentum h. Time
derivative of H is H . =[ x, y, z]T is the angular
velocity vector of the satellite. By substituting H=I+h,
Equation (1) gives
  ω  Iω  h.
h   Iω
(2)
In Equation (2), I R33 is the inertia matrix of the satellite.
Quaternion q=[q1, q2, q3, q4]T is chosen to describe the
attitude of the satellite. The quaternion consists of the
maneuver angle and desired axis of rotation r in the body
fixed frame.
2. CMG CONTROL LAW FOR PERIODIC
MANEUVERING SATELLITES
r  l m nT ,
This section introduces satellite attitude dynamics with
CMGs and formulates control law for periodic maneuvering
satellites. This section also shows simulation results with
the proposed control law.
q1
q2
q3


q4 T  l sin
2

q3  q2
 q4

ω  2 q3 q4
q1
 q2  q1 q4
2.1 Satellite Attitude Dynamics
The model in this study is a rigid body with a cluster of
four CMGs arranged in a pyramid configuration like Figure
2. In Fig. 2, ii=1,2,3,4is the ith gimbal angle and is
the pyramid skew angle.
(3)
m sin
 q 
 q1   1 
 q
 q2   2 .
 q 
 q3   3 
q4 

2
n sin

2
T

cos  ,
2
(4)
(5)
This study treats periodic maneuvering satellites. The
maneuver angle  therefore has to be satisfied with the
periodic condition, (t+T)=(t). From Equations (3)-(5), 
is also given as periodic form,
(6)
ωt  T   ωt .
In Equation (6), T is the period of the maneuver. The
following relation is given between the gimbal angle vector
x=[1,  2,  3,  4]T and the gimbal rate vector u, which
define the system state and the control input, respectively.
x  u.
(7)
For the pyramid type CMG systems, CMG angular
momentum h is sum of each CMG angular momentum
vector hi.
Fig. 2 Pyramid arrangement
199
4
h

hi
i 1
 cos sin1  cos 2  cos sin 3  cos 4 


 h  cos1  cos sin 2  cos 3  cos sin 4 ,
 sin  (sin 1  sin 2  sin 3  sin 4 ) 
(8)
where h is the constant magnitude of the axial angular
momentum of each CMG. The time derivative of Equation
(8) is,
h  A x u.
J int 
sin 2
 cos cos1

A  h   sin1
 cos cos 2
 sin  cos1
sin  cos 2
cos cos 3
 sin 4 

sin 3
cos cos 4 .
sin  cos 3 sin  cos 4 
(10)
This model is approximation of the real CMG dynamics
which has been widely used in the studies on CMG systems.
The control law for periodic maneuvering satellites is
designed based on this dynamics in the following section.
State


1 T
u u  ρ x - xopt T u.
2
(14)
(9)
In Equation (9), the matrix A has dimension of 3&times;4,
2.2
input variables that corresponds to the energy consumption,
and second term (x-xopt)Tu represents the derivative of (xxopt)T(x-xopt); the square of the difference between current
gimbal angles and optimal gimbal angles. The second term
has the potential to stabilize current gimbal angles x to
desired optimal gimbal angles at xopt. The criterion is
integration of sum of two terms in one period. The integrant
of the criterion Jint is,
Feedback
Control
Law
for
Minimum J int subject to Au-S(t)=0
Applying the Lagrange multipliers can solve this
optimization problem. Lagrangian, L is expressed using
multiplier ∈R3
1
(15)
L  uT u   x  xopt T u  λT  Au  S .
2
Periodic

Maneuvering Satellites
The control law for periodic maneuvering satellite is
formulated as follows. Equation (2), (6), and (9) give
periodic constraint for desired periodic maneuvers that has
to be satisfied at all time.
Au  S t   0,
It is possible to derive an optimal initial state and input with
minimum criterion by numerical calculation. However, the
gimbal rates between initial gimbal angles and optimal
gimbal angles needs to be optimized, because both angles
would be different in many cases. Hence, the local
minimum solution is used to design the feedback control
law to improve the energy consumption in every cycle. The
local minimum solution solves the following optimal control
problem to derive optimal input.

Necessary conditions for minimum Jint are,
(11)
L
L
 0,
 0.
u
λ
(12)
The CMG control law for periodic maneuvering satellites is
obtained by the following expression,
(16)
where,
 t   ωt  Iωt   ht .
S t   ht    Iω

n 1T
J

nT

1 T
 2 u u  ρ x - x opt


T

udt.



S  ρ  E  AT AAT

 us  ρP x  xopt ,
u  AT AAT
The matrix S follows the periodic condition, S(t+T)=S(t).
This study aims to propose minimum energy control law for
periodic maneuvering satellites. The criterion J is expressed
as,
1


 Ax  x 
1
opt
(17)
where,
(13)

P  E  AT AAT

u s  AT AA
In Equation (13), n is the cycle number and is the weight.
xopt indicates optimal gimbal trajectory for single cycle that
achieves the minimum energy maneuver. The criterion
contains two terms: the first term uTu is the square of the


T 1

1
S,
x opt  uopt dt  C 
200
A,
 A A A 
T 1
T
c
c
c
(18)
Sdt  C .
diag[10000,10000,10000][kgm2], pyramid skew angle  ,
45[deg], constant magnitude of the axial angular momentum
h , 1000[kgm2/s], and weight , 0.005. These parameters
are used throughout this paper. It is assumed that the
satellite performs a 60-deg rotation and then returns back to
its initial attitude in 60[s] around the rotation axis: r=[0.50.5
0.50.5 0]T. The maneuver is performed periodically. Let C
in Equation (18) be set as [53.4 -53.4 53.4 -53.4]T [deg].
Fig. 3 shows the simulation results. Fig. 3a) and 3b)
shows the time histories of four gimbal angles and four
gimbal rates, respectively. Fig. 3c) indicates the criterion in
each cycle. Fig. 3d) represents the unit quaternion of the
attitude of the satellite. Note that the proposed control law
achieves to stabilize the gimbal angles to the desired optimal
initial gimbal angles [53.4 -53.4 53.4 -53.4]T [deg]. The
gimbal rates are minimized accordingly. The commanded
maneuver is confirmed by the periodic results of the
quaternion that changes its value from q=[0, 0, 0, 1]T to
q=[0.354, 0.354, 0, 0.866]T, and then back to q=[0, 0, 0, 1]T
in one-cycle.
In Equation (17), E is 4&times;4 unit matrix. The matrix P can be
considered as projection matrix where P is idempotent PT=P
and self-adjoint PPT=P. In Equation (18), optimal initial
gimbal angles C is the constant value and is very important
to achieve minimum-energy periodic maneuvers because of
the following reason. Energy consumption for performing
commanded maneuvers depends crucially on the initial
gimbal angles.
This optimal gimbal angles can be
calculated numerically, where optimal angles are relative to
various conditions: maneuver axis and angle, satellites
moment of inertia, CMG pyramid skew angle, and so on. Ac
in Equation (18) is almost the same form as A in Equation
(10), but including the value of C.
2.3 Proof of Optimality
In the preceding section, optimal gimbal angles are
defined as Equation (18). This section proves the optimality
of the set of optimal gimbal angles. The discussion stated in
this section can be referred to the publication by Rao8).
The solution of Au=S can be represented in the form
u=GS where G is a generalized inverse of A that satisfies
the condition, AGA=A, and general solution is u=GS+(IGA)z where z is arbitrary. Let G be a generalized inverse of
A such that Gy is a minimum norm solution for all z and x,
that is, the following inequality holds.
z
Gy
Gy
IGA
z.

GAx
GAx
IGA
(19)
It is necessary and sufficient for the condition (19) that the
Gimbal angles [deg]
90
30
1
3
0
2
4
-30
-60
following equivalent conditions are satisfied;
GAT I  GA  0 or GAT  GA.
60
-90
0
(20)
Looking back at the control law shown in Equation (17), G
generalized inverse of A, has the form AT(AAT)-1, which
satisfies the necessary and sufficient condition (20). In
CMG system, initial gimbal angles C plays a key role to
decide the amount of energy consumption. Therefore the
solution u=AcT(AcAcT)-1S including the optimal initial
gimbal angle combination C is a minimum norm solution in
the class of all solutions of Au=S(t). It can be concluded
that the setting of desired optimal gimbal angles as Equation
(18) has the optimality property for the system.
4
8
12
Cycle
16
20
24
a) Gimbal angles
Gimbals Rate [deg/s]
3.0
2.4 Simulation Results
This section shows simulation results with the CMG
control law (17) for periodic maneuvering satellites. The
simulation parameter values are as follows: inertia matrix I ,
u1
u3
1.5
u2
u4
0.0
-1.5
-3.0
0
4
8
Cycle
b) Gimbal rates between 1st and 12th cycle
201
12
mission requirement referring to the index given by
Equation (21). The results of Fig. 3 clearly indicate the
effectiveness and utilization of the proposed control law for
periodic maneuvering satellites.
Criterion[1/s]
0.03
3. SINGULARITY PROBLEM
0.02
Singularities are major difficulty for using CMGs as
attitude controller. This chapter is on singularity problem
for using the proposed control law (17). This section
provides simulation results to indicate the problem where
singularity prevents gimbal angles from stabilizing at
desired optimal gimbal angles. Discrete Lyapunov Stability
Theory is used to analyze the singularity problem.
0.01
0
4
8
12
16
20
24
Cycle
3.1 Simulation Example of Singularity Problem
Singularity problem is well known as major difficulty for
using CMGs as attitude controller. Singularity is a gimbal
angle combination that CMG system cannot produce torque
about certain axis. To avoid the situation, singularity
avoidances have been developed for many years including
SR-inverse method, exact solution, and offline planning.
Not only singularity avoidance but also singularity analysis
and visualization9) have also been developed.
The
singularity problem can cause severe problems as shown in
the following simulation. Fig. 4 shows simulation results of
the case when maneuver axis r is [0.250.5, 0.250.5, 0.50.5]T.
The satellite performs 60-deg maneuver and then returns to
the initial attitude around the axis. Starting from the gimbal
angles [60, -60, 60, -60]T [deg], let the desired optimal initial
gimbal angle combination C be set as [13.1, -13.1, 13.1, 13.1]T [deg]. All other parameters are same as the previous
simulation results. Fig. 4a and 4b show the time history of
gimbal angles and gimbal rates, respectively. Fig. 4c and 4d
show the criterion and the unit quaternion of the attitude of
the satellite, respectively. From Fig. 4a, the results show
that the gimbal angles stabilize at [43.5, -43.5, 27.0, -27.0]T
[deg], which combination is apparently different from the
desired optimal initial gimbal angles C. Criterion in onecycle on the desired optimal combination is about 0.051; on
the contrary, the result of Fig. 4c, criterion cannot reach the
minimum value. The gimbal angle combination [43.5, -43.5,
27.0, -27.0]T [deg] therefore can be considered as nonoptimal but stable gimbal angle combination.
This
undesirable stabilization of gimbal angles appears just
before passing through a singularity [35.3, -35.3, 35.3, 35.3]T [deg] with zero-angular momentum that exists
between initial and desired optimal gimbal angles. The
singularity prevents gimbal angles from stabilizing at the
c) Criterion
1.0
Quaternion
0.8
q1
q3
0.6
q2
q4
0.4
0.2
0.0
0
4
8
12
Cycle
16
20
24
d) Quaternion
Fig. 3 Simulation results on r=[0.50.5 0.50.5 0]T
Consequently,
the satellite performs the commanded
periodic maneuver with minimum energy consumption. In
addition, the speed of stabilization of gimbal angles can be
changed by the value of the weight  in Equation (17).
There is a method to determine the weight using the
relationship between the weight  and time constant . The
relationship is described by,
1.0
(21)
1  exp     0.632   
.

The simulation results of Fig.3 assume that the time
constant is 200[s]. The  is determined from Equation (21)
to be 0.005. Higher the value of  is, faster the minimum
energy periodic maneuver can be achieved, but the energy
for the convergence increases as well.  can be changed by
202
Discrete Lyapunov Stability guarantees the stable at
steady state x0 if there is a positive definite function V(x0)≧
0 for which V(x0)=0 and V(t+⊿t)-V(t)≦0 for some time
step ⊿t＞0. Based on the theory, consider the Lyapunov
function V,
desired C. This is singularity problem for using the
proposed control law, which problem is comparatively
unique to ordinary singularity problem because of the
following reason. Gimbal motion passing through a
singularity is also a problem. This singularity problem
needs to be deeply analyzed because minimum energy
periodic maneuvers cannot be achieved.
V
3.2 Numerical Stability Analysis
It is difficult to discuss stability for complex non-linear
system. Lyapunov Stability is one of the methods to
analyze the system stability. In this study, Discrete
Lyapunov Stability is used to analyze stability of desired
optimal gimbal angles for the proposed control law (17).
This section describes why the singularity problem occurs.
1
3

T P x  x opt dt
nT
n 1T

x  x opt

(22)

T

V is positive definite in all cycles and zero if and only if
x=xopt because of the projection matrix P.
2
4
0.10
60
0.09
30
0.08
0
-30
-60
0.07
0.06
0.05
-90
0
4
8
12
Cycle
16
20
0.04
24
0
4
8
12
16
20
24
Cycle
c) Criterion
a) Gimbal angles
6
u1
u3
u2
u4
1.0
3
0.8
Quatenion
Gimbals Rate [deg/s]

P T P x  x opt dt  V n .
nT
Criterion[1/s]
Gimbal angles [deg]
90
n 1T
x  x opt
0
-3
q1
q3
0.6
q2
q4
0.4
0.2
0.0
-6
0
4
8
12
0
Cycle
b) Gimbal rates between 1st and 12th cycle
4
8
12
16
Cycle
d) Quaternion
Fig. 4 Simulation results: r=[0.250.5 0.250.5 0.50.5]T
203
20
24
V and V(n+1)-V(n) are numerically calculated to analyze
the singularity problem. Fig. 5 shows gimbal angle
combinations with zero-angular momentum. The horizontal
axis shows the 1st gimbal angle and left and right vertical
axis represent the 2nd and 3rd gimbal angle, respectively.
There are seven gimbal angle combinations: [a, -a, a, -a]T,
[a, a-180, a, a-180]T,[a, a+180, a, a+180]T, [a1, -a1, a2, a2]T, [-a2, a1, -a1, a2]T, [a2, -a2, a1, -a1]T, [-a1, a2, -a2, a1]T.
The relation between a1 and a2 is,
(23)
Singularity exists at the cross points between each line in
Fig. 5. In Fig. 5, the initial and optimal gimbal angles in
Fig. 4 are also plotted. Fig. 5 shows important factor for
control law to utilize null-motion to steer gimbal angles.
The line indicates the trajectory of the gimbal motion where
no output torque is generated to perform the transition to
optimal gimbal angles.
Lyapunov function V with the same parameters as the
case of Fig. 4 is numerically calculated and shown in Fig. 6a
and 6b. Fig. 6b is close-up view of V near singular point.
Gimbal angles are steered in the direction that the value of V
decreases, V(n+1)-V(n)＜0.
Lyapunov Function V
150
125
100
75
50
60
40
25
20
00
40
Gimbal Angle:
Singularity
―　3rd Gimbal angle
- - 2nd Gimbal angle
-180
20
60
0
1[deg]
Gimbal Angle: 3[deg]
 
   tan 1  1

a 2  sin 1 sina1  tan 1  1
 
 cos 
cos


 


 
This numerical analysis for V and V(n+1)-V(n) can verify
that the desired optimal initial gimbal angles [13.1, -13.1,
13.1, -13.1]T [deg] can be stable state, but also shows that
the system can stabilize at the non-optimal gimbal angles at
[43.5, -43.5, 27.0, -27.0]T [deg]. From Fig. 6b, there is
significant decrease of Lyapunov function to the nonoptimal angles at the singularity. Gimbal angles are easy to
steer toward [43.5, -43.5, 27.0, -27.0]T [deg] accordingly.
Consequently, as shown in Fig. 4, the gimbal angles are
stabilized at the non-optimal gimbal angles when
approaching the singularity at [35.3, -35.3, 35.3, -35.3]T
[deg]. Singularity problem, undesirable stabilization of
gimbal angles is caused by the significant decrease of V at a
singularity.
a) Lyapunov function V
180
[a -a a -a]
0
0
[a1 -a1 a2 -a2]
90
180
-180
-90
-90
0
90
-180
180
30
20
10
030
32
34
36
1st Gimbal Ang
le
38
40
[deg]: 
40
38
36
34
32
30

3rd Gimbal Angle [deg]: 3
90
Lyapunov Function V
-90
3rd Gimbal angle [deg]; a3
2nd Gimbal angle [deg]; a2
40
1
1st Gimbal angle [deg]; a1
b) Lyapunov function V around the cross point in Figure 6a
Fig. 6 Numerical analysis of Lyapunov function
Fig. 5 Gimbal angle combinations with zero-angular
momentum
204
The significant decrease of V can be considered mainly
due to the value of P(x-xopt), which is included in V. The
P(x-xopt) in Equation (17) and (22) plays an important role
in steering gimbal angles to optimal gimbal angles. Fig. 7a
shows the 1st element of P(x-xopt) on gimbal angle
combinations with zero-angular momentum. The other
elements are similar to the 1st element. It can be confirmed
by Fig. 7a that P(x-xopt) has sharp discontinuous drop when
gimbal angles change its combination to [a1, -a1, a2, -a2]T at
the singularity, which is induced by a difference between
the value of P on [a1, - a1, a2, - a2]T and P on [a, -a, a, -a]T.
W is 4&times;4 diagonal matrix. Substitute Equation (24) into the
periodic constraint Au-S(t)=0, the modified control law can
satisfy the constraint for all time. Fig. 7b shows the 1st
element of PW(x-xopt) on the gimbal angle combinations
with zero-angular momentum. The result of Fig. 7b is given
from the same parameters presented in Figure 7a except for
the W, diag[100, 100, 1, 1]. It is clear by comparing Fig. 7b
to 7a that inserting proper W setting contributes to prevent
the discontinuous drop of P(x-xopt), which leads to the
singularity problem.
To prevent the discontinuous drop, the effective W
settings tend to be the form [100, 100, 1, 1], [100, 1, 100,1],
 1 4 1 4 1 4 1 4


1 4 1 4 1 4 1 4 
on [a, -a, a, -a]T
P
 1 4 1 4 1 4 1 4


1 4 1 4 1 4 1 4 
[100, 1, 1, 100] … ; two diagonal elements in W are set
large enough to pass a singularity. One of these effective W
settings can be chosen from a singularity between initial and
 1 4 1 4 1 4 1 4 


1 4 1 4 1 4 1 4
on [a1, -a1, a2, -a2]T
P
1 4 1 4 1 4 1 4


 1 4 1 4 1 4 1 4 
optimal gimbal angles, and non-optimal convergence gimbal
angles.
1 .0
The sharp discontinuous drop of P(x-xopt) at a singularity
lead to significant decrease of V and steers gimbal angles
toward non-optimal gimbal angles. Next chapter proposes
singularity robust control law that can solve the singularity
problem based on the analysis in chapter 3.
al
An

g]
g ]:
[d e
3
[d e
0
60
g le
40
1 .0
0 .6
Gi
al
mb
PW(x-xo p )t
0 .8
3 rd
60
40
0 .4
0 .2
40
0
g]
[d e
g ]:
1 s t G im b a l A n g le [d e g ]:  1 [d e g ]
60
[d e
20
3
0 .0 0
g le
(24)
20
An

20
a) P(x-xopt)
1
opt
mb

 x  x 
Gi
 us  ρPW

S  ρ  E  W 1 AT AW 1 AT

x  xopt ,
1
20
1 s t G im b a l A n g le [d e g ]:  1 [d e g ]
4.1 Singularity Robust Control Law
The modified control law is proposed on the idea that a
modification by inserting diagonal matrix W can increase
the value of P(x-xopt) and steer gimbal angles toward desired
optimal gimbal angles.

60
40
0 .4
0 .0 0
This chapter proposes modified version of the control law
by inserting diagonal matrix to solve the singularity problem.
Inserting diagonal matrix can eliminate the sharp
discontinuous drop of P(x-xopt), which leads the system
asymptotically stable at desired optimal gimbal angles. The
modified control law results in improved performance,
which is shown in simulation results.

0 .6
0 .2
4. SINGULARITY ROBUST CONTROL LAW
u  AT AAT
3 rd
P(x-xo p )t
0 .8
where

PW  E  W 1 AT AW 1 AT

1
b) PW(x-xopt)
A.
(25)
205
Fig. 7 P(x-xopt) and PW (x-xopt) about 1st gimbal angle
0, 1]T to q=[0.25, 0.25, 0.354, 0.866]T, and then back to q=[0,
0, 0, 1]T in one-cycle, which indicates that the modified
control law with W can achieve the commanded periodic
maneuver.
Table 1 shows results for the case: rotation axis r, [1, 0,
0]T, and optimal initial gimbal angle C, [73, -73, 73, -73]T
[deg]. Table 1 also clearly indicates that the modified law
(24) can steer gimbal angles to the desired optimal angles;
by contrast, the control law (17) fails to stabilize at the
desired optimal state.
It can be concluded through
simulation results that the modified control law with proper
diagonal matrix W has significant performance; it can pass
through singularities on gimbal angle combinations with
zero-angular momentum successfully and steer gimbal
angles toward desired optimal gimbal angles.
For the modified control law, let the Lyapunov function
be redefined as,
VW
n 1T

x  x opt

T PW T PW x  x opt dt  VW n.
(26)
nT
Numerical analysis of the Lyapunov function VW
(W=diag[100, 100, 1, 1]) is shown in Fig. 8 under the same
condition of Figure 6. It is apparently different result from
Fig. 6 that the non-optimal gimbal angles [43.5, -43.5, 27.0,
-27.0]T [deg] is not stable and the desired optimal initial
gimbal angles [13.1, -13.1, 13.1, -13.1]T [deg] becomes
asymptotically stable state.
5 CONCLUSION
200
Periodic maneuvers can be utilized to widen the swath
and observe multiple targets. This study proposes the
minimum energy CMG control law for periodic
maneuvering satellites. Simulation results verify its
performance that gimbal angles can be stabilized at desired
optimal gimbal angles during maneuver. However, in case
there is a singularity between certain initial and desired
optimal gimbal angles, there is possibility that the control
law cannot stabilize at desired optimal gimbal angles.
Discrete Lyapunov Stability technique is used to analyze the
singularity problem. Numerical analysis of Lyapunov
function clarifies that gimbal angles are easy to stabilize at
non-optimal gimbal angles because of the following reason.
Lyapunov function is decreased significantly at a singularity
for non-optimal gimbal angles, which is mainly induced by
sharp discontinuous drop of P(x-xopt) when gimbal angles
change its combination with zero-angular momentum at a
singularity. The proposed control law is modified by
inserting diagonal matrix W that can eliminate the
discontinuous drop of P(x-xopt). The modified control law
makes the system asymptotically stable at desired optimal
gimbal angles. Simulation results demonstrate the useful
merits of the singularity robust control law that can stabilize
at desired optimal gimbal angles even if a singularity
appears in the stabilizing phase. As the result, the satellite
achieves commanded periodic maneuvers with minimum
energy consumption. The proposed CMG control law will
enable spacecrafts, especially for next-generation Earth
imaging satellites, to perform periodic attitude maneuvers
much more effectively.
Gimbal Angle: 3[de
g]
100
60
40
50
20
0
0
20
Gimbal Angle: 1[d
40
Lyapunov Func
tion V
150
0 60
eg]
Fig. 8 Lyapunov Function VW in case W=[100, 100, 1, 1]
4.2 Performance Comparison using Simulation Results
The performance of W is confirmed by comparing the
simulation result from the case without W. Fig. 9 shows
simulation results under the same parameter in Fig. 4 except
for W, diag[100, 100, 1, 1]. Fig. 9a and 9b show timehistory of gimbal angles and gimbal rates of the modified
control law, respectively. Fig. 9c and Fig. 9d show the
comparison between criterions in each case and quaternion,
respectively. Fig. 9e illustrates the stabilization of gimbal
angles in two cases. Comparing Fig. 4 to 9a, it can be
confirmed that the modified control law (24) can achieve to
stabilize the gimbal angles at the desired optimal initial
gimbal angles, [13.1, -13.1, 13.1, -13.1]T [deg]. The
criterion in Fig. 9c, which implies amount of energy
consumption, is obviously much less than that of Fig. 4. Fig.
9d shows the quaternion that changes its value from q=[0, 0,
206
1
3
60
0.10
2
4
30
0
-30
0.08
0.07
0.06
0.05
-60
0.04
-90
0
4
8
12
Cycle
16
20
0
24
4
8
12
16
20
24
Cycle
a) Gimbal angles
c) Comparison between criterion in Fig. 4 and Fig. 9a
6
u1
u3
u2
u4
1.0
3
0.8
Quatenion
Gimbals Rate [deg/s]
W=[100, 100, 1, 1]
W=[1, 1, 1, 1]
0.09
Criterion[1/s]
Gimbal angles [deg]
90
0
-3
q1
q3
0.6
q2
q4
0.4
0.2
0.0
-6
0
4
8
12
0
4
8
12
16
Cycle
Cycle
d) Quaternion
b) Gimbal rates between 1st and 12th cycle
W=diag[100, 100, 1, 1]
W=diag[1, 1, 1, 1]
[60, -60, 60, -60]
[35.3, -35.3, 35.3, -35.3]
[45, -45, 25, -25]
[13.1, -13.1, 13.1, -13.1]
e) Stabilization of gimbal angles in Fig. 4 and Fig. 9a
Fig. 9 Simulation results with W, [100, 100, 1, 1]
207
20
24
Table 1 Simulation results
with C, [73, -73, 73, -73]T [deg] and r, [1, 0, 0]T
Takuya Ohmura
He received his B.E. degrees from Yokohama
National University in 2009. Currently, he is
Initial gimbal
angles [deg]
Singularity
[deg]
[120,-120,
120,-120]
[30,-30,
30,-30]
Gimbal angles stabilize at
a second-year master candidate at the same
Control law
(17) [deg]
Modified control
law (23) [deg]
university. He received the “1st Place Student
[90,-90,
90,-90]
[97,-84,97,-84]
[73,-73,73,-73]
W=[100,1,100,1]
Session Award” at 18th IFAC Symposium on
[35,-35,
35,-35]
[29,-29,42,-42]
[73,-73,73,-73]
W=[100,100,1,1]
Automatic Control in Aerospace, Nara, Japan,
2010.
Takehiro Higuchi (Member)
REFERENCES
He received the B.E., M.E., and Ph.D.
degrees from Yokohama National
1） Philippe CAMPENON: PLEIADES AND SPOT 6: EARTH
OBSERVATION
IN
HIGH
University in 2000, 2002, and 2005,
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60th
AND
VARY
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Seiya Ueno (Member)
Moment Gyroscope, Journal of Guidance, Control, and
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Dynamics, 13-6, 1083/1089(1990)
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Rao,
and
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Kumar
Mitra:
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of
the
Sixth
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208
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