Gain Scheduling Control of Control Moment Gyroscope
M2015SC007 Toru INABA
Supervisor: Isao TAKAMI

Abstract
This paper presents gain scheduling (GS) control of a
Variable Speed Control Moment Gyroscope (VSCMG)
based on sum of squares (SOS). Nonlinear motion equations of the VSCMG are complicated because they contain sine and cosine of angles of gimbals. The dynamics
varies depending on an attitude of a spacecraft. In thisstudy, the diﬃculty of control design of the VSCMG issolved by two methods. First, the complex nonlinear
model is transformed to the linear parameter varying(LPV) model, which the linear control method can beapplied, to make prospect of control design easy by using a proposed approximation method. The highly accurate LPV model is reduced to polynomially parameterdependent linear matrix inequalities (PDLMIs). Second, GS controller whose gains are adjusted dependingon the angles and the attitude is applied. The polynomially PDLMIs can be relaxed to ﬁnite design conditions
based on matrix SOS polynomials. The GS controller
is designed by solving the ﬁnite SOS conditions. By using those methods, GS controller which appropriately
calculates gains depending on the nonlinearities is designed. The eﬀectiveness of the proposed controller isillustrated by simulations and experiments.

1

approximation is applied. A highly accurate LPV model
including rational function of the varying parameter isderived by the proposed approximation method. In thisstudy, the GS controller is designed to adjust gains depending on the nonlinear dynamics of the VSCMG. The
GS controller is designed based on matrix SOS polynomials, because the polynomially parameter-dependent
linear matrix inequalities (PDLMIs) are relaxed to ﬁnite
matrix SOS polynomials conditions, naturally. The conservativeness of the GS controller designed by the proposed approximation method is lower comparing with a
GS controller designed by an ordinary approximation
method. The eﬀectiveness of our proposed GS controller is shown by simulations and experiments comparing with an ordinary GS controller.

2

Attitude Control Model of a VSCMG

2.1

Hardware and System Restriction

Model 750 CMG having the variable speed wheel is
4-DOF control plant. The schematic diagram of Model
750 CMG is shown in Fig. 1. It consists of four rigid
Rotor1
Gimbal3

Gimbal2

Introduction

A Control Moment Gyroscope (CMG) is known as
an attitude control device for space crafts. The CMG
is an attractive actuator because the maximum torquegenerated by a CMG is dozens of times compared withthe torque by a conventional actuator called reaction
wheel. In this study, the control design of a singleVariable Speed CMG (VSCMG) is treated as a fundamental study for cooperation control of CMGs. The
VSCMG consists of a variable speed reaction wheel anda single gimbal. It is not easy to adopt general linear
control methods to the VSCMG, because the dynamics
of the VSCMG varies depending on an attitude of thespacecraft. Nonlinear control methods are adopted tothe controller design of the VSCMG. Recently, griddingbased gain scheduling (GS) controller design based ona linear parameter varying (LPV) system is reported tosolve the nonlinear dynamics [1], while robust stabilityof the system is guaranteed by solving many linear matrix inequalities (LMIs).
This paper presents a GS controller design of the
VSCMG based on sum of squares (SOS). The VSCMG
which has the multi-freedom actuator is able to control
two-degrees-of-freedom (DOF) attitude, but the controller design is very complicated to include many nonlinearities such as sine and cosine of the gimbal’s angles
in the motion equations. Let θ(t) be a time varying
parameter in the motion equations. Nonlinearities suchas sin θ(t) and cos θ(t) in motion equations are generally approximated by the ﬁrst-order Taylor series expansion to get a simpliﬁed linear model. The nonlinearitysin θ(t) is assumed to be a varying parameter and an
approximation depending on sin θ(t) is applied only tocos θ(t) in our study. In this study, in order to conduct
a highly accurate approximation, the third-order Padé

Axis3
Axis1
Axis2
Axis4

Gimbal4

Figure 1 Model 750 Control Moment Gyroscopebodies which are Rotor1, Gimbal2, Gimbal3, and Gimbal4. Rotor1 and Gimbal2 are a VSCMG. Gimbal3and Gimbal4 are controlled bodies. These bodies rotate
around Axis1, 2, 3, and 4. Here, let q1 , q2 , q3 , and q4 be
the angles of Rotor1, Gimbal2, Gimbal3, and Gimbal4,
respectively. And, q̇1 , q̇2 , q̇3 , and q̇4 indicate the angular velocity of Rotor1, Gimbal2, Gimbal3, and Gimbal4,
respectively. Let τ1 and τ2 be the input torque for Rotor1 and Gimbal2, respectively. Note that the VSCMG
which is used in this study has a hardware restriction
on the motion range of Gimbal2. The motion rangeof Gimbal2 is ±π/2[rad] from the vertical position to
Gimbal3. Rotor1 and Gimbal2 are driven by DC motors, while Gimbal3 and Gimbal4 have no drive source.
Gimbal3 and Gimbal4 are driven by reaction torque andgyroscopic precession, respectively. These torque are
generated by the VSCMG. The gyroscopic precession islargest when the position of Gimbal2 is tilted rapidly
around the vertical position to Gimbal3. The motion
range of Gimbal2 is assumed as follows:−ππ≤ q2 ≤ .
6
6(1)

An attitude of Gimbal3 is given by the angles q3 and q4 .
The attitude is regarded as an attitude of a spacecraft.
The motion range of Gimbal3 is restricted, because thesystem has singular points depending on the attitude

−

5π
5π
≤ q3 ≤
.
12
12

(2)

In this study, MIMO system which ensure the stabilityof the responses q3 and q4 under the motion range (1)and (2) is treated.
2.2

Nonlinear ModelIn this subsection, motion equations of Model 750
CMG are derived. If the angular velocity q̇1 is large
enough compared with other angular velocity q̇2 , q̇3 ,and q̇4 , a simpliﬁed nonlinear mathematical model isderived [1]. Thus, motion equations of Model 750 CMG
is represented as follows:

b3
0

0
b12 sin2 q2 − b1 + b4
−b3 sin q3 −b12 sin q2 cos q2 cos q3[]−b3 sin q3
q̈2
 q̈3
−b12 sin q2 cos q2 cos q3
q̈4−b12 sin2 q2 (1 − sin2 q3 ) + b5 sin2 q3 + b6[
0
q̇1 b1 sin q2−q̇1 b1 sin q2
0+
q̇1 b1 cos q2 cos q3 −q̇1 b1 sin q2 sin q3][]−q̇1 b1 cos q2 cos q3
q̇2
q̇1 b1 sin q2 sin q3
q̇3
0
q̇4
[
]
]
0
1 [
τ1
− cos q2
0
=
,
(3)
τ2
− sin q2 cos q3 0{
b12 = b1 + b2 , b1 = JD , b2 = ID − JC − JD + KC ,b3 = IC + ID , b5 = IB + IC − KB − KC ,b4 = JB + JC + JD , b6 = ID + KA + KB + KC .
Ix , Jx , and Kx (x = A, B, C, D) are the moment ofinertia. As can be seen in (3), there exist nonlinearities
such as sin qi and cos qi (i = 2, 3).
2.3

Linear Parameter Varying (LPV) ModelingIn this subsection, our model approximation method
is proposed. Assume that there exist nonlinear terms
such as sin θ(t) and cos θ(t) in motion equations. The
nonlinearity of sin θ(t) is a varying parameter such asα(t) := sin θ(t). The other√ nonlinearity of cos θ(t) isrepresented as cos θ(t) = 1 − α(t)2 . Approximation isdone only for cos θ(t) by using Taylor series expansion
or Padé approximation depending on sin θ(t). Thus, if
the nonlinearity of the sine function in (3) are deﬁned
as αi := sin qi , the nonlinearity
of cosine function are√2
represented as cos qi = 1 − αi . The square roots areapproximated by using Taylor series expansion or Padé
approximation√ under the motion range (1) and (2). The
square root 1 − α22 is approximated by second-order√
Taylor series expansion. The other square root 1 − α32
is more accurately approximated by third-order Padé
approximation. Those approximations are represented
as follows:cos q2 ≃ 1 −1 − α32
α22, cos q3 ≃.
2
1 − α32 /2(4)

Furthermore, the coeﬃcients in (4) are optimized by using the method of least-squares to conduct more accurate approximation under the range (1) and (2), respectively. Note that the approximated functions pass theupper bound value of cos qi . Consequently, our model
approximation method is proposed as follows:{
sin q2 = α2,(5)cos q2 ≃ 1 − 0.5251α22 =: αp{sin q3 = α3.
(6)1−0.8943α2
cos q3 ≃ 1−0.4401α32 =: αr3

The true values of the square root and its approximatedvalues of αp and αr are illustrated in Fig. 2. The dot1
0.8
cos qiof Gimbal3. The singular points are the positions of±π/2[rad] from the vertical position to Gimbal4. The
motion range of Gimbal3 is assumed as follows:

0.6
0.4
0.2
0
-1

1-α2i

-0.5

αp

0
sin qiαr

0.5

1Figure 2 Approximation
ted curve, long dashed short dashed curve, and the solid
curve show the true values of the square root and theapproximated value ap , and the approximated value ar
respectively. The maximum relative error between thetrue value and the approximated value of αp and αr
is 0.31% and 8.55% in the range (1) and (2), respecTT
tively. Here, let x = [q̇2 q̇3 q̇4 ] , u = [τ1 τ2 ] , andT
α = [α2 α3 ] be the state variable vector, the control
input vector, and the varying parameter vector, respectively. The linear parameter varying (LPV) system isrepresented as follows by using the proposed method:
E (α) ẋ = A (α) x + B (α) u,(7)There exist the rational function of the varying parameter α3 in the matrices E (α) , A (α) , and B (α) in (7).
It is not easy to reduce the system (7) to parameterdependent linear matrix inequalities (PDLMIs) for control design. Descriptor representation and linear fractional transformation (LFT) is applied to the system(7)to get a LPV model which do not include the rational function of the varying parameter α3 . The LPVmodel (7) is transformed to an equivalent system with
fourth-order polynomials of the varying parameters αi
by using those methods. First, the redundant descriptor
representation is applied. The varying parameters αi are
integrated into a new matrix Â by introducing a redun[]Tdant descriptor variable vector as x̂ := xT ẋT τ1 .
The redundant descriptor representation is obtained asfollows:Ê x̂˙ = Â (α) x̂ + B̂ (α) u,(8)Second, LFT is applied to the redundant descriptor system (8) including the rational function of the varying
parameter α3 to get a LPV model including fourth-orderpolynomials of the varying parameter αi . Here, the matrix Â is expressed as−1Â = An + Bδ (I − ∆Dδ )

∆Cδ .

(9)

By using the matrices An , Bδ , Cδ , Dδ , I, and ∆ satisfying the equivalent condition (9), the redundant system
is rewritten as follows:Êδ x̂˙ δ = Âδ (α) x̂δ + B̂δ (α) u(10)As can be seen in (10), there exist fourth-order polynomial functions of the varying parameter αi . The LPVmodel that is equivalent to the system (7) is obtained.

3

Controller Synthesis

3.1

Gain Scheduling Controller Design

A gain scheduling (GS) controller is designed bychoosing the scheduling parameter vector α ∈ R2 . The
purpose of our study is to achieve attitude control of
Gimbal3 and Gimbal4. The angles q3 and q4 are added
to the state variable vector x in (7). Let q3ref and q4ref
be given constant references for the angles q3 and q4 ,respectively. The output equation is deﬁned as follows:y = Cxe ,Y (α) := [ Y1 (α)

0

)
]T
)
[∫ ∞ (∫∞(,
where x̃δ := 0 q3ref − q3 dt 0 q4ref − q4 dt x̂TδTQ̃ = Q̃h Q̃h ≽ 0, R ≻ 0, x0 := xe (0). The matrices
Q and R are weighting matrices. A optimal controllerthat the initial-value response is improved is designedby minimizing γ under (13).
Design Condition

In this subsection, design conditions of the LPV model(10) are given by parameter-dependent linear matrix inequalities (PDLMIs). Let Ωα , Ωd , and Π be the parameter set of the scheduling parameters α, its derivatives
α̇, and initial-values α0 := α (0), respectively. If the
motion range and the region of given initial-values are
assumed such as |qi | ≤ q̄i and |qi0 | ≤ q̄i0 , then parameterregions are deﬁned as |αi | ≤ sin q̄i and |αi0 | ≤ sin q̄i0 , respectively. The regions of the derivatives α̇ is chosen as|α̇i | ≤ δi . Note that the set Π satisﬁes Π ⊂ Ωα . Thosesets Ωα , Ωd , and Π are deﬁned as follows:
(14)
(15)
(16)

In order to design the GS controller minimized γ under(13), suﬃcient conditions for the redundant descriptor
system (10) are introduced as follows:

02×6 ] ,Φ (α) := Âδe (α) X (α) + B̂δe (α) Y (α) .
Then, the cost function J is less than γ.
Note that ξ is the vector of the decision variables X(i) ,Y(i) , and γ. Let second-order polynomial matrices be
the matrices X (α) and Y (α).
3.3

0

]T
[
where y ref := q3ref q4ref . The GS controller consists ofnot the redundant descriptor variable vector x̂δ but theoriginal state variable vector xe .
In this study, linear quadratic (LQ) optimal control
is adopted to evaluate a initial-value response. A costfunction is introduced as follows [2]:
∫ ∞(
)
T
J=
x̃T
(13)
δ Q̃x̃δ + u Ru dt < γ,Ωα = {α ∈ R : gj (α) ≥ 0 (j = 1, 2)} ,Ωd = {α̇ ∈ R : gj (α̇) ≥ 0 (j = 3, · · · , 6)} ,Π = {α0 ∈ R : gj (α0 ) ≥ 0 (j = 7, 8)} .

minimized γ subject to ;F1 (ξ, α) := X11 (α) ≻ 0 (∀α ∈ Ωα ) ,(17)F2 (ξ, α) := ()− He [Φ (α)] − Êδe Ẋ (α)
∗
∗



Q̂h X (α)
I13×13 ∗  ≻ 0
RY (α)
02×13 R(∀ (α, α̇) ∈ Ωα × Ωd ) ,
(18)
[
]
γI7×7
∗
F3 (ξ, α) :=≻ 0 (∀α0 ∈ Π) ,(19)I7×7 X11 (α0 )[]X11 (α)07×6
X (α) :=,
X21 (α) X22 (α)

(11)

]T
[
T
where y = [q3 q4 ] , C = [I2 02×5 ] , xe := q3 q4 xT .
To follow the reference without steady-state error, servo
system is adopted as follows:
∫ ∞
( ref
)
u = K (α) xe + G (α)
y − y dt, (12)3.2

Proposition 3.1 : If there exist polynomial matrices
X11 (α) and Y1 (α) satisfying the following PDLMIs,
the redundant descriptor system is stabilized by the state−1
feedback u = Y1 (α) X11(α) x̃e .

SOS Formulation

In this subsection, the design conditions as PDLMIs(17)-(19) in subsec. 3.2 are relaxed to sum of squares(SOS) conditions. Based on the results in [3], the
PDLMIs are formulated to SOS conditions as follows:
Lemma 3.1 : If there exist polynomial matrices
X11 (α) and Y1 (α), and matrices SOS polynomials S10 ,S20 , S30 , S1j , S2k , and S3l satisfying the following
conditions (20)-(22), the redundant descriptor system is−1stabilized by the state feedback u = Y1 (α) X11(α) x̃e .
minimized γ subject to ;2∑
F1 (ξ, α) − ϵI7×7 −gj (α) S1j (α) = S10 (α) , (20)j=1

F2 (ξ, α) − ϵI28×28(21)
 ( ∑2)g (α) S2k (α)∑k=1 k∗ = S20 (α) ,−  + 6k=3 gk (α̇) S2k (α)
015×13015×15F3 (ξ, α) − ϵI14×14
[
]
07×7
∗
∑8
−
= S30 (α0 ) . (22)07×7l=7 gl (α0 ) S3l (α0 )Then, the cost function J is less than γ.

4

Simulation

In this section, the eﬀectiveness of the GS controller
designed by the proposed approximation method (5) and(6) is illustrated by simulations. Here, an ordinary approximation method is deﬁned as follows:{
sin q2 ≃ 0.9728q2,(23)
cos q2 ≃ 1 − 0.4919q22{
sin q3 ≃ 0.9945q3 − 0.1514q33.
(24)
cos q3 ≃ 1 − 0.4512q32

line and the dashed line show the responses of q3 and q4
by the proposed controller and the ordinary controller,
respectively. The thin dotted line show the reference.
As can be seen in Fig. 3 and 4, the responses of q3 andq4 by the proposed controller are improved. As a resultsof the simulations, the eﬀect of the friction disturbanceis reduced by using the proposed method.

5|q20 | ≤25π
5π, |q30 | ≤
.
180
12

(26)

Those GS controllers are designed by solving suﬃcientconditions in the lemma 3.1 under the necessary conditions (25) and (26). Thus, the upper bound value γof each cost function are minimized. Those results areshown in Table 1. As can be seen in Table 1, the upperbound value produced by proposed controller is 17.95%
lower than the upper bound produced by the ordinary
controller.
By the results in [4], eﬀect of the friction disturbance can’t be ignored. Simulations which include effect of the friction disturbance are carried out. Let
Fn (n = 1, · · · , 4) be the friction disturbance for Rotor1,
Gimbal2, Gimbal3, and Gimbal4, respectively. Let fc,nand fv,n be the coeﬃcients of coulomb friction and viscous friction. The friction disturbance Fn is deﬁned asfollows:
Fn = fc,n arctan(q̇n × 103 ) + fv,n q̇n .

(27)

1.5

1.5

1

1
q4 [rad]q3 [rad]The simulations results are shown in Fig. 3 and 4 when
the step reference q3ref = 1.0 [rad] and q4ref = 1.0 [rad] aregiven for the angles q3 and q4 , respectively. The solid

0.5
0
-0.5

proposedordinaryreference

050
time [sec]100

Figure 3 Responses of q3

0.5
0
-0.5

proposedordinaryreference

050
time [sec]100

Figure 4 Responses of q4

1

1
q4 [rad]

1.5

0.5

-0.5

experiment
simulationreference0

10
20
time [sec]30

Figure 5 Responses of q3

0.5
0
-0.5

experiment
simulationreference0

10
20
time [sec]30

Figure 6 Responses of q4

solid line and the dashed line show the responses of q3
and q4 by the simulation result and the experimental
result, respectively. As can be seen in Fig. 5 and 6, theresponses by proposed controller are stable, and follow
the references without error. The reliability of the proposed controller is veriﬁed by the experimental results
because the responses of the experiments are similar to
the responses of the simulations.

6
Table 1 Upper bound value
proposed controller γ = 22.95
ordinary controller γ = 27.97

1.5

0

π
5π
|q2 | ≤ , |q3 | ≤, |q̇2 | ≤ 1.0, |q̇3 | ≤ 1.5. (25)6
12
The range of given initial-values are chosen as follows:

Experiment

Experiments are carried out under the same situation
indicated in sec. 4. The experimental results by the
proposed controller are shown in Fig. 5 and 6. The

q3 [rad]Those trigonometric functions in (23) and (24) are approximated not the ﬁrst-order but a high-order Taylor
series expansion depending on qi . Furthermore, in thesame way as subsec. 2.3, the coeﬃcients are optimized
by using the method of least-squares to conduct moreaccurate approximation under the range (1) and (2),respectively. Let the proposed controller and the ordinary controller be the GS controllers designed by the
proposed approximation method (5) and (6), and theordinary approximation method (23) and (24). The
proposed controller is compared with the ordinary controller designed by the simulations. The ordinary controller is designed by choosing the scheduling parameters ρ := [q2 q3 ]T based on matrix sum of squares (SOS)
polynomials. Here, the following necessary conditions
for the simulations are given in the controller design.
Assume that the rotational speed of Rotor1 is constantvalue q̇1 = 40 [rad/sec]. The range of the angles q2 andq3 , and its derivatives are assumed as follows:

Conclusion

This paper presents the GS controller design basedon sum of squares (SOS) to solve nonlinear dynamics
of the VSCMG. The highly accurate linear parametervarying (LPV) model is derived by using the proposed
approximation. Design conditions are described by using PDLMIs to design the GS controller. The PDLMIs
are relaxed to the suﬃcient conditions based on matrix SOS polynomials. Finally, the eﬀectiveness of the
proposed GS controller is shown by simulations and experiments comparing with the ordinary controller.

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LPV Gain-Scheduled Control of a Control Moment
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Sum of Squares: Comparison with a SDRE Method.
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[3] H. Ichihara and M. Kawata: Attitude Control of Acrobot by Gain Scheduling Control Based on Sum
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