Research Article Satellite Attitude Control System Design considering …downloads.hindawi.com/journals/sv/2014/260206.pdf · 2019-07-31 · Research Article Satellite Attitude Control

Research ArticleSatellite Attitude Control System Design consideringthe Fuel Slosh Dynamics

Luiz Carlos Gadelha de Souza and Alain G. de Souza

National Institute for Space Research (INPE/DMC), Avenida dos Astronautas 1758, 12227-010 Sao Jose dos Campos, SP, Brazil

Correspondence should be addressed to Luiz Carlos Gadelha de Souza; [email protected]

Received 26 June 2013; Accepted 8 January 2014; Published 26 June 2014

Academic Editor: Nuno Maia

Copyright © 2014 L. C. G. de Souza and A. G. de Souza. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The design of the satellite attitude control system (ACS) becomes more complex when the satellite structure has different typeof components like, flexible solar panels, antennas, mechanical manipulators, and tanks with fuel. A crucial interaction canoccur between the fuel slosh motion and the satellite rigid motion during translational and/or rotational manoeuvre since theseinteractions can change the satellite centre of mass position damaging the ACS pointing accuracy. Although, a well-designedcontroller can suppress such disturbances quickly, the controller error pointing may be limited by the minimum time necessary tosuppress such disturbances thus affecting the satellite attitude acquisition. As a result, the design of the satellite controller needsto explore the limits between the conflicting requirements of performance and robustness. This paper investigates the effects ofthe interaction between the liquid motion (slosh) and the satellite dynamics in order to predict what the damage to the controllerperformance and robustness is. The fuel slosh dynamics is modelled by a pendulum which parameters are identified using theKalman filter technique. This information is used to design the satellite controller by the linear quadratic regulator (LQR) andlinear quadratic Gaussian (LQG) methods to perform a planar manoeuvre assuming thrusters are actuators.

1. Introduction

The problem of interaction between fluid and structure isimportant when one needs to study the dynamic behaviorof offshore and marine structures and road and railroadcontainers partially filled with a fluid [1, 2]. In space missionsthe sloshing problem appears when the spacecraft is spinningand there is liquid inside it. An example is the damp deviceinvolving fluid as the damping material, whose motion caninteract with flexible panels and/or manipulators [3]. Aninteresting approach to analyze a rigid container mountedon flexible springs interacting with a perfect fluid includingsloshing effects has been done by Lui and Lou [4].The successof a space mission can depend on taking into account theknowledge of the interaction between fluid motion (slosh)and structure dynamics since this interaction can damage theACS pointing requirements. A space structure, like rockets,geosynchronous satellites, and the space station, usuallycontains liquid in tanks that can represent more than 40%of the initial mass of the system. As a result, the first step to

design its ACS is to obtain a detailed dynamics model of thespace structure. When the fuel tanks are only partially filledand suffer a transverse acceleration and/or rotational motion,large quantities of fuel move uncontrollably inside the tanksand generate the sloshing effects. Agrawal [5] has shown thatthe dynamics interaction between the fuel motion and therigid and/or flexible body dynamics can result in some kindof control instability. For minimizing these effects the ACSmust be designed using a robust control method in order toassure stability and good performance to achieve the attitudecontrol system requirement [6].When a rigid-flexible satellitewith fuel tanks inside is subjected to large angle manoeuvreits dynamics is only captured by complex nonlinear mathe-matical model. Besides, the remaining flexible and/or liquidvibration can introduce a tracking error resulting in a mini-mum attitude acquisition time. Souza [7] has done a detailedinvestigation of the influence of the nonlinearities introducedby the panel’s flexibility into the ACS design. It was shownthat system parameters variation can degrade the controlsystem performance, indicating the necessity to improve

Hindawi Publishing CorporationShock and VibrationVolume 2014, Article ID 260206, 8 pageshttp://dx.doi.org/10.1155/2014/260206

2 Shock and Vibration

the ACS robustness. An experimental controller robustnessand performance investigation was done by Conti and Souza[8], where the estimation of the platform inertia parameterswas introduced as part of the platform ACS design. Theproblem of designing satellite nonlinear controller for rigidsatellite has been done by Souza and Gonzales [9] using thestate-dependent Riccati equation (SDRE) method which isable to deal with high nonlinear plants. Due to the complexityof modeling the fluid and/or flexible dynamic of the system itis common to usemechanical systems analogies that describethis dynamic. Besides, if one needs to know some physicalparameters related with the slosh or the flexibility dynamicsit is common to obtain them by experimental apparatus orsome kind of estimating method such as Kalman filter [10].

2. Satellite with Sloshing Model

Thephenomenon of sloshing is due to themovement of a freesurface of a liquid that partially fills a compartment and thismovement is oscillating. It depends on the shape of the tank,the acceleration of the gravity, and on the axial/rotationalacceleration of the tank. As a representative of the behaviorof the total weight of the system it is accepted that whenthe mass of the liquid oscillates the mass center of therigid body also oscillates, thereby disturbing the rigid-flexiblepart of the vehicle under consideration. As an oscillatingmovement it is natural to consider the wave generated bythe movement of the liquid as a stationary wave in alloscillation modes. Each mode of oscillation has a specialfeature of this phenomenon under study and one observes,in a quantitative sense, how much mass is displaced. Amongall the modes that cause the greatest disruption in the systemare the first and the second modes. Despite the fact thatoscillation has lower frequency it is capable of resulting inviolent shifting of the center of mass of the liquid creatingan oscillation in the system as a role. The other oscillationmodes act as a less aggressive and it may not even vary theposition of its center of mass due to the symmetry of thewave which on average causes no displacement. Due to itscomplexity, the sloshing dynamics is usually represented bymechanical equivalents that describe and reproduce faithfullythe actions and reactions due to forces and torques actingon the system. The main advantage of replacing the fluidmodel with an equivalent oscillatingmodel [11] is simplifyingthe analysis of motion in the rigid body dynamics comparedto the fluid dynamics equations. Due to the complexity ofestablishing an analytical model for the fluid moving freelywithin a closed tank, a simplified system is used, taking intoaccount the following criteria [6]: (a) small displacements,(b) a rigid tank, and (c) no viscous, incompressible, andhomogeneous liquid. Under these conditions the dynamicsof the sloshing can be approximated by mechanical systemconsisting of a mass-spring or pendulum. Consider a rigidspacecraft moving in a fixed plane with a spherical fuel tankand including the lowest frequency slosh mode. Based on theLagrange equation and the Rayleigh dissipation function onecan model systems using the mechanical mass-spring andpendulum type system, respectively. Figure 1 shows a satellite

model where slosh dynamics is represented by its pendulumanalogous mechanical system, where the mass of the satelliteand the moment of inertia, regardless of the fuel, are givenby 𝑚 and 𝐼, respectively, and the mass equivalent of fuel andits inertia moment are given by𝑀𝑓 and 𝐼𝑓, respectively. Theattitude control of the spacecraft is done by the force 𝑓 andby the pitch moment𝑀. The constant thrust 𝐹 is responsiblefor the orbital transfer of the spacecraft with respect to theinertial reference system (𝑥, 𝑦, 𝑧) and it acts on the center ofmass of spacecraft in the longitudinal axis. Also it is given thevelocity of the center of the fuel tank 𝜐𝑥, 𝜐𝑧 and the attitudeangle 𝜃 of the spacecraft with respect to a fixed reference(𝑋, 𝑌, 𝑍). Besides, 𝑉 represents the linear velocity and 𝜔

represents the angular velocity of the rigid body.The length ofthe pendulum is 𝑎, the distance from satellite center of massto the pendulum connected point is 𝑏, and the angle of thependulum with respect to the spacecraft longitudinal axis is𝜓, which is assumed in the equilibrium position (𝜓 = 0)

about the reference axis𝑋.

3. The Satellite Equations of Motion

The satellite equations of motion can be derived using theLagrange equations [10] given by

ddt

(

𝜕𝐿

𝜕𝑉

) + 𝜔× 𝜕𝐿

𝜕𝑉

= 𝜏𝑡,

ddt

(

𝜕𝐿

𝜕𝜔

) + 𝜔× 𝜕𝐿

𝜕𝜔

+ 𝑉× 𝜕𝐿

𝜕𝑉

= 𝜏𝑟,

ddt

(

𝜕𝐿

𝜕��

) −

𝜕𝐿

𝜕𝜓

+

𝜕𝑅

𝜕��

= 0.

(1)

Details of the equations of motion derivation can be found in[12], where 𝐿 is the Lagrangian of the system, the generalizedcoordinates are 𝑉 and 𝜔, 𝑅 is the Rayleigh dissipationfunction, 𝜏𝑟 is the internal torque, and 𝜏𝑡 is the externaltorque. Assume that 𝑅, 𝜏𝑟, 𝜏𝑡, 𝜔, 𝑉 are given by

𝑅 =

1

2

𝜀��2; 𝑉 =

[

[

V𝑥0

V𝑧

]

]

; 𝜔 =[

[

0

𝜃

0

]

]

;

𝜏𝑡 =[

[

𝐹

0

𝑓

]

]

; 𝜏𝑟 =[

[

0

𝑀 + 𝑓𝑏

0

]

]

.

(2)

The position vector of the satellite mass center with respect tothe inertial reference system (𝑥, 𝑦, 𝑧) (see Figure 1) is

𝑟 = (𝑥 − 𝑏) 𝑖 + 𝑧𝑘. (3)

Assuming the relations V𝑥 = ��+𝑧

𝜃 and V𝑧 = ��−𝑥

𝜃 the satellitevelocity is given by

𝑟 = V𝑥 �� + (V𝑧 + 𝑏

𝜃)

𝑘. (4)

The position of the mass of fuel with respect to the inertialreference system (𝑥, 𝑦, 𝑧) (see Figure 1) is given by

𝑟𝑓 = (𝑥 − 𝑎 cos (𝜓)) �� + (𝑧 + 𝑎 sin (𝜓)) 𝑘. (5)

Shock and Vibration 3

b

�x

�z

X

x

𝜃

𝜓𝛼

Z

z →rf

→r

M

F

f

Figure 1: Satellite model with slosh dynamics pendulum analogous mechanical system.

As a result, the mass of the fuel velocity is

𝑟𝑓 = (V𝑥 + 𝑎 sin (𝜓) (

𝜃 + ��)) ��

+ (V𝑧 + 𝑎 cos (𝜓) ( 𝜃 + ��))

𝑘.

(6)

The Lagrangian of the entire system is given by

𝐿 =

1

2

𝑚𝑟

2+

1

2

𝑚𝑓𝑟

2

𝑓 +1

2

𝐼𝑓(

𝜃 + ��)

2+

1

2

𝐼

𝜃2. (7)

Substituting (4) and (6) into (7), using the relations given by(2), and performing the derivations of (1), one obtains thesatellite equations of motion given by

(𝑚 + 𝑚𝑓) (V𝑥 + V𝑧 𝜃) + 𝑚𝑏

𝜃 + 𝑚𝑓𝑎 (�� +

𝜃) sin (𝜓)

+ 𝑚𝑓𝑎(

𝜃 + ��)

2cos (𝜓) = 𝐹,

(8)

(𝑚 + 𝑚𝑓) (V𝑧 − V𝑥 𝜃) + 𝑚𝑓𝑎 (

𝜃 + ��) cos (𝜓)

− 𝑚𝑓𝑎(

𝜃 + ��)

2sin (𝜓)𝑚𝑏

𝜃 = 𝑓,

(9)

(𝐼𝑓 + 𝑚𝑏2)

𝜃 + 𝑚𝑏 (V𝑧 − V𝑥

𝜃) − 𝜀�� = 𝑀 + 𝑏𝑓, (10)

(𝑚𝑓𝑎2+ 𝐼𝑓) (

𝜃 + ��)

+ 𝑚𝑓𝑎 ((V𝑥 + V𝑧 𝜃) sin (𝜓)

+ (V𝑧 − V𝑥 𝜃) cos (𝜓)) + 𝜀�� = 0.

(11)

Assuming the relations 𝑎𝑥 = V𝑥 + V𝑧 𝜃, 𝑎𝑧 = V𝑧 − V𝑥

𝜃 andsubstituting them into (8) and (9), one can isolate and obtainthe satellite accelerations given by

𝑎𝑥

=

𝐹 − 𝑚𝑏

𝜃 − 𝑚𝑓𝑎 (�� +

𝜃) sin (𝜓) − 𝑚𝑓𝑎(

𝜃 + ��)

2cos (𝜓)

𝑚 + 𝑚𝑓

,

𝑎𝑧

=

𝑓 − 𝑚𝑓𝑎 (

𝜃 + ��) cos (𝜓) + 𝑚𝑓𝑎(

𝜃 + ��)

2sin (𝜓) − 𝑚𝑏

𝜃

𝑚 + 𝑚𝑓

.

(12)

All equations derived previously are nonlinear. However, inorder to design LQR and LQG controllers and estimate thesloshing parameters using the Kaman filter technique one hasto get the linear set of equations of motion, which is obtainedassuming that the systemmakes smallmovements around thepoint of equilibrium very close to zero; that is, (𝜃,

𝜃, 𝜓, ��) =

(0, 0, 0, 0).Now, substituting (12) into (10) and (11) and assuming

the linearization conditions, one has the satellite equation ofmotion given by

𝜃 (𝐼𝑓 + 𝑚

∗(𝑎2− 𝑏𝑎)) + 𝜓 (𝐼𝑓 + 𝑚

∗𝑎2)

+ 𝑎𝑚∗𝑓𝐹𝜓 + 𝜀�� = −𝑎𝑚

∗𝑓𝑓,

𝜃 (𝐼 + 𝑚

∗(𝑏2− 𝑏𝑎)) − 𝑚

∗𝑎𝑏𝜓 − 𝜀�� = 𝑀 + 𝑏

∗𝑓,

(13)

where 𝑏∗ = 𝑏𝑚𝑓/(𝑚 +𝑚𝑓), 𝑚∗= 𝑚𝑚𝑓/(𝑚 +𝑚𝑓), and𝑚

∗𝑓 =

𝑚𝑓/(𝑚 + 𝑚𝑓).


4. Kalman Filter (KF)

The Kalman filter estimates the instantaneous state of adynamic system from the reading of measurements. In otherwords the Kalman filter is the optimal solution of minimumvariance, whichmeans that the equations of the Kalman filtercan be deduced from this premise [7]. The Kalman filteringcan be divided into two steps: time update and measurementupdate.

4.1. Time Update. This step propagates the states and thecovariance of the time 𝑡𝑘−1 to 𝑡𝑘. For this it just integratesthe following equations, with the boundary conditions x𝑘−1 =_x 𝑘−1 and P𝑘−1 = P𝑘−1. Herein P is the covariance matrix andx is the vector state:

x = 𝑓 (x) , (14)

where 𝑓 is a nonlinear vector function of the state x and time𝑓. Consider

P = FP + PF𝑡 + GQG𝑡, (15)

whereG is a matrix that adds noise to the system dynamic,Qis the process noise covariance, and F is the Jacobian matrixof 𝑓.

4.2. Measurement Update. This step updates the state andcovariance to the time 𝑡 due to the measurement y. Considerthe following:

K𝑘 = P𝑘H𝑡𝑘(H𝑘P𝑘H

𝑡𝑘 + R𝑘)

−1,

P𝑘 = (I − K𝑘H𝑘)P𝑘,

x𝑘 = x𝑘 + K𝑘 [y𝑘 − ℎ𝑘 (x𝑘)] ,

(16)

where 𝐾𝑘 is the filter gain, 𝐻𝑘 is the Jacobian matrix of ℎwhich in turn is a vectorial nonlinear function of the state,and y is the measurement vector.

5. Linear Quadratic Gaussian (LQG)

This method is, basically, the union with the LQR [13] andthe Kalman filter. In the LQR method one assumes that allstates are available to be feedback but in reality that is nottrue. Therefore, when there is any state that is not availableone has to use the Kalman filter to estimate it in order to bea feedback.The separation principle ensures that each step ofthis process can be made independently of each other; onemay first solve the LQR problem and then design the optimalestimator (Kalman filter), or vice versa, so that the globalsolution is always the same.

Assume a plant described by the linear state equationsgiven by

x (𝑡) = Ax (𝑡) + Bu (𝑡) + Γ𝑤,

y = Cx (𝑡) + 𝜐,

(17)

where x represents the state vector, the matrix A is the statematrix, B is the input matrix, y is the output vector, C is theoutput matrix, 𝜐 and 𝑤 are white noise, and u is the controllaw. In the LQGmethod [14] the control law gain is obtainedby the LQR method and it is given by

K𝑐 = R−1B𝑇P𝑐, (18)

where R is real symmetric positive definite matrix and P𝑐 isthe symmetrical solution of the LQR Riccati equation givenby

A𝑇P𝑐 + P𝑐A − P𝑐BR−1B𝑇P𝑐 +M𝑇QM = 0. (19)

Similarly the Kalman filter gain now is given by

K𝑓 = P𝑓C𝑇V−1, (20)

where V is real symmetric positive definite matrix and P𝑓 isthe symmetrical solution matrix of the KF Riccati equationgiven by

P𝑓A𝑇+ AP𝑓 − P𝑓C

𝑇V−1CP𝑓 + Γ𝑇WΓ = 0, (21)

where P𝑐 = P𝑇𝑐 ≥ 0 and P𝑓 = P𝑇𝑓 ≥ 0 and Q,R,V,andW are weight matrices which can be regarded as settingparameters (tuning) that must be manipulated until they findone acceptable response to the system.

A necessary and sufficient condition to guarantee theexistence of the K𝑐 and K𝑓 is if the system is completelycontrollable and observable. The LQG method is morerealistic than the LQRmethod, since the states are not alwaysavailable to be feedback or need to be measurable with theaid of sensors. The inclusion of the noise in the model whichrepresents imperfections of the system is also one advantageof the LQG method.

6. Simulations and Results

The first simulation is the design of the control law usingthe LQR and LQG control theories, for the spacecraft witha partially filled tank, to account for the sloshing dynamicsby the mechanically analog pendulum type. The pendulumphysical parameters used in the simulations are as follows:𝑚 = 600 kg, 𝑚𝑓 = 100 kg, 𝐼 = 720 kg/m2, 𝐼𝑓 = 90 kg/m2,𝑎 = 0.3m, 𝑏 = 0.3m, 𝐹 = 500N, and 𝜀 = 0.19 kgm2/s.The equations of motion that describe the dynamics of thesystem are given by (13) which need to be put in state spaceform. These equations describe the angular displacement ofthe spacecraft and the angular displacement of the pendulumand the initial conditions used are 𝜃 = 2

∘, 𝜃 = 0.57

∘/s,𝜓 = 1

∘,and �� = 0

∘.Figures 2 and 3 show that the LQR control law perfor-

mance is better than the LQG and the reason is because thefirst one considers that the sloshing variables are available tobe feedback which is not true.

The performance of the LQG control law is damagedbecause the sloshing motion is controlled indirectly and


−4

−2

4

2

0

0 50 100

𝜃(d

eg)

t (s)

(a)

−4

−2

2

0

0 50 100

𝜃(d

eg/s

)

t (s)

(b)

−10

−5

15

10

5

0

0 50 100

𝜓(d

eg)

t (s)

(c)

−10

−15

−5

10

5

0

0 50 100

𝜓

(deg

/s)

t (s)

(d)

Figure 2: Performance of the LQR control law.

−1

2

1

0

3

0 20 40 60 80 100 120

LQG

𝜃(d

eg)

t (s)

(a)

−1

−0.5

0.5

1

0

0 60 10020 40 80 120

LQG

𝜃(d

eg/s

)

t (s)

(b)

−10

−5

0

5

10

0 20 40 60 80 100 120

LQG

𝜓(d

eg)

t (s)

(c)

−10

−5

0

5

10

0 20 40 60 80 100 120

LQG

𝜓

(deg

/s)

t (s)

(d)

Figure 3: Performance of the LQG control law.


00

0.5

1

1.5

20 40 60 80 100

a + 𝜎

a − 𝜎a

a(m

)

t (s)

(a)

0.02

0.080.060.04

0.1

0 20 40 60 80 100

𝜎

t (s)

(b)

Figure 4: The estimation of the rod length.

−1

−5−6 −4 −3 −2 −1−1.5

−0.5

0

0

0.5

1

1.5

Imag

inar

y ax

is

Real axis

Pole-zero map

×10−3

Figure 5: Poles position during the estimation.

the sloshing state variables are not available to be feedbackand they need to be estimated by the Kalman filter.

Now one uses the Kalman filter to estimate the rodlength of the pendulum using a database containing thetime evolution of the variables 𝜃 and

𝜃. The rod length 𝑎 isnow considered as state variable so one has a system withfive state variables which can be estimated. Using the samecontrol LQR law, it is possible to analyze the control systemperformance during the estimation of the rod length. Besides,one inserts a white noise in the measurements 𝜃 and

𝜃, givenby the noise process ]𝑘 = 𝑁(0, 0.001).

Figure 4 shows that the estimated value of the rod lengthis 𝑎 = 0.33m, remembering that the “actual” value of the rodis 0.3m. Figure 5 shows that plant of the system depends onthe estimation of the rod, since at the time that the rod lengthis varying the position of the poles also moves.

Figures 6 and 7 show the LQR and the LQG controllersduring the rod length estimation. One observes that thesystem is controlled in less than approximately 15 secondsand that the slosh motion is more oscillatory with the LQGcontroller. This is because the sloshing motion is controlledindirectly and the sloshing state variables are being estimatedby the Kalman filter.

−1

2

1

0

0 5 10 15

3

LQR

𝜃(d

eg)

t (s)

(a)

−1

−0.5

0.5

1

0

0 5 10 15

LQR

𝜃(d

eg/s

)

t (s)

(b)

−0.5

0.5

1

0

0 5 10 15

LQR

𝜓(d

eg)

t (s)

(c)

−0.2

−0.4

−0.6

0.2

0

0 5 10 15

t(s)

LQR

𝜓

(deg

/s)

(d)

Figure 6: LQR controller during the rod estimation.


−1

2

1

0

3

0 10 20

LQG

𝜃(d

eg)

t (s)

(a)

−1

−0.5

0.5

1

0

0 10 20

LQG

𝜃(d

eg/s

)

t (s)

(b)

−1

−1.5

−0.5

0.5

1

0

0 10 20

LQG

𝜓(d

eg)

t (s)

(c)

−1

−0.5

0.5

1

0

0 2010

LQG

𝜓

(deg

/s)

t (s)

(d)

Figure 7: LQG controller during the rod estimation.

7. Conclusions

In this paper one described briefly the concepts of thesloshing phenomenon which is associated with the dynamicsof a liquid moving into at partially fills reservoir. Derivingthe equation of motion of a spacecraft with liquid insidethe sloshing phenomenon is represented by its mechanicalanalog of a pendulum type. One uses the Kalman filtertechnique to estimate the pendulum length. One showsthat the performance of the LQR control is better than theLQG control during the estimation process. The reason theLQG control degraded is because the sloshing states needto be estimated by the filter. This degradation was expected,because as it has been demonstrated by [15] the introductionof the Kalman filter degrades the stability margin of the LQG.One way to improve the performance of LQG is to makethe dynamics of the filter faster than the dynamics of LQR.However, this must be done carefully since this time willdepend on the on-board computer capacity. These resultsshow that control of spacecraft with sloshing and flexibilityis not an easy task and needs to be investigated better. Onehas also observed that the estimation of the rod length altersthe plant of the system causing the poles to walk to andfrom the imaginary system, leaving in the end the plant morestable.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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Research Article Satellite Attitude Control System Design considering …downloads.hindawi.com/journals/sv/2014/260206.pdf · 2019-07-31 · Research Article Satellite Attitude Control