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The design of the spacecraft Attitude Control System (ACS) becomes more complex when the spacecraft has different type of components like, flexible solar panels, antennas, mechanical manipulators and tanks with fuel. The interaction between the fuel slosh motion, the panel’s flexible motion and the satellite rigid motion during translational and/or rotational manoeuvre can change the spacecraft center of mass position damaging the ACS pointing accuracy. This type of problem can be considered as a Fluid-Structure Interaction (FSI) where some movable or deformable structure interacts with an internal fluid. This paper develops a mathematical model for a rigid-flexible satellite with tank with fuel. The slosh dynamics is modelled using a common pendulum model and it is considered to be unactuated. The control inputs are defined by a transverse body fixed force and a moment about the centre of mass. A comparative investigation designing the satellite ACS by the Linear Quadratic Regulator (LQR) and Linear Quadratic Gaussian (LQG) methods is done. One has obtained a significant improvement in the satellite ACS performance and robustness of what has been done previously, since it controls the rigid-flexible satellite and the fuel slosh motion, simultaneously.

The problems due to fluid-structure interaction (FSI) are crucial in the design of many engineering systems. They appear in the dynamic behaviour of offshore and marine structures, road, and railroad containers partially filled with a fluid [

The phenomenon of sloshing is due to the movement of a free surface of a liquid that partially fills a compartment and this movement is oscillating. It depends on shape of the tank, the acceleration of gravity, and the axial/rotational acceleration of the tank. As representative of the behaviour of the total weight of the system it is accepted that when the mass of the liquid oscillates the mass center of the rigid body also oscillates, thereby disturbing the rigid-flexible part of the vehicle under consideration. As an oscillating movement it is natural to consider the wave generated by the movement of the liquid as a stationary wave. Each mode of oscillation has a special feature of this phenomenon under study, and one observes, in a quantitative sense, how much mass is displaced. Among all the modes that cause the greatest disruption in the system are the first and second modes. Despite the fact that oscillation has lower frequency it is capable of resulting in violent shifting of the center of mass of the liquid creating an oscillation in the system as a role. The other oscillation modes act less aggressive and may not even vary with the position of its center of mass due to the symmetry of the wave which on average causes no displacement.

Due to its complexity, the sloshing dynamics is usually represented by mechanical equivalents that describes and reproduces faithfully the actions and reactions due to forces and torques acting on the system. The main advantage of replacing the fluid model with an equivalent oscillating model is simplifying the analysis of motion in the rigid body dynamics, compared to the fluid dynamics equations. Due to the complexity of establishing an analytical model for the fluid moving freely within a closed tank, it is used as a simplified system, taking into account the following criteria [

Consider a rigid spacecraft moving in a fixed plane, with a spherical fuel tank and including the lowest frequency slosh mode. Based on the Lagrange equation and the Rayleigh dissipation function one can model systems using the mechanical mass-spring and pendulum type system, respectively. Figure

Satellite model with slosh dynamics pendulum analogous mechanical system.

The mass of the satellite and the moment of inertia, regardless of the fuel, are given by

The satellite equations of motion for the satellite with sloshing can be derived using the Lagrange equations given by

The position vector of the satellite mass center with respect to the inertial system is

The position of the mass of fuel is given by

The Lagrangian of the entire system is given by

Assuming the relations

All equations derived previously are nonlinear. However, in order to design LQR and LQG controllers one has to get the linear set of equations of motion, which is obtained assuming that the system makes small movements around the zero point of equilibrium [

To derive the equations of motion for the satellite model with sloshing and flexibility one considers the same rigid satellite with tank partially filled plus a flexible appendage connected to the satellite as shown in Figure

Satellite model with slosh dynamics and a flexible panel.

The flexible appendage has mass

The panel kinetic, potential energy and the dissipation function of energy

Now the Lagrangian considering the slosh and theappendices’ flexibility is given by

In order to obtain the equation of motion for the satellite with sloshing and flexible panel one uses (

The LQG method is the union of the LQR problem with the Kalman filter problem. The separation principle [

Assume a plant described by the linear state equations given by

Similarly the Kalman filter gain now is given by

The first simulation is the design comparison between the LQR and LQG control law, for the satellite model with sloshing dynamics given by (^{2}/s. The initial conditions used are

The angular and sloshing control by the LQR and LQG controller.

Figure

The LQR and LQG controller effort to control the angular and sloshing motion.

The second simulation is also the comparison between the LQR and LQG control law, but now the satellite model has the sloshing dynamics plus the flexible dynamics of panel, in which the data values are ^{2}/s^{2}, and ^{2}/s. The simulations initials conditions are

Figure

Control of the angular motion, sloshing, and flexibly.

Figure

The LQR and LQG performance controlling angular motion, sloshing, and flexibly.

In this paper one has developed a new model for the planar dynamics of a rigid-flexible satellite, incorporating the axial, transverse, and pitch dynamics of the satellite, plus fuel slosh dynamics, which is considered underactuated. That model is used to investigate the effects of the interaction between the rigid motion, the liquid motion (slosh), and the flexible satellite dynamics in order to predict what the damage to the controller performance and robustness is. The control inputs are defined by a transverse body fixed force and a moment about the centre of mass. The comparative simulations, designing the satellite ACS by the linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) methods, have showed a significant improvement in the satellite ACS performance and robustness of what has been done previously, since it simultaneously controls the rigid-flexible satellite motion and the fuel slosh dynamics. But, one observes that the performance of the LQR control is better than and LQG control. The degradation of the LQG control is because this controller spends more energy associated with the force and torque, suggesting that this is due to the estimation process of the sloshing and flexible variables by the Kalman filter. However, this is the realistic situation since both the slosh and the flexible states are not measured for feedback. On the other hand, the LQG controller is more robust than the LQR since it takes into account the noises that represent the imperfections of the models acting over the system. Finally, one observes that flexibility still has small fluctuation, which is not appropriated when one needs high precise pointing. One way to deal with that problem is to apply a nonlinear control technique considering the nonlinear satellite model.

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