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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 848741, 18 pages
http://dx.doi.org/10.1155/2012/848741
Research Article

Thrust Vector Control of an Upper-Stage Rocket with Multiple Propellant Slosh Modes

Department of Physical Sciences, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA

Received 24 May 2012; Revised 4 July 2012; Accepted 4 July 2012

Academic Editor: J. Rodellar

Copyright © 2012 Jaime Rubio Hervas and Mahmut Reyhanoglu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The thrust vector control problem for an upper-stage rocket with propellant slosh dynamics is considered. The control inputs are defined by the gimbal deflection angle of a main engine and a pitching moment about the center of mass of the spacecraft. The rocket acceleration due to the main engine thrust is assumed to be large enough so that surface tension forces do not significantly affect the propellant motion during main engine burns. A multi-mass-spring model of the sloshing fuel is introduced to represent the prominent sloshing modes. A nonlinear feedback controller is designed to control the translational velocity vector and the attitude of the spacecraft, while suppressing the sloshing modes. The effectiveness of the controller is illustrated through a simulation example.

1. Introduction

In fluid mechanics, liquid slosh refers to the movement of liquid inside an accelerating tank or container. Important examples include propellant slosh in spacecraft tanks and rockets (especially upper stages), cargo slosh in ships and trucks transporting liquids, and liquid slosh in robotically controlled moving containers.

A variety of passive methods have been employed to mitigate the adverse effect of sloshing, such as introducing baffles or partitions inside the tanks [1, 2]. These techniques do not completely succeed in canceling the sloshing effects. Thus, active control methods have been proposed for the suppression of sloshing effectively.

The control approaches developed for robotic systems moving liquid filled containers [311] and for accelerating space vehicles are mostly based on linear control design methods [12, 13] and adaptive control methods [14]. The linear control laws for the suppression of the slosh dynamics inevitably lead to excitation of the transverse vehicle motion through coupling effects. The complete nonlinear dynamics formulation in this paper allows simultaneous control of the transverse, pitch, and slosh dynamics.

Some of the control design methods use command input shaping methods that do not require sensor measurements of the sloshing liquid [3, 811], while others require either sensor measurements [4, 5] or observer estimates of the slosh states [15]. In most of these approaches, only the first sloshing mode represented by a single pendulum model or a single mass-spring model has been considered and higher slosh modes have been ignored. The literature that considers more than one sloshing mode in modeling the slosh dynamics includes [2, 16].

In this paper, a mechanical-analogy model is developed to characterize the propellant sloshing during a typical thrust vector control maneuver. The spacecraft acceleration due to the main engine thrust is assumed to be large enough so that surface tension forces do not significantly affect the propellant motion during main engine burns. This situation corresponds to a “high-𝑔’’ (here 𝑔 refers to spacecraft acceleration) regime that can be characterized by using the Bond number Bo—the ratio of acceleration-related forces to the liquid propellants surface tension forces, which is given by Bo=𝜌𝑎𝑅2𝜎,(1.1) where 𝜌 and 𝜎 denote the liquid propellants density and surface tension, respectively, 𝑎 is the spacecraft acceleration, and 𝑅 is a characteristic dimension (e.g., propellant tank radius). During the steady-state high-𝑔 situation, the propellant settles at the “bottom’’ of the tank with a flat free surface. When the main engine operation for thrust vector control introduces lateral accelerations, the propellant begins sloshing. As discussed in [17], Bond numbers as low as 100 would indicate that low-gravity effects may be of some significance. A detailed discussion of low-gravity fluid mechanics is given in [2].

The previous work in [16] considered a spacecraft with multiple fuel slosh modes assuming constant physical parameters. In this paper, these results are extended to account for the time-varying nature of the slosh parameters, which renders stability analysis more difficult. The control inputs are defined by the gimbal deflection angle of a nonthrottable thrust engine and a pitching moment about the center of mass of the spacecraft. The control objective, as is typical in a thrust vector control design for a liquid upper stage spacecraft during orbital maneuvers, is to control the translational velocity vector and the attitude of the spacecraft, while attenuating the sloshing modes characterizing the internal dynamics. The results are applied to the AVUM upper stage—the fourth stage of the European launcher Vega [18]. The main contributions in this paper are (i) the development of a full nonlinear mathematical model with time-varying slosh parameters and (ii) the design of a nonlinear time-varying feedback controller. A simulation example is included to illustrate the effectiveness of the controller.

2. Mathematical Model

This section formulates the dynamics of a spacecraft with a single propellant tank including the prominent fuel slosh modes. The spacecraft is represented as a rigid body (base body) and the sloshing fuel masses as internal bodies. In this paper, a Newtonian formulation is employed to express the equations of motion in terms of the spacecraft translational velocity vector, the angular velocity, and the internal (shape) coordinates representing the slosh modes. A multi-mass-spring model is derived for the sloshing fuel, where the oscillation frequencies of the mass-spring elements represent the prominent sloshing modes [19].

Consider a rigid spacecraft moving on a plane as indicated in Figure 1, where 𝑣𝑥,𝑣𝑧 are the axial and transverse components, respectively, of the velocity of the center of the fuel tank, and 𝜃 denotes the attitude angle of the spacecraft with respect to a fixed reference. The fluid is modeled by moment of inertia 𝐼0 assigned to a rigidly attached mass 𝑚0 and point masses 𝑚𝑖,𝑖=1,,𝑁, whose relative positions along the spacecraft fixed 𝑧-axis are denoted by 𝑠𝑖. Moments of inertia 𝐼𝑖 of these masses are usually negligible. The locations 0 and 𝑖 are referenced to the center of the tank. A restoring force 𝑘𝑖𝑠𝑖 acts on the mass 𝑚𝑖 whenever the mass is displaced from its neutral position 𝑠𝑖=0. A thrust 𝐹 is produced by a gimballed thrust engine as shown in Figure 1, where 𝛿 denotes the gimbal deflection angle, which is considered as one of the control inputs. A pitching moment 𝑀 is also available for control purposes. The constants in the problem are the spacecraft mass 𝑚 and moment of inertia 𝐼, the distance 𝑏 between the body 𝑧-axis and the spacecraft center of mass location along the longitudinal axis, and the distance 𝑑 from the gimbal pivot to the spacecraft center of mass. If the tank center is in front of the spacecraft center of mass, then 𝑏>0. The parameters 𝑚0,𝑚𝑖,0,𝑖,𝑘𝑖, and 𝐼0 depend on the shape of the fuel tank, the characteristics of the fuel and the fill ratio of the fuel tank. Note that these parameters are time-varying, which renders the Lyapunov-based stability analysis more difficult.

848741.fig.001
Figure 1: A multiple-slosh mass-spring model for a spacecraft.

To preserve the static properties of the liquid, the sum of all the masses must be the same as the fuel mass 𝑚𝑓, and the center of mass of the model must be at the same elevation as that of the fuel, that is, 𝑚0+𝑁𝑖=1𝑚𝑖=𝑚𝑓,𝑚00+𝑁𝑖=1𝑚𝑖𝑖=0.(2.1) Assuming a constant fuel burn rate, we have 𝑚𝑓=𝑚ini𝑡1𝑡𝑓,(2.2) where 𝑚ini is the initial fuel mass in the tank and 𝑡𝑓 is the time at which, at a constant rate, all the fuel is burned.

To compute the slosh parameters, a simple equivalent cylindrical tank is considered together with the model described in [2], which can be summarized as follows. Assuming a constant propellant density, the height of still liquid inside the cylindrical tank is =4𝑚𝑓𝜋𝜑2𝜌,(2.3) where 𝜑 and 𝜌 denote the diameter of the tank and the propellant density, respectively. As shown in [2], every slosh mode is defined by the parameters 𝑚𝑖=𝑚𝑓𝜑tanh2𝜉𝑖/𝜑𝜉𝑖𝜉2𝑖1,(2.4)𝑖=2𝜑2𝜉𝑖𝜉tanh𝑖𝜑1cosh2𝜉𝑖/𝜑sinh2𝜉𝑖𝑘/𝜑,(2.5)𝑖=𝑚𝑖𝑔𝜑2𝜉𝑖tanh2𝜉𝑖𝜑,(2.6) where 𝜉𝑖, for all 𝑖, are constant parameters given by 𝜉1=1.841,𝜉2=5.329,𝜉𝑖𝜉𝑖1+𝜋,(2.7) and 𝑔 is the axial acceleration of the spacecraft. For the rigidly attached mass, 𝑚0 and 0 are obtained from (2.1)–(2.5). Assuming that the liquid depth ratio for the cylindrical tank (i.e., /𝜑) is less than two, the following relations apply: 𝐼0=10.85𝜑𝑚𝑓3𝜑2+16212𝑚020𝑁𝑖=1𝑚𝑖2𝑖,if𝜑𝐼<1,0=0.35𝜑𝑚0.2𝑓3𝜑2+16212𝑚020𝑁𝑖=1𝑚𝑖2𝑖,if1𝜑<2.(2.8)

Let ̂𝑖 and ̂𝑘 be the unit vectors along the spacecraft-fixed longitudinal and transverse axes, respectively, and denote by (𝑥,𝑧) the inertial position of the center of the fuel tank. The position vector of the center of mass of the vehicle can then be expressed in the spacecraft-fixed coordinate frame as 𝑟̂̂=(𝑥𝑏)𝑖+𝑧𝑘.(2.9) The inertial velocity and acceleration of the vehicle can be computed as ̇𝑟=𝑣𝑥̂𝑣𝑖+𝑧̇𝜃̂̈+𝑏𝑘,𝑟=𝑎𝑥̇𝜃+𝑏2̂𝑎𝑖+𝑧̈𝜃̂+𝑏𝑘,(2.10) where we have used the fact that (𝑣𝑥,𝑣𝑧̇̇)=(̇𝑥+𝑧𝜃,̇𝑧𝑥𝜃) and (𝑎𝑥,𝑎𝑧̇𝑣)=(𝑥+𝑣𝑧̇̇𝑣𝜃,𝑧𝑣𝑥̇𝜃).

Similarly, the position vectors of the fuel masses 𝑚0,𝑚𝑖, for all 𝑖, in the spacecraft-fixed coordinate frame are given, respectively, by 𝑟0=𝑥+0̂̂𝑖+𝑧𝑘,𝑟𝑖=𝑥+𝑖̂𝑖+𝑧+𝑠𝑖̂𝑘,𝑖.(2.11)

The inertial accelerations of the fuel masses can be computed as ̈𝑟0=𝑎𝑥0̇𝜃2+̈0̂𝑎𝑖+𝑧̇20̇𝜃0̈𝜃̂̈𝑘,𝑟𝑖=𝑎𝑥+𝑠𝑖̈𝜃𝑖̇𝜃2+̈𝑖+2̇𝑠𝑖̇𝜃̂𝑎𝑖+𝑧+̈𝑠𝑖𝑖̈𝜃𝑠𝑖̇𝜃2̇2𝑖̇𝜃̂𝑘,𝑖.(2.12)

Now Newton’s second law for the whole system can be written as 𝐹̈=𝑚𝑟+𝑁𝑖=0𝑚𝑖̈𝑟𝑖,(2.13) where 𝐹̂̂.=𝐹𝑖cos𝛿+𝑘sin𝛿(2.14)

The total torque with respect to the tank center can be expressed as 𝜏=𝐼+𝐼0+𝑁𝑖=1𝐼𝑖̈𝜃̂𝑗+̈𝜌×𝑚𝑟+𝑁𝑖=0𝜌𝑖̈×𝑚𝑟𝑖,(2.15) where ̂[]̂𝜏=𝜏𝑗=𝑀+𝐹(𝑏+𝑑)sin𝛿𝑗,(2.16) and 𝜌,𝜌0, and 𝜌𝑖 are the positions of 𝑚,𝑚0, and 𝑚𝑖 relative to the tank center, respectively, that is, ̂𝜌=𝑏𝑖,𝜌0=0̂𝑖,𝜌𝑖=𝑖̂𝑖+𝑠𝑖̂𝑘,𝑖.(2.17)

The dissipative effects due to fuel slosh are included via damping constants 𝑐𝑖. When the damping is small, it can be represented accurately by equivalent linear viscous damping. Newton’s second law for the fuel mass 𝑚𝑖 can be written as 𝑚𝑖𝑎𝑧𝑖=𝑐𝑖̇𝑠𝑖𝑘𝑖𝑠𝑖,(2.18) where 𝑎𝑧𝑖=̈𝑠𝑖+𝑎𝑧𝑖̈𝜃𝑠𝑖̇𝜃2̇2𝑖̇𝜃.(2.19)

Using (2.13)–(2.18), the equations of motion can be obtained as 𝑚+𝑚𝑓𝑎𝑥̇𝜃+𝑚𝑏2+𝑁𝑖=1𝑚𝑖𝑠𝑖̈𝜃+2̇𝑠𝑖̇̈𝜃+𝑖+𝑚0̈0=𝐹cos𝛿,(2.20)𝑚+𝑚𝑓𝑎𝑧̈+𝑚𝑏𝜃+𝑁𝑖=1𝑚𝑖̈𝑠𝑖𝑠𝑖̇𝜃2̇2𝑖̇𝜃2𝑚0̇0̇𝐼̈𝜃=𝐹sin𝛿,(2.21)𝜃+𝑁𝑖=1𝑚𝑖𝑠𝑖𝑎𝑥𝑖̈𝑠𝑖𝑠+2𝑖̇𝑠𝑖+𝑖̇𝑖̇𝜃+𝑠𝑖̈𝑖+2𝑚00̇0̇𝜃0+𝑚𝑏𝑎𝑧𝑚=𝜏,(2.22)𝑖̈𝑠𝑖+𝑎𝑧𝑖̈𝜃𝑠𝑖̇𝜃2̇2𝑖̇𝜃+𝑘𝑖𝑠𝑖+𝑐𝑖̇𝑠𝑖=0,𝑖,(2.23) where 𝑝=𝑏+𝑑 and 𝐼=𝐼+𝐼0+𝑚𝑏2+𝑚020+𝑁𝑖=1𝐼𝑖+𝑚𝑖2𝑖+𝑠2𝑖.(2.24)

The control objective is to design feedback controllers so that the controlled spacecraft accomplishes a given planar maneuver, that is a change in the translational velocity vector and the attitude of the spacecraft, while suppressing the fuel slosh modes. Equations (2.20)–(2.23) model interesting examples of underactuated mechanical systems. The published literature on the dynamics and control of such systems includes the development of theoretical controllability and stabilizability results for a large class of systems using tools from nonlinear control theory and the development of effective nonlinear control design methodologies [20] that are applied to several practical examples, including underactuated space vehicles [21, 22] and underactuated manipulators [23].

3. Nonlinear Feedback Controller

This section presents a detailed development of feedback control laws through the model obtained via the multi-mass-spring analogy.

Consider the model of a spacecraft with a gimballed thrust engine shown in Figure 1. If the thrust 𝐹 during the fuel burn is a positive constant, and if the gimbal deflection angle and pitching moment are zero, 𝛿=𝑀=0, then the spacecraft and fuel slosh dynamics have a relative equilibrium defined by 𝑣𝑧=𝑣𝑧,𝜃=̇𝜃,𝜃=0,𝑠𝑖=0,̇𝑠𝑖=0,𝑖,(3.1) where 𝑣𝑧 and 𝜃 are arbitrary constants. Without loss of generality, the subsequent analysis considers the relative equilibrium at the origin, that is, 𝑣𝑧=𝜃=0. Note that the relative equilibrium corresponds to the vehicle axial velocity 𝑣𝑥(𝑡)=𝑣𝑥0+𝑎𝑥𝑡,𝑡𝑡𝑏,(3.2) where 𝑣𝑥0 is the initial axial velocity of the spacecraft, 𝑡𝑏 is the fuel burn time, and 𝑎𝑥=𝐹𝑚+𝑚𝑓.(3.3)

Now assume the axial acceleration term 𝑎𝑥 is not significantly affected by small gimbal deflections, pitch changes, and fuel motion (an assumption verified in simulations). Consequently, (2.20) becomes ̇𝑣𝑥+̇𝜃𝑣𝑧=𝑎𝑥.(3.4) Substituting this approximation leads to the following reduced equations of motion for the transverse, pitch, and slosh dynamics: 𝑚+𝑚𝑓̂𝑎𝑧̈+𝑚𝑏𝜃+𝑁𝑖=1𝑚𝑖̈𝑠𝑖𝑠𝑖̇𝜃2̇2𝑖̇𝜃2𝑚0̇0̇𝐼̈𝜃=𝐹sin𝛿,(3.5)𝜃+𝑁𝑖=1𝑚𝑖𝑎𝑥𝑠𝑖𝑖̈𝑠𝑖+𝑠𝑖̈𝑖𝑠+2𝑖̇𝑠𝑖+𝑖̇𝑖̇𝜃+2𝑚00̇0̇𝜃0+𝑚𝑏̂𝑎𝑧𝑚=𝜏,(3.6)𝑖̈𝑠𝑖+̂𝑎𝑧𝑖̈𝜃𝑠𝑖̇𝜃2̇2𝑖̇𝜃+𝑘𝑖𝑠𝑖+𝑐𝑖̇𝑠𝑖=0,𝑖,(3.7) where ̂𝑎𝑧=̇𝑣𝑧̇𝜃𝑣𝑥(𝑡). Here 𝑣𝑥(𝑡) is considered as an exogenous input. The subsequent analysis is based on the above equations of motion for the transverse, pitch, and slosh dynamics of the vehicle.

Eliminating ̈𝑠𝑖 in (3.5) and (3.6) using (3.7) yields 𝑚+𝑚0̂𝑎𝑧+𝑚𝑏𝑚00̈𝜃2𝑚0̇0̇𝜃𝑁𝑖=1𝑘𝑖𝑠𝑖+𝑐𝑖̇𝑠𝑖=𝐹sin𝛿,𝑚𝑏𝑚00̂𝑎𝑧+𝐼𝑁𝑖=1𝑚𝑖2𝑖̈𝜃+2𝑚00̇0̇𝜃+𝐺=𝑀+𝐹𝑝sin𝛿,(3.8) where 𝐺=𝑁𝑖=1𝑚𝑖𝑎𝑥+𝑚𝑖̈𝑖+𝑘𝑖𝑖𝑠𝑖+𝑖𝑐𝑖̇𝑠𝑖+2𝑚𝑖𝑠𝑖̇𝑠𝑖̇𝜃𝑚𝑖𝑖𝑠𝑖̇𝜃2.(3.9) Note that the expressions (2.1) have been utilized to obtain (3.8) in the form above.

By defining control transformations from (𝛿,𝑀) to new control inputs (𝑢1,𝑢2): 𝑢1𝑢2=𝑚+𝑚0𝑚𝑏𝑚00𝑚𝑏𝑚00𝐼𝑁𝑖=1𝑚𝑖2𝑖1𝐹sin𝛿+2𝑚0̇0̇𝜃+𝑁𝑖=1𝑘𝑖𝑠𝑖+𝑐𝑖̇𝑠𝑖𝑀+𝐹𝑝sin𝛿2𝑚00̇0̇,𝜃𝐺(3.10) the system (3.5)–(3.7) can be written as ̇𝑣𝑧=𝑢1+̇𝜃𝑣𝑥̈(𝑡),(3.11)𝜃=𝑢2,(3.12)̈𝑠𝑖=𝜔2𝑖𝑠𝑖2𝜁𝑖𝜔𝑖̇𝑠𝑖𝑢1+𝑖𝑢2+𝑠𝑖̇𝜃2̇+2𝑖̇𝜃,𝑖,(3.13) where 𝜔2𝑖=𝑘𝑖𝑚𝑖,2𝜁𝑖𝜔𝑖=𝑐𝑖𝑚𝑖,𝑖.(3.14) Here 𝜔𝑖 and 𝜁𝑖, for all 𝑖, denote the undamped natural frequencies and damping ratios, respectively.

The main idea in the subsequent development is to first design feedback control laws for (𝑢1,𝑢2) and then use the following equations to obtain the feedback laws for the original controls (𝛿,𝑀) for 𝑡𝑡𝑏: 𝛿=sin1𝑚+𝑚0𝑢1+𝑚𝑏𝑚00𝑢22𝑚0̇0̇𝜃𝑁𝑖=1𝑘𝑖𝑠𝑖+𝑐𝑖̇𝑠𝑖𝐹,(3.15)𝑀=𝑚𝑏𝑚00𝑢1+𝐼𝑁𝑖=1𝑚𝑖2𝑖𝑢2+2𝑚00̇0̇𝜃+𝐺𝐹𝑝sin𝛿.(3.16)

Consider the following candidate Lyapunov function to stabilize the subsystem defined by (3.11) and (3.12): 𝑟𝑉=12𝑣2𝑧+𝑟22𝜃2+𝑟32̇𝜃2,(3.17) where 𝑟1,𝑟2, and 𝑟3 are positive constants so that the function 𝑉 is positive definite.

The time derivative of 𝑉 along the trajectories of (3.11) and (3.12) can be computed as ̇𝑉=𝑟1𝑣𝑧̇𝑣𝑧+𝑟2𝜃̇𝜃+𝑟3̇𝜃̈𝜃(3.18) or rewritten in terms of the new control inputs ̇𝑟𝑉=1𝑣𝑧𝑢1+𝑟1𝑣𝑥𝑣𝑧+𝑟2𝜃+𝑟3𝑢2̇𝜃.(3.19) Clearly, the feedback laws 𝑢1=𝑙1𝑣𝑧,𝑢(3.20)21=𝑟3𝑟2𝜃+𝑙2̇𝜃,(3.21) where 𝑙1,𝑙2 are positive constants and taking into account that 𝑟1𝑣𝑥𝑣𝑧̇̇𝜃𝜃22+𝑟1𝑣𝑥𝑣𝑧22,(3.22) yield ̇𝑉=𝑙1𝑟1𝑣2𝑧𝑙2̇𝜃2+𝑟1𝑣𝑥𝑣𝑧̇𝜃𝑟1𝑙1𝑟1𝑣2𝑥2𝑣2𝑧𝑙212̇𝜃2,(3.23) which satisfies ̇𝑉0 if 𝑙1>0.5𝑟1𝑣2𝑥 and 𝑙2>0.5.

The closed-loop system for (𝑣𝑧,𝜃)-dynamics can be written as ̇𝑣𝑧=𝑙1𝑣𝑧+̇𝜃𝑣𝑥̈(𝑡),(3.24)𝜃=𝐾1𝜃𝐾2̇𝜃,(3.25) where 𝐾1=𝑟2/𝑟3 and 𝐾2=𝑙2/𝑟3.

Equation (3.25) can be easily solved in the case of 𝐾22>4𝐾1 as 𝜃(𝑡)=𝐴𝑒𝜆1𝑡+𝐵𝑒𝜆2𝑡,(3.26) where 𝐴,𝐵 are integration constants and 𝜆1,𝜆2 are the eigenvalues of the linear system (3.25). Therefore, 𝜃(𝑡) and ̇𝜃(𝑡) can be upper bounded as ||||𝜃(𝑡)𝐶𝑒𝜆𝑡,||̇||𝜃(𝑡)𝐷𝑒𝜆𝑡,(3.27) respectively, where 𝐶,𝐷 are positive constants and 𝜆=min(𝜆1,𝜆2). Now, assuming that 𝜆𝑙1, (3.24) can be integrated to obtain an upper bound for 𝑣𝑧(𝑡) as ||𝑣𝑧||(𝑡)𝛼𝑒𝛽𝑡,(3.28) where 𝛼,𝛽 are positive constants. Therefore, it can be concluded that the (𝑣𝑧,𝜃)-dynamics are exponentially stable under the control laws (3.20) and (3.21).

To analyze the stability of the 𝑁 equations defined by (3.13), it will be first shown that the system described by the equation ̈𝑠𝑖+2𝜁𝑖𝜔𝑖(𝑡)̇𝑠𝑖+𝜔2𝑖(𝑡)𝑠𝑖=0(3.29) is exponentially stable.

From (2.4) and (2.6), 𝜔𝑖(𝑡)=2𝑔𝜉𝑖𝜑tanh2𝜉𝑖(𝑡)𝜑𝐶1.(3.30) The following properties can be shown to hold: 𝜔2𝑖(𝑡)𝜀211,𝑝(𝑡)=2̇𝜔𝑖(𝑡)𝜔𝑖(𝑡)+2𝜁𝑖𝜔𝑖(𝑡)𝜀22,||2𝜁𝑖𝜔𝑖||(𝑡)2𝜁𝑖2𝑔𝜉𝑖𝜑=𝑀1,||𝜔2𝑖||(𝑡)2𝑔𝜉𝑖𝜑=𝑀2,||2̇𝜔𝑖(𝑡)𝜔𝑖||(𝑡)𝑔2𝜉𝑖𝜑2=𝑀3,(3.31) where 𝜀1 and 𝜀2 are small positive parameters given the fact that the tank will never be totally empty, but a small amount of fuel will always remain inside. For this same reason, (𝑡)>0, for all 𝑡. Therefore, by Corollary A.2 in the Appendix, the system (3.29) is exponentially stable.

Now write (3.13) as 𝐴̇𝑥=1(𝑡)+𝐴2(𝑡)𝑥+𝐻(𝑡),(3.32) where 𝑥=[𝑠𝑖,̇𝑠𝑖]𝑇 and 𝐴1(𝑡)=01𝜔2𝑖(𝑡)2𝜁𝑖𝜔𝑖(𝑡),𝐴2̇𝜃(𝑡)=002(,0𝑡)0𝐻(𝑡)=̂𝑎𝑧(𝑡)+𝑖̈̇(𝑡)𝜃(𝑡)+2𝑖̇.(𝑡)𝜃(𝑡)(3.33)

Under the stated assumptions, 𝐴1(𝑡) is exponentially stable (see the Appendix) and there exist positive constants 𝜆0,𝜆1, and 𝜆2 such that 0𝐴2(𝑡)𝑑𝑡𝜆0,𝐻(𝑡)𝜆1𝑒𝜆2𝑡,𝑡0.(3.34) Hence, for any initial condition, the state of the system (3.5)–(3.7) converges exponentially to zero.

4. Simulation

The feedback control law developed in the previous section is implemented here for the fourth stage of the European launcher Vega. The first two slosh modes are included to demonstrate the effectiveness of the controller (3.15), (3.16), (3.20), (3.21) by applying to the complete nonlinear system (2.20)–(2.23). The physical parameters used in the simulations are given in Table 1.

tab1
Table 1: Physical parameters for AVUM stage of Vega.

We consider stabilization of the spacecraft in orbital transfer, suppressing the transverse and pitching motion of the spacecraft and sloshing of fuel while the spacecraft is accelerating. In other words, the control objective is to stabilize the relative equilibrium corresponding to a specific spacecraft axial acceleration and 𝑣𝑧̇=𝜃=𝜃=𝑠𝑖=̇𝑠𝑖=0,𝑖=1,2.

Time responses shown in Figures 2, 3, and 4 correspond to the initial conditions 𝑣𝑥0=3000 m/s, 𝑣𝑧0=100 m/s, 𝜃0=5, ̇𝜃0=0, 𝑠10=0.1 m, 𝑠20=0.1 m, and ̇𝑠10=̇𝑠20=0. We assume a fuel burn time of 650 seconds. As can be seen, the transverse velocity, attitude angle, and the slosh states converge to the relative equilibrium at zero while the axial velocity 𝑣𝑥 increases and ̇𝑣𝑥 tends asymptotically to 𝐹/(𝑚+𝑚𝑓). Note that there is a trade-off between good responses for the directly actuated degrees of freedom (the transverse and pitch dynamics) and good responses for the internal degrees of freedom (the slosh dynamics); the controller given by (3.15), (3.16), (3.20), (3.21) with parameters 𝑟1=8×107, 𝑟2=103, 𝑟3=500, 𝑙1=104, and 𝑙2=4×104 represents one example of this balance.

fig2
Figure 2: Time responses of 𝑣𝑥,𝑣𝑧, and 𝜃.
fig3
Figure 3: Time responses of 𝑠1 and 𝑠2.
fig4
Figure 4: Gimbal deflection angle 𝛿 and pitching moment 𝑀.

Figures 5, 6, and 7 show the results of a simulation with no control (𝑀=𝛿=0) using the same initial conditions and physical parameters as above. As expected, the fuel slosh dynamics destabilize the uncontrolled spacecraft.

fig5
Figure 5: Time responses of 𝑣𝑥,𝑣𝑧 and 𝜃 (zero control case).
fig6
Figure 6: Time responses of 𝑠1 and 𝑠2 (zero control case).
fig7
Figure 7: Gimbal deflection angle 𝛿 and pitching moment 𝑀 (zero control case).

5. Conclusions

A complete nonlinear dynamical model has been developed for a spacecraft with multiple slosh modes that have time-varying parameters. A feedback controller has been designed to achieve stabilization of the pitch and transverse dynamics as well as suppression of the slosh modes, while the spacecraft accelerates in the axial direction. The effectiveness of the feedback controller has been illustrated through a simulation example.

The many avenues considered for future research include problems involving multiple liquid containers and three-dimensional transfers. Future research also includes designing nonlinear observers to estimate the slosh states as well as nonlinear control laws that achieve robustness, insensitivity to system and control parameters, and improved disturbance rejection.

Appendix

Consider the system̈𝑠+𝑓(𝑡)̇𝑠+𝑔(𝑡)𝑠=0,(A.1) where 𝑔(𝑡)𝐶1, |𝑓(𝑡)|<𝑀1, |𝑔(𝑡)|<𝑀2, |̇𝑔(𝑡)|<𝑀3.

Theorem A.1. If 𝑔(𝑡)>𝜀21 and 𝑝(𝑡)=(1/2)(̇𝑔(𝑡)/𝑔(𝑡))+𝑓(𝑡)>𝜀22, then the origin is globally uniformly asymptotically stable.

Proof. Given the conditions above, the following bounds can be set: 𝑀1<𝑓(𝑡)<𝑀1,𝛼21<𝑔(𝑡)<𝑀2,𝑀3<̇𝑔(𝑡)<𝑀3,𝜀22𝑀<𝑝(𝑡)<32𝜀21+𝑀1.(A.2)
Consider the following candidate Lyapunov function: 1𝑉(𝑧,𝑡)=2𝑠2+2𝛽𝑠̇𝑠+𝑔(𝑡)̇𝑠2𝑔(𝑡),(A.3) where 𝑧=[𝑠̇𝑠]𝑇 is the state vector and 𝛽 is a positive constant. This function can be rewritten in a matrix form as 1𝑉(𝑧,𝑡)=21𝛽𝑠̇𝑠𝑔𝛽𝑔1𝑔𝑠̇𝑠,(A.4) which is positive definite if 𝛽<1.
Recalling that a positive definite quadratic function 𝑧𝑇𝑃𝑧 satisfies 𝜆min(𝑃)𝑧𝑇𝑧𝑧𝑇𝑃𝑧𝜆max(𝑃)𝑧𝑇𝑧,(A.5) where 𝜆min(𝑃)=1+𝑔2𝑔114𝑔1𝛽2(1+𝑔)2,𝜆max(𝑃)=1+𝑔2𝑔1+14𝑔1𝛽2(1+𝑔)2,(A.6) and thus the following hold: 𝛾1𝑧2𝑉𝛾2𝑧2,(A.7) where 𝛾1 and 𝛾2 are positive constants.
Taking the time derivative of 𝑉(𝑧,𝑡) yields ̇𝛽𝑉=𝑔(𝑡)𝑔(𝑡)𝑠2+𝑝(𝑡)𝑠̇𝑠+𝑝(𝑡)𝛽𝑔(𝑡)1̇𝑠2,(A.8) which can be rewritten as ̇𝛽𝑉=𝑔𝑔𝑝𝑠̇𝑠2𝑝2𝑝𝛽𝑔𝑠1̇𝑠<0.(A.9) Clearly, ̇𝑉<0 if 𝛽<16𝜀51𝜀2216𝑀2𝜀41+𝑀3+2𝜀21𝑀12.(A.10) Note that ̇𝑉 satisfies ̇𝛽𝑉𝑔𝜆min(𝑄)𝑧2.(A.11) It can be shown that if 𝜀𝛽<min1,221𝑀2𝑀2,16𝜀51𝜀2216𝑀2𝜀41+𝑀3+2𝜀21𝑀12,(A.12) then, using Theorem 4.10 of [24], it can be concluded that the origin is exponentially stable. Hence, the following result can be stated.

Corollary A.2. There exist 𝛼,𝛽>0 such that |𝑠|<𝛽𝑒𝛼(𝑡𝑡0),|̇𝑠|<𝛽𝑒𝛼(𝑡𝑡0),𝑡𝑡0.(A.13)
The following result is a modified version of that presented in [20].

Lemma A.3. Consider a system that is described by the linear time-varying differential equation 𝐴̇𝑥=1(𝑡)+𝐴2(𝑡)𝑥+𝐻(𝑡),𝑥𝑛.(A.14)
If the matrix 𝐴1(𝑡) is exponentially stable and there exist positive constants 𝜆0, 𝜆1, and 𝜆2 such that (i)0𝐴2(𝑡)𝑑𝑡𝜆0,(ii)𝐻(𝑡)𝜆1𝑒𝜆2𝑡,𝑡0,(A.15) then all the solutions of (A.14) approach zero exponentially as 𝑡 goes to .

Acknowledgment

The authors wish to acknowledge the support provided by Embry-Riddle Aeronautical University.

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