This paper presents a modeling approach for the liquid dynamics inside the tanks of liquid-propelled vehicles. The proposed formulation is applicable to high-g environments and planetary atmospheric flight, encompassing expendable and reusable launch vehicles as well as planetary landers. The oscillatory behavior of the liquid free surface is modeled as a superposition of n swinging pendula, each corresponding to a harmonic mode of sloshing.The coupling between the rigid body and the pendula is formulated using a classical multibody dynamics methodology. Specifically, the iterative Newton–Euler approach is applied to a general spacecraft configuration with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varvec{m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">m</mi> </mrow> </math></EquationSource> </InlineEquation> tanks and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">n</mi> </mrow> </math></EquationSource> </InlineEquation> associated sloshing modes. A detailed derivation of a symbolic three-degree-of-freedom pendulum model is presented, followed by its reduction to a two-degree-of-freedom planar model. While the full model is suitable for time-domain simulations, the reduced version is tailored for control design and stability analysis.Symbolic computations are performed with the Symbolic Multibody Dynamics toolbox developed by the Guidance, Navigation and Control Systems Department at the German Aerospace Center. This library enables analytical derivation of equations of motion for a wide range of multibody space systems and supports seamless export to simulation frameworks for the verification and validation of guidance, navigation and control systems.The model parameters for a representative space mission were derived from the geometry of a reusable launch vehicle’s propellant and oxidizer tanks. A stability analysis of the complex pole–zero pairs associated with each sloshing mode was conducted, and the locations of the equivalent sloshing masses relative to critical stability zones were examined. Finally, a closed-loop analysis was performed using an <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {H}_\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> -based controller, and the resulting stability margins were evaluated along a reference trajectory and via nonlinear simulations.

Advanced nonlinear modeling and control stability analysis of sloshing dynamics in liquid-propelled reusable launch vehicles

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2026年5月7日

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José Alfredo Macés-Hernández

摘要

This paper presents a modeling approach for the liquid dynamics inside the tanks of liquid-propelled vehicles. The proposed formulation is applicable to high-g environments and planetary atmospheric flight, encompassing expendable and reusable launch vehicles as well as planetary landers. The oscillatory behavior of the liquid free surface is modeled as a superposition of n swinging pendula, each corresponding to a harmonic mode of sloshing.

The coupling between the rigid body and the pendula is formulated using a classical multibody dynamics methodology. Specifically, the iterative Newton–Euler approach is applied to a general spacecraft configuration with $\varvec{m}$ $m$ tanks and $\varvec{n}$ $n$ associated sloshing modes. A detailed derivation of a symbolic three-degree-of-freedom pendulum model is presented, followed by its reduction to a two-degree-of-freedom planar model. While the full model is suitable for time-domain simulations, the reduced version is tailored for control design and stability analysis.

Symbolic computations are performed with the Symbolic Multibody Dynamics toolbox developed by the Guidance, Navigation and Control Systems Department at the German Aerospace Center. This library enables analytical derivation of equations of motion for a wide range of multibody space systems and supports seamless export to simulation frameworks for the verification and validation of guidance, navigation and control systems.

The model parameters for a representative space mission were derived from the geometry of a reusable launch vehicle’s propellant and oxidizer tanks. A stability analysis of the complex pole–zero pairs associated with each sloshing mode was conducted, and the locations of the equivalent sloshing masses relative to critical stability zones were examined. Finally, a closed-loop analysis was performed using an ${\mathcal {H}_\infty }$ $H_{\infty}$ -based controller, and the resulting stability margins were evaluated along a reference trajectory and via nonlinear simulations.