The impact of liquid fuel sloshing in storage tanks during rocket flight and attitude adjustment cannot be ignored. In response to this research background, this paper conducted a coupled dynamic modeling of liquid rockets and proposed an improved proportional-integral-derivative (PID) control method based on wave theory, achieving efficient dynamic modeling and sloshing suppression. Firstly, a bouncing ball model for the liquid nonlinear sloshing in the Cassini tanks was established. By comparing it with the computational fluid dynamics method, it was demonstrated that the equivalent model can efficiently calculate the sloshing force, torque, and center-of-mass motion trends of the liquid. Secondly, based on the wave theory, an improved PID control method is proposed to achieve both target tracking and sloshing suppression simultaneously. Finally, taking the working condition of the liquid rocket attitude adjustment as an example, a rigid-liquid-control coupling simulation was achieved. By comparing with the traditional PID method, the effectiveness of the control method proposed in this paper was verified.

[1] Kong W. and Tian Q., "Dynamics of Fluid-Filled Space Multibody Systems Considering the Microgravity Effects," Mechanism and Machine Theory, Vol. 148, June 2020, Paper 103809. https://doi.org/10.1016/j.mechmachtheory.2020.103809 Google Scholar

[2] Deng M. L. and Yue B. Z., "Attitude Dynamics and Control of Liquid Filled Spacecraft with Large Amplitude Fuel Slosh," Journal of Mechanics, Vol. 33, No. 3, 2017, pp. 125-136. https://doi.org/10.1017/jmech.2016.60 Google Scholar

[3] Salem M. I., Mucino V. H., Saunders E., Gautam M. and Lozano-Guzman A., "Lateral Sloshing in Partially Filled Elliptical Tanker Trucks Using a Trammel Pendulum," International Journal of Heavy Vehicle Systems, Vol. 16, Nos. 1-2, 2009, pp. 207-224. https://doi.org/10.1504/IJHVS.2009.023861 Google Scholar

[4] Calvi G. M. and Nascimbene R., Seismic Design and Analysis of Tanks, Wiley, Pavia, 2023, pp. 93-111, Chap. 2. Google Scholar

[5] Brunesi E., Nascimbene R. and Beilic D., "Understanding the Seismic Resilience of Metallic Cylindrical Tanks Through Parametric Analysis," Applied Sciences, Vol. 15, No. 1, 2025, p. 474. https://doi.org/10.3390/app15010474 Google Scholar

[6] Merino R., Nascimbene R. and Calvi G. M., "Direct Displacement-Based Seismic Design of Unanchored Cylindrical Steel Tanks for Excessive Plastic Rotations of the Base Plate During Uplift," Journal of Earthquake Engineering, Vol. 29, No. 9, 2025, pp. 1951-1979. https://doi.org/10.1080/13632469.2025.2493271 Google Scholar

[7] Nan M., Junfeng L. and Tianshu W., "Equivalent Mechanical Model of Large-Amplitude Liquid Sloshing under Time-Dependent Lateral Excitations in Low-Gravity Conditions," Journal of Sound and Vibration, Vol. 386, Jan. 2017, pp. 421-432. https://doi.org/10.1016/j.jsv.2016.08.029 CrossrefGoogle Scholar

[8] Nan M., Junfeng L. and Tianshu W., "Large-Amplitude Sloshing Analysis and Equivalent Mechanical Modeling in Spherical Tanks of Spacecraft," Journal of Spacecraft and Rockets, Vol. 53, No. 3, May 2016, pp. 393-586. https://doi.org/10.2514/1.A33394 Google Scholar

[9] Bayle O., L.'Hullier V., Ganet M., Delpy P., Francart J. L. and Paris D., "Influence of the ATV Propellant Sloshing on the GNC Performance," AIAA Guidance, Navigation, and Control Conference and Exhibit, AIAA Paper 2022-4845, Aug. 2002. https://doi.org/10.2514/6.2002-4845 LinkGoogle Scholar

[10] Zhou Z. and Huang H., "Constraint Surface Model for Large Amplitude Sloshing of the Spacecraft with Multiple Tanks," Acta Astronautica, Vol. 111, June-July 2015, pp. 222-229. https://doi.org/10.1016/j.actaastro.2015.02.023 CrossrefGoogle Scholar

[11] Vreeburg J. P. B., "Dynamics and Control of a Spacecraft with a Moving Pulsating Ball in a Spherical Cavity," Acta Astronautica, Vol. 40, No. 2-8, 1997, pp. 257-274. https://doi.org/10.1016/S0094-5765(97)00095-7 CrossrefGoogle Scholar

[12] Vreeburg J. and Chato D., "Models for Liquid Impact Onboard Sloshsat FLEVO"," Space 2000 Conference and Exposition, AIAA Paper 2000-5152, 2000. https://doi.org/10.2514/6.2000-5152 LinkGoogle Scholar

[13] Yu L., Baozeng Y. and Bole M., "Improved Moving Pulsating Ball Equivalent Model for Large-Amplitude Liquid Slosh," AIAA Journal, Vol. 60, No. 8, Aug. 2022, pp. 4478-5030. https://doi.org/10.2514/1.J061622 Google Scholar

[14] Lu Y., Yue B., Ma B., Hao B. and Upham M. P., "A Universal Nonlinear Equivalent Model and Its Stability Analysis for Large Amplitude Liquid Sloshing in Rotationally Symmetric Tanks," Nonlinear Dynamics, Vol. 113, No. 7, 2025, pp. 6,031-6,047. https://doi.org/10.1007/s11071-024-10439-8 Google Scholar

[15] Lu Y., Yue B., Hao B., Ma B. and Liu F., "Stage Separation of Recoverable Liquid Launch Vehicle by Using Moving Pulsating Ball Analogue for Propellant Sloshing," Chinese Journal of Aeronautics, Vol. 37, No. 2, Feb. 2024, pp. 360-370. https://doi.org/10.1016/j.cja.2023.09.005 CrossrefGoogle Scholar

[16] Lu Y. and Yue B., "Dynamic Modeling of Liquid-Filled Free-Floating Space Robot and Joint Trajectory Planning with Considering Liquid Positioning," Aerospace Science and Technology, Vol. 161, June 2025, Paper110133. https://doi.org/10.1016/j.ast.2025.110133 Google Scholar

[17] Pizzoli M., Saltari F., Mastroddi F., Martinez-Carrascal J. and González-Gutiérrez L. M., "Nonlinear Reduced-Order Model for Vertical Sloshing by Employing Neural Networks," Nonlinear Dynamics, Vol. 107, No. 2, 2022, pp. 1,469-1,478. https://doi.org/10.1007/s11071-021-06668-w Google Scholar

[18] Hu P. and Ren G., "Multibody Dynamics of Flexible Liquid Rockets with Depleting Propellant," Journal of Guidance, Control, and Dynamics, Vol. 36, No. 6, Nov. 2013, pp. 1565-1906. https://doi.org/10.2514/1.59686 Google Scholar

[19] Cui D. L., Yan S. Z., Guo X. S. and Gao R. X., "Parametric Resonance of Liquid Sloshing in Partially Filled Spacecraft Tanks During the Powered-Flight Phase of Rocket," Aerospace Science and Technology, Vol. 35, May 2014, pp. 93-105. https://doi.org/10.1016/j.ast.2014.03.006 CrossrefGoogle Scholar

[20] Hervas J. R. and Reyhanoglu M., "Thrust-Vector Control of a Three-Axis Stabilized Upper-Stage Rocket with Fuel Slosh Dynamics," Acta Astronautica, Vol. 98, May-June 2014, pp. 120-127. https://doi.org/10.1016/j.actaastro.2014.01.022 Google Scholar

[21] Farì S., Seelbinder D. and Theil S., "Advanced GNC-Oriented Modeling and Simulation of Vertical Landing Vehicles with Fuel Slosh Dynamics," Acta Astronautica, Vol. 204, March 2023, pp. 294-306. https://doi.org/10.1016/j.actaastro.2022.12.035 CrossrefGoogle Scholar

[22] Song X. J., Yue B. Z. and Wu W. J., "Investigation on Attitude Disturbance Control and Vibration Suppression for Fuel-Filled Flexible Spacecraft," Acta Mechanica Sinica, Vol. 31, No. 4, 2015, pp. 581-588. https://doi.org/10.1007/s10409-015-0431-8 CrossrefGoogle Scholar

[23] Baozeng Y. and Lemei Z., "Hybrid Control of Liquid-Filled Spacecraft Maneuvers by Dynamic Inversion and Input Shaping," AIAA Journal, Vol. 52, No. 3, March 2014, pp. 465-667. https://doi.org/10.2514/1.J052526 Google Scholar

[24] Zhang H. and Wang Z., "Attitude Control and Sloshing Suppression for Liquid-Filled Spacecraft in the Presence of Sinusoidal Disturbance," Journal of Sound and Vibration, Vol. 383, Nov. 2016, pp. 64-75. https://doi.org/10.1016/j.jsv.2016.08.001 CrossrefGoogle Scholar

[25] Haghparast B., Salarieh H., Pishkenari H. N., Abdollahi T., Jokar M. and Ghanipoor F., "A Cubature Kalman Filter for Parameter Identification and Output-Feedback Attitude Control of Liquid-Propellant Satellites Considering Fuel Sloshing Effects," Aerospace Science and Technology, Vol. 144, Jan. 2024, Paper 108813. https://doi.org/10.1016/j.ast.2023.108813 Google Scholar

[26] O.'Connor W. J. and Hu C., "A Simple, Effective Position Control Strategy for Flexible Systems," IFAC Proceedings, Vol. 35, No. 2, Dec. 2002, pp. 141-146. https://doi.org/10.1016/S1474-6670(17)33932-0 Google Scholar

[27] O.'Connor W. J.Wave-Echo Control of Lumped Flexible Systems," Journal of Sound and Vibration, Vol. 298, Nos. 4-5, Dec. 2006, pp. 1001-1018. https://doi.org/10.1016/j.jsv.2006.06.010 Google Scholar

[28] O.'Connor W. J., "Control of Flexible Mechanical Systems: Wave-Based Techniques," 2007 American Control Conference, Inst. of Electrical and Electronics Engineers, New York, 2007, pp. 4192-4202. https://doi.org/10.1109/ACC.2007.4283157 Google Scholar

[29] Habibi H. and O'Connor W., "Wave-Based Motion and Slewing Control of a Double-Appendage, Flexible System with Ungrounded Actuator Through Development of Direct Actuator Force Control," Mechanical Systems and Signal Processing, Vol. 137, March 2020, Paper 106175. https://doi.org/10.1016/j.ymssp.2019.05.059 Google Scholar

[30] Thompson J. W. and O.'Connor W., "Wave-Based Attitude Control of Spacecraft with Fuel Sloshing Dynamics," Archive of Mechanical Engineering, Vol. 63, No. 2, 2016, pp. 263-275. https://doi.org/10.1515/meceng-2016-0015 CrossrefGoogle Scholar

[31] Liu F., Yue B. and Ma B., "Dynamics and Control of Mass-Variable Liquid-Filled Spacecraft with Combined Nonlinear Slosh," IEEE Transactions on Aerospace and Electronic Systems, Vol. 60, No. 3, June 2024, pp. 3509-3522. https://doi.org/10.1109/TAES.2024.3363111 CrossrefGoogle Scholar

[32] Martinez-Carrascal J., Pizzoli M., Saltari F., Mastroddi F. and González-Gutiérrez L. M., "Sloshing Reduced-Order Model Trained with Smoothed Particle Hydrodynamics Simulations," Nonlinear Dynamics, Vol. 111, No. 22, 2023, pp. 21,099-21,115. https://doi.org/10.1007/s11071-023-08940-7 Google Scholar