Validated nonlinear air-water transition dynamics and provably stable cross-medium control for small multirotor hybrid aerial-aquatic vehicles

A hybrid aerial-aquatic multirotor must repeatedly cross the air-water interface, where fluid density changes by three orders of magnitude, buoyancy, drag, and added mass switch on gradually, and the free surface deforms under the rotors. The source framework (Wang and Shang, arXiv:2507.03869) handles this with a linearly depth-weighted three-zone model, a CFD-calibrated thrust coefficient $C_T(h)$ obtained from a single scaled propeller family, and a switched controller that combines twisting sliding-mode control with cascade PID under hysteresis and safe-switch guards. The paper leaves open: quantifying and reducing the linearization error and building an experimentally validated nonlinear transition model for small platforms; a rigorous global-stability guarantee for the switched closed loop under uncertain subsystems; explicit compensation of wave disturbances and inertial-fluid coupling during splash and impact; energy-aware medium-transition mission planning; and extension of the data-driven thrust model beyond its fixed grid into near-water and ventilation regimes. Adjacent 2025-2026 work has closed parts of the general questions in simulation and at rotor level — a unified controller with proven global asymptotic stability, a dwell-time proof of switched ultimate boundedness, and dedicated near-water rotor experiments — but the coupled core remains open: a full-vehicle, control-oriented, experimentally validated nonlinear transition model running inside a provably stable cross-medium controller, corroborated on a physical platform in wavy conditions, with energy-aware transition planning on top.

Background

A hybrid aerial-aquatic vehicle (HAUV, also called an AquaUAV or cross-medium vehicle) is a robot that can fly in air, submerge and maneuver underwater, and cross the water surface repeatedly in both directions. Multirotor HAUVs are attractive because quadrotor airframes are cheap, hover-capable, and mechanically simple, but the interface crossing is physically violent: water is roughly 800-1000 times denser than air, so buoyancy, drag, and added mass (the extra effective inertia a body acquires because it must accelerate the fluid around it) turn on as the vehicle submerges, the hull can experience slamming (short, large impact loads upon water entry), and the propellers interact with a deforming free surface. A specific hazard is ventilation: when a rotor operates too close to the surface, air is drawn down along the blades and thrust can collapse abruptly. The collective change in rotor thrust and torque caused by the nearby free surface is called the near-water effect and is distinct from the familiar aerodynamic ground effect.

Wang and Shang (arXiv:2507.03869) built a complete low-cost framework for such a vehicle. Their dynamics model divides the vertical operating range into three zones (air, underwater, and a hybrid transition zone of height $H$ centered on the surface) and captures the gradual switch-on of hydrodynamic effects by weighting the fully submerged added mass, buoyancy, and drag by a common linear coefficient $C(z)$ that ramps from 0 to 1 across the transition zone, which assumes the vehicle stays nearly vertical while crossing. Propeller thrust under partial submersion is modeled as $T=C_T(h)\Omega^2D^4$, where $\Omega$ is rotation rate, $D$ is propeller diameter, and the thrust coefficient $C_T$ depends on the height $h$ of the rotor center relative to the surface; $C_T(h)$ was calibrated by computational fluid dynamics (CFD) over a fixed grid of 500 combinations of $\Omega$, $D$, and $h$, all derived by scaling a single commercial three-blade propeller. Control is hierarchical: cascade PID control (nested proportional-integral-derivative loops) for tracking in the homogeneous air and water zones, twisting sliding-mode control (TWSMC, a second-order sliding-mode controller whose input switches between two levels, giving robustness to bounded uncertainty with reduced chattering) for attitude during the transition zone, and a state-dependent switching logic with hysteresis (a dead-band preventing rapid re-switching at zone boundaries) plus safe-switch guards that permit a switch only when roll and pitch angles and rates are below set limits. Experiments on a modified FPV (first-person-view) quadrotor showed height errors below 0.1 m and attitude oscillations within about 5 degrees in calm water.

The paper acknowledges three families of limitations. First, real transition physics (surface tension, slamming, ventilation, splashing, and the near-water effect on propellers) is reduced to linear interpolation, and the resulting modeling error is simply left for the controller to reject; the authors state that developing a highly complex, experimentally validated nonlinear transition model was prohibitive for a low-cost platform. Second, no stability guarantee is given for the switched closed loop; the paper notes that ensuring global stability of such a switched system, especially with uncertain subsystems, may require analysis using common Lyapunov functions (a single Lyapunov function that decreases in every mode, which certifies stability under arbitrary switching) or dwell-time conditions (which certify stability when consecutive switches are separated by a minimum time). Third, wave disturbances and inertial-fluid coupling during splash and impact are uncompensated, and mission-level planning of energy-efficient medium transitions is named as future work.

These gaps sit in an active literature. Stability-oriented switched or hybrid control of HAUVs predates the paper: Neto et al. (ICRA 2015) proved a robust switched attitude-control strategy with global stability for a hybrid unmanned aerial/underwater vehicle, and Mercado Ravell et al. (2018) used hybrid control in which linearization errors are treated as disturbances to be rejected. Field reviews identify the same open challenges: Zeng et al. (2022) single out complete dynamic modeling and cross-domain motion control under complex sea state, and Meng et al. (2026) list the lack of unified modeling frameworks, limited energy efficiency, and insufficient autonomy as key challenges; the dedicated near-water-effect review of Bai et al. (2025) lists flow-field measurement and simulation of that effect among open problems. Recent work has begun closing parts of the gap, mostly in simulation and at rotor level: Ma et al. (2025) derived a unified cross-domain model and quaternion-based dual-loop controller with rigorously proven global asymptotic stability for a micro quadrotor HAUV, arguing superiority over switching-based designs; Wu et al. (2026) proved switched closed-loop uniform ultimate boundedness (convergence of trajectories to a bounded set, UUB) under an average dwell-time condition for a mode-dependent adaptive robust controller on segmented cross-media dynamics; Honts et al. (2025) reported dedicated experiments on isolated small (~10 cm) rotors dynamically transitioning from water to air, linking abrupt lift, torque, and frequency changes to free-surface vortex dynamics; Zhan et al. (2025) validated a finite-vortex rotor thrust model against water experiments and defined a thrust-loss zone; and Qin et al. (2026) resolved full-vehicle water entry and exit, including a coupled multi-rotor near-water effect, in CFD. None of these validates a full-vehicle nonlinear transition model inside the control loop on a physical platform, and none analyzes the TWSMC/PID hysteresis architecture of the source paper.

Problem Statement

For a small, low-cost multirotor hybrid aerial-aquatic vehicle of the type built by Wang and Shang (2025) — three-zone dynamics with linearly depth-weighted added mass, buoyancy, and drag; a CFD-calibrated thrust coefficient $C_T(h)$ from a single scaled propeller family on a fixed operating grid; and a switched controller combining twisting sliding-mode attitude control with cascade PID under hysteresis and safe-switch guards — close the loop between validated interface physics and guaranteed-stable control. The common objective is safe, repeatable, and energy-efficient air-water transitions demonstrated on a physical platform, addressing the open questions that the source leaves: (1) Transition modeling: quantify the error introduced by the linear depth-weighting and near-vertical-attitude approximation, and develop a nonlinear, control-oriented model of the air-water transition for small multirotor platforms — capturing the dominant interface physics, which may include water-entry slamming, free-surface deformation, surface tension, splashing, ventilation onset, and near-water rotor effects — validated against dedicated physical experiments, together with an extension of the data-driven thrust model beyond one scaled propeller family and its fixed grid into these near-water and ventilation regimes. (2) Switched-system stability: establish a rigorous stability guarantee (global asymptotic stability or uniform ultimate boundedness) for the multi-mode control architecture under the quantified transition-model and subsystem uncertainty. (3) Disturbance compensation: design explicit compensation for wave disturbances and for inertial-fluid coupling during splash and impact. (4) Energy-aware transition planning: build mission-level planning that schedules medium transitions using the validated dynamics for energy-efficient operation. These four sub-questions are jointly required: they constitute the single common objective, and a resolution must address all four and report results that are mutually consistent on the modeled small multirotor platform, whereas a result on any one sub-question is a partial contribution rather than a resolution. The demonstration on a physical platform is required rather than idealized: the transition model and the stability guarantee must be validated against dedicated experiments on a small platform of this type, not in simulation or CFD alone.

References

1. RenKai Wang and ZhiGang Shang (2025). Modeling and control of a low-cost multirotor hybrid aerial underwater vehicle. arXiv:2507.03869, DOI 10.48550/arxiv.2507.03869, https://arxiv.org/abs/2507.03869 (source paper)
2. Armando Alves Neto, Leonardo A. Mozelli, Paulo L. J. Drews, and Mario F. M. Campos (2015). Attitude control for an Hybrid Unmanned Aerial Underwater Vehicle: A robust switched strategy with global stability. IEEE International Conference on Robotics and Automation (ICRA), pp. 395-400, DOI 10.1109/ICRA.2015.7139029, https://doi.org/10.1109/ICRA.2015.7139029
3. Diego Alberto Mercado Ravell, Marco Moreno Maia, and Francisco Javier Diez (2018). Modeling and control of unmanned aerial/underwater vehicles using hybrid control. Control Engineering Practice 76:112-122, DOI 10.1016/j.conengprac.2018.04.006, https://doi.org/10.1016/j.conengprac.2018.04.006
4. Zheng Zeng, Chenxin Lyu, Yuanbo Bi, Yufei Jin, Di Lu, and Lian Lian (2022). Review of hybrid aerial underwater vehicle: Cross-domain mobility and transitions control. Ocean Engineering 248:110840, DOI 10.1016/j.oceaneng.2022.110840, https://doi.org/10.1016/j.oceaneng.2022.110840
5. Xingzhi Bai, Mingqing Lu, Qi Zhan, Yu Wang, Daixian Zhang, Xiao Wang, and Wenhua Wu (2025). Review of the Near-Water Effect of Rotors in Cross-Media Vehicles. Drones 9(3):195, DOI 10.3390/drones9030195, https://doi.org/10.3390/drones9030195
6. Logan Patrick Honts, Yuanhang Zhu, and Daniel B. Quinn (2025). Free surface vortex dynamics during water-to-air amphibious rotor transitions. Journal of Fluid Mechanics 1022:A18, DOI 10.1017/jfm.2025.10778, https://doi.org/10.1017/jfm.2025.10778
7. Qi Zhan, Xiao Wang, Junhui Hu, Xingzhi Bai, and Pierangelo Masarati (2025). Aerodynamic modeling and analysis of aerial-aquatic rotorcraft performance near and crossing the air-water interface. Chinese Journal of Aeronautics 38(9):103511, DOI 10.1016/j.cja.2025.103511, https://doi.org/10.1016/j.cja.2025.103511
8. Hongde Qin, Yuxiang Cheng, Xiaojian Cao, Hongkun Chen, and Yuxuan Chen (2026). Water-entry and exit hydrodynamics and near-water effects of a low-density cross-media vehicle. Ocean Engineering 362:126314, DOI 10.1016/j.oceaneng.2026.126314, https://doi.org/10.1016/j.oceaneng.2026.126314
9. Zongcheng Ma, Wuji Zheng, and Zeqian Liu (2025). An integrated water-air cross-domain controller for seamless trajectory tracking of the micro quadrotor hybrid aerial underwater vehicle. Ocean Engineering 339:122152, DOI 10.1016/j.oceaneng.2025.122152, https://doi.org/10.1016/j.oceaneng.2025.122152
10. Jiayue Wu, Fangfang Dong, Xiaomin Zhao, and Ye-Hwa Chen (2026). Adaptive robust constraint-following control for a HAUV with underactuated and segmented cross-media dynamics. Ocean Engineering 364:127108, DOI 10.1016/j.oceaneng.2026.127108, https://doi.org/10.1016/j.oceaneng.2026.127108
11. Yuhang Meng, Vicenç Puig, Haoran Tang, Dong Wu, Hui Ye, Xiaofei Yang, and Zhengrong Xiang (2026). Aerial–aquatic unmanned amphibious vehicles: A review of prototypes, modeling, and control. Ocean Engineering 364:126801, DOI 10.1016/j.oceaneng.2026.126801, https://doi.org/10.1016/j.oceaneng.2026.126801
12. Junping Li, Yufei Jin, Rui Hu, Yulin Bai, Di Lu, Zheng Zeng, and Lian Lian (2024). Trajectory Tracking Control of Fixed-Wing Hybrid Aerial Underwater Vehicle Subject to Wind and Wave Disturbances. Journal of Intelligent & Robotic Systems 110(2):92, DOI 10.1007/s10846-024-02099-y, https://doi.org/10.1007/s10846-024-02099-y
13. Pedro Miranda Pinheiro, Armando Alves Neto, Ricardo Bedin Grando, Cesar Bastos da Silva, Vivian Misaki Aoki, Dayana Cardoso, Alexandre Campos Horn, and Paulo Lilles Jorge Drews-Jr (2022). Trajectory Planning for Hybrid Unmanned Aerial Underwater Vehicles with Smooth Media Transition. Journal of Intelligent & Robotic Systems 104(3):46, DOI 10.1007/s10846-021-01567-z, https://doi.org/10.1007/s10846-021-01567-z (preprint arXiv:2112.13819)