Six-degree-of-freedom trans-media motion control of multi-rotor hybrid aerial underwater vehicles under state-dependent actuator saturation: theoretical extension, wave-affected thruster capability, real-time deployment, and closed-loop validation

For the multi-rotor hybrid aerial underwater vehicle (HAUV) Nezha I, Wei et al. (2024) designed a trans-media motion controller for three degrees of freedom (pitch, roll, heave) that explicitly accounts for the measured, state-dependent collapse of thruster capability during free-surface crossing: saturation bounds identified from calm-water thrust measurements, an adaptive anti-windup compensator driven by the instantaneous saturation deficit, adaptive model-uncertainty compensation, and a Lyapunov stability proof with an explicit sufficient gain condition. The design was validated only in numerical simulation, with the direct effect of waves on the trans-media thrusters explicitly ignored. The open problem is the completion of this control solution. The source paper's conclusion defers exactly one item — the practical application of the algorithm to the Nezha I platform — and flags as its obstacle that the algorithm's complexity and the computing power it requires are relatively high; the restricted three-degree-of-freedom scope, the neglect of the direct wave effect on the trans-media thrusters, the simulation-only evidence, and the sufficiency-only gain condition are stated in the paper as limitations or facts of the present study rather than as announced future work. The completion agenda accordingly covers four jointly supporting items: extend the model, controller, and stability analysis to full six-degree-of-freedom motion under the same saturation and uncertainty assumptions (resolving the conflict between underactuated horizontal control and the attitude-first priority strategy), incorporate wave-driven, time-varying thruster capability into the saturation and disturbance models and into the stability guarantee, demonstrate real-time implementability on vehicle-class embedded hardware, and validate the controller in closed-loop trans-media experiments at sea against a comparable baseline.

Background

A hybrid aerial underwater vehicle (HAUV) is a robotic platform that can fly in air, submerge, travel underwater, and repeatedly cross the air-water interface; targeted uses include three-dimensional ocean observation and rapid emergency response at the sea-air interface. For a multi-rotor HAUV — a quadrotor-like vehicle whose four propellers operate in both media (trans-media thrusters) — the hardest phase is the crossing itself: the surrounding medium changes from air (density about 1.2 kg/m^3) to water (about 1000 kg/m^3) within seconds, the hydrodynamic loads on the hull vary sharply and nonlinearly (free-surface uplift, slamming — the impact loading as the hull strikes the surface — and residual water), and the maximum thrust each propeller can deliver collapses as it passes through the free surface. Conventional controllers assume fixed actuator limits, which forces either conservative actuator sizing or degraded performance and instability when saturation occurs mid-crossing.

Wei et al. (2024) addressed this for the Nezha I multi-rotor HAUV. Their key ingredients are:

• Dynamic input saturation: the available thrust/moment upper bound is not a constant but a state-dependent function $f_i(h_P,\theta,\varphi)$ of immersion state, pitch, and roll, identified from thrust measurements on a test rig (each thruster commanded at 100% throttle and slowly raised through the surface at 5 mm/s in calm water). In the source these bounds are constructed only for the heave, roll, and pitch channels.
• A five-degree-of-freedom rigid-body model with generalized position $\eta=(x,y,z,\varphi,\theta,0)^T$ — yaw is omitted because its influence during the crossing is judged small — given by $M\ddot{\eta}+C_{RB}\dot{\eta}+g+F_d=\tau+F_D$ where the nonlinear trans-media hydrodynamic term $F_d$ is a regression fitted to earlier trans-media resistance experiments, and $F_D$ collects wind, wave, and current disturbances. The generalized control vector $\tau$ carries zero surge, sway, and yaw components.
• A model-based sliding-mode controller (sliding-mode control, SMC, is a robust nonlinear technique that drives a sliding variable $s_i=\dot{e}_i+\lambda_i e_i$ to zero and keeps it there against bounded uncertainty) acting on only three degrees of freedom: pitch and roll (attitude) and heave (vertical position $z$). A priority strategy stabilizes attitude first and heave second, because attitude excursions amplify thruster-generated trans-media excitation forces and can cause failure of the crossing. Sway, surge, and yaw (the lateral and forward translations and the rotation about the vertical axis) are explicitly left uncontrolled during the trans-media process.
• Saturation handling: an adaptive anti-windup compensator (anti-windup means a correction that prevents accumulated controller effort from winding up while the actuator is saturated, which would cause overshoot at desaturation) driven online by the instantaneous saturation deficit $\Delta\tau_i^*=f_i-\tau_i^*$, with piecewise dynamics for the compensator state $w_i$ and a dead-zone threshold $\varepsilon_i$, plus an adaptive estimate $\hat{\rho}_i$ of the unknown disturbance bound. A Lyapunov-function proof (exhibiting an energy-like function that decreases along the closed-loop trajectories — the standard stability-certification technique of nonlinear control) yields closed-loop stability under the sufficient gain condition $\frac{1}{2}\xi_i-k_{3i}+\frac{1}{2}<0$ for each controlled degree of freedom $i$. In the source, $\xi_i>0$ and $k_{1i}>0$ are the gains of the exponential reaching law $\dot{s}_i=-\xi_i s_i-k_{1i}\,\mathrm{sgn}(s_i)$ imposed on the sliding variable, $k_{3i}>0$ is the decay gain of the piecewise anti-windup dynamics of $w_i$, $\varepsilon_i>0$ is that dynamics' dead-zone threshold, $\lambda_i>0$ weights the tracking error in $s_i$, and $k_{2i}>0$ scales the adaptive model-uncertainty compensation; all are declared to be constants greater than zero and are fixed by hand for the three controlled channels. The source states the coefficient condition as the one under which its $\dot{V}\leq 0$ argument closes, so it is sufficient rather than necessary.
• Evidence: numerical simulation only — uniform trans-media motion at $|w|=0.1$ m/s (the paper frames uniform water exit as the representative challenge) in static water and under wind/wave/current disturbances (current 0.5 m/s, wind 0.5 m/s, wave height 0.05 m, wave period 0.5 s), where the traditional SMC's peak speed overshoot (0.085 m/s) was 4.25 times that of the proposed controller (0.020 m/s). Waves act on the vehicle body, but their direct effect on the trans-media thrusters is explicitly ignored; the paper's own results show closed-loop performance depends strongly on the initial wave phase, and the paper reports four representative initial phases ($t=0$, $0.25T$, $0.50T$, $0.75T$), with better trans-media speed control at the wave equilibrium positions than at crest or trough phases.

Context on the platform and the field: the same research group demonstrated real-ocean trans-media flights with a different, folding-wing vehicle (Nezha-IV; Jin et al. 2024), so mission-level trans-media autonomy exists in the group — but the Nezha I multi-rotor control algorithm itself has never run closed-loop on hardware. Other groups have developed switched, constraint-following, and learning-based controllers for their own cross-media vehicles, and experimental trans-media control of a low-cost multi-rotor has been demonstrated without any state-dependent saturation model or stability certificate; the hydrodynamics of partially submerged propellers in waves have also been measured. What no published work provides is a full-motion (six-degree-of-freedom) trans-media controller whose stability guarantee explicitly accounts for the measured, state-dependent — and in reality wave-affected — actuator saturation, validated in closed loop on hardware.

The unresolved question is the completion of this control solution: from three controlled degrees of freedom, calm-water quasi-static saturation bounds, and simulation-only evidence, to full six-degree-of-freedom motion under the same assumptions with wave-affected thruster capability covered by the stability guarantee, real-time implementability, and closed-loop experimental validation.

Problem Statement

Starting from the state of Wei et al. (2024) — a three-degree-of-freedom (pitch, roll, heave) trans-media controller for the multi-rotor HAUV Nezha I with state-dependent actuator saturation bounds $f_i(h_P,\theta,\varphi)$ (the available thrust or moment of control channel $i$ as a function of thruster immersion depth $h_P$, pitch $\theta$, and roll $\varphi$) measured quasi-statically in calm water, adaptive anti-windup, adaptive model-uncertainty compensation, Lyapunov-verified stability under the sufficient condition $\frac{1}{2}\xi_i-k_{3i}+\frac{1}{2}<0$ (with $\xi_i$ the reaching-law gain and $k_{3i}$ the anti-windup decay gain defined in the Background), and numerical-simulation evidence only — complete and validate full-motion trans-media control of the vehicle. The single overarching objective is a deployable, guaranteed-stable, full-motion trans-media control capability for the multi-rotor HAUV under realistic actuation limits and environmental disturbances. An answer must deliver, as jointly supporting parts of that objective:

(1) Model and control extension to full six-degree-of-freedom motion: restore the omitted yaw dynamics and bring surge, sway, and yaw under active control during the trans-media process under the same state-dependent saturation and model-uncertainty assumptions, resolving how horizontal motion is commanded through attitude in an underactuated four-thruster multi-rotor while the attitude-first priority strategy (or a justified replacement) holds, with a complete Lyapunov stability analysis yielding explicit sufficient gain conditions of the same character as the source condition. Because the source defines actuation authority and saturation bounds only for heave, roll, and pitch, this item requires stating where the yaw moment comes from once the yaw dynamics are restored (for example rotor reaction or drag torque, itself immersion-dependent) and constructing the corresponding state-dependent capability bounds for the newly controlled channels. Active control is understood in the standard underactuated sense — position $(x,y,z)$ and yaw tracked with attitude acting as the virtual input — since four thrusters cannot follow six independent references.

(2) Wave-affected actuator capability: replace the calm-water, quasi-static saturation bounds with a model in which thruster capability and thrust loads vary with wave phase — the wave-thruster interaction the source paper explicitly ignores — and extend the stability analysis of item (1) to that model, so that the resulting guarantee covers the time-varying, wave-driven saturation and thrust perturbations rather than only the calm-water bounds. Closed-loop performance under the wave-driven model must be characterized across the initial wave phase, following the source's convention of enumerating representative initial phases $t=0$, $0.25T$, $0.5T$, and $0.75T$ of a documented regular wave of stated height and period, and must identify which phase is worst for each performance quantity reported (for example trans-media speed overshoot, actuator saturation duration, tracking error).

(3) Real-time implementability: implement the extended algorithm — the adaptive update laws, the anti-windup differential equations, and the online evaluation of the state-dependent saturation bounds — on the vehicle's own onboard computer rather than offboard, and demonstrate that the achieved sampling period is short relative to the closed-loop response time and to the convergence time of the adaptive estimators of that implemented controller, within the documented onboard computational budget. This addresses the computational complexity that the source paper flags as an obstacle to practical application.

(4) Closed-loop experimental validation: validate the controller in closed-loop trans-media operation (repeated water entry and exit) on the Nezha I platform or on a multi-rotor HAUV whose thrusters operate in both media and exhibit the same state-dependent collapse of available thrust at the free surface, so that capability bounds of the same character as $f_i(h_P,\theta,\varphi)$ are identifiable for it. The trials must run under documented environmental conditions — sea state, wave height and period, current and wind — that engage the wave-driven capability model of item (2), and must include a baseline comparison carried out on the same platform, the same reference trajectories, and the same documented conditions, so that any measured difference is attributable to the saturation-aware mechanism. (The source's own simulation comparison used a traditional sliding-mode controller and reported trans-media speed overshoot, actuator saturation duration, and tracking errors.)

Items (1) and (2) form the theoretical core; items (3) and (4) are its deployment and validation on the physical system class the controller was designed for. The problem is solved when all four hold for one and the same controller design on one and the same platform: the design certified in items (1) and (2) is the design implemented in item (3) and validated in item (4).

References

1. Wei, Tongjin; Bi, Yuanbo; Zeng, Zheng; Lian, Lian; Dan, Yangwen (2024). "Trans-Media Motion Control of a Hybrid Aerial Underwater Vehicle Considering Actuator Dynamic Input Saturation." Journal of Marine Science and Engineering 12(12):2169. DOI 10.3390/jmse12122169. https://doi.org/10.3390/jmse12122169
2. Wei, Tongjin; Li, Junping; Zeng, Zheng; Lian, Lian (2022). "Trans-media resistance investigation of hybrid aerial underwater vehicle base on hydrodynamic experiments and machine learning." Ocean Engineering 266:112808. DOI 10.1016/j.oceaneng.2022.112808. https://doi.org/10.1016/j.oceaneng.2022.112808
3. Wei, Tongjin; Zhou, Ziyi; Wang, Yingdong; Zeng, Zheng; Lian, Lian (2023). "Hydrodynamic study of hybrid aerial underwater vehicle in the wind, wave, and current environment." 2023 IEEE Underwater Technology (UT), pp. 1-6. DOI 10.1109/UT49729.2023.10103380. https://doi.org/10.1109/UT49729.2023.10103380
4. Jin, Yufei; Bi, Yuanbo; Lyu, Chenxin; Bai, Yulin; Zeng, Zheng; Lian, Lian (2024). "Nezha-IV: A hybrid aerial underwater vehicle in real ocean environments." Journal of Field Robotics 41(2):420-442. DOI 10.1002/rob.22274. https://doi.org/10.1002/rob.22274
5. Jin, Yufei; Zeng, Zheng; Lian, Lian (2026). "Nezha-SeaDart III: A VTOL tail-sitting vehicle with versatile dual-medium thrusters." Ocean Engineering 352:124619. DOI 10.1016/j.oceaneng.2026.124619. https://doi.org/10.1016/j.oceaneng.2026.124619
6. Wu, Shichong; Xian, Jun; Xie, Lingli (2025). "Adaptive asymptotic switched tracking control for a cross-media vehicle." Journal of the Franklin Institute 362:107750. DOI 10.1016/j.jfranklin.2025.107750. https://doi.org/10.1016/j.jfranklin.2025.107750
7. Wu, Jiayue; Dong, Fangfang; Zhao, Xiaomin; Chen, Ye-Hwa (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
8. Xu, Hengyang; Tan, Fuxiao (2026). "Phase-Aware ESO and PPO Residual Control for Trans-Medium Water Entry." 2026 5th Conference on Fully Actuated System Theory and Applications (FASTA), pp. 2152-2156. DOI 10.1109/FASTA70174.2026.11549436. https://doi.org/10.1109/FASTA70174.2026.11549436
9. Meng, Yuhang; Puig, Vicenç; Tang, Haoran; Wu, Dong; Ye, Hui; Yang, Xiaofei; Xiang, Zhengrong (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
10. Wang, RenKai; Shang, ZhiGang (2025). "Modeling and control of a low-cost multirotor hybrid aerial underwater vehicle." arXiv:2507.03869. https://arxiv.org/abs/2507.03869
11. Li, Zhitong; Zhang, Wanting; Huang, Xincheng; Ye, Xiufen; Chen, Yunsai; Jiang, Qinghua (2025). "Water-exit dynamics and system identification for a hybrid aerial underwater vehicle." Engineering Applications of Computational Fluid Mechanics 19:2512956. DOI 10.1080/19942060.2025.2512956. https://doi.org/10.1080/19942060.2025.2512956
12. Zhou, Biaojun; Zhao, Zijie; Zhang, Yufei; Wang, Anhua; Liu, Xiaohan; Dai, Qi (2026). "Aerial-aquatic vehicle transition dynamics of a cylindrical structure crossing a wave." Defence Technology. DOI 10.1016/j.dt.2026.05.017. https://doi.org/10.1016/j.dt.2026.05.017
13. Eom, Myeong-Jin; Jang, Yoon-Ho; Paik, Kwang-Jun (2021). "A study on the propeller open water performance due to immersion depth and regular wave." Ocean Engineering 219:108265. DOI 10.1016/j.oceaneng.2020.108265. https://doi.org/10.1016/j.oceaneng.2020.108265