The 2018 Control Engineering Practice study of the Naviator prototype (Mercado Ravell, Maia and Diez) formulated the multirotor unmanned aerial/underwater vehicle as a hybrid dynamical system and proved global asymptotic trajectory tracking for a hybrid controller spanning air, water, and the interface crossing. The authors explicitly identified what the results did not cover: the controllers assume exactly known parameters and no environmental disturbances (wind, currents, model mismatch); drag and hydrodynamic parameters were illustrative constants never identified experimentally; actuator dynamics and the rotor spin-up transients that perturb attitude during transition-out were unmodeled; and the experiments were semi-autonomous and tethered, with horizontal position flown manually by a pilot because lightweight, flight-capable underwater position estimation did not exist. The subsequent literature has advanced each component separately — dozens of robust, adaptive, and learning-based cross-domain controllers (mostly simulation-validated), experimental trans-media hydrodynamic measurements and identification, actuator-saturation-aware transition control, and new prototypes including a field-tested fixed-wing vehicle — yet 2026 reviews still list unstable medium transition and insufficient autonomy among the field's key unsolved challenges, and no multirotor hybrid aerial–underwater vehicle has demonstrated autonomous closed-loop full-state trajectory tracking underwater and across the interface using onboard, GPS-denied sensing under realistic disturbances. Stating that integrated objective, faithful to the source, is the open problem recorded here.

Background

An unmanned aerial–underwater vehicle (also called a hybrid aerial underwater vehicle, HAUV, or aerial–aquatic vehicle) is a single robot designed to fly in air, propel itself underwater, and cross the water surface repeatedly, so that one platform can observe the air–sea interface and respond quickly in both media; early examples include the Loon Copter and the Rutgers Naviator. The core difficulty is that water is roughly a thousand times denser than air: forces negligible for an aerial vehicle — buoyancy (the upward force g(m−ρV)g(m-\rho V) on a body of mass mm and displaced volume VV in fluid density ρ\rho), quadratic drag, and added mass (extra effective inertia from fluid dragged along with the body) — dominate underwater, and the plant parameters change abruptly, not smoothly, when the interface is crossed.

The source paper (Mercado Ravell, Maia and Diez, Control Engineering Practice 76 (2018) 112–122) modeled this multi-medium control problem in the hybrid dynamical systems framework of Goebel, Sanfelice and Teel, where a system has continuous evolution (flows) in each discrete mode plus instantaneous jumps between modes. The rigid-body Newton–Euler equations are written per medium with buoyancy and quadratic drag included, the medium density ρ∈{ρA,ρW}\rho\in\{\rho_A,\rho_W\} is a discrete state that jumps at the interface, and a transition zone with hysteresis (a dead band that keeps the mode from re-switching immediately at the boundary, preventing chattering) bounds the switching. Within one medium, a feedback-linearizing translational controller (feedback linearization: cancelling the known nonlinear model terms so the closed-loop error dynamics become linear with assignable gains) with explicit buoyancy and drag compensation, plus a PID-type attitude law with drag compensation, tracks trajectories; a dedicated transition controller with a nominal-thrust condition enforces a single directed crossing of the transition zone. A hybrid Lyapunov function together with the LaSalle invariance principle (a convergence argument based on the largest invariant set on which the Lyapunov function stops decreasing, extended to hybrid flows and jumps) gives global asymptotic tracking of the closed hybrid system — under exactly known parameters, no external disturbances, no actuator dynamics, and reference trajectories that stay out of the transition zone.

The experimental platform, the Naviator, was an X8 octo-quadcopter (eight rotors in four coaxial pairs, all used in both media) whose electronic speed controllers (ESCs, the power electronics commanding motor speed) were re-firmware'd to run at the very low speeds viable underwater, with a waterproof pressure vessel, water-detection sensors, and a pressure sensor for depth. The tests demonstrated air–water–air operation with the same rotors in both fluids, but were semi-autonomous: depth and attitude were stabilized onboard while the horizontal position was flown manually by a pilot, telemetry ran through a 30 m tether, and the experiments exposed rotor spin-up transients that perturbed attitude during transition-out and pressure-sensor accuracy differences between air and water.

The authors' conclusions list what remains to be done: (1) the proofs assume known parameters and no wind, currents, or model mismatch, so robust extensions are required; (2) the drag coefficients and several hydrodynamic parameters used in simulation were illustrative constants, not experimentally identified for the platform; (3) actuator dynamics and smoother rotor ramping must be included to avoid transition-out attitude perturbations; (4) autonomous full-state trajectory tracking is blocked by the absence of lightweight, flight-capable underwater position estimation — GPS is unavailable underwater and acoustic or vision-based solutions optimized for combined air/underwater operation remained open; (5) robust, adaptive, or observer-based control schemes and their experimental evaluation were left for future work.

Since 2018 this has become an active subfield, surveyed in 2022 and 2026. On robustness and uncertainty: adaptive dynamic surface control (a step-by-step adaptive backstepping design with filtered intermediate commands) for a HAUV with parametric uncertainty (Lu et al., 2020); second-order sliding-mode control under periodic wave forces (sliding-mode control, SMC, is a nonlinear design that constrains the system to a switching surface for robustness; Bi et al., 2022); adaptive finite-time tracking under unknown environmental disturbances (Lu et al., 2023); reinforcement-learning finite-time tracking under unknown dynamics and disturbances (Wu et al., 2024); active disturbance rejection control (ADRC, which estimates the lumped total disturbance with an observer and cancels it in the control law) with swarm-optimized parameter tuning (Li et al., 2024 — validated in simulation only); barrier-function-based SMC (a sliding-mode law augmented by a barrier function that keeps the state within prescribed bounds) under model uncertainty and disturbance (Lee and Kim, 2026); and a wave-adaptive submergence strategy for sea states (Han et al., 2026). Quaternion-based geometric controllers (attitude laws written directly on the vehicle's rotation group rather than through Euler angles) resolve the 180°-turnover attitude singularity that buoyancy induces underwater and give globally asymptotically stable cascaded cross-domain tracking (Ma et al., 2024; Ma et al., 2025), again in numerical simulation. On model validation: dedicated rigs have measured trans-media hydrodynamic forces on HAUV hulls (Wei et al., 2022) and learned multirotor trans-media resistance from experiments (Wei et al., 2022); system identification based on CFD (computational fluid dynamics) covers the water-exit process including retained-water effects (Sun et al., 2024; Li et al., 2025; Li et al., 2026); noise-reduction least-squares identification drives diving control for underwater vehicles (Zhong et al., 2025). On actuation: a trans-media model with actuator dynamic input saturation built from measured data, with Lyapunov-proven control and simulation validation, whose practical deployment its authors state as future work (Wei et al., 2024). On platforms: the TJ-FlyingFish line of aerial–aquatic quadrotors with tiltable propulsion units for underwater maneuverability (Liu et al., 2023; Huang et al., 2023); the morphable Puffin Platform (Rao et al., 2024); a low-cost multirotor with twisting sliding-mode attitude control through repeated water crossings (Wang and Shang, 2025); the tail-sitting fixed-wing Nezha-SeaDart, field-tested over ten days with autonomous seamless water exit (Jin et al., 2025); and a positive-buoyancy amphibious robot with a nonlinear disturbance observer demonstrating robust repeated transitions experimentally (Zhong et al., 2026). For underwater state estimation itself, low-cost camera-plus-IMU (inertial measurement unit) visual-inertial autonomy has been field-demonstrated on a dedicated underwater robot (Singh et al., 2025) but has not been integrated on any flight-capable cross-domain platform.

The field's reviews frame what is still missing: the 2022 review of cross-domain mobility and transitions control (Zeng et al., 2022) identifies the control system as the core enabler and persistent autonomy as the open challenge; the 2026 Journal of Field Robotics transition-strategy survey (Liu et al., 2026) reports that medium transition remains inflexible and unstable with transition-process stability control still to be solved; the 2026 Ocean Engineering review (Meng et al., 2026) lists the lack of unified modeling frameworks, limited energy efficiency, and insufficient autonomy in complex environments as key open challenges.

Problem Statement

For a multirotor-class hybrid aerial–underwater vehicle that uses the same rotors in air and in water, design and experimentally demonstrate an estimation-and-control system that removes the assumptions under which the original hybrid tracking guarantees were proved — exactly known parameters, absence of environmental disturbances, ideal actuators, and semi-autonomous operation. Concretely, the vehicle must autonomously track full-state reference trajectories (position and attitude) in air, underwater, and through repeated air–water transitions, where: (i) the closed loop remains stable and retains tracking performance under environmental disturbances — wind above the surface, waves and currents below — and under model-parameter uncertainty (drag, buoyancy, added-mass and inertia) rather than exactly known parameters; (ii) the hydrodynamic parameters used by the controller are experimentally identified for the physical platform rather than chosen for illustration; (iii) actuator dynamics — motor spin-up lag during transition-out, input saturation, and the differing rotor operating ranges in the two media — are modeled and compensated so that rotor ramping does not destabilize attitude when leaving the water; and (iv) the full vehicle state is estimated onboard without GPS, without a tether, and without external tracking, using sensing lightweight enough that the vehicle still flies; external instrumentation is permitted only as ground truth for evaluation, not as feedback. These four requirements are components of one objective — closing the loop from the 2018 semi-autonomous, tethered demonstrations to fully autonomous cross-domain trajectory tracking — and the completion boundary is an experimental demonstration on a physical multirotor hybrid aerial–underwater vehicle executing autonomous air-to-underwater-to-air mission cycles with quantitative tracking-performance reporting — the achieved position and attitude tracking errors, the disturbance conditions under which they were measured, and the number and duration of completed transitions — together with the stability and robustness analysis of the combined controller and estimator. A design validated only in numerical simulation, or a demonstration in which horizontal or depth control is flown manually or measured by external instrumentation in the feedback loop, does not complete this problem.

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