The development of Vertical Takeoff and Landing (VTOL) unmanned aerial vehicles capable of operating from water surfaces presents significant advantages for missions such as maritime monitoring, ecological surveying, and offshore infrastructure inspection. Traditional approaches often utilize tilt-rotor or tilt-wing configurations, which introduce mechanical complexity and reliability concerns in the harsh marine environment. An alternative is the tilt-body concept, where the entire airframe rotates to transition between horizontal cruise and vertical hover. While promising, the design methodology for adapting a tilt-body VTOL drone for waterborne operations, particularly concerning the hydrodynamic hull design and its impact on flight stability, remains underdeveloped. This work establishes a comprehensive design and analysis framework to address these challenges.
The primary challenge for a waterborne VTOL drone is the vertical takeoff phase. Unlike a ground-based takeoff, the vehicle must overcome significant hydrodynamic moments generated by the buoyant hull as it pitches up from a resting attitude. An improperly designed hull can produce excessive nose-down pitching moments that exceed the control authority of the propulsion system, preventing a successful lift-off. Furthermore, the introduction of a hull necessary for water buoyancy can drastically alter the aerodynamic characteristics of the VTOL drone, especially in hover, potentially leading to lateral-directional static instability. This paper details a complete process encompassing hull optimization, coupled simulation of controlled water takeoff, stability analysis, and experimental validation.
1. Hydrodynamic Hull-Fuselage Optimization
The core of the waterborne adaptation is the design of the hull-fuselage, which provides buoyancy and must facilitate efficient vertical egress from the water. The design objective is to minimize the resisting hydrodynamic pitching moment during the takeoff sequence while satisfying buoyancy and aerodynamic stability constraints.
1.1 Geometric Parameterization and Optimization Framework
The three-dimensional hull shape is defined using a parametric approach. The longitudinal profile (XZ-plane) is described by a B-spline curve. A B-spline curve $C(u)$ of degree $p$ is defined by:
$$C(u) = \sum_{i=0}^{n} N_{i,p}(u)P_i$$
where $P_i$ are the control points, $N_{i,p}(u)$ are the B-spline basis functions, and $u$ is the knot vector. This formulation allows for local control and smooth shape variations. The cross-sectional shape at any longitudinal station $x$ is defined by another B-spline in the YZ-plane, whose width is constrained by a secondary curve representing the maximum beam.
The vertical takeoff maneuver is simplified for the optimization loop. The changing waterline during the pitch-up and lift-off is modeled as a time-dependent linear equation:
$$z_{water}(t) = a(t) \cdot x + b(t)$$
where $a(t)$ represents the pitch angle (slope) and $b(t)$ represents the height (intercept) of the vehicle’s reference point relative to the water surface. Coefficients $a(t)$ and $b(t)$ are derived from an idealized or initial CFD simulation of the takeoff trajectory. For any given hull geometry and a specific time $t$, the submerged volume $V_{sub}(t)$ and its center of buoyancy $(x_b(t), z_b(t))$ are calculated via numerical integration. The resulting buoyancy force $F_b(t) = \rho g V_{sub}(t)$ acts through the center of buoyancy, creating a pitching moment about the vehicle’s center of gravity (CG):
$$M_{b,\ pitch}(t) = F_b(t) \cdot (x_{cg} – x_b(t))$$
The optimization aims to minimize the hydrodynamic resistance to pitch-up. Therefore, the objective functions are defined as the average and the maximum of the absolute value of $M_{b,\ pitch}(t)$ over the discretized takeoff sequence:
$$\text{Minimize: } J_1 = \overline{|M_{b,\ pitch}|}, \quad J_2 = \max(|M_{b,\ pitch}|)$$
A multi-objective Pareto optimization is performed to find the best trade-off between these two goals.
1.2 Design Constraints and Optimization Results
The optimization is subject to several critical constraints:
- Buoyancy Constraint: The hull must provide sufficient displacement to support the vehicle’s weight at the resting attitude. $V_{sub}(t=0) \geq V_{min}$, where $V_{min}$ is calculated from the takeoff weight.
- Aerodynamic Stability Constraints: To preemptively address hover instability introduced by the hull, constraints on the lateral projected area $S_{side}$ and the hull height $z_{height}$ at cruise attitude are imposed. A large $S_{side}$ close to the CG can create destabilizing rolling moments in crosswinds.
$$S_{side} \leq S_{max}, \quad z_{height} \leq z_{max}$$
Two distinct optimal designs, resulting from different weighting of the objectives and constraints, are summarized below. Version 1 prioritizes minimal takeoff pitching moment, resulting in a shorter, taller hull. Version 2 incorporates stronger aerodynamic stability constraints, leading to a longer, lower-profile hull.
| Design Version | Submerged Volume $V_0$ (m³) | Hull Height $z_{height}$ (mm) | Lateral Area $S_{side}$ (cm²) |
|---|---|---|---|
| Optimization 1 (Min Moment) | 0.0072 | 343 | 171.5 |
| Optimization 2 (Balanced) | 0.0073 | 289 | 164.7 |
2. Coupled Controller-Fluid Dynamics Simulation of Water Takeoff
Analyzing the dynamic water takeoff of a tilt-body VTOL drone requires simulating the tight interaction between fluid forces, vehicle dynamics, and flight control. A traditional open-loop simulation with prescribed motion cannot capture the controller’s effort to counteract hydrodynamic disturbances. We developed a method that integrates a flight controller directly into a two-phase (water-air) CFD simulation.
2.1 Controller Integration within CFD
The vehicle is modeled as a 3-Degree-of-Freedom Dynamic Fluid Body Interaction (DFBI) object in STAR-CCM+, allowing motion in heave (Z), surge (X), and pitch (θ). The Eulerian Multiphase VOF model captures the air-water interface. The key innovation is the real-time calculation of control forces within each simulation time step.
A dual-loop PID controller is implemented using field functions. The outer loop controls pitch angle $\theta$ and altitude $Z$, while the inner loop controls pitch rate $q$. The control laws for the pitch channel are:
$$q_{des} = K_{P,\theta} \cdot (\theta_{cmd} – \theta)$$
$$u_{pitch} = K_{P,q} \cdot (q_{des} – q) + K_{I,q} \cdot \int (q_{des}-q) dt + K_{D,q} \cdot \frac{d(q_{des}-q)}{dt}$$
The control output $u_{pitch}$ is distributed to the differential thrust of the fore and aft propellers on the tandem wings of the tilt-body VTOL drone. The total thrust commands for the front ($T_f$), middle ($T_m$), and rear ($T_r$) propellers are:
$$T_f = k \cdot (throttle + u_{pitch}), \quad T_r = k \cdot (throttle – u_{pitch}), \quad T_m = k \cdot throttle \cdot f(\theta)$$
where $k$ is a scaling factor and $f(\theta)$ is a ramp function activating the middle motor at higher pitch angles. These forces and moments are applied as external inputs to the DFBI object at every time step, while the CFD solver computes the aerodynamic and hydrodynamic reactions. This closed-loop simulation reveals the true control effort required and allows for a comparative performance assessment of different hull designs under identical control logic.
2.2 Simulation Results and Analysis
The coupled simulations were run for both optimized hull versions. Figure 11 shows that both designs achieved the vertical takeoff maneuver under controller command. However, the decomposition of the total pitching moment acting on the vehicle reveals significant differences.
The total resisting moment $M_{total}$ is the sum of contributions from the hull-fuselage ($M_{hull}$), wings ($M_{wing}$), and floats ($M_{float}$): $M_{total} = M_{hull} + M_{wing} + M_{float}$. Analysis shows that the hull-fuselage moment $M_{hull}$ is the dominant resisting component during the early and mid-phase of takeoff, contributing approximately 45% of the total nose-down moment. This moment arises primarily from the vertical buoyancy force, not from hydrodynamic drag.
| Component | Optimization 1 (N·m) | Optimization 2 (N·m) | Notes |
|---|---|---|---|
| Hull-Fuselage ($M_{hull}$) | -4.8 | -5.3 | Dominant source, buoyancy-induced. |
| Wings ($M_{wing}$) | -1.1 | -1.0 | Aerodynamic damping due to pitch rate. |
| Total Resisting Moment ($M_{total}$) | -5.9 | -6.3 | Optimization 1 shows ~6% lower peak moment. |
The simulation confirms that the hull optimized for minimal pitching moment (Version 1) presents a 6-10% lower peak resisting moment compared to Version 2. This translates to a lower demand on the control system, allowing for faster pitch response and ascent, which is a critical performance metric for the waterborne VTOL drone.
3. Lateral-Directional Static Stability Analysis in Hover
Introducing a hull can adversely affect the aerodynamic stability of the VTOL drone in hover. A large lateral area near the CG can make the vehicle sensitive to crosswinds. We define a framework to analyze the static stability in hover, where the aircraft is pitched to 90° (nose-up).
3.1 Parameter Definitions for Hover Analysis
A wind azimuth angle $\phi_{wind}$ is defined relative to the hovering vehicle’s body frame. $\phi_{wind}=0^\circ$ corresponds to wind coming from directly beneath the fuselage (ventral), $\phi_{wind}=90^\circ$ from the left side, and $\phi_{wind}= \pm180^\circ$ from above (dorsal). The relevant forces and moments in the body frame are side force $Y$, rolling moment $L$, and yawing moment $N$.
Static stability in roll for a given $\phi_{wind}$ is determined by the sign of the rolling moment coefficient $C_l$. If a disturbance increases $\phi_{wind}$ (e.g., a gust from the left), the resulting rolling moment should be negative (right wing down) to counteract the disturbance and restore $\phi_{wind}=0^\circ$. This is termed “wind-seeking” behavior. Conversely, if the rolling moment is positive, it will drive the vehicle towards $\phi_{wind}=180^\circ$ (dorsal to wind), indicating static instability in the conventional flying attitude. Therefore, the stability criterion for hover can be stated: For a statically stable hover in a wind-seeking attitude ($\phi_{wind} \approx 0^\circ$), the rolling moment $L$ must have the opposite sign to the wind azimuth angle $\phi_{wind}$ for small disturbances.
$$ \text{Stable (Wind-Seeking): } \text{sign}(L) = -\text{sign}(\phi_{wind}) \quad \text{for } |\phi_{wind}| < 90^\circ$$
3.2 Stability Comparison of Hull Designs
Steady-state CFD simulations were conducted for both hull versions and a baseline land configuration across a full range of $\phi_{wind}$ from $-180^\circ$ to $+180^\circ$. The rolling moment $L$ on the complete vehicle was computed.
The baseline land configuration (no hull) exhibits classic wind-seeking stability for $|\phi_{wind}| < 90^\circ$: wind from the left ($\phi_{wind}>0$) produces a negative rolling moment, and vice-versa. For $100^\circ < |\phi_{wind}| < 180^\circ$, it shows a stable “dorsal-to-wind” attitude.
Optimization 1 (Tall Hull): This design exhibits a dramatic change. For $|\phi_{wind}| < 90^\circ$, the sign of $L$ is reversed compared to the baseline. Wind from the left now produces a positive rolling moment, which will roll the vehicle further, driving it away from the wind-seeking attitude and towards the stable dorsal-to-wind point ($\phi_{wind} \approx 180^\circ$). This indicates static instability for hover in the intended attitude. The tall hull acts as a large, destabilizing fin located near the CG.
Optimization 2 (Low Hull): By constraining the hull height and lateral area, this design largely preserves the wind-seeking stability characteristics of the baseline. The rolling moment sign convention for $|\phi_{wind}| < 90^\circ$ remains correct, indicating static stability. The magnitude of the stabilizing moment is slightly reduced but sufficient.
The analysis conclusively shows that hull geometry parameters like height $z_{height}$ are more critical to hover stability than lateral area $S_{side}$ alone, as they determine the lever arm of the side force. Optimization 2’s success demonstrates that integrating stability constraints into the hydrodynamic design phase is essential for a viable waterborne VTOL drone.
4. Flight Test Validation
4.1 Water Takeoff Performance
Prototypes were built based on Optimization 1 and Optimization 2 hull designs. Vertical takeoff tests from water were conducted. With near-identical control stick inputs for throttle and pitch, the Optimization 1 prototype demonstrated a faster pitch rate and achieved a higher altitude more quickly than the Optimization 2 prototype. Specifically, the pitch-up response was approximately 0.1 seconds faster, and the altitude difference at the end of the vertical climb phase was nearly 1 meter greater for Optimization 1. This validates the simulation prediction that the hull optimized for minimal pitching moment enables more efficient vertical water egress for the VTOL drone.
4.2 Hover Flight Stability
Hover flight tests, involving low-speed forward flight at high pitch angles (60°-80°), revealed stark differences in handling qualities. The Optimization 1 prototype exhibited persistent, slowly diverging oscillations in roll and yaw, requiring constant corrective input. In contrast, the Optimization 2 prototype maintained steady, controlled hover and responded predictively to pilot commands with only mild, damped oscillations. This behavior directly correlates with the stability analysis: the Optimization 1 hull induces lateral static instability in hover, manifesting as poor disturbance rejection and a tendency to weathercock to a dorsal-to-wind attitude, while the Optimization 2 design yields a stable and controllable VTOL drone platform.
5. Conclusion
This paper has presented a holistic design and analysis methodology for a waterborne tilt-body VTOL drone. The key contributions are:
- Hydrodynamic Hull Optimization: An automated optimization framework was developed, parameterizing the hull with B-splines and minimizing buoyancy-induced pitching moment during takeoff, a dominant factor accounting for ~45% of the total resisting moment.
- Coupled Simulation for Controlled Takeoff: A novel simulation technique integrating a flight controller into a two-phase CFD solver was established. This enables realistic assessment of different hull designs under closed-loop control, providing critical insights not available from open-loop analysis.
- Hover Stability Criterion and Analysis: A stability criterion for VTOL drones in hover was defined based on the relationship between rolling moment and wind azimuth angle. It was demonstrated that unconstrained hull design can lead to lateral static instability, and appropriate geometric constraints (on height and area) must be included in the optimization to ensure stable hover flight.
- Experimental Validation: Prototype flight tests confirmed the theoretical predictions. The hull optimized for takeoff performance (Opt. 1) showed superior water egress but poor hover stability, while the stability-constrained hull (Opt. 2) provided a balanced performance, successfully completing both stable hover and efficient water takeoff.
The integrated approach—from optimized design and high-fidelity coupled simulation to stability theory and flight test—provides a comprehensive and effective pathway for the development of capable and reliable waterborne VTOL unmanned aerial systems.