# A 4 foils hydrofoiler regulation solution and experiments

Lionel Lapierre, Laurent Latorre, Loic Daridon, Denis Julien, Vincent Dufour

► To cite this version:

Lionel Lapierre, Laurent Latorre, Loic Daridon, Denis Julien, Vincent Dufour. A 4 foils hydrofoiler regulation solution and experiments. Applied Ocean Research, 2025, 160, pp.104596. 10.1016/j.apor.2025.104596. hal-05306012

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# A 4 foils hydrofoiler regulation solution and experiments

Lionel Lapierre<sup>a,b ID,*</sup>, Laurent Latorre<sup>b</sup>, Loic Daridon<sup>c ID</sup>, Denis Julien<sup>d</sup>, Vincent Dufour<sup>d</sup>

## ARTICLE INFO

**Keywords:**

Autonomous surface Craft

Foil regulation

Marine robotics

## ABSTRACT

This paper introduces the Black Pearl, a novel 4-foil electric hydrofoiler, by presenting its design, control structure, and experimental findings. The innovative aspect of our approach lies in the redundant design of the sustentation structure, which utilizes four independent foils to control three degrees of freedom: roll, pitch, and altitude above the water.

This redundancy improves actuation efficiency, significantly reducing the risk of actuation saturation. Moreover, the explicit management of this redundancy ensures robust actuation performance, even in scenarios where one foil becomes locked at a fixed angle.

The paper also presents experimental trials conducted on the Black Pearl prototype, as well as the proposal and simulation testing of novel guidance laws designed to enhance the hydrofoiler's flight performance.

## 1. Introduction

The idea of using a lifting wing with a asymmetrical profile has driven the early development of aviation but was also quickly tested in a denser fluid: water. The first documented use of this concept dates back to 1861, when Thomas Moy experimented with a towed structure featuring three lifting underwater surfaces. He observed the effects of lift and drag reduction and their relationship to flow velocity (Monsonnec, 2020). The regulation of the lifting effect, which remains a key challenge, was initially addressed with self-regulating static structures. These included “ladder”, “V”, “L”, “J”, or “Y” shapes, where the lifting effect diminishes as the submerged portion of the foiling structure decreases. However, these static designs are suboptimal, as they generate internal forces and require structural elements that introduce additional drag. Consequently, the ‘T’ foil has been identified as the optimal structure, though it requires active regulation. This can be achieved using a frontal sensor wand, which adjusts the foil’s angle of attack through a passive elastic transmission system, as demonstrated in the Moth hydrofoiler (Day et al., 2019). As size and performance objectives increased, the need for electronic control became evident. A first approach involves regulating the incidence of a double-flap stern foil to control altitude and pitch, with a passive foiling structure installed at the bow (Sang-Hyun and Hiroyuki, 2004; AlaviMehr et al., 2016), activated when the boat velocity reached a prescribed value. Another approach consists of using fixed foils at stern and regulating double-flap bow foils (Zong et al., 2019). The control objective of these

latest references was to reduce the vertical acceleration of the vessel in rough seas, as shown in Esteban et al. (2000), or to decrease the wet surface area (Suastika et al., 2022), thereby reducing boat resistance by up to 30% when the position and profile of the foil are optimized with respect to boat dynamics (Shen et al., 2022). Achieving full regulation of attainable degrees of freedom, including altitude above water, roll, and pitch, while accommodating different velocities, requires, at least, the regulation of the angle of attack of three independent foiling structures (Hörberg and Lundgren, 2023). Recently, considerable effort has been made on the question of optimizing the foil profile using a CFD approach to maximize the lifting effect while minimizing the drag component, as shown in Rayhan et al. (2023), or to actuate the surface with an oscillatory behavior as an efficient thrust mean in waves (Lopes et al., 2023).

The control algorithm and its guidance strategy greatly influence hydrofoiler performance. The significant contribution of the guidance system to a mobile robot's performance is evident in recent advances discussed in the literature, particularly in the context of the path-following control problem. For instance, as demonstrated in Lapierre and Soetanto (2007), these advancements help mitigate intrinsic singularities initially identified in Samson and Ait-Abderrahim (1991), explicitly consider actuator saturation as discussed in Lapierre and Indiveri (2007), and integrate these benefits with terrain constraints such as obstacle avoidance (Lapierre and Zapata, 2012). More recent

<footnote><sup>a</sup> ENSTA Bretagne, 2 rue François Verny, 29200 Brest, France</footnote>

<footnote><sup>b</sup> LIRMM, 161 rue Ada, 34095 Montpellier, France</footnote>

<footnote><sup>c</sup> LMGC, 161 rue Ada, 34095 Montpellier, France</footnote>

<footnote><sup>d</sup> Neocean, 13 rue d'Amsterdam, 34200 Sète, France</footnote>

<footnote>* Corresponding author at: ENSTA Bretagne, 2 rue François Verny, 29200 Brest, France.</footnote>

<footnote>E-mail addresses: lionel.lapierre@ensta-bretagne.fr, lionel.lapierre@ensta.fr, lapierre@lirmm.fr, lionel.lapierre@umontpellier.fr (L. Lapierre).</footnote>
approaches proposes to use machine learning to solve the guidance and control problem of multiple ocean vehicles for target tracking, (Song et al., 2022; Rout et al., 2020), or review the existing Line-of-Sight approaches for marine vehicles (Gu et al., 2023). As explained in the following, we will use the guidance sub-system to compute an error function that directs the hydrofoiler system through its planing phase and regulates the roll angle while turning, similar to the banking behavior in airplanes.

This paper presents the design and control structure of the new 4-foil electric hydrofoiler, named Black Pearl. The novelty of our approach lies in the redundant design of the sustentation structure of the Black Pearl: 4 independent foils to regulate 3 degrees of freedom, namely roll, pitch, and altitude above water. As shown in Fig. 2, the foils are mounted at the bottom of a 70 cm-long profiled leg, which is articulated relative to the hull. The angulation of each of these four legs (relative to the hull) is controlled by four independent servo motors, thereby providing a four-dimensional actuation system to regulate three degrees of freedom

As demonstrated later in this paper, this redundancy enables more efficient actuation allocation, reducing actuation saturation and improving responsiveness. Moreover, the explicit management of this redundancy provides actuation robustness in situations where one foil is blocked at a fixed angulation. Experiments were conducted on the Black Pearl prototype, and new guidance laws to enhance its flight were presented and tested in simulations. This study represents an initial step towards the broader goal of developing regulated hydrofoils capable of navigating rough seas. We believe that incorporating a redundant foiling actuation structure will be highly beneficial.

The paper is organized as follows. Section 2 introduces the notation used in the paper, as well as the dynamic and actuation models of the Black Pearl craft. Section 3 proposes the control structure and details the design of guidance laws. Section 4 presents some simulation results that illustrate the solution's performance and robustness. Section 5 presents experimental results that confirm the expected system performance. The final section concludes this work and discusses further developments.

## 2. System description

In the sequel, 'bold' notation is reserved for vectors and matrices, while scalars are written classically. Let {$U$} be the NED (North-East-Down) universal coordinate frame and {$B$} be the marine body frame, attached to the center of gravity of the system. We also define a set of body-frames attached to each element $i$ of the system, called element-frames and denoted {$B_i$}. In the sequel, the indices $S$ and $P$ stand for 'Starboard' and 'Port-board' and $R$ ans $F$ mean 'Rear' and 'Front', respectively.

The ASC (Autonomous Surface Craft), named 'Black Pearl', is composed of 14 different elements (cf. Fig. 1):

* Two hulls, denoted $\mathcal{H}_S$ and $\mathcal{H}_P$, whose gravity centers are located in {$B$} as $\mathbf{X}_{\mathcal{H}_S}^G$ and $\mathbf{X}_{\mathcal{H}_P}^G$.

* Four vertical legs, denoted $\mathcal{L}_{F,S}$, $\mathcal{L}_{R,S}$, $\mathcal{L}_{F,P}$, and $\mathcal{L}_{R,P}$, with their respective foils, denoted $\mathcal{F}_{F,S}$, $\mathcal{F}_{R,S}$, $\mathcal{F}_{F,P}$, and $\mathcal{F}_{R,P}$, rigidly attached at their respective leg's bottom. These legs are articulated relative to the hulls, along the $v$ axis of {$B$} with angles $\delta_{F,S}$, $\delta_{R,S}$, $\delta_{F,P}$, and $\delta_{R,P}$. These articulations are located in {$B$} at $\mathbf{X}_{\mathcal{F},S}^A$, $\mathbf{X}_{\mathcal{F},P}^A$ and $\mathbf{X}_{\mathcal{R},P}^A$. The position of the center of mass of the 4 legs, in {$B$}, are expressed as $\mathbf{X}_{\mathcal{L}_{F,S}}^G(\delta_{F,S})$, $\mathbf{X}_{\mathcal{L}_{R,S}}^G(\delta_{R,S})$, $\mathbf{X}_{\mathcal{L}_{F,P}}^G(\delta_{F,P})$ and $\mathbf{X}_{\mathcal{L}_{R,P}}^G(\delta_{R,P})$. The center of mass of the 4 foils are denoted $\mathbf{X}_{\mathcal{F}_{F,S}}^G(\delta_{F,S})$, $\mathbf{X}_{\mathcal{F}_{R,S}}^G(\delta_{R,S})$, $\mathbf{X}_{\mathcal{F}_{F,P}}^G(\delta_{F,P})$ and $\mathbf{X}_{\mathcal{F}_{R,P}}^G(\delta_{R,P})$.

* A rudder, denoted $\mathcal{R}$, is articulated along the $w$ axis of {$B$} with an angle $\delta_R$. The position of the rudder's articulation is expressed as $\mathbf{X}_R^A$, in {$B$}. The rudder's center of mass is located as $\mathbf{X}_R^G(\delta_R)$, in {$B$}.

* A Thruster $\mathcal{T}$, rigidly attached at the bottom of a static vertical leg $\mathcal{L}_T$, producing a thrust $T$. The center of mass of the leg is located at $\mathbf{X}_{\mathcal{L}_T}^G$, in {$B$} and the center of mass of the thruster is located at $\mathbf{X}_T^G$, which is also the application point of the thrust force $T$, of norm $T$.

* A load, denoted $\mathcal{L}$, representing the equivalent inertial effect of other system components (electronics, batteries, wires...) is mounted on the system at point $\mathbf{X}_\mathcal{L}^G$.

We also define a set of 14 element frames {$B_i$}, where

$$i = \{\mathcal{H}_S, \mathcal{H}_P, \mathcal{F}_{F,S}, \mathcal{F}_{R,S}, \mathcal{F}_{F,P}, \mathcal{F}_{R,P}, \mathcal{L}_{F,S}, \mathcal{L}_{R,S}, \mathcal{L}_{F,P}, \mathcal{L}_{R,P}, \mathcal{R}, \mathcal{L}, \mathcal{L}_T, \mathcal{T}\} \quad (1)$$

attached to the center of mass of each system's elements and described at Fig. 1. The control variables are represented by the vector $\zeta$ as:

$$\zeta = [\delta_{F,S}, \delta_{R,S}, \delta_{F,P}, \delta_{R,P}, \delta_{R}, T]^T \quad (2)$$

### 2.1. Notation

In the sequel, $\eta$ denotes the system's situation (position, denoted $\mathbf{X}$, and attitude, denoted $\mathbf{\Theta}$) in {$0$}, $v$ is the system's velocities in {$B$}, where $u, v, w$ are surge, sway and heave velocities and $p, q, r$ denotes roll, pitch and yaw angular velocities in the body frame, as stated on Eqs. (3) and illustrated at Fig. 1.

$$\left\{ \begin{array}{l} \mathbf{X} = [x, y, z]^T \\ \mathbf{\Theta} = [\phi, \theta, \psi]^T \\ \mathbf{\eta} = [\mathbf{X}^T, \mathbf{\Theta}^T]^T \end{array} \right. , \quad \left\{ \begin{array}{l} \mathbf{v} = [u, v, w]^T \\ \mathbf{\omega} = [p, q, r]^T \\ \mathbf{v} = [v^T, \omega^T]^T \end{array} \right. \quad (3)$$

Let $\mathbf{R}_A^B(\phi, \theta, \psi)$ denote the 3D rotation matrix from frame {$A$} to frame {$B$} with a relative attitude denoted by Euler angles $\phi$ (roll), $\theta$ (pitch) and $\psi$ (yaw). Note that $\mathbf{R}_A^B$ is generally invertible and its inverses is denoted $\mathbf{R}_B^A = (\mathbf{R}_A^B)^{-1}$. The jacobian relation that links the Euler angle velocity $(\phi, \theta, \psi)$ with body-frame rotational velocity $w$ is denoted $\mathbf{J}_A^B(\phi, \theta)$, which remains well-posed since $|\theta| < \pi/2$, as the regulation, defined and experimented hereafter, will show.

$$\mathbf{R}_B^0(\mathbf{\Theta}) = \begin{bmatrix} \cos \psi \cos \theta & \cos \psi \sin \theta \sin \phi - \sin \psi \cos \phi & \sin \psi \sin \theta + \cos \psi \sin \theta \cos \phi \\ \sin \psi \cos \theta & \sin \psi \sin \theta \sin \phi + \cos \psi \cos \phi & \sin \psi \sin \theta \cos \phi - \cos \psi \sin \phi \\ -\sin \theta & \cos \theta \sin \phi & \cos \theta \cos \phi \end{bmatrix} \quad (4)$$

$$\mathbf{J}_B^0(\mathbf{\Theta}) = \begin{bmatrix} 1 & \sin(\phi) \tan(\theta) & \cos(\phi) \tan(\theta) \\ 0 & \cos(\phi) & -\sin(\phi) \\ 0 & \frac{\sin(\phi)}{\cos(\theta)} & \frac{\cos(\phi)}{\cos(\theta)} \end{bmatrix} \quad (5)$$

The system's kinematic model is classically written as:

$$\dot{\eta} = \begin{bmatrix} \mathbf{R}_B^0(\mathbf{\Theta}) & \mathbf{0}_{3 \times 3} \\ \mathbf{0}_{3 \times 3} & \mathbf{J}_B^0(\mathbf{\Theta}) \end{bmatrix} \cdot \mathbf{v} \quad (6)$$

where $\mathbf{0}_{i \times j}$ denotes the null ($i \times j$) matrix. Since $\mathbf{J}_B^0(\mathbf{\Theta})$ and $\mathbf{R}_B^0(\mathbf{\Theta})$ are invertible, i.e. $\mathbf{J}_B^0(\mathbf{\Theta}) = (\mathbf{J}_B^0(\mathbf{\Theta}))^{-1}$ and $\mathbf{R}_B^0(\mathbf{\Theta}) = (\mathbf{R}_B^0(\mathbf{\Theta}))^{-1}$, we can write the inverse kinematic relations:

$$\begin{aligned} \mathbf{v} &= \mathbf{R}_B^B(\mathbf{\Theta}) \cdot \dot{\mathbf{X}} \\ \mathbf{\omega} &= \mathbf{J}_B^0(\mathbf{\Theta}) \cdot \dot{\mathbf{\Theta}} \end{aligned} \quad (7)$$

We also need to write the acceleration relations:

$$\dot{\mathbf{v}} = \mathbf{R}_0^B(\mathbf{\Theta}) \cdot \ddot{\mathbf{X}} + \dot{\mathbf{R}}_0^B(\mathbf{\Theta}) \cdot \dot{\mathbf{X}} \quad (8)$$

$$\dot{\mathbf{\omega}} = \mathbf{J}_0^B(\mathbf{\Theta}) \cdot \ddot{\mathbf{\Theta}} + \dot{\mathbf{J}}_0^B(\mathbf{\Theta}) \cdot \dot{\mathbf{\Theta}} \quad (9)$$

At last, $\mathbf{I}_d$ denotes the ($i \times i$) identity matrix and $\mathbf{S}([x, y, z]^T)$ is the skew symmetrical matrix:

$$\mathbf{S}([x, y, z]^T) = \begin{bmatrix} 0 & z & -y \\ -z & 0 & x \\ y & -x & 0 \end{bmatrix} \quad (10)$$
<figure><img src="image_3.png" /><figcaption>Fig. 1. Notation and frames definition.</figcaption></figure>

## 2.2. Dynamic modeling

A simplified dynamic model of the system is introduced for control design and initial validation through simulations. This modeling effort identifies the appropriate guidance and control structures and highlights the effects of actuation on the system's dynamics. The final validation of the approach is conducted through sea trials, as discussed in Section 5.

### 2.2.1. Gravity and buoyancy

The global system's gravity center location $\mathbf{X}^G$ and the resulting weight force $\mathbf{G}$, both expressed w.r.t. $\{B\}$, can be computed as follows

and is illustrated at Fig. 2:

$$ \mathbf{X}^G = \frac{\sum_i (m_i \cdot \mathbf{X}_i^G)}{M}, \quad \mathbf{G} = \mathbf{R}_0^B \cdot [0, 0, M \cdot g]^T \quad (11) $$

where $g = 9.81 \text{ m/s}^2$ is the gravitational acceleration, $m_i$ denotes the mass of the element $i$ and $M = \sum_i m_i$ where $i$ is defined at Eq. (1).

Buoyancy acts on the immersed part of all the elements $i$ of the system. It is expressed as the force of norm $b_i = v_i \cdot \rho \cdot g$, where $v_i$ denotes the volume of the immersed part of the element $i$ and $\rho$ is the water density. Its action point is located at the volumetric center of the immersed part of the element $i$ and is denoted $\mathbf{X}_i^B$, where $i$ is defined at Eq. (1). Hence the buoyancy can be computed as:

$$ \mathbf{X}^B = \frac{\sum_i (b_i \cdot \mathbf{X}_i^B)}{B}, \quad \mathbf{B} = \mathbf{R}_0^B \cdot [0, 0, -B]^T \quad (12) $$
<figure><img src="image_4.png" /><figcaption>a) Gravity and buoyancy centers definition</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>b) Foil's parameters definition</figcaption></figure>

<caption>Fig. 2. Forces applying on the system.</caption>

where $B = \sum_i b_i$.

The immersed volume of each component that may intersect the air-water interface is computed using an approximate geometric model: cylinders for the hulls and elliptical cylinders for the legs. This approach offers the advantage of an analytical solution for calculating the immersed volume and the center of buoyancy of each component, i.e., the volumetric center of the submerged part.

### 2.2.2. Hydrodynamic damping and thrust

Assuming that each system element *i* undergoes laminar interaction with the fluid, the hydrodynamic damping effect can be decomposed into two components: drag and lift. Drag acts in the opposite direction of the flow while Lift lives perpendicularly to it. For each element *i*, the magnitude of both these forces, denoted $D_i$ and $L_i$ depend on:

* The relative velocity of the water with respect to the immersed part of the element *i*. Considering that hydrodynamic forces apply at the center of buoyancy of *i*, we state that the relevant hydrodynamic velocity of *i* is denoted $v_i = [u_i, v_i, w_i]^T$, in $\{B\}$, with $v_i = v + \omega \times X_i^B$. This element velocity, expressed in the local element-frame $\{B_i\}$ is denoted $v_i^B = [u_i^B, v_i^B, w_i^B]^T$, allowing to define attack angle $\alpha_i = \arctan(w_i^B/u_i^B)$ and side-slip angle $\beta_i = \arctan(v_i^B/u_i^B)$, as illustrated in Fig. 2.

We notice that:

- for the elements that are rigidly attached to the system, i.e. $j = \{H_S, H_P, L, L_T, T\}$, $v_j^B = R_B^B \cdot v_j$, with $R_B^B = \mathbf{1}_3$ and where $\mathbf{1}_3$ denotes the $(3 \times 3)$ identity matrix.

- For the legs of the foils and the foils, i.e. $k = \{F_{F,S}, F_{R,S}, F_{F,P}, F_{R,P}, L_{F,S}, L_{R,S}, L_{F,P}, L_{R,P}\}$, $v_k^B = R_B^B \cdot v_k$, where $R_B^B = R_B^B(0, \delta_k, 0)$.

- For the rudder, $v_R^B = R_B^B \cdot v_R$, where $R_B^B = R_B^B(0, 0, \delta_R)$.

* Drag effect on element *i* depends on its shape, attack angle $\alpha_i$ and side-slip angle $\beta_i$ with respect to the flow direction. This dependance is captured within the hydrodynamic diagonal coefficients matrix $C_i(\alpha_i, \beta_i)$, decomposing the damping action along each components of $v_i^B$, i.e.

$$ C_i = \begin{bmatrix} C_i^u & 0 & 0 \\ 0 & C_i^v & 0 \\ 0 & 0 & C_i^w \end{bmatrix} \qquad (13) $$

where $C_i^u$ is the drag coefficient of element *i* and $C_i^v$ and $C_i^w$ are either drag or lift coefficient, according to the function and situation of element *i*, as described at Fig. 2 and Table 3.

* Hydrodynamic damping force, denoted $H_i$, applies on each system component at their buoyancy center $X_i^B$ as:

$$ X_i^B, \quad H_i = -1/2 \cdot \rho \cdot S_i \cdot R_{B_i}^B \cdot C_i \cdot \begin{bmatrix} |u_i^B| & 0 & 0 \\ 0 & |v_i^B| & 0 \\ 0 & 0 & |w_i^B| \end{bmatrix} \cdot v_i^B \qquad (14) $$

where $S_i$ denotes the immersed surface of organ *i*.

* Thrust $T$ induced by the thruster, and its input $c_T$, acts at point $X_T^G$. The thruster's characteristic $T(c_T)$ is given below.

$$ X_T^G, \quad T = [T, 0, 0]^T \qquad (15) $$

Hence, total resulting force undergone by the system is expressed as vector $\mathbf{F}$:

$$ \mathbf{F} = \mathbf{G} + \mathbf{B} + \mathbf{T} + \sum_i \mathbf{H}_i \qquad (16) $$

These forces generate torques with respect to the gravity center, resulting in vector $\Gamma$, expressed in $\{B\}$ as:

$$ \Gamma = \mathbf{B} \times (\mathbf{X}^B - \mathbf{X}^G) + \mathbf{T} \times (\mathbf{X}_T^G - \mathbf{X}^G) + \sum_i (\mathbf{H}_i \times (\mathbf{X}_i^B - \mathbf{X}^G)) \qquad (17) $$

for *i* defined at Eq. (1). Finally, according to the classic Fossen's formalism (Fossen, 2011), system dynamic model is written with respect to the body frame $\{B\}$ as

$$ \begin{bmatrix} \mathbf{F} \\ \Gamma \end{bmatrix} = \mathbf{M}(\mathbf{X}^G) \cdot \dot{\mathbf{v}} + \mathbf{C}(\mathbf{v}) \cdot \mathbf{v} \qquad (18) $$

where $\mathbf{F} = [F_u, F_v, F_w]^T$, $\Gamma = [\Gamma_p, \Gamma_q, \Gamma_r]^T$ and $\mathbf{M}$ and $\mathbf{C}$ are defined according to:

$$ \mathbf{M} = M \cdot \begin{bmatrix} M \cdot \mathbf{I}_3 & -M \cdot \mathbf{S}(\mathbf{X}^G) \\ M \cdot \mathbf{S}(\mathbf{X}^G) & \mathbf{I}/M \end{bmatrix} \qquad (19) $$

$$ \mathbf{C} = M \cdot \begin{bmatrix} \mathbf{S}(\omega) & -\mathbf{S}(\omega) \cdot \mathbf{S}(\mathbf{X}_G) \\ -\mathbf{S}(\omega) \cdot \mathbf{S}(\mathbf{X}_G) & -\mathbf{S}(\mathbf{I} \cdot \omega)/M \end{bmatrix} \qquad (20) $$

where $\mathbf{S}(\omega)$ denotes the skew-symmetric matrix defined in Eq. (10) and $\mathbf{I} = \text{diag}(I_x, I_y, I_z)$ is the (assumed diagonal) inertia matrix. Note that the added-mass effect has been neglected here, based on the argument that the inertia of the displaced volume of water is negligible compared to the system's dry mass and inertia, especially when the vehicle is foiling.
## 2.3. Modeling the actuation system

The actuation system relates forces and torques generated by the 6 actuators (4 foils, 1 rudder and 1 thruster) with the resulting action on the controlled degrees of freedom, expressed in the body frame. We choose to dispatch the control problem as follows:

* the 4 foils angulation regulates the system altitude (distance above water) roll and pitch angles.

* the rudder regulates the yaw angle,

* the thruster acts along u axis of {B} and is set to a constant reasonable regime.

Note that the dynamics of the actuators are not considered in the system model. Regarding the main thruster, the effect of its time response is considered negligible because, in the following simulations and experimental sea trials, the thruster will be operated in open-loop with a constant setting. However, the response time of the actuators controlling the foils' angulation is more significant. Identifying the appropriate servos has been a key focus, as discussed in Section 5. Nevertheless, the response time of the foils' actuators is not accounted for in the simulation.

We extract from the dynamic model (Eqs. (16) and (17)) the terms related to the specific actions of the actuators along the regulated degrees of freedom.

$$ 
\begin{aligned}
\mathbf{F} &= \mathbf{F}_0 + \mathbf{F}_F + \mathbf{T} \\
\Gamma &= \Gamma_0 + \Gamma_F + \Gamma_R
\end{aligned} 
\quad (21) 
$$

where

$$ 
\begin{aligned}
\mathbf{F}_0 &= \mathbf{G} + \mathbf{B} + \sum_j \mathbf{H}_j + \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \cdot \sum_f \mathbf{H}_f \\
\Gamma_0 &= \mathbf{B} \times (\mathbf{X}^B - \mathbf{X}^G) + \mathbf{T} \times (\mathbf{X}_T^G - \mathbf{X}^G) + \sum_k (\mathbf{H}_k \times (\mathbf{X}_k^B - \mathbf{X}_G)) + \\
& \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \cdot \sum_f (\mathbf{H}_f \times (\mathbf{X}_f^B - \mathbf{X}_G)) \\
& \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \cdot (\mathbf{H}_R \times (\mathbf{X}_R^B - \mathbf{X}_G))
\end{aligned} 
\quad (22) 
$$

and

$$ 
\begin{aligned}
\mathbf{F}_F &= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \cdot \sum_f \mathbf{H}_f = [0, 0, F_F^w]^T \\
\Gamma_F &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \cdot \sum_f (\mathbf{H}_f \times (\mathbf{X}_f^B - \mathbf{X}_G)) = [\Gamma_F^p, \Gamma_F^q, 0]^T \\
\Gamma_R &= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \cdot (\mathbf{H}_R \times (\mathbf{X}_R^B - \mathbf{X}_G)) = [0, 0, \Gamma_R^T]^T \\
\mathbf{F}_T &= [F_T^u, 0, 0]^T
\end{aligned} 
\quad (23) 
$$

where $j = \{\mathcal{H}_S, \mathcal{H}_P, \mathcal{L}_{F,S}, \mathcal{L}_{R,S}, \mathcal{L}_{F,P}, \mathcal{L}_{R,P}, \mathcal{R}, \mathcal{L}, \mathcal{L}_T\}$, $k = \{\mathcal{H}_S, \mathcal{H}_P, \mathcal{L}_{F,S}, \mathcal{L}_{R,S}, \mathcal{L}_{F,P}, \mathcal{L}_{R,P}, \mathcal{L}, \mathcal{L}_T\}$ and $f = \{\Gamma_{F,S}, \Gamma_{R,S}, \Gamma_{F,P}, \Gamma_{R,P}\}$. $F_F^w$ denotes the resulting force induced by the foils action along w axis, $\Gamma_F^p$ and $\Gamma_F^q$ denote the resulting torque induced by the foils around p and q axis, T denotes the thrust along u and $\Gamma_R^r$ denotes the rudder's induced torque around w axis.

Hence, as a first stage, the actuation system is designed as the relation between the individual actuators and their resulting effect on the regulated degrees of freedom in {B}. The individual action of the foils along w in {B} is denoted $F_F^w$, written as:

$$ F_F^w = -1/2 \cdot \rho \cdot S_f \cdot \cos \delta_f \cdot (u_f^2 + w_f^2) \cdot C_f^w(\alpha_f) \cdot \frac{Asp_R}{Asp_R + 2} \quad (24) $$

where $\alpha_f = \arctan(u_f/w_f) + \delta_f$ and with $f = \{\Gamma_{F,S}, \Gamma_{R,S}, \Gamma_{F,P}, \Gamma_{R,P}\}$. $C_f^w(\alpha_f)$ is the lift coefficient of the foils given at Fig. 3 and $Asp_R$ is the Aspect Ratio of the foil. This aspect ratio can be easily computed for simple rectangular foils as $Asp_R = b/c$ (= 5.78 in our case), where b is the foils's span (width from tip to tip) and c is the chord length (Anderson, 1999). The resulting force and torque on the regulated d.o.f. of {B} induced by the foils action is written as:

$$ 
\begin{aligned}
F_F^w &= \sum_f F_f^w \\
\Gamma_F^p &= \sum_f \begin{bmatrix} 0 & 1 & 0 \end{bmatrix} \cdot (\mathbf{X}_f^B - \mathbf{X}_G) \cdot F_f^w \\
\Gamma_F^q &= \sum_f \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} \cdot (\mathbf{X}_f^B - \mathbf{X}_G) \cdot F_f^w
\end{aligned} 
\quad (25) 
$$

which can be written as:

$$ 
\begin{bmatrix} F_F^w \\ \Gamma_F^p \\ \Gamma_F^q \end{bmatrix} = A_F \cdot \begin{bmatrix} C_{F,S}^w (\alpha_{F,S}) \\ C_{F,R,S}^w (\alpha_{F,R,S}) \\ C_{F,F,P}^w (\alpha_{F,F,P}) \\ C_{F,R,P}^w (\alpha_{F,R,P}) \end{bmatrix} = A_F \cdot C_F^w \quad (26) 
$$

where $A_F$ is a $(3 \times 4)$ matrix and denotes the foils' actuation system and $C_F^w$ is a $(4 \times 1)$ vector composed with the 4 foils' lift coefficients. One should note here the actuation redundancy expressed by the structure of $A_F$. This property will be used in the sequel.

The contribution of the rudder action on r axis of {B} is computed as follows:

$$ 
\begin{aligned}
\Gamma_R^r &= \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} \cdot (\mathbf{X}_R^B - \mathbf{X}_G) \cdot \underbrace{(-1/2 \cdot \rho \cdot S_R(\eta) \cdot (u_R^2 + v_R^2) \cdot C_R^v(\alpha_R))}_{F_{F_R}^v} \\
&= A_R \cdot C_R^v(\alpha_R)
\end{aligned} 
\quad (27) 
$$

where

$$ \alpha_R = \arctan(u_R/v_R) + \delta_R \quad (28) $$

and $C_R^v(\alpha_R)$ is the lift coefficient of the symmetrical rudder's profile (cf Fig. 3). Thrusters controlled action is along u of {B} and simply expressed as:

$$ F_T^u = T \quad (29) $$

Finally, the global actuation system can be written as:

$$ 
\begin{bmatrix} F_T^u \\ F_F^w \\ \Gamma_F^p \\ \Gamma_F^q \\ \Gamma_R^r \end{bmatrix} = \begin{bmatrix} 1 & \mathbf{0}_{1 \times 4} & 0 \\ \mathbf{0}_{3 \times 1} & A_F & \mathbf{0}_{3 \times 1} \\ 0 & \mathbf{0}_{1 \times 4} & A_R \end{bmatrix} \cdot \begin{bmatrix} T \\ C_F^w \\ C_R^v(\alpha_R) \end{bmatrix} \quad (30) 
$$

The control design, addressed in the sequel, will require an inversion of the relation (30), which will be done on each decoupled components.

## 3. Control design

As introduced earlier, the control objectives are decoupled with the following strategy:

* the 4 foils act along the 3 D.O.F w, p, q, w.r.t {B}, in order to regulate a desired altitude above water $z_d$, and 2 desired roll and pitch angles $\phi_d$ and $\theta_d$, respectively.

* the rudder acts on the yaw dynamics and is open loop controlled,

* the thruster is set to a constant regime and produces the force T along u axis of {B}.

The control structure is chosen as classic PID control. Consider the following expressions, in which the subscripts .d and .c stand for desired
<figure><img src="image_6.png" /><figcaption>Fig. 3. Foil characteristics (Foils' profile: Eppler 817 (left) and rudder's (right) profile: NACA 0012, from http://www.airfoiltools.com/airfoil/).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Inputs of the actuators, resulting from the implementation of the control law (40).</figcaption></figure>

and control reference values, as desired closed-loop acceleration for the system:

$$ 
\begin{aligned}
\ddot{z}_c &= \ddot{z}_d + K_P^z \cdot (z_d - z) + K_I^z \cdot \int_{\tau=0}^{t} (z_d(\tau) - z(\tau))d\tau + K_D^z \cdot (\dot{z}_d - \dot{z}) \\
\dot{\phi}_c &= \dot{\phi}_d + K_P^\phi \cdot (\phi_d - \phi) + K_I^\phi \cdot \int_{\tau=0}^{t} (\phi_d(\tau) - \phi(\tau))d\tau + K_D^\phi \cdot (\dot{\phi}_d - \dot{\phi}) \\
\dot{\theta}_c &= \dot{\theta}_d + K_P^\theta \cdot (\theta_d - \theta) + K_I^\theta \cdot \int_{\tau=0}^{t} (\theta_d(\tau) - \theta(\tau))d\tau + K_D^\theta \cdot (\dot{\theta}_d - \dot{\theta})
\end{aligned}
\quad (31) $$

where $K_{P,I,D}^{z,\phi,\theta}$ are positive gains. Clearly, if the system tracks these desired acceleration, second order convergence of $z_d - z$, $\phi_d - \phi$ and $\theta_d - \theta$ is achieved. These references $\ddot{z}_c$, $\dot{\phi}_c$, $\ddot{\theta}_c$ are expressed in the universal frame. Their equivalent in the body frame can be obtained using relation (8), left-multiplying by $R_B^0$, and using the properties: (i) for a rotation matrix $R$, we have: $\dot{R} = R \cdot S(\omega)$ and (ii) $S(\omega)^T = -S(\omega)$. We then obtain the following relation:

$$ \dot{\mathbf{X}} = \mathbf{R}_B^0 \cdot \dot{\mathbf{v}} + \mathbf{R}_B^0 \cdot \mathbf{S}(\omega) \cdot \mathbf{v} \quad (32) $$

from which we extract the third line to finally obtain:

$$ \dot{w}_c = \frac{\dot{z}_c + \dot{u} \cdot \sin \theta - \dot{v} \cdot \cos \theta \cdot \sin \phi - [\mathbf{R}_B^0 \cdot \mathbf{S}(\omega) \cdot \mathbf{v}]^T \cdot [0 \ 0 \ 1]^T}{\cos \theta \cdot \cos \phi} \quad (33) $$

Following the same idea with Eq. (9), we obtain:

$$ 
\begin{aligned}
\dot{p}_c &= \ddot{\phi}_c - \dot{q} \cdot \sin \phi \cdot \tan \theta - \dot{r} \cdot \cos \phi \cdot \tan \theta - [\mathbf{J}_B^0 \cdot \mathbf{J}_0^B \cdot \mathbf{J}_B^0 \cdot \mathbf{\omega}]^T \cdot [0 \ 1 \ 0]^T \\
\dot{q}_c &= \frac{1}{\cos \phi} \cdot (\ddot{\theta}_c + \dot{r} \cdot \sin \phi - [\mathbf{J}_B^0 \cdot \mathbf{J}_0^B \cdot \mathbf{J}_B^0 \cdot \mathbf{\omega}]^T \cdot [0 \ 0 \ 1]^T)
\end{aligned}
\quad (34) $$

where it is assumed that $|\phi| < \pi/2$ and $|\theta| < \pi/2$ (Euler representation singularity). We now apply a classic 'computed torque' approach. The dynamic model along the $w, p$ and $q$ D.O.F. in $\{B\}$ can be extracted

from (16) and (17). For the sake of simplicity of the control design, we assume that $\mathbf{M} = \text{diag}\{M_u, M_v, M_w\}$ and $\mathbf{I} = \text{diag}\{I_p, I_q, I_r\}$ are diagonal matrices. It is then direct to state that the desired acceleration references (34) are tracked by the system if:

$$ C_F^w = A_F^+ \cdot \begin{bmatrix} M_w & 0 & 0 \\ 0 & I_p & 0 \\ 0 & 0 & I_q \end{bmatrix} \cdot \begin{bmatrix} \dot{w}_c \\ \dot{p}_c \\ \dot{q}_c \end{bmatrix} - \begin{bmatrix} \mathbf{F}_0^w \\ \mathbf{F}_0^\phi \\ \mathbf{F}_0^q \end{bmatrix} \quad (35) $$

where $\mathbf{F}_0^w$ is the third component of $\mathbf{F}_0$, $\mathbf{\Gamma}_0^p$ is the first component of $\mathbf{\Gamma}_0$ and $\mathbf{\Gamma}_0^q$ is the second component of $\mathbf{\Gamma}_0$ ($\mathbf{F}_0$ and $\mathbf{\Gamma}_0$ being defined at Eq. (21)) and $A_F^+$ is the Moore-Penrose pseudo-inverse of $A_F$. Assuming that $u > 0$, always, the desired foil angle is then computed as:

$$ \delta_f = (C_F^w)^{-1} \cdot \begin{bmatrix} A_F^+ \cdot \begin{bmatrix} M_w & 0 & 0 \\ 0 & I_p & 0 \\ 0 & 0 & I_q \end{bmatrix} \cdot \begin{bmatrix} \dot{w}_c \\ \dot{p}_c \\ \dot{q}_c \end{bmatrix} - \begin{bmatrix} \mathbf{F}_0^w \\ \mathbf{F}_0^\phi \\ \mathbf{F}_0^q \end{bmatrix} \end{bmatrix} - \arctan w_f / u_f \quad (36) $$

where $f = \{F_{F,S}, F_{R,S}, F_{F,P}, F_{R,P}\}$ and $(C_F^w)^{-1}$ denotes the inverse characteristic of the lift coefficient for the foil $f$. Since the four foils are identical, they share the same characteristics, $C_{F,F,S}^w(\cdot) = C_{F,R,S}^w(\cdot) = C_{F,F,P}^w(\cdot) = C_{F,R,P}^w(\cdot) = C_F^w(\cdot)$. $C_F^w(\cdot)$ is given as a piece-wise polynomial expression interpolating the look-up table of the chosen foil characteristics given by the Xfoil reference.<fnref n="1" /> $(C_F^w)^{-1}$ is obtained similarly considering the inverted entries of the Xfoil's look-up table, cf Fig. 3.

<footnote n="1">¹ https://web.mit.edu/drela/Public/web/xfoil/</footnote>
<figure><img src="image_8.png" /><figcaption>Fig. 5. Comparison of the simulator responses with respect to real system, in response of the actuation inputs of Fig. 4.</figcaption></figure>

### 3.1. Guidance strategy design

The guidance system aims at building the error function in order to cope with system's capabilities and to shape a desired transitory regime. In our study, the strategy is threefold:

* Consider that foil efficiency increases with velocity. During take-off, when velocity $u$ is small, a small error will easily push actuators to saturation. Assuming that the desired altitude above water is denoted $h_d$, that will be reached at a desired forward

velocity $u_d$, the following expression implements this behavior.

$$ 
\begin{aligned}
z_d &= \frac{h_d}{\tanh(k_z \cdot u_d)} \cdot \tanh(k_z \cdot u) \\
\tilde{z} &= z_d - z
\end{aligned} 
\qquad (37) 
$$

where $\tilde{z}$ stands for the altitude error and $k_z$ is a positive gain shaping the slope at origin of the tanh function.

* Regulate a transitory positive pitching during take-off. Here, we are inspired by the natural strategies of pitching positively the system to improve the take-off transitory regime, as kite-surfers
<figure><img src="image_9.png" /><figcaption>Fig. 6. Actuation inputs used for the following simulations in order to characterize the advantages of the guidance laws proposed in the sequel.</figcaption></figure>

System's states evolution, without foil

<figure><img src="image_10.png" /><figcaption>Fig. 7. System states evolution (up) and trajectory (down), without foils.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 8. System states evolution (up), trajectory (down-left) and actuator activity (down-right), without guidance.</figcaption></figure>

do or motor trimming generally allows on rubber boats. As permanent regime is achieved, a null pitch angle is expected.

$$ \begin{aligned} \theta_d &= \theta_{\max} \cdot \tanh(k_{\theta} \cdot \bar{z}) \\ \bar{\theta} &= \theta_d - \theta \end{aligned} \qquad (38) $$

where $\bar{\theta}$ denotes the pitching error, $\theta_{\max}$ is the maximum desired pitching angle and $k_{\theta}$ is a positive gain.

* Consider a 'banking turn' effect, as planes while changing heading, for passengers comfort. The desired banking angle is known (REF) and used as shown above.

$$ \begin{aligned} \phi_d &= \arctan\left(\frac{r \cdot u}{g}\right) \\ \tilde{\phi} &= \phi_d - \phi \end{aligned} \qquad (39) $$

where $\tilde{\phi}$ denotes the roll angle error.

Finally the foil control expression is written with:

$$ \begin{aligned} \ddot{z}_c &= \ddot{z}_d + K_p^z \cdot \bar{z} + K_I^z \cdot \int_0^t \bar{z}(\tau) d\tau + K_D^z \cdot \dot{\bar{z}} \\ \dot{\tilde{\phi}}_c &= \dot{\tilde{\phi}}_d + K_p^\phi \cdot \tilde{\phi} + K_I^\phi \cdot \int_0^t \tilde{\phi}(\tau) d\tau + K_D^\phi \cdot \dot{\tilde{\phi}} \\ \dot{\bar{\theta}}_c &= \dot{\bar{\theta}}_d + K_p^\theta \cdot \bar{\theta} + K_I^\theta \cdot \int_0^t \bar{\theta}(\tau) d\tau + K_D^\theta \cdot \dot{\bar{\theta}} \end{aligned} \qquad (40) $$

used in Eqs. (34)–(36) to obtain the 4 desired foil angles $\delta_l$. This ensures that the desired closed-loop error on the three regulated degrees of freedom ($z$, $\phi$ and $\theta$) behave with the second order stable dynamics of (40).

## 3.2. Redundancy management

Since the actuation system is modeled with the matrix $A_F$, which contains more columns than lines, the system has potentially a manageable redundancy. Indeed, 4 foils are used to regulate 3 D.O.F. Moreover,
System's states evolution, with guidance

<figure><img src="image_12.png" /><figcaption>Fig. 9. System states evolution (up), trajectory (down-left) and actuator activity (down-right), with guidance. The guidance references (for z, φ and θ) are drawn in red.</figcaption></figure>

one can easily verify that

$$ \mathrm{rank}(\mathbf{A}_F) = 3 \qquad (41) $$

since the choice of system parameters (cf. Fig. 3) and for any attainable values of foil angles δᵢ. Hence, as done in Ropars et al. (2018), an element Nₜ of the kernel of Aₜ is identified and used as a projector in the null space of Cₗ, as exposed in the sequel.

$$ N_F \in \mathrm{Ker}(\mathbf{A}_F) / \mathbf{A}_F \cdot \mathbf{N}_F \cdot r_m = 0 \qquad (42) $$

where Nₜ is a (4×1) element of the null space of Aₜ, and for any value of the scalar rₘ. Hence, Eq. (35) can be modified according to:

$$ C_F^w = A_F^+ \cdot \underbrace{\begin{bmatrix} M_w \cdot \dot{w}_c - F_L^w \\ I_p \cdot \dot{p}_c - \Gamma_L^p \\ I_q \cdot \dot{q}_c - \Gamma_L^q \end{bmatrix}}_{\mathbf{F}_{F,B}^d} + N_F \cdot r_m = \begin{bmatrix} A_F^+ \cdot N_F \end{bmatrix} \cdot \begin{bmatrix} \mathbf{F}_{F,B}^d \\ r_m \end{bmatrix} \qquad (43) $$

using Eq. (36) which allows to compute the corresponding foil angles δᵢ, and where Fₜ^d denotes the desired resulting action of the foil's effect. By application of Eq. (44), previous choice results in the following action, expressed in the {B} frame as:

$$ \begin{bmatrix} F_F^w \\ \Gamma_F^p \\ \Gamma_F^q \end{bmatrix} = A_F \cdot A_F^+ \cdot \begin{bmatrix} M_w \cdot \dot{w}_c - F_L^w \\ I_p \cdot \dot{p}_c - \Gamma_L^p \\ I_q \cdot \dot{q}_c - \Gamma_L^q \end{bmatrix} + \underbrace{A_F \cdot N_F \cdot r_m}_{=0} \Leftrightarrow \begin{bmatrix} \dot{w} \\ \dot{q} \end{bmatrix} = \begin{bmatrix} \dot{w}_c \\ \dot{q}_c \end{bmatrix} \qquad (44) $$

assuming a perfect knowledge of the system's dynamical model, and where the desired accelerations defined at Eq. (34) ensures the tracking of the prescribed guidance references. Hence, following the classic task function approach (Nakamura et al., 1987) a specific value of rₘ can be computed in order to respect an added criterion, without affecting the primary control objective (34). An example where a desired value is imposed on a foil is given below.
<figure><img src="image_13.png" /><figcaption>Fig. 10. System states evolution (up), trajectory (down-left) and actuator activity (down-right), imposing $\delta_{F,R}^* = -0.1 + 0.03 \cdot \sin(0.4 \cdot t)$ rad. The guidance references (for z, $\phi$ and $\theta$) are drawn in red.</figcaption></figure>

### 3.2.1. Imposing a desired value of a foil angle

Imagine that the foil $i$ is driven to follow an angle $\delta_i^*$, independently chosen from the control objective (this angle could remain constant if the foil is blocked with a fixed angulation). Hence, the redundancy management will allow to perform the control objective with the 3 remaining active foils, while the foil $i$ follows its prescribed reference $\delta_i^*$. Let's impose the prescribed value for the foil angle $\delta_i^*$. The corresponding attack angle $\alpha_i^*$ is computed according to Eq. (28) and (24) provides the induced lift force $F_i^*$. Let's now computed the corresponding $r_m$ in order to produce desired lift force $F_i^*$ on the $i^{th}$ of $F_L^c$, by inversion of relation (43) as:

$$r_m = (N_F(i))^{-1} \cdot \left( F_i^* - A_F(i, :) \cdot \begin{bmatrix} M_{\omega} \cdot \dot{w}_c - F_L^{\omega} \\ I_p \cdot \dot{p}_c - \Gamma_L^{\phi} \\ I_q \cdot \dot{q}_c - \Gamma_L^{\dot{q}} \end{bmatrix} \right) \quad (45)$$

where $N_F(i)$ is the $i^{th}$ component of the $(1 \times 4)$ matrix $N_F$ and $A_F(i, :)$ is the $i^{th}$ line of $A_F$. It is then expected, as simulation shows in the sequel, that using (45) in the control expression (43) will produce the desired acceleration on the system, as Eq. (44) states, despite the fact that the foil $i$ remains at its prescribed angulation $\delta_i^*$. The global control allocation among the 4 foils is then done using Eq. (44) and the expression (45) for $r_m$.

## 4. Simulation results

To illustrate the performance of the proposed solution, we performed simulations using the parameters reported in Table 3, where it is assumed that the charge has no hydrodynamic effect. The foils use the Eppler 817 profile, while the thruster, foils’ legs, and rudder use the symmetrical NACA 0012 profile, the characteristics of which are shown in Fig. 3.
System's states evolution, comparison of the responses

<figure><img src="image_14.png" /><figcaption>Fig. 11. Comparison of the system response without foils (red), with foils but without guidance (green), and with guidance (blue).</figcaption></figure>

As will be confirmed later, it is expected that the foiling structure, if regulated correctly, will reduce the overall drag effect when the system is planing, compared to the Archimedean situation (when the hulls are floating).

4.1. Simulator validation

We have developed a Matlab simulator based on the models described in previous section. Since this simulator will be used in the sequel to propose some improvement of the method, we need to validate its behavior with respect to the real Black Pearl vehicle. First we have implemented the control expression (40) onboard the vehicle. This regulation produced the actuation input reported at Fig. 4.

We have applied the actuation inputs from Fig. 4 to the simulator. These inputs correspond to part of a real experiment, during which the evolution of surge, foil activity, altitude above water, pitch and roll, and the CPS trace were recorded. We compare the experimental

system's responses to those of the simulator in Fig. 5, where real measurements are represented by thin lines and simulated results by bold lines. Please note that our goal here is not to develop a digital twin, which would require a precise estimation of the system's parameters. Since the simulation does not account for external disturbances, parameter misestimations, or sensor noise, we do not expect a perfect match between the simulation and reality.

In the following, we present simulations results with an identical scenario: for the first 10 s, the thruster produces a constant force T = 30 N, then for t = 10 s to t = 20 s the rudder is used with δR = 0.3 rad, and for t = 20 s to t = 30 s the rudder is oriented to δR = -0.3 rad, as illustrated at Fig. 6.

In order to show the advantage of using a foiling structure, we first perform a simulation of the system without foiling structures. State evolution are given at Fig. 7-up. The system reaches a constant velocity u∞ = 1.63 m/s. The torque produced by the thrust is clearly visible and
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 12.** Experimentations on the Black Pearl robot (left) which was teleoperated using Thrust and Rudder controls. The trajectory (right) shows a lot of different curves, where the system is expected to maintain the regulation of the other degrees of freedom: roll, pitch and altitude above water.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 13.** The system commands, provided by the operator, are the thruster's regime (Thruster cmd) and the Rudder's control (Rudder Cmd). The resulting activity on the impacted degrees (Surge velocity *u* and Yaw angle) of freedom is shown.</figcaption></figure>

induces a positive static pitch angle $\theta_{\infty} = 0.054$ rad. At least the system performed the trajectory depicted at Fig. 7-down.

The second simulation shows the system behavior without guidance function, with a constant $z_d = -0.2$ m, $\theta_d = 0$ and $\phi_d = 0$. The control gains have been chosen according to Table 1. The results are shown at Fig. 8.

The analysis of Fig. 8 shows that the system is achieving regulation during the first 10 s, while the rudder is fixed to $\delta_r = 0$. The velocity reaches a velocity of $u = 7.1$ m/s, showing the efficiency of the foiling strategy. The system takes-off at $t = 4.13$ s, after an actuation saturation period as the foils' activity, i.e. evolution of the foils' angles with time, Figures indicates, 8-down-right. The first rudder action ($\delta_R = 0.3$ rad at

<table><caption>**Table 1**
Control gains used in simulation.</caption><thead><tr><th></th><th>K<sub>P</sub></th><th>K<sub>I</sub></th><th>K<sub>D</sub></th></tr></thead><tbody><tr><td>z<sub>c</sub></td><td>100</td><td>500</td><td>10</td></tr><tr><td>φ<sub>c</sub></td><td>800</td><td>0</td><td>100</td></tr><tr><td>θ<sub>c</sub></td><td>800</td><td>100</td><td>200</td></tr></tbody></table>

$t = 10$ s) induces a roll angle dynamic which produces a constant roll of $\phi = 7e^{-3}$ rad, while the system maintains regulation. The second rudder change ($\delta_R = -0.3$ rad at $t = 20$ s) induces a bigger effects which pushes
<figure><img src="image_18.png" /><figcaption>Fig. 14. Foils activity during the experimentation.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 15. Evolutions of the regulated degrees of freedom: altitude above water, roll and pitch angle.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Fig. 16. The Black Pearl prototype before tests at sea.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 17. Frame definition.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 18. Notation, Starboard view.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Fig. 19. Notation, upper view.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 20. Notation, front view.</figcaption></figure>

<table><caption>**Table 2**
Parameters of the guidance functions.</caption><thead><tr><td>k<sub>z</sub></td><td>θ<sub>max</sub></td><td>k<sub>θ</sub></td></tr></thead><tbody><tr><td>0.8</td><td>0.2</td><td>5</td></tr></tbody></table>

actuation to saturation and a bad system response to this disturbance. Finally, the system trajectory is shown at Fig. 8-down-left.

The third simulation includes the guidance functions of Eqs. (37), (38) and (39), with the control gains of Table 1 and guidance parameters of Table 2. Results are shown at Fig. 9.

The system follows the guidance references, depicted in red at Fig. 9-up. As shown on the actuation activity Fig. 9-down-right, actuators are pushed to saturation only at the beginning, while the forward velocity $u$ is small. Clearly system response is better as the guidance references are followed. At least system trajectory is shown at Fig. 9-down-left and exhibits a longer trajectory as the system efficiency is improved. Notice also that the achieved curvature is also improved, as one could expect from the banking strategy. The fourth simulation implements the redundancy management proposed at Section 3.2.1. We follow previous scenario with a null $r_m$ up to $t = 6s$, where a desired $\delta_{F,P}^* = -0.1 + 0.03 \cdot \sin(0.4 \cdot t)$ rad is imposed on Front-Portside foil. Then Eq. (45) is implemented and used as in (43). Results are shown at Fig.

10. Analyzing the results in Fig. 10, it is clear that, as expected, the constraint $\delta_{F,P}^* = -0.1 + 0.03 \cdot \sin(0.4 \cdot t)$ rad does not affect the system's response but induces an increase in the activity of the other three (still regulated) foils. Nevertheless, it should be noted that the constraint has been chosen to oscillate around the average value of the foil $\delta_{F,P}$ activity during the steady regime of the simulation in Fig. 9. This is a favorable situation for illustrating redundancy management. It is worth noting that this situation increases the activity of the other foils. It is therefore expected that, in this scenario where one foil structure leaves the regulated angulation, the other actuators will be pushed to saturation, thereby degrading the regulation capabilities.

The guidance laws have so far only been tested in simulation. Additional sea trials need to be scheduled to verify that the expected improvements in performance are effectively achieved. It is important to note that these guidance laws operate in alignment with the system's natural behaviors. Specifically, the system's propulsion naturally induces a positive pitch, which the guidance law (38) drives the system along. Similarly, the proposed guidance law (39) enables the system to bank along a curved trajectory (as airplanes do), following the system's natural behavior due to the fact that the foils on the inside of the curve move at a lower velocity than those on the outside.

We compare the system's different simulated responses in Fig. 11, where the same thruster and rudder activities, as shown in Fig. 6,
<figure><img src="image_25.png" /><figcaption>Fig. 21. Notation, view of the front-starboard foil.</figcaption></figure>

have been considered. These guidance laws are anticipated to enhance
system performance in the following ways:

* The blue curves, compared to the green and red ones, show that Guidance (37) reduces the foil's angulation when the forward velocity is low. This reduction in angulation decreases drag and improves system acceleration, leading to better takeoff performance. This is illustrated in Fig. 11, where the curves showing the evolution of $u(m/s)$ (Fig. 11, first line, left) indicate that the system's surge velocity (blue curve) reaches a higher steady-state value when guidance (37) is implemented. In this case, the system achieves a steady surge velocity of 3.43, m/s, whereas without guidance, the velocity reaches 3.24, m/s and only 1.85, m/s without the foil. Another effect of implementing (37) is that the desired altitude is linked to the system's velocity, resulting in a small vertical error when the surge velocity is low. This error increases as the surge velocity rises. This relationship helps prevent the foil's angulation from reaching saturation when the surge velocity is not yet high enough to generate significant lift but is producing considerable drag. This is why guidance (37) leads to a faster surge velocity response and ultimately induces a higher final surge velocity. One should also have noticed that this guidance solution (37) induces an undesired behavior, as the desired altitude is related to surge velocity. Indeed surge velocity variation, which occurs during turns at $t = 20$ s and $t = 40$ s, naturally impacts the desired altitude, shown at (Fig. 9-3rd line-right) and (Fig. 11-3rd line-right).

* Guidance (38) follows the natural effect of positive pitching induced by the propeller and leverages this positive pitch to trigger the system's elevation. This pitch evolution, represented by the blue curve of $\theta$ (rad) in Fig. 11, 4th line, center, is regulated to follow the desired value, shown as the red curve in Fig. 9, 4th line, center. As a result, the vertical velocity response, $w$ (m/s), is smoother compared to the red curve, where the guidance law

(38) is not implemented. This leads to the positive outcome of reduced foil activity and a faster vertical response in $z$ (m), as shown in Fig. 11, 3rd line, right.

* Guidance (39) enables the system to bank into curves, aligning with the natural behavior induced by the reduced foil velocity within the curve. This results in a faster heading response, $r$ (rad/s), as shown in Fig. 11, 2nd line, right, and $\psi$ (rad), as shown in Fig. 11, 2nd line, right (blue curves compared to the red and green ones). This affords the system the ability to make sharper turns, as illustrated by the system's trajectory in Fig. 11-bottom.

At last, in order to show the advantages of the foiling structure in terms of system reactivity, we compare the system responses in different configurations (system without foils, with foils and without guidance strategy and, with guidance strategy), at Fig. 11. The advantages of the foiling structure is clearly shown with the attained forward velocity which reaches this higher value because the drag is reduced. The rotational velocity profile exhibits also a higher dynamics, specially when using the guidance strategy proposed at Section 3.1.

## 5. Experimentation

We conducted experimental validation on the Autonomous Surface Craft introduced in Chapter 2, known as the Black Pearl. The experiments took place at the Thau Lagoon in Sète, France (43° 25'04.3" N, 3° 41'55.3" E), a site sheltered from significant swells and ocean currents. These experimental conditions were favorable, as evidenced in the video of the trials.<fnref n="2" /> The system was regulated to maintain zero pitch and roll angles with a specified altitude above the water, while thrust and rudder control remained under manual operation. After confirming stable behavior in straight-line motion, the operator guided the craft through various curves to assess the system's performance under stress

We see, at Fig. 12, that the operator performed many rotations on systems trajectory, while the system's altitude above water was maintained around the desired value, while roll and pitch angles remained small (Fig. 15). As expected, and implemented in the actuation model with matrix $A_F$ in Eq. (26) and its pseudo-inverse $A_F^+$ at Eq. (35), the foils are responding to a behavior consisting in a common regime to regulate the vertical DDF (altitude above water), a differential regime between front and rear foils to regulated pitch angle and a differential regime between left and right foils to regulate roll angle. This behavior is clearly illustrated during the experimentation around 90 s, where the operator asked for an important rudder angulation which induces a decrease of the altitude above water and a peak of the roll evolution (underlined by the boxes at Figs. 13–15). The consequence is an important reaction of the two right foils (frames at Fig. 14), which are pushed to saturation, in order to reduce the roll angle. Outside this limit situation, the regulation of the 3 degrees of freedom is achieved. It should be noted that an important factor was the reactivity and overall capability of the foil's angulation actuators. Initially, we used waterproof Savöx SW-0240MG servos, capable of producing a torque of 35 kg-cm; however, this proved to be insufficient. We then replaced these actuators with waterproof QYRC Q30-1C-180 servos, capable of producing a torque of 80 kg-cm, with higher reactivity. Then, stable flight was quickly achieved.

## 6. Conclusion and future work

This paper has presented the Black Pearl, a novel electric hydrofoiler distinguished by its innovative 4-independent-foil sustentation system. This redundant design allows for the regulation of three degrees of freedom: roll, pitch, and altitude above water. The key contribution of this work lies in the development and experimental validation of a

<footnote n="2">² https://seafile.lirmm.fr/f/a362bef0cbc343e28445/</footnote>
<figure><img src="image_26.png" /><figcaption>Fig. 22. Notation, gravity and buoyancy centers.</figcaption></figure>

<table><caption>Table 3
Main parameters of the Black Pearl surface craft.</caption><thead><tr><th></th><th>Pos/{B} (m)</th><th>Dimensions (m)</th><th>Mass (kg)</th><th>Volume (l)</th><th>Type</th></tr></thead><tbody><tr><td>Hull Portboard</td><td>X<sup>G</sup><sub>H<sub>R</sub></sub> = [0, -6, 0]<sup>T</sup></td><td>(1.8 x .2 x .35)</td><td>3.2</td><td>126</td><td>—</td></tr><tr><td>Hull Starboard</td><td>X<sup>G</sup><sub>H<sub>S</sub></sub> = [0, .6, 0]<sup>T</sup></td><td>(1.8 x .2 x .35)</td><td>3.2</td><td>126</td><td>—</td></tr><tr><td>Foil's leg Front Left</td><td>X<sup>G</sup><sub>L<sub>F,R</sub></sub> (0) = [5, -35, .3]<sup>T</sup></td><td>(.1 x -x.7)</td><td>1</td><td>1</td><td>NACA 0012</td></tr><tr><td>Foil's leg Front Right</td><td>X<sup>G</sup><sub>L<sub>F,S</sub></sub> (0) = [5, .35, .3]<sup>T</sup></td><td>(.1 x -x.7)</td><td>1</td><td>1</td><td>NACA 0012</td></tr><tr><td>Foil's leg Rear Left</td><td>X<sup>G</sup><sub>L<sub>R,R</sub></sub> (0) = [-5, -35, .3]<sup>T</sup></td><td>(.1 x -x.7)</td><td>1</td><td>1</td><td>NACA 0012</td></tr><tr><td>Foil's leg Rear Right</td><td>X<sup>G</sup><sub>L<sub>R,S</sub></sub> (0) = [-5, .35, .3]<sup>T</sup></td><td>(.1 x -x.7)</td><td>1</td><td>1</td><td>NACA 0012</td></tr><tr><td>Foil Front Left</td><td>X<sup>G</sup><sub>T<sub>F,R</sub></sub> (0) = [5, -35, .6]<sup>T</sup></td><td>(.1 x .5 x -)</td><td>.6</td><td>.5</td><td>EPPLER 817</td></tr><tr><td>Foil leg Front Right</td><td>X<sup>G</sup><sub>T<sub>F,S</sub></sub> (0) = [5, .35, .6]<sup>T</sup></td><td>(.1 x .5 x -)</td><td>.6</td><td>.5</td><td>EPPLER 817</td></tr><tr><td>Foil leg Rear Left</td><td>X<sup>G</sup><sub>T<sub>R,R</sub></sub> (0) = [-5, -35, .6]<sup>T</sup></td><td>(.1 x .5 x -)</td><td>.6</td><td>.5</td><td>EPPLER 817</td></tr><tr><td>Foil leg Rear Right</td><td>X<sup>G</sup><sub>T<sub>R,S</sub></sub> (0) = [-5, .35, .6]<sup>T</sup></td><td>(.1 x .5 x -)</td><td>.6</td><td>.5</td><td>EPPLER 817</td></tr><tr><td>Thruster's Leg</td><td>X<sup>G</sup><sub>L<sub>T</sub></sub> = [-5, 0, 2]<sup>T</sup></td><td>(.1 x -x.4)</td><td>.6</td><td>.6</td><td>NACA 0012</td></tr><tr><td>Thruster</td><td>X<sup>G</sup><sub>T</sub> = [-5, 0, 4]<sup>T</sup></td><td>(.1 x -x.4)</td><td>4</td><td>1.4</td><td>Watt & Sea</td></tr><tr><td>Rudder</td><td>X<sup>G</sup><sub>R</sub> (0) = [-8, 0, 2]<sup>T</sup></td><td>(.3 x -x.4)</td><td>2</td><td>2</td><td>NACA 0012</td></tr><tr><td>Load</td><td>X<sup>G</sup><sub>L</sub> = [0, 0, 0]<sup>T</sup></td><td>—</td><td>10</td><td>—</td><td>—</td></tr></tbody></table>

control solution for this unique configuration. The paper introduced a new control law that effectively regulates the flight of the Black Pearl above water using its four independent foils. Furthermore, novel guidance laws were proposed and simulated to enhance the hydrofoiler's performance during take-off, curve banking, and altitude regulation. A significant aspect of the developed control strategy is its robustness, demonstrating the ability to maintain regulation even when one of the four foils deviates from regulation and follows an independent profile. This explicit management of redundancy improves actuation efficiency and significantly reduces the risk of actuator saturation. Experimental trials conducted on the Black Pearl prototype in real-world conditions at the Thau Lagoon provided a valuable assessment of the proposed regulation solution. These experiments confirmed the stable behavior of the hydrofoiler in maintaining a desired altitude and near-zero roll and pitch angles during straight-line motion and various turning maneuvers. The responsiveness of the foil actuators was identified as a critical factor for achieving stable flight, leading to an upgrade of the servo motors. The observed behavior of the foils during the experiments aligned with the actuation model, demonstrating a

coordinated response to regulate the vertical degree of freedom, pitch, and roll. Simulation results further illustrated the potential benefits of the proposed guidance laws. The simulations indicated that the guidance strategies contribute to a faster take-off, sharper turns through banked maneuvers, and an overall improvement in system efficiency by reducing foil angulation at low speeds and leveraging the natural pitching motion. The redundancy management strategy was also successfully simulated, showing the ability to impose a desired angle on one foil without compromising the regulation of the other degrees of freedom, albeit potentially increasing the activity of the remaining actuators. Looking ahead, the next phase of this research will focus on implementing and optimizing the proposed guidance laws on the physical Black Pearl system. Future work also includes testing the robustness of the regulation system in more challenging, disturbed environments, including the presence of waves, wind, and a moving payload. These tests will be crucial in understanding the physical limitations of the system with respect to sea state and further validating the benefits of the redundant 4-foil actuation system for achieving stable and efficient hydrofoiling. This study represents a significant
step towards the development of regulated hydrofoils capable of
navigating rough sea conditions.

CRediT authorship contribution statement

**Lionel Lapierre:** Writing – review & editing, Writing – original draft, Validation, Formal analysis, Data curation, Conceptualization. **Laurent Latorre:** Software, Resources, Project administration, Data curation, Conceptualization. **Loic Daridon:** Writing – review & editing, Project administration, Methodology, Investigation, Funding acquisition. **Denis Julien:** Software, Resources. **Vincent Dufour:** Project administration, Conceptualization.

Declaration of competing interest

The authors declare the following financial interests/personal rela-
tionships which may be considered as potential competing interests:
Loic Daridon reports financial support was provided by Occitanie Re-
gion and Feder Funds, with project number: ASV-81074SRRM07. Lionel
Lapierre reports a relationship with ENSTA that includes: employment.
Lionel Lapierre has patent pending to No patent. If there are other
authors, they declare that they have no known competing financial in-
terests or personal relationships that could have appeared to influence
the work reported in this paper.

Acknowledgments

The authors would like to express their warm gratitude to the Re-
gion Occitanie and the Feder funds (project number : ASV -81074SRR
M07), for supporting this work.

Annex

See Figs. 16–22.

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