<table><caption>Table 4.2 – Simulation parameters - Ideal case Hovercraft</caption><thead><tr><th>Item</th><th>Value</th><th>Item</th><th>Value</th></tr></thead><tbody><tr><td>Trajectory</td><td>Lawn Mower</td><td>Nominal Speed</td><td>1m.s<sup>-1</sup></td></tr><tr><td>Simulation time sample</td><td>0.05ms</td><td>Vehicle Shape</td><td>Circular</td></tr><tr><td colspan="4">Conventional Flatness-based controller</td></tr><tr><td>k<sub>p</sub></td><td>11.1</td><td>k<sub>i</sub></td><td>3.3</td></tr><tr><td>k<sub>d</sub></td><td>6.6</td><td>k<sub>p</sub><sup>ψ</sup></td><td>100</td></tr><tr><td>k<sub>i</sub><sup>ψ</sup></td><td>10</td><td>k<sub>d</sub><sup>ψ</sup></td><td>20</td></tr><tr><td colspan="4">iPID</td></tr><tr><td>α<sub>x</sub></td><td>1</td><td>α<sub>y</sub></td><td>1</td></tr><tr><td>K<sub>P</sub><sup>x</sup></td><td>11.1</td><td>K<sub>P</sub><sup>y</sup></td><td>11.1</td></tr><tr><td>K<sub>D</sub><sup>x</sup></td><td>6.6</td><td>K<sub>D</sub><sup>y</sup></td><td>6.6</td></tr><tr><td>k<sub>p</sub><sup>ψ</sup></td><td>100</td><td>k<sub>i</sub><sup>ψ</sup></td><td>10</td></tr><tr><td>k<sub>d</sub><sup>ψ</sup></td><td>20</td><td></td><td></td></tr></tbody></table>

Without disturbances, and applying the controllers to the hovercraft system, the adaptive functions $F_x$ and $F_y$ of the iPD have noting to estimate and converge to zero very quickly. Once they reach zero, both controllers are perfectly equivalent.

This example shows that the flatness-based structure of the controllers acts as a guidance method on the underactuated hovercraft. The calculation of the command angle $\psi_c$ creates the compensation mechanism of underactuation mentioned in previous examples this time relying on the natural flatness of the plant.

### 4.5.2.2 External disturbance and Measurement noise

In this experiment, a constant irrotational sea current and white measurement noise are added. The current has a constant speed of $-0.5 \text{ m s}^{-1}$ and is aligned with $\mathbf{x}_o$ axis of the inertial frame. The vehicle is therefore directly facing the current during the first segment, has it askew on the second segment and in diagonal all along the last segment.

The magnitude of the angle measurement is voluntary chosen to be much more consequential than the position measurement noise to emulate the difference in accuracy
<table><caption>Table 4.3 – Simulation parameters - Hovercraft with sea current</caption><thead><tr><th>Item</th><th>Value</th><th>Item</th><th>Value</th></tr></thead><tbody><tr><td>Trajectory</td><td>Lawn Mower</td><td>Nominal Speed</td><td>1 m s<sup>-1</sup></td></tr><tr><td>Simulation time sample</td><td>0.05 ms</td><td>Vehicle Shape</td><td>Circular</td></tr><tr><td>Noise deviation in position</td><td>0.1 m</td><td>Noise deviation in orientation</td><td>0.1rad</td></tr><tr><td>Current velocity in R<sub>0</sub></td><td>[-0.5 m s<sup>-1</sup>, 0, 0]<sup>T</sup></td><td></td><td></td></tr><tr><td colspan="4">Conventional Flatness-based controller</td></tr><tr><td>k<sub>p</sub></td><td>11.1</td><td>k<sub>i</sub></td><td>3.3</td></tr><tr><td>k<sub>d</sub></td><td>6.6</td><td>k<sub>p</sub><sup>ψ</sup></td><td>100</td></tr><tr><td>k<sub>i</sub><sup>ψ</sup></td><td>10</td><td>k<sub>d</sub><sup>ψ</sup></td><td>20</td></tr><tr><td colspan="4">iPID</td></tr><tr><td>α<sub>x</sub></td><td>1</td><td>α<sub>y</sub></td><td>1</td></tr><tr><td>K<sub>P</sub><sup>x</sup></td><td>11.1</td><td>K<sub>P</sub><sup>y</sup></td><td>11.1</td></tr><tr><td>K<sub>D</sub><sup>x</sup></td><td>6.6</td><td>K<sub>D</sub><sup>y</sup></td><td>6.6</td></tr><tr><td>k<sub>P</sub><sup>ψ</sup></td><td>100</td><td>k<sub>i</sub><sup>ψ</sup></td><td>10</td></tr><tr><td>k<sub>D</sub><sup>ψ</sup></td><td>20</td><td></td><td></td></tr></tbody></table>

between a position sensor like a GPS and the angular measurement obtained with an IMU or a magnetic compass.

Figure 4.11 shows that the two controllers are robust to the constant current and noise on the three segments. As shown on figure 4.12, both error signals of the two vessels oscillate around 0 because of the measurement noise but are stable.

This experiment shows the behavior of the adaptive functions $F_x$ and $F_y$ of the iPID when the system is disturbed. Figure 4.13 shows that the $F_x$ function converges towards the estimated values of the current velocity multiplied by the damping coefficient $\hat{F}_x \rightarrow -0.5\beta = -6$. In the conventional flatness-based controller, the integral term of the control law compensates the effects of the sea current. Figure 4.13 confirms that on simple case like this one, the adaptive functions of iPIDs have a similar behavior as the integral term of a conventional PID.
<figure><img src="image_47.png" /><figcaption>Figure 4.11 – Comparison of the trajectory of the flatness-based controllers with sea current - Blue: conventional flatness, Red: iPID, Black: Reference</figcaption></figure>
<figure><img src="image_48.png" /><figcaption>Figure 4.12 – Comparison of the tracking errors of the flatness-based controllers with sea current - a: $\boldsymbol{x}_o$ axis, b: $\boldsymbol{y}_o$ axis, Blue: conventional flatness, Red: iPID</figcaption></figure>
<figure><img src="image_49.png" /><figcaption>Figure 4.13 – Adaptive functions *F*<sub>*x*</sub> and *F*<sub>*y*</sub> on the hovercraft with constant sea current.
Red: Adaptive functions iPID - Blue: Integral term conventional PID</figcaption></figure>
<figure><img src="image_50.png" /><figcaption>Figure 4.14 – Comparison of the trajectory of the flatness-based controllers on the SV with sea current - Blue: conventional flatness, Red: iPD, Black: Reference</figcaption></figure>

The experiments performed on the hovercraft confirm the natural robustness of flatness-
based controllers to both external disturbances and measurement noise.

**4.5.2.3 Application to the surface vessel**

In this section, the two controllers are applied to a generic surface vessel model with no
modification of the control parameters. The same sea current and measurement noise are
applied to the vehicle as in the previous section. The added mass and damping parameters
of the vessel are given in table D.3. The surface vessel does not meet the hypothesis of
circular shape of the hovercraft, the sway added mass and damping parameters $Y_v$ and
$Y_\nu$ are much larger than the surge counter-parts.

Figure 4.14 and figure 4.15 shows that both controllers still give good results when applied to the surface vessel in presence of current and noise. The natural robustness to model uncertainties of the flatness-based conventional controller is sufficient to keep the
<figure><img src="image_51.png" /><figcaption>Figure 4.15 – Comparison of the tracking errors of the flatness-based controllers on the SV with sea current - a: $x_o$ axis, b: $y_o$ axis, Blue: conventional flatness, Red: iPD</figcaption></figure>
<figure><img src="image_52.png" /><figcaption>Figure 4.16 – Comparison of the tracking errors of the flatness-based controllers on the SV with a 15 m initial error - a: $\boldsymbol{x}_o$ axis, b: $\boldsymbol{y}_o$ axis, Blue: conventional flatness, Red: iPD</figcaption></figure>

SV on track even though it is not meeting the hovercraft hypothesis at all.

### 4.5.2.4 Application to the surface vessel with initial error

As shown in the fully-actuated case in Section 4.5.1, the main weakness of flatness-based control (and feedforward linearization) is the behavior of the system when it gets too far from the desired trajectory. Feedforward linearization does not allow coming back on track if the vehicle exists a safety layer around the task. To test the behavior of the two controllers in such a case, they are evaluated on a single $\boldsymbol{x}_o$ axis segment with an initial error of 15 m on the $\boldsymbol{y}_o$ axis.

Figure 4.16 shows that the conventional flatness-based controller completely diverges from the trajectory with the initial error whereas the iPD allows converging back to the path.
<table><caption>Table 4.4 – Simulation parameters - Surface Vessel with sea current</caption><thead><tr><th>Item</th><th>Value</th><th>Item</th><th>Value</th></tr></thead><tbody><tr><td>Trajectory</td><td>Lawn Mower</td><td>Nominal Speed</td><td>1 m s<sup>-1</sup></td></tr><tr><td>Simulation time sample</td><td>0.05 ms</td><td>Vehicle Shape</td><td>Surface Vessel</td></tr><tr><td>Noise deviation in position</td><td>0.1 m</td><td>in orientation</td><td>0.1rad</td></tr><tr><td>Current velocity in R<sub>0</sub></td><td>[-0.5 m s<sup>-1</sup>, 0, 0]<sup>T</sup></td><td></td><td></td></tr><tr><td colspan="4">Vehicle parameters</td></tr><tr><td>m</td><td>18 kg</td><td>I<sub>z</sub></td><td>0.61 kg m</td></tr><tr><td>X&#770;<sub>u</sub></td><td>2</td><td>Y&#770;<sub>v</sub></td><td>50</td></tr><tr><td>Y&#770;<sub>r</sub></td><td>2</td><td>N&#770;<sub>r</sub></td><td>0.5</td></tr><tr><td>X<sub>u</sub></td><td>12.8</td><td>Y<sub>v</sub></td><td>100</td></tr><tr><td>Y<sub>r</sub></td><td>2</td><td>N<sub>r</sub></td><td>4</td></tr><tr><td colspan="4">Conventional Flatness-based controller</td></tr><tr><td>k<sub>p</sub></td><td>11.1</td><td>k<sub>i</sub></td><td>3.3</td></tr><tr><td>k<sub>d</sub></td><td>6.6</td><td>k<sub>p</sub><sup>&psi;</sup></td><td>100</td></tr><tr><td>k<sub>i</sub><sup>&psi;</sup></td><td>10</td><td>k<sub>d</sub><sup>&psi;</sup></td><td>20</td></tr><tr><td colspan="4">iPD</td></tr><tr><td>&alpha;<sub>x</sub></td><td>1</td><td>&alpha;<sub>y</sub></td><td>1</td></tr><tr><td>K<sub>P</sub><sup>x</sup></td><td>11.1</td><td>K<sub>P</sub><sup>y</sup></td><td>11.1</td></tr><tr><td>K<sub>D</sub><sup>x</sup></td><td>6.6</td><td>K<sub>D</sub><sup>y</sup></td><td>6.6</td></tr><tr><td>k<sub>p</sub><sup>&psi;</sup></td><td>100</td><td>k<sub>i</sub><sup>&psi;</sup></td><td>10</td></tr><tr><td>k<sub>d</sub><sup>&psi;</sup></td><td>20</td><td></td><td></td></tr></tbody></table>
4.6 Partial conclusion on flatness-based control

This chapter applies flatness-based control to both fully-actuated and underactuated
marine craft. It shows that flatness-based control is an interesting alternative for the
control of sch vehicles as it is efficient and robust.

The flatness-based controller designed for underactuated systems naturally embeds a
form of guidance as it allows turning position references into force and moment control
inputs. In fact, the equations of flatness could also be used as a guidance principle associ-
ated with any controller as it notably allows calculating surge and yaw references for the
vehicle to track.

Flatness-based control is also associated to intelligent PID to control underactuated surface vessels. Using the hovercraft hypothesis, a simplified flat model of the surface vessel is used to calculate a flatness-based controller. Then, an adaptive layer based on iPD is added to increase the robustness of the controller. The idea is to be able to successfully apply this controller to any surface vessel.

The simulation experiments performed on the hovercraft and surface vessel show that the flatness-based controller alone is already robust to model uncertainties, disturbance and measurement noise. Flatness also appears to be a good guidance solution as it allows calculating proper angular references for position trajectory tracking of an underactuated surface vessel. Yet, the last experiment performed on the surface vessel shows that conventional flatness-based control reaches its limit when the system is initialized far from the desired path. Nevertheless, this last experiment also shows that the association of an iPD layer increases the already robust flatness-based controller and compensates for its weakness with great errors allowing the vehicle to converge back to the path after being initialized with a large position error.

Overall, flatness-based control is proven to be effective and robust for both fully-
actuated and underactuated marine vehicles. Further work will be required to extend the
results obtained on underactuated surface vessels to uneractuated AUVs. More studies on
physical systems will be conducted.
# VECTOR THRUST FOR AUVS

## Contents

<table><tr><td>5.1</td><td>Vector thrust effects</td><td>194</td></tr><tr><td>5.1.1</td><td>Coupled force effects</td><td>194</td></tr><tr><td>5.1.2</td><td>Impacts on the dynamic model</td><td>196</td></tr><tr><td>5.1.3</td><td>Controller calculations for coupled actuators</td><td>202</td></tr><tr><td>5.1.4</td><td>Force control with vector thrusters</td><td>204</td></tr><tr><td>5.2</td><td>PlaSMAR</td><td>210</td></tr><tr><td>5.2.1</td><td>Reconfiguration ring</td><td>210</td></tr><tr><td>5.2.2</td><td>Thrust allocation of PlaSMAR</td><td>215</td></tr><tr><td>5.2.3</td><td>Control of PlaSMAR</td><td>218</td></tr><tr><td>5.3</td><td>Simulations</td><td>219</td></tr><tr><td>5.3.1</td><td>Seabed scanning</td><td>219</td></tr><tr><td>5.3.2</td><td>Roll compensation</td><td>221</td></tr><tr><td>5.4</td><td>Conclusion</td><td>225</td></tr></table>

This chapter studies effects of vector thrust on AUVs and proposes a new concept of thrust reconfiguration. Notably, it presents potential side-effects of conventional off-centered vector thrusters like those found in [43]. It shows that reconfigurable thrusters can have damaging coupled effects on AUVs and notably on smaller vehicles. Although it is physically bounded, the coupled forced effect created by off-center vector thrusters can jeopardize the control objectives of a vehicle.

The coupled effects of conventional vector thrusters are studied in details. Simulation studies show that they can be neglected on larger and heavier vehicles but not on smaller and lighter ones. More than the actual weight of the system, its shape can make it more or less naturally robust to these coupled effects making vector thrusters not viable for some systems.
To counteract these coupled effects, a new concept of vector thrust is introduced. With this concept, pairs of parallel thrusters are mounted on a ring rotating around the hull. Actuation of the ring allows directing the force of the pair of thrusters while keeping a decoupled moment or choosing the axis of the moment they generate keeping the force on a constant axis. A proof of concept has been designed: the PlaSMAR robot. Simulations show that the PlaSMAR robot has increased capabilities thanks to the reconfiguration ring.

**5.1 Effects of vector thrusters**

Vector thrust is one of the most common solutions for increasing the maneuverability of a vehicle with a minimal amount of additional weight and expense. The first example of vector thrust in the marine context is the off-board engine of most private boats. The engine rotates around a vertical axis to generate thrust in the whole horizontal plane. Here, the rotation is often generated by the pilot or can be actuated.

For underwater vehicles, several technologies allow mimicking this principle in two or three dimensions [44, 43]. Underwater reconfigurable thrusters are usually composed of the main motor and one or two additional servos. Through mechanical contraptions, the servos actuate the rotation of the main motor or of the propeller axis to direct the thrust vector.

Vector thrusters generate off-center forces that can be used to control the orientation of the vehicle through their coupled moments. Yet, in general, only the moments generated by the actuators are taken into account both in the modeling and in the control of underactuated vessels. This section shows that the impact of the coupled force and moments on a vehicle depend on the physical properties of this vehicle. It also shows that even though the coupled force effects are considered as disturbances in general, they can also have an interest in applications like docking.

**5.1.1 Coupled force effects**

This section shows the coupled effects of off-centered vector thrusters and how they
may interfere in the control calculations. The reconfigurable *uqr*-architecture introduced
in section 1.2.2 is used as an example. A schematic of the vehicle is recalled in figure 5.1.
The vehicle is equipped with a single 3-DOF stern thruster generating forces on the *x*<sub>B</sub>,
<figure><img src="image_53.png" /><figcaption>Figure 5.1 – Schematic of the example uqr-vehicle.</figcaption></figure>

$y_B$ and $z_B$ axes. As exposed in section 1.2.2, the vector of forces and moments created by the 1D3 thruster $\tau$ is calculated using the TCM denoted $B$:

$$ \tau = Bu \qquad (5.1) $$

where $u$ is the vector of elementary forces created by the reconfigurable thruster. In this case, the 3-DOF thruster is equivalent to three fixed thrusters aligned with the $x_B$, $y_B$ and $z_B$ axes respectively. The vector of forces is therefore $u = [u_x, u_y, u_z]^T$ with:

$$ u_x = \cos(\theta_i) \cos(\psi_i)U \qquad (5.2a) $$

$$ u_y = \cos(\theta_i) \sin(\psi_i)U \qquad (5.2b) $$

$$ u_z = -\sin(\theta_i)U \qquad (5.2c) $$

where $\theta_i$ and $\psi_i$ are respectively the $y_B$ and $z_B$ reconfiguration angles of the vector thruster and $U = \sqrt{u_x^2 + u_y^2 + u_z^2}$ is the command thrust.

Considering that the force application point of the reconfigurable thruster is positioned at the very back of the hull of the vehicle and that the vector thruster rotates around this
point, the fixed-thruster equivalent TCM is:

$$ B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & -L \\ 0 & -L & 0 \end{bmatrix} \qquad (5.3) $$

where L represents half of the vehicle hull length.

The vector of generated forces and moments is then of shape:

$$ \tau = B \begin{bmatrix} u_x \\ u_y \\ u_z \end{bmatrix} = \begin{bmatrix} u_x \\ u_y \\ u_z \\ 0 \\ -Lu_z \\ -Lu_y \end{bmatrix} \qquad (5.4) $$

Equation (5.4) shows clearly the pairs of coupled forces and moments generated by the reconfigurable thruster. The sway force and yaw moment are coupled and the heave force and pitch moment are coupled.

From a controller perspective, one of the two coupled efforts is selected and controlled. Yet, the other coupled effort of each pair cannot be neglected as seen in the following.

### 5.1.2 Impacts on the dynamic model

The effects of coupled actuators are described on a simplified dynamic model representing an underwater vehicle equipped with a stern 3-D vector thruster. The compacted
model is:

$$ M\dot{\nu} + N(\nu, \eta) = \tau \qquad (5.5) $$

where $N(\nu, \eta)$ represents the Coriolis and centripetal effects acting on the system as well as the damping, gravitational and buoyancy effects. $M$ and $N$ include added-mass effects.

Considering that the vehicle has a cylindrical shape (negligible off-diagonal added-mass parameters), the simplified dynamic model is given in component form:

$$ m_1\dot{u} + n_1(\nu, \eta) = u_x \qquad (5.6a) $$

$$ m_2\dot{v} + n_2(\nu, \eta) = u_y \qquad (5.6b) $$

$$ m_3\dot{w} + n_3(\nu, \eta) = u_z \qquad (5.6c) $$

$$ m_4\dot{p} + n_4(\nu, \eta) = 0 \qquad (5.6d) $$

$$ m_5\dot{q} + n_5(\nu, \eta) = -Lu_z \qquad (5.6e) $$

$$ m_6\dot{r} + n_6(\nu, \eta) = -Lu_y \qquad (5.6f) $$

where $m_1, ..., m_6$ are the diagonal coefficients of the mass and inertia matrix $M$ and $n_1, ..., n_6$ are the components of the vector $N$.

In this example, the pitch and yaw moments generated by the vector thruster are chosen as control inputs. The sway and heave forces generated by the vector thruster $u_y$ and $u_z$ appearing respectively in equation (5.6b) and equation (5.6c) are therefore considered disturbances.

Equation (5.6b) and (5.6c) show that the disturbing coupled forces are mainly mitigated by the mass and added mass parameters on the sway and heave axes $m_2$ and $m_3$. They are also counter-acted by the damping effects contained in $n_2$ and $n_3$:

$$ \dot{v} = \frac{1}{m_2}(u_y - d_2v - \bar{n}_2(\nu, \eta)) \qquad (5.7a) $$

$$ \dot{w} = \frac{1}{m_3}(u_z - d_3w - \bar{n}_3(\nu, \eta)) \qquad (5.7b) $$

where the linear damping terms $d_2v$ and $d_3w$ have been extracted from $n_2(\dot{\nu}, \eta) = \bar{n}_2(\nu, \eta) + d_2v$ and $n_3(\dot{\nu}, \eta) = \bar{n}_3(\nu, \eta) + d_3w$.

The impact of the disturbing sway and heave forces is therefore mitigated by the shape of the vehicle. The robustness of the natural sway and heave stability of the vehicle due to the shape of the hull and the mass distribution could allow neglecting the sway and heave disturbing forces created by the vector thruster. Also, different hull shapes create
different coupling terms in the mass and damping matrices counter-acting the coupled forces.

As an example, figure 5.2 shows the behavior of three cylindrical vehicles with different $m_2$ and $m_3$ mass parameters. The first vehicle is a close representation of the Remus100 (presented in details in section 3.5.2) (Blue lines on figure 5.2 and figure 5.3), the second one (Red lines) has been artificially made ten times heavier on the sway and heave axes and the third one (Green lines) has been made ten times lighter on the same axes. The three vehicles are controlled in surge, pitch and yaw using the same $H$-matrix based controller on the seabed scanning task. Note that, as shown in the next section, the construction of the $H$ based controller as presented in chapter 3 indirectly assumes that the sway and heave disturbing forces are negligible. Nonetheless, as shown in this example, reasonably low sway and heave velocities are sufficient to ensure stability of the vehicle.

As expected, figure 5.3 shows that the coupled sway and heave forces generate less sway and heave velocity on the heaviest vehicle than on the two lighter ones. This natural stability results in better tracking performance as shown on figure 5.2 even though the lighter vehicle still shows reasonably good tracking performances too. It stays stable but oscillates around the desired trajectory.

Following the same idea, the $H$-matrix controller has been applied to the RSM vehicle (presented in chapter 1), the Remus100 and a mock up of the ODIN. All three vehicles are equipped with a single rear 3-D vector thruster and tested on the seabed scanning task. Numerical values of the model coefficients for the three vehicles can be found in appendix D.

Figure 5.4 represents the position of the three vehicles. It shows that while the RSM (Blue line) and Remus100 (Red line) perform well on the task, the ODIN (Green line) oscillates around the desired trajectory. The spherical shape of ODIN does not offer a robust enough stability on the sway and heave axes to counteract the coupled effect of the vector thruster. While the vehicle oscillates in a stable tube around the desired trajectory, the tracking mission can hardly be considered successful.

This example goes to show that even though the hypothesis of negligible sway and heave disturbing forces is not formally correct, the natural stability of some systems is sufficient to counter-act the disturbing effects of the coupled forces. Note that, as demonstrated in [66, 64] and detailed in the following section, because the thrust of the propeller is physically bounded, the disturbing forces are as well. Hence, the sway and heave speeds of the vehicle cannot completely diverge. Yet, from a control perspective,
<figure><img src="image_54.png" /><figcaption>Figure 5.2 – Comparison of tracking errors of the 3 Remus-like vehicles - (a): **x**<sub>o</sub> axis,
(b) **y**<sub>o</sub> axis, (c) **z**<sub>o</sub> axis - Blue: Normal, Red: Heavy, Green: Light</figcaption></figure>
<figure><img src="image_55.png" /><figcaption>Figure 5.3 – Comparison of linear velocities of the 3 Remus-like vehicles - (a): x_B axis,
(b) y_B axis, (c) z_B axis - Blue: Normal, Red: Heavy, Green: Light</figcaption></figure>
<figure><img src="image_56.png" /><figcaption>Figure 5.4 – Comparison of the position of the 3 vehicles on the Seabed Scanning task -
(a): **x**<sub>o</sub> axis, (b) **y**<sub>o</sub> axis, (c) **z**<sub>o</sub> axis - Blue: RSM, Red: Remus100, Green: ODIN</figcaption></figure>
the sway and heave behavior of some systems can jeopardize the whole tracking mission as it is the case for ODIN in the last experiment.

### 5.1.3 Controller calculations for coupled actuators

Calculating the controller for a non-decoupled system is not trivial. As shown in the previous section and in appendix C, the coupled effects of the input can disturb both the controlled and uncontrolled DOF of the system to a point where the tracking task cannot be considered successful. Some of the controllers introduced in chapter 2 require the assumption of negligible coupled force when the moment is used for control of the position of the vehicle through kinematic or dynamic coupling or any other guidance principle. This is notably the case for the sliding mode underactuated controller described in [40] and used in simulations in section 3.4 and for the $\mathcal{H}$ based controller introduced in chapter 3.

In the case of a vehicle equipped with one stern vector thruster, the underactuated SMC controller would use the first order derivative of the sway and heave dynamical equations to exploit the dynamic couplings of the system and control the sway and heave modes with the yaw and pitch inputs respectively. Considering a simplified model of an underwater vehicle (diagonal mass and damping matrices, linear damping), the sway, heave, pitch and yaw equations of the dynamic model taking the coupled forces into account are:

$$ \dot{v} = \frac{1}{m - Y_{\dot{v}}} (u_y + (m - X_{\dot{u}})ur - (m - Z_{\dot{w}})wp + Y_v v) \quad (5.8a) $$

$$ \dot{w} = \frac{1}{m - Z_{\dot{w}}} (u_z + (m - Y_{\dot{v}})vp - (m - X_{\dot{u}})uq + Z_w w) \quad (5.8b) $$

$$ \dot{q} = \frac{1}{I_y - M_{\dot{q}}} (-Lu_z + (Z_{\dot{w}} - X_{\dot{u}})uw + (I_x - I_z + N_{\dot{r}} - K_{\dot{p}})pr + M_q q) \quad (5.8c) $$

$$ \dot{r} = \frac{1}{I_z - N_{\dot{r}}} (-Lu_y + (X_{\dot{u}} - Y_{\dot{v}})uv + (I_y - I_x + K_{\dot{p}} - M_{\dot{q}})pq + N_r r) \quad (5.8d) $$

Taking the example of the sway equation (5.8a), the second order time derivative of $v$
gives:

$$
\begin{align}
\ddot{v} &= \frac{1}{m - Y_{\dot{v}}} (\dot{u}_y + (m - X_{\dot{u}})(\dot{u}r + u\dot{r}) - (m - Z_{\dot{w}})(\dot{w}p + w\dot{p}) + Y_v \dot{v}) \nonumber \\
&= \frac{1}{m - Y_{\dot{v}}} \left[ \dot{u}_y + (m - X_{\dot{u}})(\dot{u}r + \frac{u}{I_z - N_r} (-Lu_y + (X_{\dot{u}} - Y_{\dot{v}})uv + N_r r)) \right. \nonumber \\
&\qquad \left. -(m - Z_{\dot{w}})(\dot{w}p + w\dot{p}) + Y_v \dot{v} \right] \tag{5.9}
\end{align}
$$

where both $u_y$ and $\dot{u}_y$ appear due to the coupled force effect of the vector thruster. The
sway dynamics of the vehicle are now expressed as a generalized system: the first order
derivative of the input appears in the equation(see [47] and appendix B for more details).
Hence, the strategy used to establish the SMC on the decoupled system cannot be used
as such in this case.

In the same way, the $\mathcal{H}$ matrix based controller introduced in chapter 3 indirectly assumes that the coupled force effects on the sway and heave are negligible. This time, it is in fact the sway and heave control speeds that are assumed to be 0 in the kinematic stage of the controller to establish the relation $\nu_c = \mathcal{H}\mathbf{T}^{-1}\mathbf{J}(\eta)^{-1}\dot{\eta}_E$, where $\dot{\eta}_E$ is the sum of the desired velocity and the PI control law. Without assuming $v_c = w_c = 0$, the kinematic relations $v_c = \bar{v}_E - \varepsilon_x r_c$ and $w_c = \bar{w}_E + \varepsilon_x q_c$ cannot be used to cancel the sway and heave errors using the yaw and pitch speeds respectively as they were in section 3.3.

Nonetheless, some controller designs can take the coupling effects of vector thrusters
into account. This is the case of the adaptive and integral LOS guidance solutions of [61,
62] described in section 2.1. In these controllers, the sway force (and heave force if applied
to underwater vehicles) is not directly taken into account but is instead compensated
through the calculations of the adaptive crab angle. Using the adaptive crab angle in the
LOS calculations allows considering the surge and sway speeds ratio and couteracting
the disturbing sway speed. The coupled effect is therefore estimated as a disturbance and
compensated in the guidance calculations.

Also, the work of [66] introduces another controller taking the coupled force effects of
an underactuated surface vessel equipped with a vector thruster into account. The con-
troller is derived using the back stepping technique and Lyapunov theory. It includes the
uncontrolled sway modes and shows that the system is stable even though it is disturbed
by the coupled forces. In facts, this work shows that because the inputs of the system are
physically bounded by the maximum thrust of the motor, the system is stable.

Some more theoretical work like [115] demonstrate the mathematical stability of cou-
pled cascaded systems like the AUV with vector thrusters using the property of *forward completeness* (see [5] for in depth studies of forward completeness). Yet, the mathematical stability of the system demonstrated in this work does not ensure tracking performances. Even though the system cannot be completely destabilized, the uncontrolled sway modes disturb the trajectory tracking mission. The use of reconfigurable thrust is to consider for naturally stable and robust systems but is not ideal for lighter vehicles.

### 5.1.4 Force control with vector thrusters

On of the interest of vector thrust is the possibility to control either the forces generated by the thruster or the coupled moments. As depicted before in this section and all along this work, the traditional way of controlling marine craft is to use the moment generated by an off-center motor and propeller or control surface to control the steering of the vehicle. Doing so, the force component is either neglected or considered a disturbance. In this section, simulation experiments performed on the Remus100 vehicle equipped with a single 3-D reconfigurable thruster show that the sway and heave force components generate by the thruster can be used directly in the controller in specific use cases like docking and station keeping.

Of course, intuition and experience show that controlling a free body in water with rear forces is unstable, the vehicle would turn around so that the forces are actually pulling it instead of pushing. This instability is due to the hydrostatic and hydrodynamic effects acting on the hull of the craft. Consequently, in this section, the reconfigurable thruster is positioned at the front of the vehicle. This is equivalent to having a vehicle going “backwards” with a tracking point positioned at the rear.

One major issue when “pulling” the vehicle is that the water flow generated by the thruster is directly hitting the hull. This is energetically inefficient and potentially damaging for the vehicle in the long term. It must therefore be limited to short range applications. One noteworthy example where force control is most useful and more efficient that moment control is the docking maneuver.

The docking maneuver is a really complex case study for the guidance and control of autonomous marine craft that has been solved in many different ways like conventional control or machine learning [119, 144]. Here, a very simplified docking maneuver is presented. It is similar to the maneuver performed in ports with boats. A virtual dock parallel to the $x_o$ axis is simulated and the vehicle must approach the dock, turn to align with it and then maintain its position alongside the virtual dock. The task is composed of one
<figure><img src="image_57.png" /><figcaption>Figure 5.5 – Top view of the desired docking trajectory in the inertial frame</figcaption></figure>

approach segment at a 135° angle to the dock, a full turn to align the vessel with the
dock and a station keeping subtask when the vessel reaches the virtual dock. The desired
docking trajectory is displayed on figure 5.5. The blue circle represents the initial position
of the vehicle where the docking maneuver is initialized and the red dot represents the end
position. The vehicle takes around 10 seconds to reach the red dot and then maintains its
position on the red dot for as long as the simulation lasts. The virtual dock is materialized
by the horizontal axis on figure 5.5.

The position of the tracking point of the *uqr*-vehicle is controlled using the *H*-based
controller with the corresponding forces generated by the thruster. No compensation mech-
anism is necessary so the *H* matrix in this case is the identity. The controller is almost
equivalent to a single stage PID controller defined in the inertial frame (see Chapter 3 for
more details).

The attitude of the vehicle is left uncontrolled. The hydrodynamic effects acting on the
hull naturally align the vehicle to the water flow surrounding it. Hence, with no current
the vehicle aligns with the velocity vector at the tracking point. With currents, the vehicle finds an equilibrium between the direction of the velocity and the direction of the current. In the docking maneuver, the turn before station keeping aligns the vessel with the virtual dock. In the presence of sea current, the vehicle naturally aligns with the current field when station keeping.

The docking experiment is performed with a constant sea current of −0.5 m s⁻¹ velocity aligned with the $y_0$. Two control strategies are compared on the Remus100 equipped with a rear reconfigurable thruster. The position of the first vehicle is controlled conventionally, using the moments generated by the thruster. Kinematic couplings are used in the controller to compensate the lack of sway and heave actuation. The second vehicle is controlled directly with the force generated by the thruster. No compensations mechanism is involved. The second vehicle is initiated backwards on the approach segment for stability.

Recalling the formalism introduced before, the pair of DOF vectors in the first case is:

$$h_{O_B}^1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 1 \\ 1 \end{bmatrix} \qquad h_E = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \qquad (5.10)$$
while in the second case with force control it is

$$h_{O_B}^2 = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \quad h_E = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \qquad (5.11)$$

The first vehicle is considered to be actuated in surge, pitch and yaw and the three position coordinates of the tracking point are controlled through a guidance mechanism (in this example based on the *H* matrix). The second vehicle is considered to be actuated in surge, sway and heave and the three position coordinates of the tracking point are controlled. This second case does not require any guidance or compensation mechanism.

The position tracking errors of the two vehicles are displayed on figure 5.6. The force-controlled vehicle (Red line) appears much more stable than the moment-controlled one (Blue lines). Notably, figure 5.6 shows that force control allows much better station keeping for this vehicle. In moment-control, the vehicle circles around the final point in a stable ellipsoid as seen on figure 5.7.
<figure><img src="image_58.png" /><figcaption>Figure 5.6 – Comparison of position tracking errors on the docking task - a: **x**<sub>o</sub> axis, b: **y**<sub>o</sub> axis, c: **z**<sub>o</sub> axis - Blue: Moment control, Red: Force control</figcaption></figure>
<figure><img src="image_59.png" /><figcaption>Figure 5.7 – Top view of the trajectory of the two vehicles on the docking task - Blue: Moment control, Red: Force control</figcaption></figure>
## 5.2 New vector thrust concept: PlaSMAR

This section introduces a new concept of reconfigurable thrust. The idea is to direct and reposition a pair of thrusters to allow generating a directed force and a decoupled moment. This concept is notably suited for cylindrical-hull vehicles as it relies on a ring carrying the thrusters around the hull.

It has been developed on the test platform PlaSMAR<fnref n="1" />. PlaSMAR is an easily reconfigurable robot designed to easily emulate any fixed-thrusters arrangement as well as to be equipped with the reconfiguration ring. The first version of this vehicle can be seen on figure 5.8 in a four fixed-thrusters configuration equivalent to the RSM vehicle.

This first prototype has been mainly 3D printed. It is inspired by the *Blue ROV* (Blue Robotics company) and by the printable AUV designed in [146]. On figure 5.8, the vehicle is equipped with two 3D printed black rings holding two thrusters each. The fixed rings allow positioning and orienting pairs of thrusters to reproduce commercial propulsive arrangements and facilitate testing fixed configurations.

This section details the concept of the rotating ring used for reconfiguration of a pair of parallel thrusters. Notably, it presents the thrust allocation used on two configurations of PlaSMAR and control solutions.

### 5.2.1 Reconfiguration ring

Conventional thruster reconfiguration is pushed a step further with the PlaSMAR concept. The idea is to direct a pair of parallel thrusters to control both the force and the moment they generate without introducing new coupling terms. To do so, two parallel thrusters are mounted on a ring rotating around the hull of the vehicle. They can be mounted either tangent to the ring or parallel to its axis of rotation as seen on figure 5.9.

The first configuration of figure 5.9 allows directing the summed force of the two thrusters in the plane of the ring while maintaining the moment actuation around the ring axis created by differential thrust. In the second configuration, the force of the two thruster is kept parallel to the axis of rotation of the ring but the axis of the moment they generate can be directed.

Of course, in order to equip the ring on a vehicle, it must have a cylindrical hull or at least a cylindrical section of hull around which the ring can rotate. On a cylindrical-hull vehicle, the ring can be positioned anywhere along the hull and in the $(y_B, z_B)$ plane.

<footnote n="1">1. Plateforme Sous-Marine Autonome Reconfigurable</footnote>
<figure><img src="image_60.png" /><figcaption>Figure 5.8 – Picture of PlaSMAR during one of the first pool tests. Configuration equivalent to *RSM* with four fixed thrusters.</figcaption></figure>

<figure><img src="image_61.png" /><figcaption>Figure 5.9 – Schematics of the rotating ring principle. Black ring: hull of the AUV, Red ring: Rotating ring</figcaption></figure>
<figure><img src="image_62.png" /><figcaption>Figure 5.10 – 3-D renders of the first configuration of PlaSMAR. The rotating ring is placed at the front (rendered on Fusion360)</figcaption></figure>

<figure><img src="image_63.png" /><figcaption>Figure 5.11 – 3-D renders of the second configuration of PlaSMAR. The rotating ring is placed at the rear (rendered on Fusion360)</figcaption></figure>

It rotates around the **x**<sub>B</sub> axis of the cylinder. The two configurations of figure 5.9 are displayed on 3-D renders of PlaSMAR on figure 5.10 and figure 5.11 respectively.

Figure 5.10 and figure 5.11 shows two stages of the design of PlaSMAR. Initially, as shown on figure 5.10, four metallic bars were used to hold the different rings around the hull. The bars were only pushed inside the two black end pieces holding on the front and back flanges. The latest version shown on figure 5.11 uses two threaded rods to hold the two end pieces appearing in red together. The ring are mounted on two points and secured with nuts. The rotating rings presented on figure 5.10 and figure 5.11 are simplified proof of concepts of the system. The motor is bigger than needed and should be encased. Also, the front ring on both configurations should be in the middle of the hull to avoid coupled pitch effects as it is on figure 5.8.
In the first configuration(figure 5.10), the two front thrusters are mounted on the reconfiguration ring. They generate a force in the ($y_B, z_B$) plane parameterized by the rotation angle of the ring and a roll moment independent from their configuration. This contraption therefore allows actuation in sway and heave whilst maintaining decoupled roll. With the ring in its neutral 0° configuration (figure 5.10a), the thrusters are aligned with the $z_B$ axis and generate a pure heave force. At a 90 degrees angle (figure 5.10b), the thrusters are aligned with the $y_B$ axis and generate a pure sway force. In between the two extreme configurations, they can generate any diagonal force in the ($y_B, z_B$) plane. With the two rear fixed-thrusters, this vehicle is actuated in surge, sway, heave, roll and yaw. Only the pitch moment is not controlled.

In the second configuration (figure 5.11), the two back thrusters are mounted on the reconfiguration ring. The two thrusters are parallel to the $x_B$ of the vehicle so they generate a surge force in any configuration of the ring. The main interest of the reconfiguration ring in this case is the possibility to direct the axis of the moment generated by the two thrusters. At a 0 degree angle (figure 5.11a), the thrusters are in the ($x_B, y_B$) plane aligned with $x_B$ and generate a yaw moment while at a 90 degrees angle (figure 5.11b), they are in the ($x_B, z_B$) plane and generate a pitch moment. In the intermediary configurations, they can generate a moment around any axis of the vertical plane ($y_B, z_B$). With the two front fixed-thrusters, this vehicle is actuated in surge, heave, roll, pitch and yaw. Note that the rear rotating ring alone constitute an alternative to the common uqr-vessels. The two rear thrusters mounted on the rotating ring can independently generate a surge force, a pitch and a yaw moment. With the proper guidance, these three DOF are sufficient to perform any underwater position tracking task. This solution is a viable alternative to the usual fixed-thruster and control surfaces actuation of torpedo-shaped vehicles.

Overall, a pair of thrusters mounted on the reconfiguration can be utilized to generate an adjustable force vector whilst maintaining the moment or to direct the axis of the moment whilst maintaining the force. Effectively, the reconfiguration ring creates a new actuated DOF on the vehicle.

Several solutions can be designed to actuate the ring. The first and most simple one is to use a small servomotor with gear reduction placed on the outside of the hull. This is by far the simplest solution mechanically but it requires water proof casing to protect the electronic components exposed to water. Also, adding components on the hull modifies its hydrodynamic profile and can be detrimental. It could also be difficult to fix the motor assembly to the hull without risks of leakage. A second solution could be to magnetically
connect the outside ring to another ring inside the hull holding magnets and actuate
the inside ring. This way, no external components are needed. This kind of magnetic
contraption has been used for example in [168, 73, 44] to transmit force through the
hull of a craft with no hole. Finally, and notably in the first configuration presented on
figure 5.10a, the thrusters themselves could be used to rotate the ring. Unlatching the ring
from the hull, the roll moment generated by the two thrusters of the first configuration
would rotate the ring. A mechanical latch on the outside would be necessary or one could
imagine some form of magnetic latching from the inside. This solution implies that the
thursters are reconfigured at times when no roll is needed in the application since the ring
will be rotating freely around the hull.

One of the other interests of the reconfiguration ring is to compensate for roll distur-
bances. In facts, section 3.4.1 shows that roll actuation can be very useful in association
with heave to compensate for the lack of sway. Yet, this example shows one of the main
problem of staying at a non-zero roll angle. Because most vehicles are designed to operate
at a zero degree roll angle and because tasks are usually defined in the inertial frame, a
non-zero roll angle equilibrium can jeopardize the actuation of another DOF of the task.
In the example of section 3.4.1, the vehicle is rolling on the side to compensate for sway
but doing so, the two rear thrusters responsible for heave are misaligned with the vertical
axis of the inertial frame. At the extreme, the vehicle is not able to rotate around the ver-
tical axis of the inertial frame anymore because a 90 degrees roll angle puts the two rear
thrusters in the vertical plane of the inertial frame. The vehicle loses a DOF with respect
to the task (rotation around the vertical axis of R₀). Therefore, the reconfiguration ring
could be used to compensate for this phenomenon and maintain the rear thrusters in the
horizontal plane of R₀. Using the ring to rotate of the opposite of the roll angle of the
vehicle, the thrusters mounted on the ring can be kept fixed in the horizontal plane of the
inertial frame. This behavior can be used to compensate for disturbances in roll as well
as for consequences of sway compensation using roll.

Then, in the second configuration presented on figure 5.10b, compensating for the roll
angle would allow the two rear thrusters to generate a moment around the vertical axis of
the inertial frame independently from the attitude of the vehicle (considering that pitch
is different from 90°). In this use case, the reconfiguration ring does not really add a new
DOF to the vehicle but makes it more robust to roll disturbance.

The roll compensation behavior is not as useful in the first configuration of figure 5.10
because roll itself is actuated and can be stabilized. Yet, one could imagine scenarios
where the vehicle needs to dive vertically whilst suffering from roll disturbances. In this case, it would be useful to be able to keep the two thrusters vertical w.r.t. the inertial frame to maintain full heave capacity.

### 5.2.2 Thrust allocation of PlaSMAR

Considering the first configuration of PlaSMAR figure 5.10a, this section exposes the thrust allocation of the reconfigurable ring. In this case the front thrusters are mounted on a ring placed around the cylindrical hull of the vehicle, approximately in the middle of its length. In section 1.2.1 and section 1.2.2, thrust allocation is performed using the TCM matrix denoted $B$. The TCM is used to calculate the necessary thrust of all the thrusters of the vehicles for a given command vector $\tau_c$:

$$ \tau_c = Bu \qquad (5.12) $$

where $u$ is the vector containing all the elementary thrusts. The relation (5.12) needs to be inverted to perform thrust allocation.

For fixed thrusters, $B$ is constant. In the fully-actuated case without redundancy, $B$ is invertible and thrust allocation is performed using:

$$ u = B^{-1}\tau_c \qquad (5.13) $$

In the underactuated case, two methods are detailed in section 1.2.1: the Moore-Penrose pseudo-inverse used notably for decoupled propulsive arrangement and the space reduction which consists in inverting $B$ in a reduced subspace where the vehicle can be considered fully-actuated.

For vector thrusters, the same methods are used considering an equivalent TCM representing a virtual fixed-thruster equivalent of the reconfigurable arrangement [30]. Then, a 3-DOF vector thruster is represented with 3 orthogonal fixed virtual thrusters and the TCM is calculated considering the virtual thrusters. The main interest of this solution is that it provides a constant TCM independent from the reconfiguration angles of the vector thrusters. The angles and thrust are reconstructed after thrust allocation is performed on the fixed-thrusters equivalent.

Here, with the thrusters mounted on the ring their position is not constant. It is therefore difficult to find a fixed-thrusters equivalent. In the first configuration of PlaSMAR,
the two front thrusters are used to generate a force in the (yB, zB) plane and a roll moment without coupling. One fixed-thrusters solution emulating the generation of a force in the (yB, zB) plane is to consider two orthogonal thrusters placed at the center of the hull. Yet, this solution does not allow generating roll moment. One or several additional virtual thrusters would be necessary to generate roll which would modify the dimension of the problem. A configuration with three thrusters placed in the vertical plane all around the hull with 120° angle one with the other could be used to generate sway, heave and yaw.

In a first approximation, the ring is considered to have a perfect tracking of the com-
manded angle $\alpha = \alpha_c$. Therefore, the TCM can be considered locally constant to perform
thrust allocation. The controller outputs a control effort vector and notably the sway and
heave forces as well as the roll moment, $Y_c$, $Z_c$ and $K_c$ respectively. The control angle of
the ring $\alpha_c$ is calculated geometrically for a given pair $Y_c, Z_c$:

$$
\alpha_c = \arctan_2(Z_c, Y_c) \tag{5.14}
$$

The arctan₂ function is used for a complete four quadrants definition of the angle α_c. More
studies are needed to establish the optimal solution of α_c considering the rotation speed of
the ring and the variation rate of the thrusters. Practically, if the thrusters are considered
to be able to switch direction and to vary its thrust fast enough, α could be limited to a
[0, π/2] interval and cover all possible forces of the plane. For now, the thrust allocation
procedure favors rotating the ring over inverting the thrust of the two thrusters. As seen
in the following simulation, this can result in a light oscillation due to numerical delay.

The TCM can then be calculated for a given $\alpha_c$ as:

$$
\mathbf{B}_1 =
\begin{bmatrix}
1 & 1 & 0 & 0 \\
0 & 0 & \sin(\alpha_c) & \sin(\alpha_c) \\
0 & 0 & -\cos(\alpha_c) & -\cos(\alpha_c) \\
0 & 0 & -R & R \\
0 & 0 & 0 & 0 \\
-R & R & 0 & 0
\end{bmatrix}
\qquad (5.15)
$$
where R is the radius of the hull. The two first columns correspond to the two rear thrusters actuating surge and yaw and columns three and four correspond to the two thrusters mounted on the ring. Equation (5.15) confirms that when the ring angle α_c is 0 degrees, the two front thrusters generate a pure heave force and when the angle is 90 degrees, they generate a pure sway force. Then, between 0 and 90 degrees, they generate a diagonal force in the (yB, zB) plane. Equation (5.15) also shows that the roll moment generated by the two thrusters mounted on the ring is independent from the angle of the ring and is decoupled from the sway and heave forces.

Because the sway and heave modes are coupled with α_c different from 0 or 90 degrees, space reduction is not suited for inverting B_1. The Moore-Penrose pseudo-inverse is used in this case (see section 1.2.1). Simulation results presented in the following show that the Moore-Penrose pseudo-inverse behaves properly in this case but this result is hardly general. More investigation on other inverting methods based on quadratic programming would be needed to ensure that the inversion of B_1 is always successful. The Moore-Penrose pseudo-inverse of the TCM is:

$$
\mathbf{B}_1^\dagger = \begin{bmatrix} \frac{1}{2} & 0 & 0 & 0 & 0 & -\frac{1}{2R} \\ \frac{1}{2} & 0 & 0 & 0 & 0 & \frac{1}{2R} \\ 0 & \frac{\sin(\alpha_c)}{4} & -\frac{\cos(\alpha_c)}{4} & -\frac{1}{2R} & 0 & 0 \\ 0 & \frac{\sin(\alpha_c)}{4} & -\frac{\cos(\alpha_c)}{4} & \frac{1}{2R} & 0 & 0 \end{bmatrix} \quad (5.16)
$$

A more rigorous approach would be to extend the model of the vehicle with the
dynamics of the ring. This way, the input of the system would no be τ_c anymore but the
forces of each thruster and the angle of the ring. Doing so would also allow adding one
more control layer on the behavior on the ring.

Of course, similar calculations can be realized in the second configuration presented
on figure 5.10b where the rear thrusters are mounted on the ring and used to generate
a surge force and a moment around any axis of the plane (yB, zB). When the two rear
thrusters are mounted on the reconfigurable ring, the TCM becomes:

$$ B_2 = \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & -R & R \\ R \sin(\alpha_c) & -R \sin(\alpha_c) & 0 & 0 \\ -R \cos(\alpha_c) & R \cos(\alpha_c) & 0 & 0 \end{bmatrix} \quad (5.17) $$

In the third mode where the ring is used to compensate the roll angle of the vehicle, the ring angle is set to $\alpha_c = -\phi$ where $\phi$ is the current roll angle of the vehicle.

### 5.2.3 Control of PlaSMAR

Thanks to the allocation method detailed in the previous section, pretty much any controller presented in this work can be used on PlaSMAR. In the first configuration, the vehicle has five independent DOF: surge, sway, heave, roll and yaw. Because the three translations are independently actuated, no compensation mechanism is needed in position tracking tasks. Moreover, roll and yaw are actuated and pitch is usually naturally stabilized because of the mass distribution. The vehicle is therefore capable of keeping a neutral roll angle allowing rotations around the vertical axis of the inertial plane. Also, the association of roll and yaw allows the vehicle to roll to the side and get into an upright position allowing vertical diving. For the control, PID based controllers with a linearization mechanism or SMC can be used.

Same goes for the second configuration. The vehicle has 5 actuated DOF: surge, heave, roll, pitch and yaw. Then, for a position tracking task like seabed scanning, roll or yaw can be used in association with heave or surge to compensate for sway. In such a case, the $H$-matrix based controller or the SMC controller for underactuated craft can be used. If roll is used for compensation and pitch is not actually part of the task, then the reconfiguration ring can be used to cancel out the roll angle and maintain the rear thrusters in the horizontal plane of the inertial frame. This way, the vehicle can be controlled in yaw at all times. (for any roll angle)
In the following, the vehicle in the first configuration is controlled using a very simple PID-based controller. The control effort vector is calculated in the inertial frame as:

$$ \tau_c = \tilde{M}(\eta) (\ddot{\eta}^* + K_d \dot{e}_\eta + K_p e_\eta + K_i \int_0^t e_\eta d\zeta) + \tilde{C}(\eta, \dot{\eta}) \dot{\eta} + \tilde{D}(\eta, \dot{\eta}) \dot{\eta} + g(\eta) \quad (5.18a) $$

$$ \tilde{M}(\eta) = M J(\eta)^{-1} \quad (5.18b) $$

$$ \tilde{C}(\eta, \dot{\eta}) = M \dot{J}(\eta, \dot{\eta})^{-1} + C(\nu) J(\eta)^{-1} \quad (5.18c) $$

$$ \tilde{D}(\eta, \dot{\eta}) = D(\nu) J(\eta)^{-1} \quad (5.18d) $$

where $K_d$, $K_p$ and $K_i$ are diagonal strictly definite gain matrices, $\ddot{\eta}^*$ is the second order derivative of the desired state vector $\eta^*$ and $e_\eta = \eta^* - \eta$ is the error signal. The model matrices with the subscript $\tilde{\cdot}$ are expressed in the inertial frame using the kinematic relation $\dot{\eta} = J(\eta)\nu$. Because the vehicle is not actuated in pitch, the fifth line of $\tau_c$ is discarded.

## 5.3 Simulations

PlaSMAR is evaluated in simulation on several study cases. It is compared with the RSM AUV used before in this work and controlled using the $H$-matrix based controller (see chapter 3 for more details) or a SMC inspired by [40]. In this SMC, the guidance action is created using sway error in the calculation of the yaw sliding surface. More details about this controller can be found in section 2.4.2.

### 5.3.1 Seabed scanning

In this first scenario, the first configuration of PlaSMAR is tested on the seabed scanning task. The vehicle is equipped with the front reconfigurable ring holding two parallel thrusters and two horizontal rear thrusters on the fixed ring. In this configuration, the vehicle is actuated in surge, sway, heave, roll and yaw and thus over-actuated with respect to the task which does not require roll. It is compared to the RSM vehicle which is ill-actuated on the task. It must use yaw to compensate for sway.

Figure 5.12 shows that both vehicles have perfect position tracking on the whole task. The main difference between the two vehicles appears in figure 5.13 where PlaSMAR (red line) appears to have a much smaller heading error than RSM (blue line). In fact, because
<figure><img src="image_64.png" /><figcaption>Figure 5.12 – Comparison of the position signals of the two vehicles. Blue: RSM, Red: PlaSMAR – a: $\boldsymbol{x}_o$ axis, b: $\boldsymbol{y}_o$ axis, c: $\boldsymbol{z}_o$ axis</figcaption></figure>

<figure><img src="image_65.png" /><figcaption>Figure 5.13 – Comparison of the orientation errors of the two vehicles. Blue: RSM, Red: PlaSMAR – a: roll, b: pitch, c: yaw</figcaption></figure>
<figure><img src="image_66.png" /><figcaption>Figure 5.14 – Simulation snapshots of the two vehicles on the seabed scanning with $\psi^* = 0$ – Blue axis: $\boldsymbol{x}_B$, Green axis: $\boldsymbol{y}_B$, Black axis: $\boldsymbol{z}_B$</figcaption></figure>

the sway of PlaSMAR is directly actuated, the vehicle does not need any compensation mechanism and can truly control its yaw angle. This is not the case of RSM which uses yaw to compensate for sway. Numerical delay creates a slight heave and pitch oscillation on PlaSMAR.

To accentuate the difference between the two vehicles, they are evaluated on a second version of the seabed scanning task where the heading angle is constraint to $\psi^* = 0$.

Figure 5.14 shows the two different behaviors of the vehicles. On figure 5.14a, RSM is tangent to the path aligned with the $\mathbf{y}_o$ axis. Because yaw is used for compensation of sway, the heading constraint is neglected and the heading angle naturally stabilizes tangent to the velocity vector. On figure 5.14b on the other hand, PlaSMAR is aligned with the $\mathbf{x}_o$ axis while traveling along the $\mathbf{y}_o$ axis. The new DOF provided by the ring allows the vehicle to translate in sway while controlling its heading and maintaining constant depth.

## 5.3.2 Roll compensation

In this section, the reconfigurable ring of PlaSMAR is used for compensation of the roll of the vehicle. The second configuration shown on figure 5.10b is used where the two rear thrusters are mounted on a ring. The vehicle is therefore actuated in surge, heave,
<figure><img src="image_67.png" /><figcaption>Figure 5.15 – Comparison of the attitude of the two vehicles. Blue: RSM, Red: PlaSMAR - a: roll, b: pitch, c: yaw</figcaption></figure>

roll, pitch and yaw. In this example, it is evaluated on the seabed scanning. Because sway
is not actuated, roll is used for compensation using the $\mathcal{H}_p$ matrix controller introduced
in section 3.4.1. Section 3.4.1 shows that compensation mechanisms can disturb the other
DOF of the systems. In particular in this case, using roll disturbs the heading control.
With a non-zero roll angle, the capability of the vehicle to rotate around the vertical axis
of the inertial frame is reduced. At the extreme, when $\phi = \pi/2$ and $\theta = 0$, the $\mathbf{z}_B$ axis of
the vehicle is in the horizontal plane of the inertial frame. Yaw actuators can no longer
be used to control the heading of the craft.

To counteract this effect, the two rear thrusters are mounted on a rotating ring and
maintained as fast as possible in the horizontal plane of the inertial frame. The angle
command of the ring is at all time $\alpha_c = -\phi$.

This solution is compared with the *RSM* vehicle using the $\mathcal{H}_p$ based controller on the
seabed scanning trajectory with the heading constraint $\psi^* = 0$.

Figure 5.15 shows the attitude of the two vehicles on the task. In the first turn around
15s, both vehicles roll to the side to generate the needed sway velocity at the tracking
point but, because it stabilizes around $\phi = \pi/2$, the first vehicle (blue line) is not able to
control its heading anymore. The two rear thrusters are in the vertical plane of $\mathcal{R}_0$. Then,
the yaw control trying to maintain the heading of the vehicle destabilizes the system. On
the other hand, the PlaSMAR vehicle stabilizes with $\phi = \pi/2$ and, thanks to the rotating
ring, is still able to really control its heading.

Comparing figure 5.14b and figure 5.16 shows that both solutions are almost equiv-
alent. The first vehicle keeps a neutral roll angle and directs the two front thrusters to
generate sway while the second vehicle rolls to the side to be able to push along the **y**_o
axis while maintaining the two rear thrusters in the horizontal plane. This comparison
also shows that one solution for this task with RSM or PlaSMAR would be to integrate
the desired roll behavior of the vehicle directly in the task. Because both vehicles are
actuated in roll and roll is not part of the task requirements, a roll trajectory mimicking
the behavior of the vehicles obtained here could be calculated. The vehicle would then
track this roll angle and naturally align its thrusters to the the **y**_o axis for instance to nav-
igate along this axis without direct sway control. Such a solution is more about trajectory
planning than it is about control.
<figure><img src="image_68.png" /><figcaption>Figure 5.16 – Simulation snapshots of PlaSMAR with rear reconfigurable ring (black cylinder) on the seabed scanning with $\psi^* = 0$. Blue axis: $\boldsymbol{x}_B$, Green axis: $\boldsymbol{y}_B$, Black axis: $\boldsymbol{z}_B$</figcaption></figure>
5.4 Partial conclusions on vector thrust

A new concept of reconfigurable thrust has been introduced in this chapter: the re-
configurable ring. PlaSMAR is presented as a proof of concept of this new propulsive
solution. The study of the TCM corresponding to the two configurations of PlaSMAR
and simulation results exposed in this chapter show that the rotating ring allows adding a
new DOF to the vehicle without creating new coupling. This new reconfiguration concept
seems to be a better suited solution for smaller vehicles as it does not introduce any new
disturbing effects.

The two configurations of PlaSMAR give new solutions to traditional control problem
of underwater vehicles. The first configuration adds sway as an actuated DOF decou-
pled from the others. It facilitates position tracking as the vehicle does not require any
compensation or guidance mechanism to track the three position components of the space.

The second configuration adds pitch. In this case, this new DOF can be used either as
such, to generate a pitch moment on the vehicle, or to compensate the roll angle of the
craft and maintain the rear thrusters in the horizontal plane of the inertial frame. These
two solutions enhance the performances of the vehicle on the tasks presented here. It is
worth noting that the rear rotating ring with two thrusters allows reproducing the surge,
pitch and yaw actuation used on many usual underwater vehicles.

Further studies are required on this new technology. Notably, the actuation of the
rotating ring is yet to be sorted out. Also, thrust allocation is not trivial on this system.
More work on the propeller and reconfiguration dynamics is required to optimize the
allocation.
CONCLUSION

This work follows on from the long line of work on the design and control of ma-
rine vehicles. Many types of different control laws and guidance principles for surface
and underwater vehicles have been introduced and studied in details. All the solutions
presented in the literature show equivalent theoretical performances. All of them ensure
global or semi-global stability of the plant even in presence of external disturbances and
noisy measurements.

Further work will be needed to properly compare all the different methods both math-
ematically and on experimental vehicles. Ranking the methods is very difficult as their
performances will likely depend on the system itself. Other parameters will be to con-
sider like computation time, effects of numerical sampling or the impact of poor-quality
measurements.

Then, a new kinematic guidance principle based on the handy matrix $\mathcal{H}$ has been in-
troduced. This new principle allows the control of an underactuated or ill-actuated vehicle
using kinematic couplings between a non actuated translation and an actuated rotation.
The control is shown to be stable and robust to external disturbance and measurement
noise. It will be easily extended to other types of systems thanks to the design rules and
matrix calculation algorithm provided. These results also allow evaluating the capabilities
of a given AUV and finding the tasks for which it is suited.

The controller has been successfully applied in both trajectory tracking and path
following scenarios. Path following was performed on Remus100, a wide spread AUV rep-
resenting the class of torpedo shape *uqr*-vessels. The *H*-matrix based controller allows
solving the path following problem in several manners. Using the *H*-matrix based con-
troller allows precisely choosing the behavior of the tracking point of the vehicle which is
very valuable. In path following for instance, it allows finding simple PID-based solutions
to the Lyapunov stability criterion. In future works, the controller could be used in other
back-stepping applications to complete different types of tasks.

One of the interesting use cases of the path following example is the leader-follower sce-
nario. In this case, the *H*-matrix controller is used in the follower vehicle. The construction
of the closed-loop system allows tracking the leader vehicle and adapting the follower's
behavior to ensure that both tasks are successful. Such applications could notably be
interesting in pipeline surveillance where one vehicle, the leader, would be measuring or
taking images of a submerged structure while the other, the follower, would go back and
forth between the leader and the surface or a support ship.

Applying the controller to a physical vehicle would be valuable to assess its perfor-
mances in conditions. Notably, it is very hard to anticipate the behavior of the control
law with respect to numerical sampling, latency of the embedded calculator or unsteady
measurement errors.

A flatness-based controller has also been developed using the more simple case of
surface vessels. Flatness-based control appears to be a very strong contestant for the
control of marine vehicles as it is shown to be very robust to model approximation and
external disturbances. In this work, a flatness-based controller has been derived for surface
vessels using the simplified model of the hovercraft. The controller has been developed
under the hypothesis of a circular hull shape and even mass distribution but has been
successfully applied to a generic surface vessel.

The flatness-based controller naturally includes a guidance principle. It allows calcu-
lating a surge force and a yaw moment out of two position signals. The flatness equations
obtained in this example could also be used as a guidance principle with another type of
controller.

For increased robustness and to counteract the potentially damaging behavior of
flatness-based control away from the desired trajectory, it has been associated with an
intelligent proportional derivative controller. The iPD increases the already impressive
robustness of the flatness-based controller and effectively solves the problems caused by
feedforward linearization when the vehicle is outside the acceptable bound around the
desired trajectory. The combined controller has been shown to be stable and robust to
external disturbances and noisy measurements. In this case again, the next step is to ap-
ply the controller on a physical system. Investigations on the application of the controller
on a scale-model of the hovercraft have been started.

Also, some more calculation work is required to apply the controller to underwater
vehicles. From the results obtained on the surface vessels, it is safe to assume that the
controller will easily scale up to the underwater case. It seems notably well suited for
*uqr*-vessels in position tracking applications.

Finally some new results on vectoring thrust applied to small scale AUVs have been
presented. Notably, simulations have been conducted to show that the use of off-centered
vector thrusters is not suited for every type of vehicle. Although they are bounded, the
coupling force effects generated by vector thrusters can jeopardize the tracking mission of
smaller vehicles or of vehicles without natural robustness to force disturbances.

A new use case of vector thrusters on AUVs has also been exposed. While vector
thrusters are usually used for moment control through dynamic couplings, simulation
results show that they can also be used for force control. This new force control mode is
shown to be especially useful for missions where moment control is naturally not suited.
This is notably the case of autonomous missions like docking, hovering and station keeping
missions.

The proof of concept for a new type of vectoring thrust system has also been proposed. The new system relies on pairs of thrusters mounted on a ring able to rotate around cylindrical sections of the hull of a vehicle. This rotating motion can be used to direct the force vector generated by the pair of thrusters while maintaining a stationary moment axis. When the thrusters are aligned with the axis of rotation of the ring, it allows directing the axis of the moment generated by the pair of thrusters while keeping the force vector aligned with the body of the vehicle. This second mode is shown to effectively unlock a new DOF for the vehicle as well as allowing compensation of roll disturbances.

On this matter, the next step is to design and build a second version of the test plat-
form and investigate the possible solutions for actuation of the ring. Optimized magnetic
couplings will be investigated as they would allow piloting the rotation of the ring from
the inside of the hull therefore avoiding the need of holes in the hull.

Some more work will be conducted on the controller of this vehicle. A controller
taking the dynamics of the ring into account would be interesting and might appear to
be necessary. The model of the vehicle would have to be extended to take the behavior of
the ring into account and include the ring angle as a new control input. In the same way,
further investigations on optimizing the allocation procedure are required for increased
performances.

Overall, this work proposes several solutions to increasing the maneuverability of un-
deractuated marine craft through control and new mechanical designs.
# Appendices
# INTEGRAL SLIDING MODE CONTROL

This appendix demonstrates that steady state can appear when using a sliding mode controller on a system subjected to external unknown disturbances or model approximations. Then, *Integral Sliding Mode* is presented as a solution against this steady state error. There are not many references mentioning integral SMC in the literature but some hints can be found in [108, 150]. Note that *super-twisting* SMC as described for instance in [151] has an integral behavior and cancels steady state error.

A demonstration based on the study of the equilibrium of the closed loop system is provided. This phenomena appears to be rather absent from the literature and notably in the context of marine craft although it can cause problems in many real-life applications. In this example, the demonstration is performed on a fully-actuated system but similar effects can be shown on underactuated vehicles.

## A.1 Apparition of steady state error

To demonstrate the apparition of steady state error, the model of a marine craft expressed in the inertial frame and in presence of an unknown external disturbance is introduced.

$$ \tau = \bar{M}(\eta)\ddot{\eta} + \bar{C}(\eta, \nu)\dot{\eta} + \bar{D}(\eta, \nu)\dot{\eta} + g(\eta) + \Delta \quad (A.1) $$

Matrices $\bar{M}$, $\bar{C}$ and $\bar{D}$ are the model matrices modified to express the model in the inertial frame. They contain the transformation based on the $J(\eta)$ matrix. Vector $\Delta$ marks the unknown external disturbance effect.

Following the reasoning introduced in the examples of section 2.4, the sliding surface for this second order system is defined with respect to $\eta$ as:

$$ \sigma = \left( \frac{d}{dt} + \lambda \right)^{(2-1)} e_{\eta} = \dot{e}_{\eta} + \lambda e_{\eta} \quad (A.2) $$
A simple sliding condition is used in this example as the super-twisting and terminal sliding mode conditions exposed in section 2.4 do not make much difference on this issue. To avoid chattering, a saturation function is used instead of the intuitive signum function. Note that the signum function does not solve the steady state error. The sliding condition is then:

$$ \dot{\sigma} = -K\mathrm{sat}(\sigma) \qquad (A.3) $$

Using the method described in section 2.4, the control is established from the sliding condifiton and the model equation:

$$ \tau_c = \bar{M}(\eta)(\ddot{\eta}^* + \lambda\dot{e}_\eta + K\mathrm{sat}(\sigma)) + \bar{C}(\eta, \nu)\dot{\eta} + \bar{D}(\eta, \nu)\dot{\eta} + g(\eta) \qquad (A.4) $$

Of course, because $\Delta$ is an unknown disturbance term, it cannot be used in the control calculations. The closed-loop system is therefore:

$$ \ddot{e}_\eta + \lambda\dot{e}_\eta + K\mathrm{sat}(\sigma) = \bar{M}^{-1}\Delta \qquad (A.5) $$

A quick way to demonstrate the appearance of a steady state error in to study the values taken by the error vector $e_\eta$ when an equilibrium $\ddot{e}_\eta = \dot{e}_\eta = 0$ is reached. To do so, the system is considered in the neighborhood of the sliding surface meaning $\mathrm{sat}(\sigma) = \sigma$. In such conditions, the closed-loop system becomes :

$$ K\lambda e_\eta = \bar{M}^{-1}\Delta \qquad (A.6) $$

In this conditions, it appears clearly that a steady state error depending on the disturbance magnitude appears at the error equilibrium. This steady state can be made as small as necessary tuning the gain parameters $K$ and $\lambda$ but cannot be canceled. In facts, inside the boundary layer, the SMC behaves similarly to a Proportional Derivative controller hence the steady state error.

## A.2 Integral Sliding Mode

When working with Proportional and Proportional Derivative controllers, the solution to cancel steady state errors is to add an integral term. The same idea is used in *Integral Sliding Mode Control (ISMC)*. Several definitions of the ISMC can be found but it is rarely seen in the context of marine vehicles. The ISMC used in this example consists in
defining the sliding surface with respect to the integral of the position and orientation error instead of the error. Same reasoning can be seen in some underactuated examples of section 2.4 where the integral of the speed signal is used to take constant position offsets into account. The system (A.3) is then considered of the third order with respect to the integral of $\eta$. The sliding surface is therefore defined as:

$$ \sigma = \left(\frac{d}{dt} + \lambda\right)^{(3-1)} \int_{0}^{t} e_{\eta}(\zeta) d\zeta = \dot{e}_{\eta} + 2\lambda e_{\eta} + \lambda^{2} \int_{0}^{t} e_{\eta}(\zeta) d\zeta \quad (A.7) $$

The same sliding condition is used as in equation (A.3):

$$ \dot{\sigma} = -K\mathrm{sat}(\sigma) = \ddot{e}_{\eta} + 2\lambda\dot{e}_{\eta} + \lambda^{2}e_{\eta} \quad (A.8) $$

In these conditions and following the same logic as before, the controller is then:

$$ \tau_c = \bar{M}(\eta)(\ddot{\eta}^* + 2\lambda\dot{e}_{\eta} + \lambda^2 e_{\eta} + K\mathrm{sat}(\sigma)) + \bar{C}(\eta, \nu)\dot{\eta} + \bar{D}(\eta, \nu)\dot{\eta} + g(\eta) \quad (A.9) $$

In this case, the closed-loop system becomes:

$$ \ddot{e}_{\eta} + 2\lambda\dot{e}_{\eta} + \lambda^2 e_{\eta} + K\mathrm{sat}(\sigma) = \bar{M}^{-1}\Delta \quad (A.10) $$

As before, the equilibrium in the neighborhood of the sliding surface is studied: $\mathrm{sat}(\sigma) = \sigma$, $\ddot{e}_{\eta} = \dot{e}_{\eta} = 0$. The closed-loop system is rewritten as:

$$ e_{\eta} = (\lambda + 2K\lambda)^{-1} \left( \bar{M}^{-1}\Delta - K\lambda^2 \int_{0}^{t} e_{\eta}(\zeta) d\zeta \right) \quad (A.11) $$

Equation (A.11) shows that the state error $e_{\eta}$ can be canceled at the equilibrium thanks to the integral term. The ISMC formulation therefore allows to cancel steady state errors due to external unknown disturbances.
# ALTERNATIVE REPRESENTATION OF THE
GENERALIZED STATE SYSTEM

This appendix gives an alternative representation to the generalized state system described in section 4.3.3. The system is recalled here:

$$ \dot{x} = u \cos \psi - v \sin \psi \qquad (B.1a) $$

$$ \dot{y} = u \sin \psi + v \cos \psi \qquad (B.1b) $$

$$ \dot{u} = \tilde{\tau}_u + v\dot{\psi} + \beta u \qquad (B.1c) $$

$$ \dot{v} = -u\dot{\psi} + \beta v \qquad (B.1d) $$

The inputs of system (B.1) are the surge force $\tilde{\tau}_u$ and the yaw angle $\psi$ whom first order derivative appears in equation (B.1c) and equation (B.1d).

Following the method exposed in [37, 38], a generalized state transformation is calculated to cancel the dependency in $\dot{\psi}$. The infinite prolongation [184] is introduced for system (B.1):

$$ \begin{aligned} \frac{d}{dt} = (u \cos \psi - v \sin \psi) \frac{\partial}{\partial x} &+ (u \sin \psi + v \cos \psi) \frac{\partial}{\partial y} \\ &+ (\tilde{\tau}_u + v \dot{\psi} + \beta u) \frac{\partial}{\partial u} + (-u \dot{\psi} + \beta v) \frac{\partial}{\partial v} \\ &+ \sum_{k=0}^{\infty} \tilde{\tau}_u^{(k+1)} \frac{\partial}{\partial \tilde{\tau}_u^{(k)}} + \sum_{k=0}^{\infty} \psi^{(k+1)} \frac{\partial}{\partial \psi^{(k)}} \end{aligned} \qquad (B.2) $$

The new state $\zeta = [\zeta_1 \ \zeta_2 \ \zeta_3 \ \zeta_4]^T$ must satisfy the following condition expressed in terms of Lie bracket:

$$ \left[ \frac{d}{dt}, \frac{\partial}{\partial \dot{\psi}} \right] (\zeta) = 0 \qquad (B.3) $$
This partial differential equation ensures that the first time derivative of the new state $\zeta$ does not depend on $\dot{\psi}$. The Lie bracket thus reads :

$$ \left[ \frac{d}{dt}, \frac{\partial}{\partial \dot{\psi}} \right] = v \frac{\partial}{\partial u} - u \frac{\partial}{\partial v} + \frac{\partial}{\partial \psi} \qquad (B.4) $$

The four components of the new state $\zeta$ are chosen as independent functions of the state variables. The natural choice for the first two components $\zeta_1$ and $\zeta_2$ is $x$ and $y$ respectively. The position coordinates obviously respect condition (B.3):

$$ (v \frac{\partial}{\partial u} - u \frac{\partial}{\partial v} + \frac{\partial}{\partial \psi})(x) = 0 \qquad (B.5a) $$

$$ (v \frac{\partial}{\partial u} - u \frac{\partial}{\partial v} + \frac{\partial}{\partial \psi})(y) = 0 \qquad (B.5b) $$

Then, two more independent functions must be found for $\zeta_3$ and $\zeta_4$. A multitude of solutions exist for this problem so two fairly simple solutions are selected:

$$ \zeta_3 = \frac{1}{2}(u^2 + v^2) \qquad (B.6a) $$

$$ \zeta_4 = \psi + \operatorname{atan}\left(\frac{v}{u}\right) \qquad (B.6b) $$

The two solutions $\zeta_3$ and $\zeta_4$ of equation (B.6) are shown to be suitable with respect to condition (B.3):

$$ \left[ \frac{d}{dt}, \frac{\partial}{\partial \dot{\psi}} \right] (\zeta_3) = vu - uv = 0 \qquad (B.7a) $$

$$ \left[ \frac{d}{dt}, \frac{\partial}{\partial \dot{\psi}} \right] (\zeta_4) = - \frac{v^2}{u^2 + v^2} - \frac{u^2}{u^2 + v^2} + 1 = 0 \qquad (B.7b) $$

The state $\zeta$ is then given as:

$$ \zeta_1 = x \qquad (B.8a) $$

$$ \zeta_2 = y \qquad (B.8b) $$

$$ \zeta_3 = \frac{1}{2}(u^2 + v^2) \qquad (B.8c) $$

$$ \zeta_4 = \psi + \operatorname{atan}\left(\frac{v}{u}\right) \qquad (B.8d) $$
and the associated new state space representation is:

$$ \dot{\zeta}_1 = \dot{x} \qquad (B.9a) $$

$$ \dot{\zeta}_2 = \dot{y} \qquad (B.9b) $$

$$ \dot{\zeta}_3 = (\dot{\zeta}_1 \cos \psi + \dot{\zeta}_2 \sin \psi)\tau_u + 2\beta\zeta_3 \qquad (B.9c) $$

$$ \dot{\zeta}_4 = -\frac{\dot{\zeta}_1 \sin \psi + \dot{\zeta}_2 \cos \psi}{2\zeta_3}\tau_u \qquad (B.9d) $$

To conclude, the new state space representation is indeed standard, the state transformation canceled all dependency in the input derivative $\dot{\psi}$. The standard representation could then be used to perform the controller calculations.

Note that, in this specific case, the Brunovsky state transformation detailed in section 4.3.3.2 also cancels the dependencies in $\dot{\psi}$ because of the symmetries in the plant but this is not a general result.
# CONTROL OF UNDERACTUATED SYSTEMS WITH COUPLED ACTUATORS

This appendix presents a side effect of the control of some underactuated systems independently from the control method used. In these examples, a feedback linearizing controller is used as it is the most frequently used in this work. This phenomena notably appears when coupled actuators are used to control underactuated systems. In such cases, one of the two coupled effects generated by an actuator (most often one force and one coupled moment) is chosen for the control of the system. The other one can create non negligible disturbance on the uncontrolled DOF or even on the whole system.

Most examples of such systems rely on the natural stability of the uncontrolled DOF (for instance autonomous boats). Yet, the passive stability of the free DOF may not be robust enough to counteract the effect of the coupled actuator. It might notably be the case for small scale vehicle equipped with vector thrusters or vehicles with simple hull shapes (cylinder, box shape).

Consider the following control system:

$$ \dot{x} = f(x) + g(x)u \qquad (C.1) $$

where $x \in \mathbb{R}^3$ is the state vector, $f(x) \in \mathbb{R}^3$ is a vector field, $g(x) \in \mathbb{R}^{3,3}$ is a known matrix and $u \in \mathbb{R}^3$ is the input vector.

Considering that $g(x)$ is invertible, an intuitive model-based feedback linearizing controller can be designed as:

$$ u = g(x)^{-1} (\dot{x}_d + \lambda(e_x) - f(x)) \qquad (C.2) $$

where $\lambda(e_x)$ is a control law based on the tracking error $e_x$ and $\dot{x}_d$ is the desired velocity used as a feedforward term. Plugging the controller (C.2) into the model equation (C.1)
gives the closed-loop system:

$$ \dot{e}_x = \lambda(e_x) \qquad (C.3) $$

the non-linear terms of the model are canceled using feedback linearization and choosing function $\lambda$ properly, stability of the equilibrium $e_x = 0$ can be ensured for all three DOF of the space.

Now, let us consider the same system but in the underactuated context. In this second example, the second component of the control vector $u$ is always 0 because of underactuation:

$$ u = \begin{bmatrix} u_1 & 0 & u_3 \end{bmatrix}^T \qquad (C.4) $$

The controller is built again using the intuitive feedback linearizing solution equation (C.2). This time though, because of underactuation, the three DOF of the system behave differently. Let us define the vectors and matrices of the system as:

$$ x = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}^T \qquad (C.5a) $$

$$ f(x) = \begin{bmatrix} f_1(x) & f_2(x) & f_3(x) \end{bmatrix}^T \qquad (C.5b) $$

$$ \lambda(e_x) = \begin{bmatrix} \lambda_1(e_x) & \lambda_2(e_x) & \lambda_3(e_x) \end{bmatrix}^T \qquad (C.5c) $$

$$ g(x) = \begin{bmatrix} g_1(x) & 0 & 0 \\ 0 & g_2(x) & 0 \\ 0 & 0 & g_3(x) \end{bmatrix} \qquad (C.5d) $$

Note that all three components of $f$ can depend on the complete state vector $x$. In this first case, $g(x))$ has been chosen diagonal. All three DOF are decoupled.

Then, applying the controller (C.2) with the hypothesis (C.4) on the system (C.1)
considering (C.5) gives:

$$ \mathbf{u} = \begin{bmatrix} u_1 \\ 0 \\ u_3 \end{bmatrix} = \begin{bmatrix} g_1^{-1}(\boldsymbol{x}) & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & g_3^{-1}(\boldsymbol{x}) \end{bmatrix} \left( \begin{bmatrix} \dot{x}_{1,d} \\ 0 \\ \dot{x}_{3,d} \end{bmatrix} \begin{bmatrix} \lambda_1(\boldsymbol{e}_x) \\ 0 \\ \lambda_3(\boldsymbol{e}_x) \end{bmatrix} - \begin{bmatrix} f_1(\boldsymbol{x}) \\ 0 \\ f_3(\boldsymbol{x}) \end{bmatrix} \right) \quad (C.6a) $$

$$ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} = \begin{bmatrix} f_1(\boldsymbol{x}) \\ f_2(\boldsymbol{x}) \\ f_3(\boldsymbol{x}) \end{bmatrix} + \begin{bmatrix} g_1(\boldsymbol{x}) & 0 & 0 \\ 0 & g_2(\boldsymbol{x}) & 0 \\ 0 & 0 & g_3(\boldsymbol{x}) \end{bmatrix} \begin{bmatrix} u_1 \\ 0 \\ u_3 \end{bmatrix} \quad (C.6b) $$

Note that the second line of $\mathbf{u}$ has been artificially set to zero to respect the constraint of underactuation. The closed-loop system becomes:

$$ \dot{x}_1 = \dot{x}_{1,d} + \lambda_1(\boldsymbol{e}_x) \quad (C.7a) $$

$$ \dot{x}_2 = f_2(\boldsymbol{x}) \quad (C.7b) $$

$$ \dot{x}_3 = \dot{x}_{3,d} + \lambda_3(\boldsymbol{e}_x) \quad (C.7c) $$

Equation (C.7) shows that the first and third components of $\boldsymbol{x}$ can indeed be driven towards the desired value using the $\lambda_1$ and $\lambda_3$ functions but that the second component $x_2$ is left uncontrolled. Moreover, since the function $f_2(\boldsymbol{x})$ can depend on the complete state $\boldsymbol{x}$ (natural coupling terms), the behavior of $x_2$ is hardly predictable, could be unstable and could destabilize the rest of the system.

Furthermore, the matrix $g(\boldsymbol{x})$ could contain off-diagonal coupling terms:

$$ g(\boldsymbol{x}) = \begin{bmatrix} g_1(\boldsymbol{x}) & 0 & 0 \\ 0 & g_{2,2}(\boldsymbol{x}) & g_{2,3}(\boldsymbol{x}) \\ 0 & 0 & g_3(\boldsymbol{x}) \end{bmatrix} \quad (C.8) $$
In such a case, the system becomes:

$$ \mathbf{u} = \begin{bmatrix} u_1 \\ 0 \\ u_3 \end{bmatrix} = \begin{bmatrix} g_1^{-1}(\boldsymbol{x}) & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & g_3^{-1}(\boldsymbol{x}) \end{bmatrix} \left( \begin{bmatrix} \dot{x}_{1,d} \\ 0 \\ \dot{x}_{3,d} \end{bmatrix} \begin{bmatrix} \lambda_1(\boldsymbol{e}_x) \\ \lambda_2(\boldsymbol{e}_x) \\ \lambda_3(\boldsymbol{e}_x) \end{bmatrix} - \begin{bmatrix} f_1(\boldsymbol{x}) \\ f_2(\boldsymbol{x}) \\ f_3(\boldsymbol{x}) \end{bmatrix} \right) \quad (C.9a) $$

$$ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} = \begin{bmatrix} f_1(\boldsymbol{x}) \\ f_2(\boldsymbol{x}) \\ f_3(\boldsymbol{x}) \end{bmatrix} + \begin{bmatrix} g_1(\boldsymbol{x}) & 0 & 0 \\ 0 & g_{2,2}(\boldsymbol{x}) & g_{2,3}(\boldsymbol{x}) \\ 0 & 0 & g_3(\boldsymbol{x}) \end{bmatrix} \begin{bmatrix} u_1 \\ 0 \\ u_3 \end{bmatrix} \quad (C.9b) $$

thus the closed-loop system becomes:

$$ \dot{x}_1 = \dot{x}_{1,d} + \lambda_1(\boldsymbol{e}_x) \quad (C.10a) $$

$$ \dot{x}_2 = f_2(\boldsymbol{x}) + g_{2,3}(\boldsymbol{x})g_3(\boldsymbol{x})^{-1}(\dot{x}_{3,d} + \lambda_3(\boldsymbol{e}_x) - f_3(\boldsymbol{x})) \quad (C.10b) $$

$$ \dot{x}_3 = \dot{x}_{3,d} + \lambda_3(\boldsymbol{e}_x) \quad (C.10c) $$

Equation (C.10) shows that the third control component $u_3$ could be used to control the second DOF of the system $x_2$ through the coupling term $g_{2,3}$ but at the cost of the control over the third DOF $x_3$ which would be left uncontrolled. If $x_3$ is controlled then $u_3$ creates disturbance on $x_2$ that could make it even more unstable.

In many systems, the stability of the non-actuated DOF is assumed to be robust enough to counteract disturbances created by the actuators. It is notably the case for most boats where the hull shape, mass distribution and hydrodynamic effects damp any sway motion and favor surge and yaw displacements. Therefore, even if the vehicle is actuated in surge and yaw only, the natural sway stability allows controlling either the yaw angle of the vehicle or the two position coordinates without risking instability. Nonetheless, other systems like the small cylindrical hull underwater vehicle used in some examples of this work (see chapter 1) are not guaranteed to be stable enough. Coupled actuators may create instability of the uncontrolled DOF. The physical bounds of the actuators ensure that the disturbances created by coupled actuators are bounded too.
# MODEL COEFFICIENTS OF THE VEHICLES

This appendix gives the numerical values of the model coefficients of the robots used in this work: RSM, Remus100 and ODIN.

<table><caption>Table D.1 – Simulation parameters - RSM</caption><thead><tr><th colspan="4">RSM</th></tr><tr><th>Item</th><th>Value</th><th>Item</th><th>Value</th></tr></thead><tbody><tr><td><i>m</i></td><td>18 kg</td><td><i>I</i><sub>x</sub></td><td>0.096</td></tr><tr><td><i>I</i><sub>y</sub></td><td>0.611</td><td><i>I</i><sub>z</sub></td><td>0.608</td></tr><tr><td><b><i>P</i></b><sub>G</sub></td><td>[0, 0, 0.012]<sup>⊤</sup></td><td><b><i>P</i></b><sub>B</sub></td><td>[0, 0, 0]<sup>⊤</sup></td></tr><tr><td><i>X</i><sub>ü</sub></td><td>-1.88</td><td><i>Y</i><sub>ü</sub></td><td>-18.81</td></tr><tr><td><i>Z</i><sub>ü</sub></td><td>-18.81</td><td><i>K</i><sub>ü</sub></td><td>0</td></tr><tr><td><i>M</i><sub>ü</sub></td><td>-0.56</td><td><i>N</i><sub>ü</sub></td><td>-0.56</td></tr><tr><td><i>X</i><sub>u</sub></td><td>-12.85</td><td><i>Y</i><sub>v</sub></td><td>-71.85</td></tr><tr><td><i>Z</i><sub>w</sub></td><td>-71.85</td><td><i>K</i><sub>p</sub></td><td>-0.38</td></tr><tr><td><i>M</i><sub>q</sub></td><td>-3.88</td><td><i>N</i><sub>r</sub></td><td>-3.88</td></tr></tbody></table>
<table><caption>Table D.2 – Simulation parameters - Remus100</caption><thead><tr><th colspan="4">Remus100</th></tr><tr><th>Item</th><th>Value</th><th>Item</th><th>Value</th></tr></thead><tbody><tr><td>m</td><td>31 kg</td><td>I<sub>x</sub></td><td>0.17</td></tr><tr><td>I<sub>y</sub></td><td>6.70</td><td>I<sub>z</sub></td><td>6.70</td></tr><tr><td><b>P</b><sub>G</sub></td><td>[0, 0, 0.02]<sup>T</sup></td><td><b>P</b><sub>B</sub></td><td>[0, 0, 0]<sup>T</sup></td></tr><tr><td>X<sub>ü</sub></td><td>-0.83</td><td>Y<sub>ü</sub></td><td>-29.43</td></tr><tr><td>Z<sub>ü</sub></td><td>-29.43</td><td>K<sub>ü</sub></td><td>-0.033</td></tr><tr><td>M<sub>ü</sub></td><td>-3.42</td><td>N<sub>ü</sub></td><td>-3.42</td></tr><tr><td>X<sub>u</sub></td><td>-1.59</td><td>Y<sub>u</sub></td><td>-3.02</td></tr><tr><td>Z<sub>w</sub></td><td>-3.02</td><td>K<sub>p</sub></td><td>-0.58</td></tr><tr><td>M<sub>q</sub></td><td>-10.78</td><td>N<sub>r</sub></td><td>-1.49</td></tr><tr><td>X<sub>uu</sub></td><td>-12.85</td><td>Y<sub>vv</sub></td><td>-1.91</td></tr><tr><td>Z<sub>ww</sub></td><td>-1.91</td><td>K<sub>pp</sub></td><td>-0.38</td></tr><tr><td>M<sub>qq</sub></td><td>-1.96</td><td>N<sub>rr</sub></td><td>-1.96</td></tr></tbody></table>
<table><caption>Table D.3 – Simulation parameters - ODIN</caption><thead><tr><th colspan="4">ODIN</th></tr><tr><th>Item</th><th>Value</th><th>Item</th><th>Value</th></tr></thead><tbody><tr><td>m</td><td>84 kg</td><td>I<sub>x</sub></td><td>3.79</td></tr><tr><td>I<sub>y</sub></td><td>2.52</td><td>I<sub>z</sub></td><td>2.52</td></tr><tr><td><b>P</b><sub>G</sub></td><td>[0, 0, 0]<sup>†</sup></td><td><b>P</b><sub>B</sub></td><td>[0, 0, 0]<sup>†</sup></td></tr><tr><td>X<sub>ü</sub></td><td>-62.50</td><td>Y<sub>ü</sub></td><td>-62.50</td></tr><tr><td>Z<sub>ü</sub></td><td>-62.501</td><td>K<sub>ü</sub></td><td>0</td></tr><tr><td>M<sub>ü</sub></td><td>-30</td><td>N<sub>ü</sub></td><td>-30</td></tr><tr><td>X<sub>u</sub></td><td>-48</td><td>Y<sub>u</sub></td><td>-48</td></tr><tr><td>Z<sub>u</sub></td><td>-48</td><td>K<sub>u</sub></td><td>-80</td></tr><tr><td>M<sub>q</sub></td><td>-80</td><td>N<sub>q</sub></td><td>-80</td></tr></tbody></table>
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**Titre :** Analyse et commande des robots sous-marins autonomes à propulsion vectorielle re-configurable.

**Mot clés :** Guidage, Contrôle, AUV sous-actionnés, Propulsion Vectorielle

**Résumé :** Ce manuscrit présente des avancées en matière de contrôle et de conception de véhicules marins autonomes. Il se concentre sur l'amélioration de la mobilité des véhicules marins afin de débloquer de nouvelles applications telles que l'inspection et la maintenance des systèmes de production d'énergie verte en mer. Les deux axes principaux de ce travail sont le contrôle des véhicules marins sous-actionnés et la conception de nouvelles solutions de propulsion vectorielle. Deux nouveaux contrôleurs ont été mis au point pour les véhicules sous-actionnés. Le premier contrôleur repose sur un nouveau principe de guidage cinématique. Une nouvelle matrice de gain non diagonale basée

sur le modèle, appelée $\mathcal{H}$, est introduite pour exploiter les relations cinématiques du modèle. Elle permet de contrôler un véhicule dans une tâche de suivi de chemin. Le deuxième contrôleur présenté dans ce travail est basé sur la platitude différentielle. Il est calculé sur une version plate du modèle de navire de surfac. Le contrôleur est associé à un iPD pour une plus grande robustesse et un meilleur comportement. La loi de commande est adaptée à la commande des véhicules de surface. Un nouveau système de poussée vectorielle basé sur deux propulseurs parallèles montés sur un anneau rotatif est également présenté pour améliorer la mobilité des AUV sans créer de nouveaux effets de couplage.

**Title:** Analysis and control of autonomous underwater vehicles with reconfigurable vectoring thrust.

**Keywords:** Guidance, Control, Underactuated AUVs, Vectoring Thrust

**Abstract:** This work presents advances in the control and design of autonomous marine vehicles. It is focused on enhancing the mobility of marine vehicles to unlock new applications like the inspection and maintenance of off-shore green energy production plants or the observation and protection of marine environments. The two main axes of this work are the control of underactuated marine vehicles and the design of new vectoring thrust solutions. Notably, two new controllers have been developed for underactuated marine craft. The first controller relies on a new kinematic guidance principle. A new non diagonal model-based gain matrix denoted $\mathcal{H}$ is introduced to exploit

kinematic relations of the model. It is shown to allow control of a torpedo shaped vehicle on a path following task. The second controller presented in this work is based on differential flatness. It is calculated on a flat version of the surface vessel model where the usual defects preventing flatness are canceled. The controller is associated with an iPD for increased robustness and better behavior away from the trajectory. The resulting control law is shown to be suited for the control of a generic surface vessel. A new vectoring thrust system based on two parallel thrusters mounted on a rotating ring is also presented to enhance the mobility of AUVs without creating new coupled effects.