1. Introduction

2. Problem Formulation

Table 1. Notations in the UUV dynamic model.

2.1. Kinematic Model

The UUV’s motion states are described using two reference frames: the body-fixed reference frame (BRF) and the inertial reference frame (IRF). The BRF is fixed to the vehicle, with the center of gravity serving as the origin, and the body axes aligned with the principal axes of inertia. The longitudinal axis, denoted as the $x_b$, extends from aft to fore. The transversal axis, referred to as the $y_b$, extends from port to starboard. Per the right-hand rule, the $z_b$ axis is perpendicular to both the $x_b$ and $y_b$ axes. The UUV’s motion is characterized as the motion of the BRF relative to the IRF. The IRF is employed to track the vehicle’s trajectory and define the control objectives. In this work, the IRF is aligned with the north-east-down (NED) coordinate system. This choice is motivated by the prevalence of expressing position vectors in NED coordinates within various navigation applications and simulation environments. The axes in IRF are denoted as $x_i$, $y_i$, and $z_i$, as shown in Figure 1.

Figure 1. The reference frames of the UUV.

Figure 1. The reference frames of the UUV.

In the UUV’s model states, the linear and angular velocities are described in the BRF as $\mathbf{v}={[u,v,w,p,q,r]}^{\mathrm{T}}$, while the linear and angular position are expressed in the IRF as $\mathit{\eta }={[x,y,z,\varphi ,\theta ,\psi ]}^{\mathrm{T}}$. Since the velocity vector and position vector are expressed in different reference frames, the rotation matrix ${\mathbf{R}}_b^i$ are required for describing the relationship between them. Let ${\mathbf{v}}^b={[u,v,w]}^{\mathrm{T}}$ represents the linear velocity in BRF, ${\mathbf{v}}^i$ represents the linear velocity in IRF. thus, the ${\mathbf{v}}^b$ and ${\mathbf{v}}^i$ can be related by equation:

$${\mathbf{v}}^i={\mathbf{R}}_b^i(\Theta ){\mathbf{v}}^b$$

(1)

where $\Theta$ includes the Euler angles (roll $\varphi$, pitch $\theta$ and yaw $\psi$). Then the rotation matrix can be computed based on $\Theta$ as:

$${\mathbf{R}}_b^i(\Theta )=\left[\begin{array}{ccc}cos\psi cos\theta & cos\psi sin\theta sin\varphi -sin\psi cos\varphi & cos\psi cos\theta sin\varphi +sin\psi sin\varphi \\ sin\psi cos\theta & sin\psi sin\theta cos\varphi +cos\psi cos\varphi & sin\psi sin\theta cos\varphi -cos\psi sin\varphi \\ -sin\theta & cos\theta sin\varphi & cos\theta cos\varphi \end{array}\right].$$

(2)

For the transformation of angular states, let ${\mathit{\omega }}^b={[p,q,r]}^{\mathrm{T}}$ represents angular velocity in the BRF relative to the IRF then:

$$\dot{\Theta }=\mathbf{T}(\Theta ){\mathit{\omega }}^b$$

(3)

where $\dot{\Theta }=[\dot{\varphi },\dot{\theta },\dot{\psi }]$ is the Euler angle rate. The transformation matrix $\mathbf{T}(\Theta )$ which represents the relationship between angular states in the BRF and IRF can be expressed as:

$$\mathbf{T}(\Theta )=\left[\begin{array}{ccc}1& sin\varphi tan\theta & cos\varphi tan\theta \\ 0& cos\varphi & -sin\varphi \\ 0& sin\varphi /cos\theta & cos\varphi /cos\theta \end{array}\right].$$

(4)

Therefore, the relationship between UUV’s velocity and position is given by:

$$\dot{\mathit{\eta }}=\left[\begin{array}{c}\dot{\mathbf{p}}\\ \dot{\Theta }\end{array}\right]=\left[\begin{array}{cc}{\mathbf{R}}_b^i(\Theta )& {\mathbf{0}}_{3\times 3}\\ {\mathbf{0}}_{3\times 3}& \mathbf{T}(\Theta )\end{array}\right]\left[\begin{array}{c}{\mathbf{v}}^b\\ {\omega }^b\end{array}\right]=\mathbf{J}\left(\mathit{\eta }\right)\mathbf{v}$$

(5)

$$J\left(\mathit{\eta }\right)=\left[\begin{array}{cc}{\mathbf{R}}_b^i(\Theta )\hfill & {\mathbf{0}}^{3\times 3}\hfill \\ {\mathbf{0}}^{3\times 3}\hfill & T\left(\mathit{\eta }\right)\hfill \end{array}\right]$$

(6)

where $\mathbf{p}={[x,y,z]}^{\mathrm{T}}$ represents the linear position of UUV in the IRF.

2.2. Kinetic Model

To simplify the process of deriving the dynamic equations for UUV motion, it is common to assume that the vehicle behaves as a rigid body. This assumption eliminates the need to analyze the interactions between individual mass elements. The overall dynamic model can be expressed as follows:

$$M\dot{\mathbf{v}}+C\left(\mathbf{v}\right)\mathbf{v}+D\left(\mathbf{v}\right)\mathbf{v}+g\left(\mathit{\eta }\right)=\mathit{\tau }+\mathit{w}$$

(7)

where $M\dot{\mathbf{v}}$ is the mass matrix, $C\left(\mathbf{v}\right)\mathbf{v}$ is the Coriolis and centripetal matrix, $D\left(\mathbf{v}\right)\mathbf{v}$ is the hydrodynamic damping matrix, $g\left(\mathit{\eta }\right)$ is the vector of the gravitational and buoyancy forces, the $\mathit{\tau }={[X,Y,Z,K,M,N]}^{\mathrm{T}}$ is the total propulsion forces and moments, and $w$ is the external disturbance. The M and $C\left(\mathbf{v}\right)\mathbf{v}$ contain terms for both the rigid body and hydrodynamic added mass as:

$$\left\{\begin{array}{c}M=M_{RB}+M_A\hfill \\ C\left(\mathbf{v}\right)=C_{RB}\left(\mathbf{v}\right)+C_A\left(\mathbf{v}\right).\hfill \end{array}\right.$$

(8)

2.2.1. Rigid-body dynamics

By using Newtonian physics, the rigid-body dynamics of the UUV, may be deduced as follows:

$$M_{RB}\dot{\mathbf{v}}+C_{RB}\left(\mathbf{v}\right)\mathbf{v}={\mathit{\tau }}_{RB}.$$

(9)

Assuming that the origin of the BRF is positioned at the geometric center of the UUV and the vehicle exhibits symmetry in both the port-starboard and fore-aft plane, the rigid-body mass matrix can be simplified. As a result, the rigid-body mass $M_{RB}$ can be expressed as:

$$M_{RB}=\left[\begin{array}{cccccc}m& 0& 0& 0& m{z}_g& 0\\ 0& m& 0& -m{z}_g& 0& 0\\ 0& 0& m& 0& 0& 0\\ 0& -m{z}_g& 0& I_x& 0& 0\\ m{z}_g& 0& 0& 0& I_y& 0\\ 0& 0& 0& 0& 0& I_z\end{array}\right]$$

(10)

where m is the mass of the vehicle, $I_x$, $I_y$ and $I_z$ are the inertia moments about the $x_b$, $y_b$, and $z_b$ axes in BRF, measured in $kg{m}^2$; $z_g$ is the position of the CG in relation to the centre of the vehicle in $z_b$ axis, measured in meters. Subsequently, using the skew-symmetric cross-product operation on ${\mathbf{M}}_{RB}$ yields the result of the rigid-body Coriolis and centripetal matrix $C_{RB}\left(v\right)$ is given by:

$$C_{RB}\left(v\right)=\left[\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill mw& \hfill -mv\\ \hfill 0& \hfill 0& \hfill 0& \hfill -mw& \hfill 0& \hfill mu\\ \hfill 0& \hfill 0& \hfill 0& \hfill mv& \hfill -mu& \hfill 0\\ \hfill 0& \hfill mw& \hfill -mv& \hfill 0& \hfill I_{z}r& \hfill -I_{y}q\\ \hfill -mw& \hfill 0& \hfill mu& \hfill -I_{z}r& \hfill 0& \hfill I_{x}p\\ \hfill mv& \hfill -mu& \hfill 0& \hfill I_{y}q& \hfill -I_{x}p& \hfill 0\end{array}\right]$$

(11)

2.2.2. Hydrodynamics

The hydrodynamics studied in this work encompass both hydrodynamic added mass and hydrodynamic damping. Hydrodynamic added mass can be perceived as a virtual mass that is introduced to a system when a body accelerates or decelerates, requiring the movement of a certain volume of the surrounding fluid. This concept is formulated based on Kirchhoff’s equation.

In an ideal fluid, for a rigid body that is either at rest or moving at a forward speed $U⩾0$, the hydrodynamic system inertia matrix $M_A$ is positive semi-definite. The hydrodynamic coefficients are defined as the partial derivatives of the added mass force with respect to the corresponding acceleration. In many practical applications, the non-diagonal elements of $M_A$ are significantly smaller than the diagonal elements. As a result, the off-diagonal terms of $M_A$ can be disregarded, leading to a simplified form of $M_A$ as follows:

$$M_A=-\left[\begin{array}{cccccc}{X}_{\dot{u}}& 0& 0& 0& 0& 0\\ 0& Y_{\dot{v}}& 0& 0& 0& 0\\ 0& 0& Z_{\dot{w}}& 0& 0& 0\\ 0& 0& 0& K_{\dot{p}}& 0& 0\\ 0& 0& 0& 0& M_{\dot{q}}& 0\\ 0& 0& 0& 0& 0& N_{\dot{r}}\end{array}\right]$$

(12)

Accordingly, the nonlinear hydrodynamic Coriolis and centripetal matrix $C_A\left(v\right)$ can be calculated as:

$$C_A\left(\mathbf{v}\right)=\left[\begin{array}{cccccc}0& 0& 0& 0& z_{\dot{w}}w& 0\\ 0& 0& 0& -z_{\dot{w}}w& 0& -X_{\dot{u}}u\\ 0& 0& 0& -Y_{\dot{v}}v& X_{\dot{u}}u& 0\\ 0& -z_{\dot{w}}w& Y_{\dot{v}}v& 0& -N_{\dot{r}}r& M_{\dot{q}}q\\ z_{\dot{w}}w& 0& -X_{\dot{u}}u& N_{\dot{r}}r& 0& -K_{\dot{p}}p\\ -Y_{\dot{v}}v& X_{\dot{u}}u& 0& -M_{\dot{q}}q& K_{\dot{p}}p& 0\end{array}\right]$$

(13)

The primary components of hydrodynamic damping for UUVs are potential damping, skin friction, wave drift damping, and damping caused by vortex shedding. Typically, potential damping and wave drift damping are disregarded in UUVs model. These various damping factors contribute to both linear and quadratic damping, as follows:

$$D\left(\mathbf{v}\right)=D_L+D_{NL}\left(\mathbf{v}\right)$$

(14)

The linear damping term ($D_L$) is a result of skin friction, while the nonlinear damping matrix ($D_{NL}$) is caused by quadratic damping and higher-order terms. Due to decoupling, the damping matrix is derived to be diagonal. As a result, the linear damping matrix is represented by Equation 15, and the quadratic damping matrix is represented by Equation 16.

$$D_L=-diag\left[X_u,Y_v,Z_w,K_p,M_q,N_r\right]$$

(15)

$$D_{NL}\left(\mathbf{v}\right)=-diag\left[X_{u\left|u\right|}\left|u\right|,Y_{v\left|v\right|}\left|v\right|,Z_{w\left|w\right|}\left|w\right|,K_{p\left|p\right|}\left|p\right|,M_{q\left|q\right|}\left|q\right|,N_{r\left|r\right|}\left|r\right|\right].$$

(16)

2.2.3. Hydrostatics

Archimedes [28] established the foundational principles of fluid statics, which form the basis of hydrostatics today. Considering the mass of the UUV as m that measured in $kg$, acceleration due to gravity as g, water density as $\rho$ that measured in $kg/m^3$, and the volume of fluid displaced by the UUV as ∇ that measured in $m^3$. Therefore, the weight of the UUV can be expressed as $W=mg$, while the buoyancy force B is expressed as $B=\rho g\nabla$.

Given the center of buoyancy of the UUV is $r_b$. Assume the centre of the vehicle’s body frame is placed at the center of buoyancy (CB), then $r_b={[0,0,0]}^T$. Since the vehicle has symmetry in the xz-plane and xy-plane, the position of the CG of the vehicle $r_g$ becomes $r_g={\left[x_g,y_g,z_g\right]}^T={\left[0,0,z_g\right]}^T$. Thus, the overall restoring force vector $g\left(\eta \right)$ can be calculated using Euler angle transformation as:

$$g\left(\mathit{\eta }\right)=\left[\begin{array}{c}(W-B)sin\theta \\ -(W-B)cos\theta sin\varphi \\ -(W-B)cos\theta cos\varphi \\ z_{g}Wcos\theta sin\varphi \\ z_{g}Wsin\theta \\ 0\end{array}\right].$$

(17)

2.2.4. Propeller Model and Control Allocation

A realistic model of the propeller is taken into account in the dynamic model of the UUV. The control inputs, denoted as $\mathit{u}={[u_1,u_2,u_3,u_4]}^{\mathrm{T}}$, where $u_1$, $u_2$, $u_3$, $u_4$ correspond to the forces and moments in the surge, sway, heave, and yaw directions of the UUV, respectively. The BlueROV2 is equipped with six propellers to facilitate movement in 4 degrees of freedom, resulting in an over-actuation scenario. Hence, the control inputs u are allocated to each propeller using the control allocation matrix. The thrust vector, represented as $\mathit{t}={[t_1,t_2,t_3,t_4,t_5,t_6]}^{\mathrm{T}}$, is determined as follows:

$$\mathit{t}=A\mathit{u}=\left[\begin{array}{cccc}\hfill -1& \hfill 1& \hfill 0& \hfill 1\\ \hfill -1& \hfill -1& \hfill 0& \hfill -1\\ \hfill 1& \hfill 1& \hfill 0& \hfill -1\\ \hfill 1& \hfill -1& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill -1& \hfill 0\\ \hfill 0& \hfill 0& \hfill -1& \hfill 0\end{array}\right]\mathit{u}$$

(18)

where t is the thrust produced by all propellers, and the layout of six propellers is shown in Figure 2. The blue propellers rotate clockwise, the green propellers rotate counterclockwise, and the red arrow indicates the positive surge direction.

Figure 2. The propeller configuration of the BlueROV2, propeller 1, 2, 3, and 4 provide thrust in x-y plane, while propeller 5 and 6 provide thrust in vertical direction.

Figure 2. The propeller configuration of the BlueROV2, propeller 1, 2, 3, and 4 provide thrust in x-y plane, while propeller 5 and 6 provide thrust in vertical direction.

The propulsion matrix, denoted as K, is utilized to calculate the combined force and moment that propel the UUV. This matrix represents the spatial distribution of propellers on the BlueROV2’s structure:

$$\mathit{\tau }=K\mathit{t}.$$

(19)

For calculating the propulsion matrix, given that $l_{xi}$ is the distance between the centre of propeller i and the CG in $x_b$ direction, and $t_{xi}$ is the force projection to x direction, therefore, the forces and moments produced by propeller 1 are:

$$\begin{array}{cc}\hfill & {\mathit{\tau }}_{\mathbf{1}}=\left[\begin{array}{c}{t}_{x1}\\ t_{y1}\\ t_{z1}\\ t_{z1}{l}_{y1}-t_{y1}{l}_{z1}\\ t_{x1}{l}_{z1}-t_{z1}{l}_{x1}\\ t_{y1}{l}_{x1}-t_{x1}{l}_{y1}\end{array}\right]=\left[\begin{array}{c}{t}_{1}cos\alpha \\ -t_{1}sin\alpha \\ 0\\ t_{1}sin\alpha ·l_{z1}\\ t_{1}cos\alpha ·l_{z1}\\ t_1\left(-sin\alpha ·l_{x1}-cos\alpha ·l_{y1}\right)\end{array}\right]\hfill \\ \hfill & =\left[\begin{array}{c}cos\alpha \\ -sin\alpha \\ 0\\ sin\alpha ·l_{z1}\\ cos\alpha ·l_{z1}\\ -sin\alpha ·l_{x1}-cos\alpha ·l_{y1}\end{array}\right]t_1\hfill \\ \hfill \end{array}$$

(20)

where $\alpha$ denotes the angle at which the propeller is positioned with respect to the UUV’s forward direction, measured in radians. Using the value $\alpha =\pi /4rad$ and $l_{x1}=0.156m$ , $l_{y1}=0.111m$ , $l_{z1}=0.072m$ for propeller 1. Therefore, the first column in K can be obtained. The rest column can be calculated in the same way, thus the propulsion matrix becomes:

$$K=\left[\begin{array}{cccccc}0.707& 0.707& -0.707& -0.707& 0& 0\\ -0.707& 0.707& -0.707& 0.707& 0& 0\\ 0& 0& 0& 0& 1& 1\\ 0.051& -0.051& 0.051& -0.051& 0.111& -0.111\\ 0.051& 0.051& -0.051& -0.051& 0.002& -0.002\\ -0.167& 0.167& 0.175& -0.175& 0& 0\end{array}\right].$$

(21)

2.2.5. Nonlinear Model

In summary, for achieving motion control of the UUV, the system state is defined as $\mathbf{x}=[\mathit{\eta };\mathbf{v}]$. From the Equation 5 and Equation 7, the general form of the UUV is obtained as:

$$\begin{array}{cc}\hfill \dot{\mathbf{x}}& =\left[\begin{array}{c}\dot{\mathit{\eta }}\\ \dot{\mathbf{v}}\end{array}\right]=f\left(\mathbf{x},\mathit{\tau },\mathit{w},t\right)\hfill \\ \hfill & =\left[\begin{array}{c}J\left(\mathit{\eta }\right)\mathbf{v}\\ M^{-1}\left[\mathit{\tau }+\mathit{w}-C\left(\mathbf{v}\right)\mathbf{v}-D\left(\mathbf{v}\right)\mathbf{v}-g\left(\mathit{\eta }\right)\right]\end{array}\right].\hfill \end{array}$$

(22)

3. Extended Active Observer

4. Disturbance Observer-Based Model Predictive Control (DOBMPC)

Figure 3. Control loop of the MPC, which mainly includes an optimizer and a prediction model.

Figure 3. Control loop of the MPC, which mainly includes an optimizer and a prediction model.

Figure 4. Block diagram of the proposed DOBMPC scheme with disturbance compensation incorporated.

Figure 4. Block diagram of the proposed DOBMPC scheme with disturbance compensation incorporated.

5. Results and Discussion

Table 2. BlueROV2 rigid-body parameters defined in URDF file.

Table 3. BlueROV2 hydrodynamic parameters defined in URDF file.

Table 4. MPC parameters utilized in this work.

Table 5. PID parameters utilized in this work.

5.1. Dynamic Positioning Results

For dynamic positioning, the UUV encounters two types of disturbances: simulated periodic wave effects and constant current effects. These disturbances manifest as forces and moments acting on the UUV. In the case of periodic disturbances, sinusoidal waves with random force amplitudes ranging from 10 - 16 N are applied in the $x_i$, $y_i$, and $z_i$ directions, while sinusoidal waves with random moment amplitudes ranging from 1 - 2 $Nm$ is applied around $z_i$ axis. The UUV is set to remain stationary at the position $[0,0,-20]$ in the IRF while keeping a yaw angle of 0 degrees, in the presence of disturbances. Figure 5 displays the estimated disturbances alongside the generated disturbances. As shown in the figure, the estimated disturbances closely match the generated disturbances, and the estimation delay is shorter than the sample time, proving the feasibility of the proposed EAOB.

Figure 5. Disturbances estimation of periodic waves with random force and moment amplitudes.

Figure 5. Disturbances estimation of periodic waves with random force and moment amplitudes.

After compensating the estimated disturbances into the MPC’s prediction model, it improves the disturbance rejection ability of the controller significantly compared to the PID and the MPC, as shown in Figure 6. A 2D diagram which indicates the dynamic positioning results of three controllers are also demonstrated in Figure 7, while the blue trajectory shows the DOBMPC can stay at the reference point more stably.

Figure 6. Tracking errors of the proposed DOBMPC, baseline MPC and PID controllers under periodic disturbances.

Figure 6. Tracking errors of the proposed DOBMPC, baseline MPC and PID controllers under periodic disturbances.

Figure 7. Trajectories of dynamic positioning results of the proposed DOBMPC, baseline MPC and PID controllers under periodic disturbances.

Figure 7. Trajectories of dynamic positioning results of the proposed DOBMPC, baseline MPC and PID controllers under periodic disturbances.

To generate constant disturbances, a 10 N force is applied in the $x_i$, $y_i$, and $z_i$ directions at time $t=10s$, and a 5 $Nm$ torque is applied around $z_i$ axis at the same time. In Figure 8, a slight overshoot can be observed when the disturbances abruptly change from 0 N to 10 N. This overshoot is caused by the significant difference in estimated states between two time steps. The EAOB takes less than 0.5 seconds to accurately adjust the estimated states using recursive iteration. Figure 9 displays the tracking errors of the proposed DOBMPC, baseline MPC, and PID controllers when subjected to constant currents. It clearly demonstrates the varying levels of disturbance rejection capability among the three controllers. The DOBMPC controller precisely estimates disturbances and compensates for them, resulting in errors converging to zero in each direction. The x-y diagram in Figure 10 further illustrates the dynamic positioning performance of the three controllers. By overlaying disturbances in the same direction, it becomes evident that the proposed DOBMPC significantly enhances disturbance rejection ability.

Figure 8. Disturbances estimation of constant currents in x, y, z directions.

Figure 8. Disturbances estimation of constant currents in x, y, z directions.

Figure 9. Tracking errors of the proposed DOBMPC, baseline MPC, and PID controllers under constant currents.

Figure 9. Tracking errors of the proposed DOBMPC, baseline MPC, and PID controllers under constant currents.

Figure 10. Trajectories of dynamic positioning results of the proposed DOBMPC, baseline MPC, and PID controllers under constant currents.

Figure 10. Trajectories of dynamic positioning results of the proposed DOBMPC, baseline MPC, and PID controllers under constant currents.

5.2. Trajectory Tracking Results

To assess the performance of trajectory tracking, two kinds of movement scenarios are employed. Firstly, a circular path with a radius of 2 m is employed. The UUV’s yaw angle is defined to be relative to the surge direction. To assess the system’s robustness, a 10 N force is applied in the $x_i$, $y_i$, and $z_i$ directions, along with a 5 $Nm$ torque around the $z_i$ axis. Figure 11 illustrates the comparison between the estimated disturbances and the generated disturbances. As the disturbance term incorporates unmodeled components from the dynamic model, the disparity between the estimated and generated disturbances can be attributed to the nonlinear damping force experienced by the UUV during circular trajectory movement. The tracking error of PID, MPC, and the proposed DOBMPC is shown in Figure 13, demonstrating a significant reduction in tracking error with the proposed DOBMPC. Additionally, Figure 12 and Figure 14 provide a visual representation of the trajectory tracking results.

Figure 11. Disturbances estimation of constant currents during circular trajectory tracking.

Figure 11. Disturbances estimation of constant currents during circular trajectory tracking.

Figure 12. Circular trajectory tracking results of the proposed DOBMPC, baseline MPC and PID controllers in x, y, z, and yaw directions.

Figure 12. Circular trajectory tracking results of the proposed DOBMPC, baseline MPC and PID controllers in x, y, z, and yaw directions.

Figure 13. Tracking errors of the proposed DOBMPC, baseline MPC and PID controllers under constant currents during circular trajectory tracking.

Figure 13. Tracking errors of the proposed DOBMPC, baseline MPC and PID controllers under constant currents during circular trajectory tracking.

Figure 14. Three-dimensional circular trajectory tracking results of the proposed DOBMPC, baseline MPC and PID controllers.

Figure 14. Three-dimensional circular trajectory tracking results of the proposed DOBMPC, baseline MPC and PID controllers.

To evaluate the effectiveness of the proposed control method in tracking a highly nonlinear trajectory, a lemniscate trajectory with a 2 m amplitude is utilized. Throughout the movement, the yaw angle remains constant at 0 degrees. Additionally, periodic wave effects are incorporated into the testing process. These waves have random force amplitudes ranging from 10-16 N and random moment amplitudes ranging from 2-4 $Nm$. Figure 15 provides the comparison between generated disturbances and estimated disturbances. The results demonstrate that the disturbance forces in the $x_i$, $y_i$, and $z_i$ directions can still be accurately estimated when following a lemniscate trajectory. However, challenges arise when accurately estimating the disturbance moment around $z_i$ during the tracking of this trajectory. This can lead to occasional deviations or noise around the actual disturbance value. The complexity of unmodeled nonlinear hydrodynamics, which becomes more prominent when tracking a nonlinear trajectory like the lemniscate, could be a contributing factor to this issue. In Figure 16, the system states of tracking a lemniscate trajectory under periodic waves are compared between PID, MPC, and the proposed DOBMPC. While the system states of PID and MPC exhibit irregularities due to time-varying massive disturbances, the states of DOBMPC remain relatively smooth. The tracking error, depicted in Figure 17, shows a significant reduction with the proposed DOBMPC compared to PID and MPC. Lastly, Figure 18 provides a visual representation of the trajectory tracking results in three dimensions.

Figure 15. Disturbances estimation of periodic waves during lemniscate trajectory tracking.

Figure 15. Disturbances estimation of periodic waves during lemniscate trajectory tracking.

Figure 16. Lemniscate trajectory tracking results of the proposed DOBMPC, baseline MPC and PID controllers in x, y, z, and yaw directions.

Figure 16. Lemniscate trajectory tracking results of the proposed DOBMPC, baseline MPC and PID controllers in x, y, z, and yaw directions.

Figure 17. Tracking errors of the proposed DOBMPC, baseline MPC and PID controllers under constant currents during lemniscate trajectory tracking.

Figure 17. Tracking errors of the proposed DOBMPC, baseline MPC and PID controllers under constant currents during lemniscate trajectory tracking.

Figure 18. Three-dimensional lemniscate trajectory tracking results of the proposed DOBMPC, baseline MPC and PID controllers.

Figure 18. Three-dimensional lemniscate trajectory tracking results of the proposed DOBMPC, baseline MPC and PID controllers.

5.3. Results Analysis

Table 6 presents an analysis of the performance of dynamic positioning and trajectory tracking of the UUV. The evaluation is based on the root mean square error (RMSE). RMSE is a commonly used metric in control systems literature and has been widely accepted as a measure of performance. The use of RMSE allows for easy comparison and interpretation of results across different controllers. It provides a quantitative measure that can be used to assess and rank the performance of different control strategies. Furthermore, RMSE is less sensitive to outliers compared to other metrics, such as Mean Absolute Error (MAE). It considers the squared errors, which amplifies the impact of larger errors, making it suitable for capturing the performance of control systems where extreme errors may occur.

The lowest RMSE in each row is highlighted. The results clearly demonstrate that the utilization of the proposed DOBMPC significantly enhances the system’s ability to reject disturbances.

Table 6. RMSE of the proposed DOBMPC, baseline MPC, and PID controllers in dynamic positioning and trajectory tracking.

Table 6. RMSE of the proposed DOBMPC, baseline MPC, and PID controllers in dynamic positioning and trajectory tracking.

Motion	Disturbance	Direction	PID (m)	MPC (m)	DOBMPC (m)
		X	0.1374	0.1689	0.0537
Dynamic	Periodic	Y	0.1095	0.1934	0.0605
Positioning	wave effects	Z	0.0871	0.0896	0.0350
		Yaw	0.0536	0.1108	0.0282
		X	0.7893	0.3099	0.0521
Dynamic	Constant	Y	0.7544	0.2882	0.0482
Positioning	current effects	Z	0.2032	0.1508	0.0469
		Yaw	0.7858	0.4547	0.0491
Circular		X	2.3626	0.8012	0.2924
Trajectory	Constant	Y	1.6433	0.5521	0.2629
Tracking	current effects	Z	0.1763	0.1510	0.0233
		Yaw	0.4355	0.6325	0.2582
Lemniscate		X	0.9854	0.3764	0.2306
Trajectory	Constant	Y	0.7732	0.3844	0.1504
Tracking	current effects	Z	0.0877	0.1133	0.0510
		Yaw	0.2059	0.2413	0.1457

6. Conclusions

References

1. Hou, Shitong and Jiao, Dai and Dong, Bin and Wang, Haochen and Wu, Gang. Underwater inspection of bridge substructures using sonar and deep convolutional network. Advanced Engineering Informatics 2022, 52, 101545. [Google Scholar] [CrossRef]
2. Hu, Sijie and Feng, Rendong and Wang, Zhanglin and Zhu, Chengcheng and Wang, Zhikun and Chen, Ying and Huang, Haocai. Control system of the autonomous underwater helicopter for pipeline inspection. Ocean Engineering 2022, 266, 113190. [Google Scholar] [CrossRef]
3. Barker, Laughlin DL and Jakuba, Michael V and Bowen, Andrew D and German, Christopher R and Maksym, Ted and Mayer, Larry and Boetius, Antje and Dutrieux, Pierre and Whitcomb, Louis L. Scientific challenges and present capabilities in underwater robotic vehicle design and navigation for oceanographic exploration under-ice. Remote Sensing 2020, 12, 2588. [Google Scholar] [CrossRef]
4. Mogstad, Aksel Alstad and degård, yvind and Nornes, Stein Melvær and Ludvigsen, Martin and Johnsen, Geir and Sørensen, Asgeir J and Berge, Jørgen. Mapping the historical shipwreck figaro in the high arctic using underwater sensor-carrying robots. Remote Sensing 2020, 12, 997. [Google Scholar] [CrossRef]
5. Summers, Natalie and Johnsen, Geir and Mogstad, Aksel and Løvås, Håvard and Fragoso, Glaucia and Berge, Jørgen. Underwater hyperspectral imaging of Arctic macroalgal habitats during the polar night using a novel mini-ROV-UHI portable system. Remote sensing 2022, 14, 1325. [Google Scholar] [CrossRef]
6. Preuß, Henrich and Cherewko, Valeska and Wollstadt, Johann and Wendt, Anne and Renkewitz, Helge. Crawfish Goes Swimming: Hardware Architecture of a Crawling Skid for Underwater Maintenance with a BlueROV2. OCEANS 2022, Hampton Roads, 2022, pp. 1–5. [CrossRef]
7. Mai, Christian and Benzon, Malte von and Sørensen, Fredrik F. and Klemmensen, Sigurd S. and Pedersen, Simon and Liniger, Jesper. Design of an Autonomous ROV for Marine Growth Inspection and Cleaning. 2022 IEEE/OES Autonomous Underwater Vehicles Symposium (AUV), 2022, pp. 1–6. [CrossRef]
8. Mao, Juzheng and Song, Guangming and Hao, Shuang and Zhang, Mingquan and Song, Aiguo. Development of a Lightweight Underwater Manipulator for Delicate Structural Repair Operations. IEEE Robotics and Automation Letters 2023, 8, 6563–6570. [Google Scholar] [CrossRef]
9. Woolsey, C. Review of Marine Control Systems: Guidance, Navigation, and Control of Ships, Rigs and Underwater Vehicles. Journal of Guidance, Control, and Dynamics 2005, 28, 574–575. [Google Scholar] [CrossRef]
10. Liu, Lu and Zhang, Lichuan and Pan, Guang and Zhang, Shuo. Robust yaw control of autonomous underwater vehicle based on fractional-order PID controller. Ocean Engineering 2022, 257, 111493. [Google Scholar] [CrossRef]
11. Hasan, Mustafa Wassef and Abbas, Nizar Hadi. Disturbance Rejection for Underwater robotic vehicle based on adaptive fuzzy with nonlinear PID controller. ISA Transactions 2022, 130, 360–376. [Google Scholar] [CrossRef] [PubMed]
12. Bingul, Zafer and Gul, Kursad. Intelligent-PID with PD Feedforward Trajectory Tracking Control of an Autonomous Underwater Vehicle. Machines 2023, 11, 300. [Google Scholar] [CrossRef]
13. Healey, A.; Lienard, D. Multivariable sliding mode control for autonomous diving and steering of unmanned underwater vehicles. IEEE Journal of Oceanic Engineering 1993, 18, 327–339. [Google Scholar] [CrossRef]
14. Yan, Zheping and Wang, Man and Xu, Jian. Robust adaptive sliding mode control of underactuated autonomous underwater vehicles with uncertain dynamics. Ocean Engineering 2019, 173, 802–809. [Google Scholar] [CrossRef]
15. Lv, Tu and Zhou, Junliang and Wang, Yujia and Gong, Wei and Zhang, Mingjun. Sliding mode based fault tolerant control for autonomous underwater vehicle. Ocean Engineering 2020, 216, 107855. [Google Scholar] [CrossRef]
16. Carreras, M.; Batlle, J.; Ridao, P. Hybrid coordination of reinforcement learning-based behaviors for AUV control. Proceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the the Next Millennium (Cat. No.01CH37180), 2001, Vol. 3, pp. 1410–1415 vol.3. [CrossRef]
17. Elhaki, Omid and Shojaei, Khoshnam. A robust neural network approximation-based prescribed performance output-feedback controller for autonomous underwater vehicles with actuators saturation. Engineering Applications of Artificial Intelligence 2020, 88, 103382. [Google Scholar] [CrossRef]
18. Patre, B.M.; Londhe, P.S.; Waghmare, L.M.; Mohan, S. Disturbance estimator based non-singular fast fuzzy terminal sliding mode control of an autonomous underwater vehicle. Ocean Engineering 2018, 159, 372–387. [Google Scholar] [CrossRef]
19. Lakhekar, Girish Vithalrao and Waghmare, Laxman M. and Roy, Rupam Gupta. Disturbance Observer-Based Fuzzy Adapted S-Surface Controller for Spatial Trajectory Tracking of Autonomous Underwater Vehicle. IEEE Transactions on Intelligent Vehicles 2019, 4, 622–636. [CrossRef]
20. Kamel, M.; Stastny, T.; Alexis, K.; Siegwart, R. Model predictive control for trajectory tracking of unmanned aerial vehicles using robot operating system. Robot Operating System (ROS) The Complete Reference (Volume 2). [CrossRef]
21. Veksler, A.; Johansen, T.A.; Borrelli, F.; Realfsen, B. Dynamic Positioning With Model Predictive Control. IEEE Transactions on Control Systems Technology 2016, 24, 1340–1353. [Google Scholar] [CrossRef]
22. Medagoda, L.; Williams, S.B. Model predictive control of an autonomous underwater vehicle in an in situ estimated water current profile. 2012 Oceans - Yeosu, 2012, pp. 1–8. [CrossRef]
23. Shen, C.; Shi, Y.; Buckham, B. Trajectory Tracking Control of an Autonomous Underwater Vehicle Using Lyapunov-Based Model Predictive Control. IEEE Transactions on Industrial Electronics 2018, 65, 5796–5805. [Google Scholar] [CrossRef]
24. Cao, Y.; Li, B.; Li, Q.; Stokes, A.A.; Ingram, D.M.; Kiprakis, A. A Nonlinear Model Predictive Controller for Remotely Operated Underwater Vehicles With Disturbance Rejection. IEEE Access 2020, 8, 158622–158634. [Google Scholar] [CrossRef]
25. Arcos-Legarda, J.; Gutiérrez, Á. Robust Model Predictive Control Based on Active Disturbance Rejection Control for a Robotic Autonomous Underwater Vehicle. Journal of Marine Science and Engineering 2023, 11, 929. [Google Scholar] [CrossRef]
26. Robotics, B. BlueROV2: The World’s Most Affordable High-Performance ROV. BlueROV2 Datasheet; Blue Robotics: Torrance, CA, USA 2016. [Google Scholar]
27. Fossen, T.I. Handbook of marine craft hydrodynamics and motion control; John Wiley & Sons, 2011; pp. 167–183. [Google Scholar]
28. Chondros, T.G. Distinguished figures in mechanism and machine science: Their contributions and legacies Part 1; Springer, 2007; pp. 1–30. [Google Scholar]
29. Frogerais, Paul and Bellanger, Jean-Jacques and Senhadji, Lotfi. Various ways to compute the continuous-discrete extended Kalman filter. IEEE Transactions on Automatic Control 2011, 57, 1000–1004. [Google Scholar] [CrossRef]
30. Chan, Linping and Naghdy, Fazel and Stirling, David. Extended active observer for force estimation and disturbance rejection of robotic manipulators. Robotics and Autonomous Systems 2013, 61, 1277–1287. [Google Scholar] [CrossRef]
31. Kalman, Rudolph E and Bucy, Richard S. New results in linear filtering and prediction theory. Journal of Fluids Engineering 1961, 84. [Google Scholar] [CrossRef]
32. Diehl, M.; Bock, H.G.; Diedam, H.; Wieber, P.B. Fast direct multiple shooting algorithms for optimal robot control. Fast motions in biomechanics and robotics: optimization and feedback control. [CrossRef]
33. Verschueren, R.; Frison, G.; Kouzoupis, D.; van Duijkeren, N.; Zanelli, A.; Quirynen, R.; Diehl, M. Towards a modular software package for embedded optimization. IFAC-PapersOnLine 2018, 51, 374–380. [Google Scholar] [CrossRef]
34. Manhães, M.M.M.; Scherer, S.A.; Voss, M.; Douat, L.R.; Rauschenbach, T. UUV Simulator: A Gazebo-based package for underwater intervention and multi-robot simulation. OCEANS 2016 MTS/IEEE Monterey. IEEE, 2016. [CrossRef]
35. von Benzon, Malte and Sørensen, Fredrik Fogh and Uth, Esben and Jouffroy, Jerome and Liniger, Jesper and Pedersen, Simon. An Open-Source Benchmark Simulator: Control of a BlueROV2 Underwater Robot. Journal of Marine Science and Engineering 2022, 10, 1898. [Google Scholar] [CrossRef]
36. Li, Zhaobin and Deng, Ganbo and Queutey, Patrick and Bouscasse, Benjamin and Ducrozet, Guillaume and Gentaz, Lionel and Le Touzé, David and Ferrant, Pierre. Comparison of wave modeling methods in CFD solvers for ocean engineering applications. Ocean Engineering 2019, 188, 106237. [Google Scholar] [CrossRef]