Kinematic analysis and advanced control of a vectored thruster based on 3RRUR parallel manipulator for micro-size AUVs

Abstract

Autonomous underwater vehicles (AUVs) have played a pivotal role in advancing ocean exploration and exploitation. However, traditional AUVs face limitations when executing missions at minimal or near-zero forward velocities due to the ineffectiveness of their control surfaces, considerably constraining their potential applications. To address this challenge, this paper introduces an innovative vectored thruster system based on a 3RRUR parallel manipulator tailored for micro-sized AUVs. The incorporation of a vectored thruster enhances the performance of micro-sized AUVs when operating at minimal and low forward speeds. A comprehensive exploration of the kinematics of the thrust-vectoring mechanism has been undertaken through theoretical analysis and experimental validation. The findings from theoretical analysis and experimental confirmation unequivocally affirm the feasibility of the devised thrust-vectoring mechanism. The precise control of the vector device is studied using Physics-informed Neural Network and Model Predictive Control (PINN-MPC). Through the adoption of this pioneering thrust-vectoring mechanism rooted in the 3RRUR parallel manipulator, AUVs can efficiently and effectively generate the requisite motion for thrust-vectoring propulsion, overcoming the limitations of traditional AUVs and expanding their potential applications across various domains.

1. Introduction

Our planet, Earth, stands as a captivating blue jewel in the vast expanse of the universe, with its oceanic ecosystems enveloping a substantial 70% of its surface and harboring a staggering 97% of the world’s water resources, as noted in the Tara Oceans expedition [1]. In a world where terrestrial resources have been exhaustively explored and cannot satiate the burgeoning needs of human development, our attention is increasingly drawn toward the untapped potential of other resources. The Earth’s oceans, with their extensive expanse, offer a cornucopia of riches, from marine biological diversity to valuable minerals and oil reserves, while also serving as a vital arena for maritime transport and the intricate network of submarine communication cables.

Thus, the exploration, development, and sustainable utilization of oceanic resources have ascended to the forefront of global development and technological advancement [2]. The cornerstone of this endeavor is sea exploration, upon which oceanic research, development, and responsible utilization pivot. As the demand for marine exploration continues to surge, scientists and engineers are innovating with an array of cutting-edge instruments and equipment, including remotely operated vehicles (ROVs) and autonomous underwater vehicles (AUVs) [3, 4]. Elevating the capacity for ocean exploration stands as an integral mission for these dedicated professionals, one driven by the imperative to expand our understanding and harness the abundant resources offered by our oceans [5, 6].

AUVs represent a class of underwater robots capable of independent underwater navigation, operating without the need for real-time human intervention and free from external energy sources, as detailed in Chen et al.’s work [7]. Equipped with a diverse array of instruments and equipment, AUVs have assumed a pivotal role in various oceanic endeavors, offering a versatile platform for harnessing and exploiting oceanic resources. Their continually advancing performance has established them as indispensable tools in a myriad of intricate underwater missions spanning scientific exploration, military operations, and commercial applications.

AUVs exhibit the remarkable capacity to undertake a wide spectrum of tasks within both civilian and military domains. These tasks include monitoring marine contaminants [8], precision mine hunting [9, 10], the exploration of marine biology [11], pipeline tracking and inspection [12, 13], military support operations [14], and anti-submarine warfare efforts [15, 16]. Recent years have witnessed a proliferation of AUV types tailored for diverse oceanic exploration and resource utilization. These include the Autosub Long Range AUV [17, 18], the Remus series AUVs [19, 20], the Bluefin series AUVs [21, 22], the OceanServer Iver AUV [23, 24], and the MBARI AUV [25].

Despite the rapid expansion of AUV applications and the significant improvements in their performance, the technical requirements on AUVs continue to increase, making them a challenging area for scientists and engineers to design better [26]. In general, common torpedo-shaped AUVs are designed with a streamlined shell, a main propeller, and several control surfaces at the tail cone for propulsion and control. However, these types of AUVs are unable to accomplish a variety of underwater missions at zero or low forward speeds because the control surfaces of the AUVs become ineffective in such conditions as the control forces depend on forward speeds [27, 28]. These limitations have severely restricted the applications of AUVs.

To address the challenge of diminished maneuverability at extremely low or zero speeds encountered by traditional AUVs, a common strategy involves equipping these vehicles with multiple thrusters, which furnish supplementary control forces. The synchronized operation of these multiple propellers enables dynamic alteration of the vehicle’s motion state, facilitating the precise motion control even at negligible or zero speeds [29, 30]. However, this approach has its drawbacks, particularly when the AUV is navigating at higher speeds. The utilization of multiple thrusters generates considerable hydrodynamic drag, significantly curtailing the operational range, a constraint accentuated by the finite onboard energy reserves. Consequently, this augmentation strategy is primarily implemented in AUV designs prioritizing low-speed, highly maneuverable operations. Current AUVs, therefore, encounter the challenge of harmoniously balancing high-speed cruising with low-speed hovering capabilities.

An alternative solution to mitigate the issues arising at very low or zero speeds is the deployment of vector propulsion technology. Vectored thrusters, when integrated into AUVs, serve as the propulsion mechanism, offering both thrust-vectoring for propulsion and precise control forces during low-speed operations [31, 32]. A pivotal advantage of vectored thrusters lies in their ability to replace the conventional control rudder of the vehicle, rendering them a highly effective solution for the problem of diminished maneuverability in underwater vehicles at minimal or zero speeds [33, 34].

In the realm of pertinent research, both the Bluefin and MBARI AUVs have garnered notable achievements and contributed invaluable insights to the utilization and exploration of vectored thrusters, as documented in studies by Panish et al. [35] and Caress et al. [36]. However, it is imperative to acknowledge that these aforementioned vectored thruster designs have a limitation: their structural complexity renders them unsuitable for integration into micro-sized AUVs. When factoring in considerations such as response time, physical dimensions, and practicality, the choice of a suitable thrust-vectoring mechanism assumes critical significance in enhancing the maneuverability of vectored thrusters.

To advance vector propulsion technology and enhance the maneuverability of underwater vehicles at low speeds, it is imperative to delve deeper into the research of various mechanism configurations aimed at creating superior vector propulsion systems. Consequently, the study of mechanism configurations assumes ever-growing importance. In comparison to commonly used serial manipulators, parallel manipulators offer a multitude of advantages, including a compact structure, rapid response time, heightened positioning accuracy, superior stiffness, enhanced load-bearing capacity, improved dynamic performance, and reduced power consumption, as evidenced by Khalil et al. [37], Parikh et al. [38], Liang et al. [39], and Wu et al. [40–43], Simoni et al. [44], Erastova et al. [45], Huang et al. [46], Liu et al. [47]. Given the advantages of parallel manipulators and the specific requirements of vectored thruster AUVs, a 3RRUR parallel manipulator has been chosen as the optimal thrust-vectoring mechanism. This selection is primarily attributed to its compact structural design and exceptional positioning accuracy. These advancements in vectored thruster designs using parallel manipulators highlight the ongoing efforts to overcome the operational limitations of traditional AUVs. The 3-RPS structure [2], known for its simplicity, is particularly suited for medium to large-sized AUVs due to its straightforward drive mechanism. However, this design’s dimensions are somewhat constrained by the driving method. The 3SPS-S parallel manipulator enhances its load-bearing capacity by modifying spherical joint bearings. This makes the 3SPS-S design particularly suitable for high-speed AUVs [32].

Although the 3RRUR parallel manipulator offers numerous advantages, there remains an opportunity to further enhance its performance as a thrust-vectoring mechanism. To seize this challenge, a series of comprehensive studies focused on vectored thrusters built upon the 3RRUR parallel manipulator have been conducted. This paper makes some contributions to the field through the following key accomplishments:

1. 1. The paper undertakes a thorough theoretical kinematic analysis of the thrust-vectoring mechanism, shedding insightful light on its fundamental principles.

2. 2. The paper presents an experimental model test, offering empirical evidence that substantiates the validity and efficacy of the proposed approach, bridging the gap between theoretical postulations and practical implementations.

3. 3. It designs an MPC strategy based on the PINN model and rigorously verifies its effectiveness, demonstrating the potential for precise control and optimization of the vectored thruster system.

This research demonstrates that a vectored thruster utilizing a 3RRUR parallel manipulator is capable of facilitating precise motion required for thrust-vectoring propulsion. Both simulations and experimental results support the effectiveness of the design scheme and kinematic analysis, showing low margins of error and enhancing confidence in the robustness and reliability of the proposed approach.

2. Position analysis of the thrust-vectoring mechanism

In this paper, we present a vectored thruster design based on a 3RRUR parallel manipulator, tailored to meet the operational requirements of a micro-sized torpedo-like AUV with a 150 mm diameter, which is shown in Figure 1. Considering the AUV’s compact size, this innovative vector propulsion actuator is designed with a $\pm 15^{^{\circ}}$ deflection capability. Affixed to the rear of the AUV’s hull, this specially crafted vectored thruster serves the primary function of delivering both propulsive and control forces directly and efficiently to empower micro-sized AUVs.

Figure 1. The vectored thruster design based on a 3RRUR parallel manipulator.

The designed vectored thruster comprises three principal components: the thrust-vectoring mechanism, the propeller thruster, and the housing along with supplementary parts. To enhance propulsion efficiency and increase resistance to wind and wave disturbances, this design employs a ducted propeller as the vector propulsion method. An electric motor actuates the propeller to generate propulsive thrust, while the thrust-vectoring mechanism plays a pivotal role in adjusting the orientation of the propeller’s rotational axis, thereby creating vector thrust to propel the AUV. The moving platform serves the dual function of housing and safeguarding the propeller power system, as well as bearing external water pressure.

Figure 2 shows 3RRUR parallel manipulator and each branch chain structure. Each RRUR kinematic chain comprises three revolute joints (R) and one universal joint (U). In line with the structural composition, these kinematic joints are denoted as RRUR kinematic joints. Within the RRUR kinematic chain, the universal joint is affixed to the end of the moving platform through a revolute joint. One of the revolute joints within the chain is linked to the driving motor mounted on the fixed platform, while the other connects to the universal joint and another revolute joint via a connecting rod.

Figure 2. 3RRUR parallel manipulator and each branch chain structure.

In order to better study the structural characteristics of the 3RRUR parallel mechanism, it is necessary to study the degrees-of-freedom of the parallel mechanism, which is the most basic analysis of the feasibility of studying parallel mechanisms. The degree-of-freedom of a parallel mechanism refers to the number of linear and rotational movements that the mechanism can complete within a spatial range. As the 3RRUR parallel mechanism studied in this chapter belongs to a three-dimensional spatial motion mechanism, the modified Kutabach-Grubler degree-of-freedom calculation formula is used to calculate the spatial degree-of-freedom of the mechanism. According to the calculation, the 3RRUR parallel mechanism has three degrees-of-freedom. According to the analysis, the parallel mechanism has one displacement degree-of-freedom and two rotational degrees-of-freedom. According to the analysis of the functional characteristics of the vector propulsion actuator of underwater vehicles, it can be concluded that it only requires two rotational degrees-of-freedom to meet the requirements of vector propulsion. Therefore, in applications as a vector propulsion actuator, the displacement degree-of-freedom is the redundant degree-of-freedom of the parallel mechanism, which needs to be restricted during the application process.

Since a portion of the thrust generated by the vectored thruster is allocated to control forces during operation, AUVs equipped with such thrusters obviate the need for additional control surfaces to manage vehicle motion. Although rudders or elevators are not installed, the ability to adjust the deflection angle of the vectored thruster is essential for controlling the AUV. Accurate measurement of the tilt angle of the vectored thruster is a crucial, yet challenging, aspect to ensure the reliability and safety of vehicle operations. While the most straightforward and effective method involves sensor installation on the rotating coupling body, this approach is often infeasible due to the constraints of the underwater environment and limited space availability. An alternative method for determining the tilt angles is through kinematic solutions, which calculate the angles based on the measurements of the three servo motors‘ angular changes. Although this method yields accurate angle information, it is relatively more complex compared to direct sensor-based measurements. Consequently, to realize reliable and precise control of the thrust-vectoring mechanism, high expectations are placed on the design of the automated control system. Hence, the kinematic analysis of the thrust-vectoring mechanism emerges as a pivotal element in ensuring the optimal functionality of the vectored thruster.

2.1. Definition of coordinate system

The study of kinematic analysis holds profound significance, serving as the foundational cornerstone of robotics. Given the intricate nature of the thrust-vectoring mechanism, the process of resolving the position solution for the vectored thruster represents a complex and crucial focal point. In order to comprehensively elucidate the dynamics governing the pose, encompassing both position and orientation, of the upper moving platform, it is imperative to establish coordinate systems within the thrust-vectoring mechanism to articulate relative motion. As illustrated in Figure 3, we have introduced two coordinate systems within the thrust-vectoring mechanism.

Figure 3. Schematic of coordinate systems in the thrust-vectoring mechanism.

As depicted in Figure 3, the reference frame $O_{xyz}$ is affixed to the central point $O$ of the fixed base. In this frame, the $x$ -axis aligns itself parallel to the direction of $A_{3}A_{2}$ , while the $z$ -axis extends normal to the fixed base. Conversely, the moving reference frame $P_{ijk}$ is anchored to point $P$ , located on the lower surface of the moving platform. Within this frame, the $i$ -axis runs parallel to $C_{3}C_{2}$ , while the $k$ -axis is oriented perpendicular to the lower surface of the moving platform.

The pose state ${\boldsymbol{q}}$ of the moving platform within the parallel manipulator can be succinctly represented in relation to the fixed coordinate system. Given that the thrust-vectoring mechanism encompasses one translational degree-of-freedom and two rotational degrees-of-freedom, the generalized vector ${\boldsymbol{q}}$ emerges as a pivotal construct. Within this vector, each element corresponds to one of the six variables judiciously selected to describe both the position and orientation of the upper moving platform with precision. The essential motion parameters encompassed in defining the pose state are as follows:

(1) \begin{equation} \begin{array}{c} {\boldsymbol{q}}=\left[x_{P},y_{P},z_{P},\alpha, \beta, \gamma \right]^{\mathrm{T}}\end{array} \end{equation}

where $\alpha$ , $\beta$ , and $\gamma$ signify the $i$ -axis, $j$ -axis, and $k$ -axis rotation angles of the moving reference frame $P_{ijk}$ in relation to the fixed reference frame $O_{xyz}$ , respectively. $x_{P}$ , $y_{P}$ , and $z_{P}$ represent the displacements of point $P$ within the fixed reference frame. Hence, the pose state of the moving platform can be unequivocally ascertained through the six parameters delineated in Equation 1. When the moving coordinate system on the moving platform of the parallel mechanism undergoes rotation in relation to the fixed coordinate system, the rotation matrix of the moving coordinate system within the mechanism in the fixed coordinate system can be aptly expressed as [2]:

(2) \begin{equation} {\boldsymbol{R}}=\left[\begin{array}{c@{\quad}c@{\quad}c} c\beta c\gamma & {s}{\alpha }s\beta c\gamma -c\alpha s\gamma & {c}{\alpha }s\beta c\gamma +s\alpha s\gamma \\[5pt] c\beta s\gamma & {s}{\alpha }s\beta s\gamma +c\alpha c\gamma & {c}{\alpha }s\beta s\gamma -s\alpha c\gamma \\[5pt] -s\beta & s\alpha c\beta & c\alpha c\beta \end{array}\right] \end{equation}

where $s(\!\cdot\!)$ represents the sine function, and $c(\!\cdot\!)$ represents the cosine function.

In cases where the origin points of the moving coordinate system and the fixed coordinate system within the parallel mechanism do not align, and their respective poses exhibit disparities, the position of point $S$ within the fixed coordinate system ${ }^{{\boldsymbol{O}}}{{\boldsymbol{S}}}{}$ can be aptly articulated by the moving coordinate ${ }^{{\boldsymbol{P}}}{{\boldsymbol{S}}}{}$ :

(3) \begin{equation} { }^{{\boldsymbol{O}}}{{\boldsymbol{S}}}{}={\boldsymbol{R}}\cdot { }^{{\boldsymbol{P}}}{{\boldsymbol{S}}}{}+{ }^{{\boldsymbol{O}}}{{\boldsymbol{P}}}{} \end{equation}

where ${ }^{{\boldsymbol{O}}}{{\boldsymbol{P}}}{}=[x_{P},y_{P},z_{P}]^{\mathrm{T}}$ denotes the position vector representing the origin point $P$ of the moving coordinate system within the fixed coordinate system.

2.2. Inverse position analysis of the designed thrust-vectoring mechanism

When the positional conditions and structural parameters of the moving platform are provided, determining the variation values in each kinematic pair under these positional constraints constitutes the process referred to as the inverse kinematics solution. As depicted in Figure 3, three kinematic branches interconnected between the fixed base and the moving platform form an equilateral triangle. The distance from the origin $O$ of the fixed coordinate system to the center of the rotating hub is designated as $r_{1}$ , representing the circumradius of the equilateral triangle formed by the three points $A_{1}$ , $A_{2}$ , $A_{3}$ . Similarly, the circumradius of the equilateral triangle formed by the vertices $C_{1}$ , $C_{2}$ , $C_{3}$ on the moving platform is denoted as $r_{2}$ .

It’s worth noting that the plane formed by the points $C_{1}$ , $C_{2}$ , $C_{3}$ shares the same orientation as the plane formed by the points $D_{1}$ , $D_{2}$ , $D_{3}$ . The key distinction lies in the origin of the dynamic coordinate system, which is established by the points $C_{1}$ , $C_{2}$ , $C_{3}$ and differs from the center point formed by the points $D_{1}$ , $D_{2}$ , $D_{3}$ solely in the $k$ -axis direction. Consequently, to streamline the kinematic analysis process, this paper exclusively scrutinizes the moving coordinate system formed by the points $C_{i}$ , from which the orientation of the entire moving platform can be deduced.

As illustrated in Figure 3, the angle between the rotation arm $A_{i}B_{i}$ and the plane formed by $A_{1}$ , $A_{2}$ , $A_{3}$ is defined as $\theta _{Ai}$ , signifying the rotation angle of the drive motor. Given the positional and orientational specifications of the moving platform within the thrust-vectoring mechanism, the kinematic inverse analysis furnishes the rotation angle of the servo motor. We further define the angle formed by the plane $O_{xy}$ and the connecting rod $B_{i}C_{i}$ as $\theta _{Bi}$ .

The structural design of the thrust-vectoring mechanism showcases an arrangement of three equidistant kinematic chains connecting the two platforms, with each chain plane orthogonal to the rotational axis of its respective revolute pair. Thanks to these structural and kinematic attributes, each revolute pair possesses a single degree-of-freedom, culminating in the emergence of three constraint equations at the central point of the thrust-vectoring mechanism. These constraint equations, pertinent to each joint within the kinematic branch chain, can be succinctly expressed in mathematical terms as follows:

(4) \begin{equation} \left\{\begin{array}{l} x_{1}=0\\[5pt] x_{2}=\sqrt{3}y_{2}\\[5pt] x_{3}=-\sqrt{3}y_{3} \end{array} \right.\end{equation}

Through the utilization of Equation 3, the position transformation relationship for point $C_{1}$ within the fixed coordinate system can be established as follows:

(5) \begin{equation} \begin{array}{c}{\,}^{O}\left[x_{C1},y_{C1},z_{C1}\right]^{\mathrm{T}}={\boldsymbol{R}}\cdot{\,}^{P}\left[0,-r_{2},0\right]^{T}+^{O}\left[x_{P},y_{P},z_{P}\right]^{\mathrm{T}}\end{array}\end{equation}

Consequently, the position of point $C_{1}$ within the fixed coordinate system can be derived as follows:

(6) \begin{equation}\begin{array}{c@{\quad}c} {}^{O}\left[x_{C1},y_{C1},z_{C1}\right]^{\mathrm{T}}= & { }^{O}\left[-r_{2}\left({s}{\alpha }s\beta c\gamma -c\alpha s\gamma \right),-r_{2}\left({s}{\alpha }s\beta s\gamma +c\alpha c\gamma \right),-r_{2}s\alpha c\beta \right]^{\mathrm{T}}\\[5pt] & +^{O}\left[x_{P},y_{P},z_{P}\right]^{\mathrm{T}}\end{array} \end{equation}

Given the structural characteristics of the thrust-vectoring mechanism and as per the constrained Equation 4, the $x$ -axis component of point $C_{1}$ in the fixed coordinate system consistently remains zero. Therefore, the determination of the $x$ -axis direction of point $C_{1}$ can be succinctly expressed as:

(7) \begin{equation} \begin{array}{c} { }^{O}{x_{C1}}{}=-r_{2}\left({s}{\alpha }s\beta c\gamma -c\alpha s\gamma \right)+^{O}x_{P}=0 \end{array} \end{equation}

Consequently, the coordinates of the origin point $P$ within the moving coordinate system ${ }^{O}{x_{p}}{}$ can be expressed as:

(8) \begin{equation} \begin{array}{c} { }^{O}{x_{p}}{}=r_{2}\left({s}{\alpha }s\beta c\gamma -c\alpha s\gamma \right) \end{array} \end{equation}

In accordance with the analysis detailed in Equation 8, the alteration in the position of the central point $P$ along the $x$ -axis direction is associated with the orientation angles $\alpha$ , $\beta$ , $\gamma$ and the radius $r_{2}$ .

Similarly, the position coordinates of points $C_{2}$ and $C_{3}$ within the fixed coordinate system can be formulated as follows:

(9) \begin{equation} \left\{\begin{array}{l} { }^{O}{{\boldsymbol{C}}}{}_{2}={\boldsymbol{R}}\cdot { }^{P}{{\boldsymbol{C}}}{}_{2}+{ }^{{\boldsymbol{O}}}{{\boldsymbol{P}}}{}\\[5pt] { }^{O}{{\boldsymbol{C}}}{}_{3}={\boldsymbol{R}}\cdot { }^{P}{{\boldsymbol{C}}}{}_{2}+{ }^{{\boldsymbol{O}}}{{\boldsymbol{P}}}{} \end{array} \right.\end{equation}

By employing the coordinate transformation equation as detailed in Equation 9, and taking into account the constraint conditions for points $C_{2}$ and $C_{3}$ as expressed in Equation 4, we can deduce the connection between the alteration in the position of the origin point $P$ along the $y$ -axis direction and the orientation angles $\alpha$ , $\beta$ , $\gamma$ as well as the radius $r_{2}$ as follows:

(10) \begin{equation} \begin{array}{c} { }^{O}{y}{}_{p}=\frac{r_{2}}{2}\left(c\beta c\gamma -{s}{\alpha }s\beta s\gamma -c\alpha c\gamma \right) \end{array} \end{equation}

The orientation angle $\gamma$ arising from the rotation of the moving platform around the $z$ -axis of the fixed coordinate system can be expressed as:

(11) \begin{equation} \begin{array}{c} \gamma =\arctan \left[\dfrac{s\alpha s\beta }{c\beta +c\alpha }\right] \end{array} \end{equation}

With the known parameters ( $z_{P}$ , $\alpha$ , $\beta$ ) of the moving platform, the pose parameters ( $x_{P}$ , $y_{P}$ , $\gamma$ ) can be determined by utilizing Equation 8 through $11$ . The inverse kinematics of the thrust-vectoring mechanism is examined based on these pose parameters of the moving platform. As shown in Figure 3, set $| \overrightarrow{A_{{i}}B_{{i}}}| =l_{ab}$ , $| \overrightarrow{B_{{i}}C_{{i}}}| =l_{bc}$ , and the vector $OC_{i}$ can be expressed as:

(12) \begin{equation} \begin{array}{c} \overrightarrow {OC_{{i}}}=\overrightarrow {OP}+\overrightarrow {PC_{{i}}}=\overrightarrow {OA_{{i}}}+\overrightarrow {AB_{{i}}}+\overrightarrow {BC_{{i}}} \end{array} \end{equation}

On the branch of $A_{1}-B_{1}-C_{1}$ , Equation 12 can be expressed as:

(13) \begin{equation} \begin{array}{c} \left[\begin{array}{c} x_{P}\\[5pt] y_{P}\\[5pt] z_{P} \end{array}\right]+{\boldsymbol{R}}\left[\begin{array}{c} 0\\[5pt] -r_{2}\\[5pt] 0 \end{array}\right]=\left[\begin{array}{c} x_{c1}\\[5pt] y_{c1}\\[5pt] z_{c1} \end{array}\right]=\left[\begin{array}{c} 0\\[5pt] -r_{1}\\[5pt] 0 \end{array}\right]+\left[\begin{array}{c} 0\\[5pt] l_{ab}{\cos } \theta _{A1}\\[5pt] l_{ab}{\sin } \theta _{A1} \end{array}\right]+\left[\begin{array}{c} 0\\[5pt] l_{bc}{\cos } \theta _{B1}\\[5pt] l_{bc}{\sin } \theta _{B1} \end{array}\right] \end{array} \end{equation}

Consider the constraint $x_{1}=0$ , the first equation can be eliminated, and the system of equations can be simplified as:

(14a) \begin{equation} \begin{array}{c} \begin{cases} l_{ab}{\cos } \theta _{A1}+l_{bc}{\cos } \theta _{B1}=b_{11}\\[5pt] l_{ab}{\sin } \theta _{A1}+l_{bc}{\sin } \theta _{B1}=b_{21} \end{cases} \end{array} \end{equation}

with:

(14b) \begin{equation} \begin{array}{c} \begin{cases} b_{11}=y_{P}-r_{2}\left({s}{\alpha }s\beta s\gamma +c\alpha c\gamma \right)\\[5pt] b_{21}=z_{P}-r_{2}\left(s\alpha c\beta \right) \end{cases} \end{array} \end{equation}

Similarly, on the branches of $A_{2}-B_{2}-C_{2}$ and $A_{3}-B_{3}-C_{3}$ , the following equations are found:

(15a) \begin{equation} \begin{array}{c} \begin{cases} l_{ab}{\cos } \theta _{A2}+l_{bc}{\cos } \theta _{B2}=b_{12}\\[5pt] l_{ab}{\sin } \theta _{A2}+l_{bc}{\sin } \theta _{B1}=b_{22} \end{cases} \end{array} \end{equation}

(16a) \begin{equation} \begin{array}{c} \begin{cases} l_{ab}{\cos } \theta _{A3}+l_{bc}{\cos } \theta _{B3}=b_{13}\\[5pt] l_{ab}{\sin } \theta _{A3}+l_{bc}{\sin } \theta _{B3}=b_{23} \end{cases} \end{array} \end{equation}

with:

(15b) \begin{equation} \begin{array}{c} \begin{cases} b_{12}=r_{1}-2\left[y_{p}+\dfrac{\sqrt{3}}{2}r_{2}\left(c\beta s\gamma \right)+\dfrac{1}{2}r_{2}\left({s}{\alpha }s\beta s\gamma +c\alpha c\gamma \right)\right]\\[10pt] b_{22}=z_{P}+\dfrac{\sqrt{3}}{2}r_{2}\left(-s\beta \right)+\dfrac{1}{2}r_{2}\left(s\alpha c\beta \right) \end{cases} \end{array} \end{equation}

(16b) \begin{equation} \begin{array}{c} \begin{cases} b_{13}=r_{1}-2\left[y_{p}-\dfrac{\sqrt{3}}{2}r_{2}\left(c\beta s\gamma \right)+\dfrac{1}{2}r_{2}\left({s}{\alpha }s\beta s\gamma +c\alpha c\gamma \right)\right]\\[10pt] b_{23}=z_{P}-\dfrac{\sqrt{3}}{2}r_{2}\left(-s\beta \right)+\dfrac{1}{2}r_{2}\left(s\alpha c\beta \right) \end{cases} \end{array} \end{equation}

Figure 4. Rotation angle of moving platform $\alpha = {15}^{\circ}$ (0.2617 rad).

Based on the kinematic analysis of the thrust-vectoring mechanism discussed above, the rotation angles $\theta _{Ai}$ of the three driving servo motors can be computed. However, owing to the properties of trigonometric functions, the calculated rotation angles $\theta _{Ai}$ and $\theta _{Bi}$ are not unique. For example, given the pose of the moving platform with $\alpha =15^{^{\circ}}$ , $\beta =0^{^{\circ}}$ , $z_{P} =105\text{ mm}$ , there are two different methods for actuating the servo motors, both of which can satisfy the pose requirements of the moving platform. In Figure 4(a), the rotation angle $\theta _{A1}$ at servo motor $A_{1}$ is $143.7678^{^{\circ}}$ , and the rotation angle $\theta _{B1}$ is $66.5273^{^{\circ}}$ ; as shown in Figure 4(b), the corresponding angles are $\theta _{A1}=17.3618^{^{\circ}}$ , $\theta _{B1}=94.6023^{^{\circ}}$ . To account for the practical constraints and spatial limitations of the thrust-vectoring mechanism, in this design, the rotation angle $\theta _{Ai}$ has been selected to be greater than $90^{^{\circ}}$ , that is $\theta _{Ai}\gt 90^{^{\circ}} (i=1,2,3)$ , thereby circumventing issues such as non-unique rotation angles and abrupt changes. At this point, the calculation results for angles $\theta _{Ai}$ and $\theta _{Ai}$ are given by Equation 17.

(17a) \begin{equation} \begin{array}{c} \begin{cases} \theta _{A1}=\pi -{\arcsin }\dfrac{{b_{11}}^{2}+{b_{21}}^{2}+{l_{ab}}^{2}-{l_{bc}}^{2}}{2l_{ab}\sqrt{{b_{11}}^{2}+{b_{21}}^{2}}}-{\arctan }\dfrac{b_{11}}{b_{21}}\\[10pt] \theta _{B1}={\arcsin }\dfrac{{b_{11}}^{2}+{b_{21}}^{2}+{l_{bc}}^{2}-{l_{ab}}^{2}}{2l_{ab}\sqrt{{b_{11}}^{2}+{b_{21}}^{2}}}-{\arctan }\dfrac{b_{11}}{b_{21}} \end{cases} \end{array} \end{equation}

(17b) \begin{equation} \begin{array}{c} \begin{cases} \theta _{A2}=\pi -{\arcsin }\dfrac{{b_{12}}^{2}+{b_{22}}^{2}+{l_{ab}}^{2}-{l_{bc}}^{2}}{2l_{ab}\sqrt{{b_{12}}^{2}+{b_{22}}^{2}}}-{\arctan }\dfrac{b_{12}}{b_{22}}\\[10pt] \theta _{B2}={\arcsin }\dfrac{{b_{12}}^{2}+{b_{22}}^{2}+{l_{bc}}^{2}-{l_{ab}}^{2}}{2l_{ab}\sqrt{{b_{12}}^{2}+{b_{22}}^{2}}}-{\arctan }\dfrac{b_{12}}{b_{22}} \end{cases} \end{array} \end{equation}

(17c) \begin{equation} \begin{array}{c} \begin{cases} \theta _{A3}=\pi -{\arcsin }\dfrac{{b_{13}}^{2}+{b_{23}}^{2}+{l_{ab}}^{2}-{l_{bc}}^{2}}{2l_{ab}\sqrt{{b_{13}}^{2}+{b_{23}}^{2}}}-{\arctan }\dfrac{b_{13}}{b_{23}}\\[10pt] \theta _{B3}={\arcsin }\dfrac{{b_{13}}^{2}+{b_{23}}^{2}+{l_{bc}}^{2}-{l_{ab}}^{2}}{2l_{ab}\sqrt{{b_{13}}^{2}+{b_{23}}^{2}}}-{\arctan }\dfrac{b_{13}}{b_{23}} \end{cases} \end{array} \end{equation}

2.3. Forward position analysis of the designed thrust-vectoring mechanism

When the rotation angles of each joint are specified, the process of determining the pose of the moving platform is referred to as the forward kinematics of the thrust-vectoring mechanism. Research on forward kinematics analysis is valuable for subsequent investigations into control and structural optimization. In this paper, the forward kinematics process of the thrust-vectoring mechanism is employed to compute the relative pose parameters of the moving platform using the known rotation angles $\theta _{A1}$ , $\theta _{A2}$ , and $\theta _{A3}$ of the servo motors.

From Eq. (15), the position of point $C_{i}$ can be described as:

(18) \begin{equation} \left\{\begin{array}{l} {\boldsymbol{C}}_{1}={ }^{O}\left[0,-\left(r_{1}-l_{ab}c\left(\theta _{A1}\right)+l_{bc}c\left(\theta _{B1}\right)\right),l_{ab}s\left(\theta _{A1}\right)+l_{bc}s\left(\theta _{B1}\right)\right]^{T}\\[5pt] {\boldsymbol{C}}_{2}={ }^{O}\left[\sqrt{3}\left(r_{1}-l_{ab}c\left(\theta _{A2}\right)+l_{bc}c\left(\theta _{B2}\right)\right)/2,\left(r_{ab}c\left(\theta _{A2}\right)+l_{bc}c\left(\theta _{B2}\right)\right)/2,l_{ab}s\left(\theta _{A2}\right)+l_{bc}s\left(\theta _{B2}\right)\right]^{T}\\[5pt] {\boldsymbol{C}}_{3}={ }^{O}\left[-\sqrt{3}\left(r_{1}-l_{ab}c\left(\theta _{A3}\right)+l_{bc}c\left(\theta _{B3}\right)\right)/2,\left(r_{1}-l_{ab}c\left(\theta _{A3}\right)+l_{bc}c\left(\theta _{B3}\right)\right)_{1}/2,l_{ab}s\left(\theta _{A3}\right)+l_{bc}s\left(\theta _{B3}\right)\right]^{T} \end{array} \right.\end{equation}

As the moving platform takes the form of an equilateral triangle with a circumradius of $r_{2}$ , the connection between the three points $C_{1}$ , $C_{2}$ , and $C_{3}$ on the moving platform can be expressed as $|C_{1}C_{2}|=|C_{1}C_{3}|=|C_{2}C_{3}|=\sqrt{3}r_{2}$ . Meanwhile, the position coordinates of point $P$ in the fixed coordinate system can be articulated as:

(19) \begin{equation} \begin{array}{c} \left.\begin{cases} { }^{O}{x_{P}}{}={ }^{O}{x_{C1}}{}+{ }^{O}{x_{C2}}{}+{ }^{O}{x_{C3}}{}\\[5pt] { }^{O}{y_{P}}{}={ }^{O}{y_{C1}}{}+{ }^{O}{y_{C2}}{}+{ }^{O}{y_{C3}}{}\\[5pt] { }^{O}{z_{P}}{}={ }^{O}{z_{C1}}{}+{ }^{O}{z_{C2}}{}+{ }^{O}{z_{C3}}{} \end{cases} \right. \end{array} \end{equation}

Simultaneously, the normal vector ${\boldsymbol{n}}$ of the moving platform can be described as:

(20) \begin{align} {\boldsymbol{n}}=\overrightarrow {C_{1}C_{2}}\times \overrightarrow {C_{1}C_{3}} & =\left| \begin{array}{c@{\quad}c@{\quad}c} i & j & k\\[5pt] x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1}\\[5pt] x_{3}-x_{1} & y_{3}-y_{1} & z_{3}-z_{1} \end{array}\right| \nonumber\\[5pt] &= ai+bj+ck=\left(a,b,c\right) \end{align}

Hence, the alteration in the attitude angle $(\alpha, \beta, \gamma )$ of the moving platform can be defined as:

(21) \begin{equation} \left\{\begin{array}{l} \alpha =\mathrm{c}^{-1}\left(\dfrac{{\boldsymbol{n}}_{1}\cdot {\boldsymbol{m}}_{1}}{\left| {\boldsymbol{n}}_{1}\right| \cdot \left| {\boldsymbol{m}}_{1}\right| }\right)\\[10pt] \beta =\mathrm{c}^{-1}\left(\dfrac{{\boldsymbol{n}}_{2}\cdot {\boldsymbol{m}}_{2}}{\left| {\boldsymbol{n}}_{1}\right| \cdot \left| {\boldsymbol{m}}_{1}\right| }\right)\\[10pt] \gamma =\mathrm{c}^{-1}\left(\dfrac{{\boldsymbol{n}}_{3}\cdot {\boldsymbol{m}}_{3}}{\left| {\boldsymbol{n}}_{1}\right| \cdot \left| {\boldsymbol{m}}_{1}\right| }\right) \end{array} \right.\end{equation}

where ${\boldsymbol{n}}_{1}$ , ${\boldsymbol{n}}_{2}$ , and ${\boldsymbol{n}}_{3}$ denote the direction vectors of the $i$ , $j$ , and $k$ axes of the moving coordinate system, respectively. Meanwhile, ${\boldsymbol{m}}_{1}$ , ${\boldsymbol{m}}_{2}$ , and ${\boldsymbol{m}}_{3}$ represent the direction vectors of the $x$ , $y$ , and $z$ axes of the fixed coordinate system. Based on the Equations 18 through 21, the position vector $[x_{P},y_{P},z_{P}]^{\mathrm{T}}$ and the attitude vector $[\alpha, \beta, \gamma ]^{\mathrm{T}}$ of the moving platform in the fixed coordinate system can be determined through mathematical calculations.

3. Simulation and trial of the thrust-vectoring mechanism

To validate the feasibility and effectiveness of the designed thrust-vectoring mechanism for underwater vehicles, a series of pertinent experiments should be conducted using machined model parts. In all tests involving the thrust-vectoring mechanism model, it is crucial to carry out kinematic analysis experiments to ensure that the mechanism functions as intended for vectored thruster AUVs. For precise measurement of the rotation angle of the moving platform within the thrust-vectoring mechanism, real-time monitoring of the change in the moving platform’s attitude angle can be accomplished by mounting an angular sensor on the moving platform, as illustrated in Figure 5.

Figure 5. Attitude measurement of the moving platform.

When the moving platform moves along the trajectory equation given by $\alpha =15^{^{\circ}}\sin (2\pi /5\cdot t)$ , $\beta =0^{^{\circ}}$ , $z_{P} = 105\text{ mm}$ , the kinematic inverse solution is obtained using the theoretical analysis method presented in this paper. In this section, the motion of the moving platform is driven by the solution result. The changes in the set attitude value and the measured value of the moving platform are shown in Figure 6.

Figure 6. The set and measured attitude value of the moving platform.

The simulation results presented in Figure 6 reveal that there are some discrepancies between the measured and set attitude values of the moving platform in the thrust-vectoring mechanism when the top platform is following the prescribed motion trajectory. The maximum deviation is observed in the rotation angle $\alpha$ , approximately $1.8^{^{\circ}}$ . The angles $\beta$ and $\gamma$ exhibit deviations within the range of $0.5^{^{\circ}}$ . The sources of the moving platform’s attitude error can be attributed to discrepancies in the machining and assembly processes, as well as errors in the angle sensor. To effectively mitigate the deviations between the set and measured values of the moving platform’s attitude, adjustments to the kinematic solution algorithm of the thrust-vectoring mechanism should be made to account for these deviations.

Based on extensive analyses and experimental tests, this paper introduces a compensation method for adjusting the input rotation angles of the three driving servomotors. By increasing the kinematic solving rotation angle by 17%, the experimental results of this compensation method are illustrated in Figure 7, and the relationship between the theoretical rotation angle and the actual angle at joint $A_{i}$ after compensation is presented in Figure 8.

Figure 7. The post-compensated attitude settings and measured values of the moving platform.

Figure 8. The theoretical rotation angle and the actual angle at the rotation joint $A_{i}$ .

The experimental results presented in Figure 7 clearly demonstrate that compensating the kinematic solution for the thrust-vectoring mechanism leads to a significant reduction in the deviation between the measured and set attitude of the moving platform. Specifically, the deviation of the rotation angle $\alpha$ is reduced to approximately $0.6^{^{\circ}}$ , while the rotation angles $\beta$ and $\gamma$ are reduced to within about $0.2^{^{\circ}}$ . It’s noteworthy that the actual rotation angles are noticeably larger than the theoretical solution results, as shown in Figure 8, with a maximum compensation of about $1.5^{^{\circ}}$ for the rotation angle $\theta _{A1}$ , and about $0.5^{^{\circ}}$ for the rotation angles $\theta _{A2}$ and $\theta _{A3}$ .

In a similar manner, when the moving platform operates according to the motion trajectory equations $\alpha =0^{^{\circ}}$ , $\beta =15^{^{\circ}}\cos (2\pi /5\cdot t)$ , $z_{P} = 105\text{ mm}$ , the kinematic solution algorithm for the thrust-vectoring mechanism is compensated. The experimental results of the attitude change of the moving platform after compensation are presented in Figure 8.

The experimental results, as illustrated in Figure 9, demonstrate a close agreement between the attitude changes of the moving platform and the set values after the kinematic solution algorithm is compensated for the thrust-vectoring mechanism. While there are minor deviations between the measured and set attitude values of the moving platform, these discrepancies are small, with rotation angles $\alpha$ , $\beta$ , and $\gamma$ being controlled to within less than $0.5^{^{\circ}}$ . Nevertheless, due to irregular gaps in the machining and assembly processes, as well as the inherent accuracy errors in the angle sensors, it remains challenging to entirely eliminate the deviation between the set and measured values of the moving platform through experimental compensation alone.

Figure 9. The theoretical rotation angle and the actual angle at the rotation joint $A_{i}$ with equation $\alpha =0^{^{\circ}}$ , $\beta =15^{^{\circ}}{\cos } (2\pi /5\cdot t)$ , $z_{P} = 105\,mm$ .

To accurately and effectively investigate and analyze the thrust generated by a propeller, a series of experiments were conducted using a ducted propeller model to obtain precise thrust values. Since the thrust generated by the propeller cannot be measured directly, a thrust measurement system was designed in this study, as depicted in Figure 10(a). And the experimental environment was designed as shown in Figure 10(b).

Figure 10. Attitude measurement system and experimental environment of moving platform.

In practical applications, the propeller thruster in this study is intended to provide vectored thrust for an underwater vehicle. During operation, the thruster will be deflected at a maximum angle of $\pm 15^{^{\circ}}$ relative to the vehicle axis. To measure the axial and lateral thrust values of the propeller thruster during deflection, adjustments to the mount and slide rail are made to achieve a thruster deflection of $15^{^{\circ}}$ and $75^{^{\circ}}$ . Meanwhile, in the thruster thrust measurement experiment, the propeller speed can be adjusted by changing the high-level time of the driving motor controller, consequently altering the magnitude of the thrust generated by the thruster. Following this approach of adjusting the propeller speed, multiple thrust measurement experiments were carried out with $0^{^{\circ}}$ , $15^{^{\circ}}$ and $75^{^{\circ}}$ deflection of the thruster, and the experimental results are shown in Figure 11.

Figure 11. Thrust generated by propeller.

Based on the thrust measurement results presented in Figure 11(a), when the thruster has no deflection, it is evident that when the controller’s high-level time is set to $1.825\text{ ms}$ , the propeller thruster generates a maximum thrust of $2.32\text{ Kg}$ , equivalent to approximately $22.75\mathrm{N}$ . Comparing the thrust values in (b) and (c), when the thruster is deflected by $15^{^{\circ}}$ from the axis, the average thrust is about 91.48% of the average thrust without deflection. When the thruster is deflected by $75^{^{\circ}}$ from the axis, the average thrust is about 23.19% of the average thrust without deflection. Although the two thrust coefficients, 0.9142 and 0.2319, deviate slightly from the theoretical coefficients of $\cos 15^{^{\circ}}$ (0.9659) and $\sin 15^{^{\circ}}$ (0.2588), the maximum deviation does not exceed 5.4%, indicating the high accuracy and reliability of the experimental measurements.

Based on the analysis of the thrust-vectoring mechanism motion solution experiment and thruster thrust measurement experiment results, it is evident that the thrust-vectoring mechanism designed in this paper effectively adjusts the thruster deflection angle. The propeller can generate the necessary driving thrust for the vector thruster and provide the lateral force required for controlling the underwater vehicle. It has successfully achieved the goal of vector propulsion for the underwater vehicle.

4. Controller design and simulation

Owing to the compact dimensions of the vector propulsion actuator, integrating a sensing apparatus to obtain precise measurements of the output shaft’s orientation proves challenging. Consequently, in the design of the motion controller for vector actuators, the inverse kinematics calculation methodology employed in parallel mechanisms is adopted to ascertain the accurate pose of the vector propulsion actuator, while disregarding structural inconsistencies. The implementation of the inverse kinematics calculation technique precludes the necessity for installing pose sensors within the moving platform, thereby effectively diminishing the overall complexity of the vector propulsion actuator’s design.

In this pivotal section, our primary endeavor lies in delving into the intricate process of developing a sophisticated vectored thruster controller, harnessing the potent capabilities of Model Predictive Control (MPC). This approach endows us with the capacity to anticipate and constrain the system’s behavioral patterns proactively. Furthermore, to augment the accuracy and reliability of our controller, we seamlessly integrate a physics-informed neural network (PINN) into the paradigm.

PINNs are neural networks that embed physical laws and principles into their learning process, rendering them uniquely suited for modeling intricate systems that adhere to known physical precepts. Within this context, the PINN undergoes training to model the intricate control equations governing the vectored thruster, enabling us to render precise predictions for MPC. By synergistically combining the predictive prowess of MPC with the physics-informed modeling proficiency of PINN, we forge a controller that exhibits both robustness and precision.

In summary, this section delves into the development of a state-of-the-art vector thruster controller, leveraging the potent synergy between Model Predictive Control and Physics-Informed Neural Networks. This integration ensures that our controller meets the demands of vectored thruster control.

4.1. PINN-MPC for the vectored thruster

According to Equation 17, the rotation angle of revolute joint $\theta _{Ai}$ can be concisely represented as $\theta _{Ai}=f_{i}(\alpha, \beta, \gamma (\alpha, \beta ))$ . Subsequently, the derivative of $\theta _{Ai}$ can be derived as:

(22) \begin{equation} \dot{\theta }_{Ai}=\left(\frac{\partial f_{i}}{\partial \alpha }+\frac{\partial f_{i}}{\partial \gamma }\frac{\partial \gamma }{\partial \alpha }\right)\dot{\alpha }+\left(\frac{\partial f_{i}}{\partial \beta }+\frac{\partial f_{i}}{\partial \gamma }\frac{\partial \gamma }{\partial \beta }\right)\dot{\beta } \end{equation}

To facilitate analysis and control design, let ${\boldsymbol{x}}=[\alpha, \beta ]^{\mathrm{T}}$ represent the state vector encompassing the attitude angles, and ${\boldsymbol{u}}=[\dot{\theta }_{A1},\dot{\theta }_{A2},\dot{\theta }_{A3}]^{\mathrm{T}}$ represent the input vector consisting of the rates of change of the revolute joint angles. The state equation can then be expressed in a compact form as:

(23) \begin{equation} \begin{array}{c} {\boldsymbol{u}}=\left[\begin{array}{c} \dot{\theta }_{A1}\\[10pt] \dot{\theta }_{A2}\\[10pt] \dot{\theta }_{A3} \end{array}\right]=\left[\begin{array}{c@{\quad}c} \left(\dfrac{\partial f_{1}}{\partial \alpha }+\dfrac{\partial f_{1}}{\partial \gamma }\dfrac{\partial \gamma }{\partial \alpha }\right) & \left(\dfrac{\partial f_{1}}{\partial \beta }+\dfrac{\partial f_{1}}{\partial \gamma }\dfrac{\partial \gamma }{\partial \beta }\right)\\[10pt] \left(\dfrac{\partial f_{2}}{\partial \alpha }+\dfrac{\partial f_{2}}{\partial \gamma }\dfrac{\partial \gamma }{\partial \alpha }\right) & \left(\dfrac{\partial f_{2}}{\partial \beta }+\dfrac{\partial f_{2}}{\partial \gamma }\dfrac{\partial \gamma }{\partial \beta }\right)\\[10pt] \left(\dfrac{\partial f_{3}}{\partial \alpha }+\dfrac{\partial f_{3}}{\partial \gamma }\dfrac{\partial \gamma }{\partial \alpha }\right) & \left(\dfrac{\partial f_{3}}{\partial \beta }+\dfrac{\partial f_{3}}{\partial \gamma }\dfrac{\partial \gamma }{\partial \beta }\right) \end{array}\right]\left[\begin{array}{c} \dot{\alpha }\\[10pt] \dot{\beta } \end{array}\right]=\boldsymbol{\varphi }\left({\boldsymbol{x}}\right){\boldsymbol{x}} \end{array} \end{equation}

which can also be written as:

(24) \begin{equation} \begin{array}{c} \dot{{\boldsymbol{x}}}=\left(\boldsymbol{\varphi }^{\mathrm{T}}\left({\boldsymbol{x}}\right)\boldsymbol{\varphi }\left({\boldsymbol{x}}\right)\right)^{-1}\boldsymbol{\varphi }^{\mathrm{T}}\left({\boldsymbol{x}}\right){\boldsymbol{u}}={\boldsymbol{B}}\left({\boldsymbol{x}}\right){\boldsymbol{u}} \end{array} \end{equation}

PINNs uniquely integrate empirical data with the governing physical laws, expressed as Partial Differential Equations (PDEs), into the training process. In order for the PINN to accurately fit the state Equation 24, it is imperative to devise a well-suited loss function for the optimized network. This loss function, denoted as $\mathcal{L}$ , serves as a composite metric that combines both supervised and unsupervised losses. The supervised loss $\mathcal{L}_{data}$ , captures the discrepancy between the network’s predictions and actual data measurements, while the unsupervised loss $\mathcal{L}_{phys}$ , enforces the satisfaction of the governing PDEs. And the loss function $\mathcal{L}$ can be expressed as:

(25) \begin{equation} \begin{array}{c} \mathcal{L}=w_{data}\mathcal{L}_{data}+w_{phys}\mathcal{L}_{phys} \end{array} \end{equation}

where $w_{data}$ and $w_{phys}$ are weighting factors that judiciously balance the contributions of the data-driven loss $\mathcal{L}_{data}$ and the physics-informed loss $\mathcal{L}_{phys}$ , respectively. These weights are tuned to ensure that both empirical data and physical laws are adequately represented in the training process.

The supervised loss $\mathcal{L}_{data}$ typically measures the difference between the network’s output and the corresponding target values from the available data. This can be achieved using various metrics such as mean squared error or mean absolute error. On the other hand, the unsupervised loss $\mathcal{L}_{phys}$ ensures that the network’s predictions satisfy the governing PDEs. This is achieved by incorporating the PDEs into the loss function using techniques like automatic differentiation. The PDEs define relationships between different variables and their derivatives, and the unsupervised loss penalizes any violations of these relationships by the network’s predictions. The specific form of the PDE violation measure depends on the nature of the PDEs and how they are encoded into the loss function. It typically involves computing the residuals of the PDEs using the network’s predictions and then summing or integrating these residuals over the relevant domain. Therefore, the losses of $\mathcal{L}_{data}$ and $\mathcal{L}_{phys}$ can be defined as:

(26) \begin{equation} \mathcal{L}_{data}=\frac{1}{N_{data}}\sum _{i=1}^{N_{data}}\left\| {\boldsymbol{x}}-\hat{{\boldsymbol{x}}}\right\| ^{2} \end{equation}

(27) \begin{equation} \mathcal{L}_{phys}=\frac{1}{N_{phys}}\sum _{i=1}^{N_{phys}}\left\| {\boldsymbol{B}}\left({\boldsymbol{x}}\right){\boldsymbol{u}}-\dot{\hat{{\boldsymbol{x}}}}\right\| ^{2} \end{equation}

where $N_{data}$ and $N_{phys}$ respectively represent the number of data points applied to $\mathcal{L}_{data}$ and $\mathcal{L}_{phys}$ , and $\hat{{\boldsymbol{x}}}$ and ${\boldsymbol{x}}$ represent network predicted values and true values, respectively. By minimizing the composite loss function $\mathcal{L}$ , the PINN is trained to not only fit the available data but also adhere to the governing physical laws expressed by the PDEs. This ensures that the network’s predictions are both accurate and physically meaningful, making it suitable for use in systems like vector thruster controllers.

A precisely defined cost function is pivotal for evaluating the performance of the control system in MPC. The cost function, commonly denoted by $J$ , aims to simultaneously achieve two primary objectives: minimizing deviations from the reference trajectory and optimizing the utilization of control inputs within the prediction horizon. The mathematical expression representing this cost function can be refined and polished as follows:

(28) \begin{equation} \begin{array}{c} \min J=\int _{0}^{Nt}\left(\left\| \hat{{\boldsymbol{x}}}\left(s\right)-{\boldsymbol{x}}_{ref}\right\| _{Q}^{2}+\left\| \hat{{\boldsymbol{u}}}\right\| _{R}^{2}\right)\text{ds } \end{array} \end{equation}

where $N$ is the length of the predicted time domain, ${\boldsymbol{x}}_{ref}$ represents the target trajectory, $Q$ and $R$ are weight matrices judiciously employed to balance state bias and the relative importance of control inputs. Meanwhile, $\hat{{\boldsymbol{x}}}$ in the optimization equation is obtained through the trained PINN model. The design of this cost function considers both the deviation between the system state and the reference trajectory, as well as the utilization of control inputs. By minimizing the cost function $J$ , the MPC controller can find an optimal sequence of control inputs that enable the system to operate along the desired trajectory while minimizing the consumption of control inputs. This optimization approach helps improve the performance and efficiency of the control system.

The control constraints of the vector thruster principally consist of two parts, the state ${\boldsymbol{x}}$ and input ${\boldsymbol{u}}$ , which can be mathematically expressed as follows:

(29a) \begin{equation} \begin{array}{c} {\boldsymbol{u}}_{max}=-{\boldsymbol{u}}_{min}=\left[5^{^{\circ}}/s,5^{^{\circ}}/s,5^{^{\circ}}/s\right]^{\mathrm{T}} \end{array} \end{equation}

(29b) \begin{equation} \begin{array}{c} {\boldsymbol{x}}_{max}=-{\boldsymbol{x}}_{min}=\left[15^{^{\circ}},15^{^{\circ}}\right]^{\mathrm{T}} \end{array} \end{equation}

Equation 29a specifies the maximum and minimum permissible values for each component of the input vector u, ensuring that the control inputs do not exceed a certain angular velocity limit. Equation 29b defines the maximum and minimum allowable values for each component of the state vector x, limiting the range of possible attitude angles for the moving platform. By adhering to these constraints, the control system ensures that the vector thruster operates within safe and effective parameters, preventing excessive or unsafe maneuvers.

4.2. Simulation analysis

For the training of this network, the loss function $\mathcal{L}$ has been calibrated using $N_{data}=1000$ data points for the data-driven loss $\mathcal{L}_{data}$ and $N_{phys}=2000$ . for the physics-informed loss $\mathcal{L}_{phys}$ . The weights assigned to these components are $w_{data}=1$ and $w_{phys}=0.1$ , respectively, ensuring a judicious balance between data and physics in the loss computation. This careful balancing of the loss function terms is crucial, as it allows the neural network to effectively learn from the available data while adhering to the underlying physical principles governing the system dynamics. By properly tuning these weights, the model can achieve an optimal tradeoff between fitting the empirical observations and satisfying the theoretical equations, resulting in enhanced accuracy and robustness.

The controller operates with a sampling period of $t=0.1$ second, and a prediction horizon of $N=10$ steps is employed to optimize the control decisions. The weighting matrices in the MPC algorithm have been calibrated as $Q=\text{diag}(200,200)$ and $R=\text{ diag}(0.1,0.1,0.1)$ , reflecting the relative importance of minimizing deviations from the target trajectory and controlling the magnitude of control inputs. The judicious selection of these weight matrices is essential for achieving the desired control performance, as it determines the relative emphasis placed on tracking accuracy and control effort minimization. By carefully balancing these competing objectives, the controller can navigate the tradeoff between precise trajectory following and efficient resource utilization, thereby maximizing overall system effectiveness. For the convenience of calculation, the Pytorch toolkit is utilized for derivative calculation.

To assess the performance of this control method, a simulation experiment has been designed. We opt for the following trajectory:

(30) \begin{equation} {\boldsymbol{x}}_{ref}= \left\{\begin{array}{l}\alpha =15^{^{\circ}}\cos \left(\dfrac{\pi }{50t}\right)\\[10pt] \beta =15^{^{\circ}}\sin \left(\dfrac{\pi }{50t}\right) \end{array}\right. \end{equation}

As depicted in Figure 12, subgraph (a) presents the trajectory tracking outcomes in the $\alpha$ and $\beta$ planes, offering a clear visualization of the platform’s movement. Subgraph (b) exhibits the integral value of the input ${\boldsymbol{u}}$ , which corresponds to the rotation angle of the rotary joint, providing insights into the dynamic behavior of the system. Additionally, subgraphs (c) and (d) showcase the tracking results and tracking errors in the $\alpha$ and $\beta$ directions, respectively, enabling a comprehensive evaluation of the tracking performance. During the initial tracking phase, which can be seen in (a) and (c), due to the mismatch between the initial attitude and the desired target value’s, the attitude angle must promptly converge toward the trajectory of the target value’s evolution and subsequently achieve a stable tracking outcome. This initial transient response is a natural consequence of the system’s dynamics and highlights the controller’s ability to rapidly compensate for deviations and synchronize with the desired trajectory.

Figure 12. The movement state change of the thrust-vectoring mechanism.

As depicted in (d), after 20 s, the tracking task is essentially in a stable stage. During this stabilized phase, the maximum tracking error for $\alpha$ is $0.103^{^{\circ}}$ , with an average error of $0.064^{^{\circ}}$ . Analogously, the maximum error for $\beta$ is $0.107^{^{\circ}}$ , averaging at $0.058^{^{\circ}}$ . These outcomes demonstrate the precision of the tracking performance and a satisfactory convergence toward the desired trajectory, indicating effective control and accuracy in achieving the tracking objectives. The low magnitude of the tracking errors, both in terms of maximum deviations and average values, underscores the controller’s capability to maintain precise and consistent trajectory following once the initial transient has subsided.

To enhance the understanding and analysis of the control effectiveness of this control method on vector thrusters, a motion trajectory is delineated for the moving platform. This trajectory, as detailed in Equation 31, represents an 8-shaped path, allowing for a comprehensive assessment of the control method’s capabilities. By studying this trajectory, we can gain insights into how the control method performs in maneuvering the platform through complex and challenging patterns. The inclusion of such intricate trajectories in the simulation is crucial, as it enables a thorough evaluation of the controller’s robustness and adaptability under various operational scenarios, which is essential for real-world applications where unpredictable and dynamic conditions may be encountered.

(31) \begin{equation} {\boldsymbol{x}}_{ref}=\left\{\begin{array}{l} \alpha =-15^{^{\circ}}\sin \left(\frac{2\pi }{50t}\right)\\[5pt] \beta =15^{^{\circ}}\cos \left(\frac{\pi }{50t}\right) \end{array} \right.\end{equation}

When examining the figure-8 trajectory tracking results in Figure 13, the controller demonstrates outstanding performance. Despite the increased complexity of this motion pattern, the controller maintains exceptional tracking precision during the stable phase (after 20 s). For the $\alpha$ angle component, the maximum tracking error is $0.252^{\circ}$ , with an average error of $0.128^{\circ}$ ; for the $\beta$ angle component, the maximum error is $0.161^{\circ}$ , with an average error of $0.065^{\circ}$ . These data reflect the control method’s ability to achieve excellent tracking performance even when handling intricate trajectories, ultimately allowing the moving platform’s attitude to converge accurately to the desired values.

Figure 13. The movement state change of the thrust-vectoring mechanism.

Notably, although the tracking errors are slightly elevated compared to the simpler sinusoidal trajectory, their magnitudes remain within an acceptable and reasonable range. This situation confirms the controller’s remarkable adaptability, as its performance only degrades moderately when faced with more challenging motion patterns, rather than suffering severe deterioration. The controller’s capability to flexibly handle various complex trajectories fully demonstrates its robust and reliable nature.

Overall, through the simulated figure-8 trajectory trial, we clearly observe the control strategy’s outstanding performance. Even under stringent conditions, the controller can rapidly and accurately track the target trajectory, achieving the desired attitude control effect. This lays a solid foundation for the widespread application of this method in practical engineering scenarios.

Figure 14. The movement state change of the thrust-vectoring mechanism.

When the moving platform of the parallel mechanism tracks the infinite motion trajectory defined by Equation 32, the control results obtained are presented in Figure 14. This scenario introduces an additional layer of complexity, as the trajectory’s non-periodic nature poses a more demanding challenge for the controller to maintain accurate tracking over an extended duration.

(32) \begin{equation} {\boldsymbol{x}}_{ref}=\left\{\begin{array}{l} \alpha =15^{^{\circ}}\cos \left(\dfrac{\pi }{50t}\right)\\[10pt] \beta =15^{^{\circ}}\sin \left(\dfrac{2\pi }{50t}\right) \end{array}\right. \end{equation}

In Figure 14, the tracking precision is rigorously evaluated after the initial 20-second period. Specifically, the maximum tracking error for the α component is measured to be a modest $0.117^{^{\circ}}$ , with an average error of only $0.063^{^{\circ}}$ . In contrast, the $\beta$ component exhibits a slightly higher maximum error of $0.202^{^{\circ}}$ , averaging at $0.127^{^{\circ}}$ . Comparing the results presented in Figure 14 with those in Figure 13, it becomes evident that the tracking error tends to be more pronounced for angle components undergoing more rapid changes, as opposed to those experiencing slower variations. This behavior is expected and aligns with theoretical predictions, as abrupt changes in the desired trajectory pose a greater challenge for the controller to respond and compensate promptly.

Despite the added complexity of the non-periodic trajectory, it is noteworthy that the tracking errors remain well within an acceptable range, effectively demonstrating the proposed method’s capability in accurately tracking the target trajectory of the vector propulsion device, even for complex, aperiodic motions. This underscores the practical feasibility and reliability of the control strategy, rendering it a viable option for real-world applications where the system may encounter diverse and unpredictable motion profiles. The consistent and robust performance across various trajectory complexities highlights the controller’s versatility and resilience, essential qualities for practical deployment in dynamic and demanding environments.

The controller’s ability to maintain precise tracking, even when faced with the compounded challenges of non-periodicity and rapid trajectory changes, is a testament to the efficacy and sophistication of the proposed control approach. This remarkable performance not only validates the theoretical foundations but also instills confidence in the method’s potential for practical implementation, where the ability to handle diverse and unpredictable conditions is paramount.

5. Conclusion and Future Work

This paper presents an in-depth exploration of the design and control of a vectored thruster system for AUVs, utilizing a 3RRUR parallel manipulator architecture. The study focuses on a theoretical analysis to elucidate the complex kinematics of the thrust-vectoring mechanism, deriving both inverse and forward solutions that reveal the mathematical intricacies governing precise thruster orientation control. Extensive numerical simulations are conducted under multi-degree-of-freedom motions to validate the accuracy of these kinematic models, providing dynamic insights into the system’s behavior and reinforcing theoretical foundations. A kinematic compensation method is proposed and experimentally validated to minimize inherent motion errors associated with parallel manipulators, bridging the gap between theory and practical implementation. Additionally, this study introduces a novel control strategy integrating MPC with PINNs, enhancing trajectory tracking precision and optimizing control input utilization across various motion profiles. The convergence of theoretical rigor, numerical analysis, experimental validation, and advanced control techniques underscores the feasibility and effectiveness of the vectored thruster system for AUV applications. This comprehensive study establishes a robust multidisciplinary foundation, advancing understanding of vectored thruster kinematics and control without subjective language. It sets the stage for innovative solutions in underwater exploration and robotics, driving practical applications and theoretical advancements in this pivotal technology.