Intelligent Alignment Control for Floating Raft Air Spring Mounting System Under Coupled Hull–Raft Deformation

Abstract

Shaft alignment is adversely affected by the increasingly severe coupled hull–raft deformation in deep-diving, highly integrated submersibles, thereby compromising operational safety and potentially amplifying vibration noise. To address to this issue, this paper investigates an intelligent alignment control method for the floating raft air spring mounting system (ASMS) applied to marine propulsion unit (MPU) under coupled hull–raft deformation conditions. A multi-objective alignment control algorithm was developed based on the NSGA-II optimization method within an N-step receding horizon optimal control framework, enabling simultaneous achievement of shaft alignment attitude adjustment, hull deformation compensation, raft deformation suppression, and pneumatic energy consumption. Experimental validation was conducted on two distinct ASMS prototypes to evaluate the control algorithm. Tests performed on the ASMS for MPU (MPU-ASMS) prototype demonstrated effective compensation of hull-induced deformations, maintaining shaft alignment offsets within ±0.3 mm and angularities within ±0.5 mm/m. Concurrently, experiments on the floating raft ASMS for the stern compartment (SC-FR-ASMS) achieved precise control of axial offsets within ±0.3 mm, angularities within ±0.5 mm/m, and vertical displacements of critical monitoring points within ±1 mm. The adaptive control strategy additionally proved effective in suppressing raft deformation while simultaneously optimizing pneumatic energy consumption. This research provides robust theoretical and technical foundations for intelligent vibration isolation systems in deep-sea equipment to accommodate extreme-depth-induced hull deformation and large-scale raft deformation.

1. Introduction

The intensification of global maritime competition is driving increasingly stringent requirements for the acoustic stealth performance of submersibles. Given that propulsion system noise represents the primary source of radiated noise [1,2,3], its control critically governs stealth effectiveness. Spatial constraints and displacement limitations impose stringent dimensional requirements on critical equipment such as the marine propulsion system in the stern compartment. Consequently, floating rafts frequently adopt high-aspect-ratio “slender” and lightweight designs to accommodate compartment geometry [4]. However, this configuration compromises the structural stiffness, leading to significant vertical bending and torsional deformations. Critically, such raft elastic deformations exhibit high sensitivity to propulsion shaft misalignment [5,6], defined as the condition where the centerlines of the drive shaft (e.g., MPU output shaft) and driven shaft (e.g., propeller input shaft) fail to maintain an ideal collinear relationship under static or dynamic operating conditions [7]. Furthermore, structural deformations in the pressure hull will induce local buckling or strain concentration under deep-submergence conditions. These hull deformations further aggravate propulsion shaft misalignment [8,9].

Although floating raft vibration isolation technology has become the mainstream solution for naval vibration control owing to its exceptional vibration energy attenuation capacity and compact layout (Figure 1) [10], prevailing shaft alignment strategies remain constrained by rigid hull/raft assumptions [11,12]. Critically, these approaches neglect two deformation mechanisms: (i) hull deformations induced by depth variations and wave loads, and (ii) raft elastic deformations arising from load distribution, vessel motions, and reaction torques. To address these limitations, this study develops an intelligent alignment control method targeting floating raft ASMS for MPU (MPU-FR-ASMS) under coupled hull–raft deformation. The proposed approach synergistically compensates for both hull and raft deformation types, ensuring safe and stable propulsion system operation.

In research on hull deformation, the Finite Element Method (FEM) serves as the primary analytical approach. When wave loading on large surface vessels and depth variations affect deep-diving submersibles, relative displacements will occur at critical alignment reference points such as main engine flexible mounts, shaft intermediate bearing supports, and stern thruster bearing seats. These displacements degrade the initial alignment conditions and induce shaft misalignment [13]. This misalignment intensifies particularly in long-span shafting systems and is magnified by increasing vessel dimensions and operational depths. Current studies predominantly utilize FEM simulations to predict alignment-critical deformations [8,11,14,15,16,17,18]. Ruan et al. [14] computed shaft bracket deformations up to 10 mm in an 8530 TEU container ship. Liu et al. [19] applied the Runge–Kutta discontinuous Galerkin method (RKDG) to calculate explosion loads from spherical charges, then used ABAQUS nonlinear FEM to solve for hull girder deformation. Choi et al. [20] employed a reinforcement learning approach to develop a surrogate model for predicting shaft deformation resulting from stern hull deformation, based on measurement data and finite element modeling of a medium-sized oil/chemical tanker. Zhou et al. [21] decomposed hull deformation into global (single-span beam model) and local (grillage beam model) components, employing matrix displacement methods to reduce computation time. However, both aforementioned methods are incapable of achieving online monitoring capabilities. In shaft alignment monitoring research, current techniques are broadly classified into indirect approaches and direct approaches:

(1)

Indirect approaches infer misalignment through proxy signals: vibration spectra [7,22,23], current signatures [24,25], temperature variations [26,27], acoustic characteristics [28,29], or torque fluctuations [30,31].

(2)

Direct approaches measure displacements at key locations using dial gauges [1,32], laser sensors [13,33,34,35,36,37,38], or displacement transducers [12,39,40,41,42].

The laser displacement sensing method in [13] enables real-time monitoring of shaft misalignment induced by hull deformation in the MPU-ASMS. However, all other aforementioned approaches lack the capability for dynamic shaft offsets while accounting for hull deformation effects. Furthermore, alternative monitoring methodologies continue to evolve. Yang et al. [43] developed a multi-camera relay measurement system capable of monitoring hull deformations at hundred-meter scales. Qu et al. [44] analyzed the relationship between hull angular motion and deformation during ship turning, incorporated sea-trial-derived correction coefficients of the ship turning left and right, and provided technical support for high-precision distributed digital referencing in inertial navigation transfer alignment and heading angle transfer of other shipborne equipment. Li et al. [45] enhanced inertial matching hull deformation measurement by compensating cross-correlation errors between the ship and dynamic deformation angular velocities based on the least squares estimation and correlation theory. However, these techniques were specifically designed for surface vessels and demonstrate limited applicability in underwater environments, particularly under deep-sea conditions.

In research on raft elastic deformation monitoring for floating raft ASMS (FR-ASMS), the primary methodologies are classified into two categories: direct measurement using displacement sensors [6,46,47] and indirect measurement using strain sensors [10,48,49].

(1)

For direct measurement using displacement sensors, Qin et al. [6,46] proposed a dual-reference displacement system that constructs a datum from two key reference points to compute raft elastic deformations at monitoring locations. Cheng et al. [47] achieved online full-field displacement monitoring of raft by integrating displacement gradient field analysis with surface spline interpolation algorithms.

(2)

For indirect measurement using strain sensors, Zhi [48] reconstructed displacements for clamped-boundary raft through modal method utilizing multi-point strain data. Cheng et al. [10] improved modal method by fusing strain data with limited displacement measurements, formulating a dimensionless least-squares function to resolve coupled rigid-elastic deformation monitoring under elastic-boundary. Cheng et al. [49] integrated the curvature-displacement relationship into a deep neural network, developing a multilevel-domain segmented physics-informed neural network (MSPINNs) tailored for warship raft structures undergoing deformation under uncertain loads and complex multipoint elastic constraints. While mature strain-based reconstruction methods exist in aerospace and other fields like modal method [48,50,51], KO method (named after its originators) [52,53], surface reconstruction [54,55], data-driven algorithms [48,56,57,58,59], inverse Finite Element Method (iFEM) [60,61,62,63], these methods face adaptability challenges on elastic-boundary in FR-ASMS [10]. The improved modal method provides a solution for displacement reconstruction adapting to elastic-boundary [10].

In research on shaft alignment attitude control for FR-ASMS, the field remains technically fragmented. While significant advances have been achieved in shaft alignment precision, hull deformation compensation, and raft deformation suppression as isolated domains, no holistic framework exists for coordinated control addressing coupled hull–raft deformation dynamics.

• In research on shaft alignment precision, Shi et al. [41] established an optimal control algorithm to minimize flange face angular deviations and radial displacements for alignment adjustment. Subsequently, an optimal pressure-distribution alignment strategy was developed for MPU [64]. Xu et al. [65] experimentally validated the disturbance control feasibility of ASMS against disturbances including ship inclination/oscillation, output counter-torque reaction, coupling additional force and so on. Bu et al. [66] proposed a pseudo-sensitivity analysis-based control strategy for propulsion shaft alignment, and subsequently investigated the alignment controllability of ASMS [67]. To address the high-precision attitude balancing requirements in dual layer ASMS, Bu et al. [68] further developed a multi-objective coordinated attitude control method. For high power density conditions, Bu et al. [69] established a hierarchical control approach by integrating hard and soft constraints, which significantly enhanced the alignment controllability and stability. In addressing the issue that mass variations in large liquid tanks can induce displacement changes, Shi et al. [70] proposed an adaptive variable-load control method to achieve raft attitude balance and load equalization optimization under the variable mass condition of the FR-ASMS. Recent studies have demonstrated significant advancements in shaft alignment and control technologies. Zhang et al. [71] developed an automated alignment system for multi-support shafting to improve alignment efficiency and accuracy. Fan et al. [72] proposed a multi-channel cross-decoupling control algorithm for multiple target parameters, achieving decoupling of vibration displacement between control channels and coordinated control between target parameters. Liu et al. [73,74] further introduced a neural network-driven digital-twin approach for alignment control, establishing data-driven mappings between the air spring pressures and shafting alignment state. However, these studies share a critical limitation in their failure to account for the coupled effects of hull/raft deformation dynamics in their system designs and control strategies.
• In research on hull deformation compensation, Cheng et al. [75] proposed a self-recovery offset compensation method utilizing structural adaptation for effective hull deformation mitigation in MPU-ASMS.
• In research on raft deformation suppression, initial uniform pressure designs risked localized overloading under uneven mass distributions, inducing elastic distortions [64]. Qin et al. [6,46] dynamically assessed raft attitude and deformation control performance using displacement sensor data. They optimized air spring selection strategies through air spring pressure parameter identification. Synchronized control of the raft attitude and elastic deformation was achieved by regulating the air spring height differentials.

The precise shaft alignment control in MPU-FR-ASMS constitutes a complex multivariable cooperative control challenge when coordinating hull deformation compensation with raft elastic deformation suppression. This challenge inherently involves strong couplings across multiple physical domains, including mechanical dynamics, structural response, and active control, necessitating the development of highly efficient and robust optimization and control strategies. Therefore, it is essential to draw upon advanced methodologies and concepts from related fields to address the multivariable cooperative control challenge. Zhang et al. [76] proposed an enhanced multi-objective JAYA (EMOJAYA) algorithm, which utilizes mixed-integer programming for dynamic task reassignment. This approach significantly mitigated the multi-objective conflicts arising from preventive maintenance, achieving an 18% reduction in cycle time and a 23% decrease in maintenance costs within an automotive production line application. Zhu et al. [77] developed a tri-population evolutionary algorithm based on dynamic constraint boundaries (TPDCB) algorithm, featuring an innovative dual dynamic constraint boundary strategy. This method enables the efficient exploration of fragmented feasible regions encountered in complex multi-objective optimization problems. To address stringent scheduling constraints, Liang et al. [78] constructed a finite-time hierarchical control framework for robust cooperative control under disturbances. This framework integrates kinematic and dynamic models to ensure finite-time convergence for multi-formation tasks. The exceptional performance of these methods provides valuable insights for addressing the inherent multi-objective control challenges.

Current research on alignment control methods for FR-ASMS focuses primarily on monitoring and mitigating air spring isolator height variations and their resulting misalignment, caused by external disturbances and fluctuations in air spring working pressure. However, the critical challenge of controlling alignment deviations arising from complex hull deformations, raft elastic deformations, and their coupled effects during navigation remains systematically unexplored in existing literature. The core contribution of this work lies in bridging this research gap, specifically manifested through two innovative advancements:

(1)

A shaft alignment response model is established under coupled hull–raft deformation. This model overcomes the limitations of rigid-body hull–raft assumptions in existing approaches by analyzing the coupling effects of hull deformation and raft elastic deformation on air spring isolator deformation. It precisely describes the dynamic relationships between shaft alignment response and structural deformations, establishing a theoretical foundation for multi-objective control under coupled hull–raft deformation scenarios.

(2)

A multi-objective control algorithm is proposed for coupled hull–raft deformation compensation. This algorithm constructs an optimization model that integrates precise shaft alignment adjustment, hull deformation compensation, and raft deformation suppression based on coupled hull–raft deformation response characteristics. An N-step receding horizon optimal control framework with NSGA-II and hybrid coding was developed to overcome the challenges of unknown control sequence length and high-dimensional variables, with experimental validation demonstrating the effectiveness of the approach.

This paper is organized as follows. Section 1 reviews hull/raft deformation monitoring and control research. Section 2 establishes a control response model under coupled hull–raft deformation. Section 3 develops a multi-objective optimization model integrating shaft alignment adjustment, hull deformation compensation, and raft deformation suppression, with dynamic control implemented through NSGA-II. Section 4 validates the algorithm’s effectiveness through hull compensation and raft suppression tests on independent platforms. Section 5 consolidates key contributions and outlines prospective refinements for the control framework.

2. Shaft Alignment Response Model Under Coupled Hull–Raft Deformation

2.1. Existing Shaft Alignment Response Model with Rigid-Body Hull–Raft Assumptions

The schematic force diagram in Figure 2 represents an abstract mechanical representation of the physical system shown in Figure 1, where $F$ is the external force vector exerted on FR-ASMS, $x_{\mathrm{c}}$ describes the displacement vector at the centroid of the FR-ASMS (representing the shaft alignment attitude) which includes translational and angular terms, $K_i$ signifies the stiffness matrix of the #i air spring isolator, and $G_i$ represents the displacement transformation matrix of the #i air spring isolator. As depicted in Figure 2, the equations of motion for FR-ASMS can be expressed as Equation (1) under the assumption of rigid-body hull–raft conditions where structural deformations are neglected [68].

$$M{\ddot{x}}_{\mathrm{c}}+{\displaystyle \sum _{i=1}^{n_k}{G}_i^T{K}_i{G}_i{x}_{\mathrm{c}}}=F$$

(1)

where $M$ denotes the mass matrix of the FR-ASMS, and $n_k$ is the total number of air spring isolators. The displacement vector $x_{\mathrm{c}}$ can be written as:

$$x_{\mathrm{c}}=\left[x_c,y_c,z_c,\alpha ,\beta ,\chi \right]$$

(2)

where $x_c$, $y_c$ and $z_c$ are translational displacements of the center-of-gravity, and $\alpha$, $\beta$ and $\chi$ are angular displacements. In the field of shaft alignment research, this study adopts the terminology system of reference [33]: $x_c$ is the horizontal offset, $z_c$ is the vertical offset, $\alpha$ is the vertical angularity, and $\chi$ is the horizontal angularity. Additionally, $\beta$ represents self-rotation.

The inflation/deflation control process of air spring isolators can be considered quasi-static. Neglecting second-order derivative terms, Equation (1) simplifies to:

$${\displaystyle \sum _{i=1}^{n_k}{G}_i^T{K}_i{G}_i{x}_{\mathrm{c}}}=F$$

(3)

Based on the initial centroid displacement vector $x_{\mathrm{c}0}$, the inflation/deflation control process of the air spring isolator can be equivalently modeled as the regulation of an applied equivalent force $\Delta F_i$ acting along the direction of the primary support force. The relationship between this equivalent force and the resultant centroid displacement change $\Delta x_{\mathrm{c}}$ is characterized by Equation (4) [68]. Iterative air pressure adjustments drive the centroid displacement vector $x_{\mathrm{c}}$ to converge within the target domain, achieving precise shaft alignment control.

$$\left\{\begin{array}{l}{\displaystyle \sum _{i=1}^{n_k}{G}_i^T{K}_i{G}_i\Delta x_{\mathrm{c}}}=\Delta F_i=G_i^T{[0,0,\Delta F_i,0,0,0]}^T\\ \Delta F_i=\Delta p_i(1-{\eta }_1\Delta r_i)S_{e0i}·\mathrm{sgn}\Delta r_i\\ \Delta p_i={\displaystyle \frac{0.1q_{z}t(P_i+0.1)}{(P_i+0.1-{\eta }_2\Delta r_i)V_{0i}}}·\mathrm{sgn}\Delta r_i\end{array}\right.$$

(4)

where subscript “i” represents the serial number of the air spring isolator undergoing inflation/deflation control, $q_z$ denotes the airflow rate within the pneumatic line satisfying supersonic flow conditions, $t$ is the duration of the inflation/deflation control cycle, $P_i$ indicates the air pressure of the i-th air spring isolator, $\Delta r_i$ signifies the deformation magnitude along the primary support direction for the i-th air spring isolator, ${\eta }_1$ and ${\eta }_2$ are empirically derived influence coefficients capturing the effect of deformation on the effective area and internal volume, $S_{e0i}$ and $V_{0i}$ correspond to the effective area and internal volume of the i-th air spring isolator at its nominal operating height, and $\mathrm{sgn}\Delta r_i$ represents the sign function which adopts the value +1 during inflation control and −1 during deflation control.

Crucially, the stiffness matrix $K_i$ requires real-time updating throughout the analysis. The nonlinear dynamic characteristics of air spring isolators primarily affect the vibration response analysis of the system, while having limited impact on the predictive calculations pertinent to this study’s focus—quasi-static multi-objective cooperative control, including shaft alignment, hull deformation, and raft frame deformation. Moreover, due to the current lack of a unified and accurate stiffness model capable of comprehensively characterizing their frequency- and amplitude-dependent properties, this research adopts the classical linear pressure-stiffness relationship model for air springs [79]. This model ensures both engineering feasibility and physical interpretability, while providing reference predictions that meet the accuracy requirements for closed-loop control. To further mitigate errors introduced by model simplification, a reinforcement learning-based prediction discount factor $\gamma$ is introduced in Section 3.2.2, which attenuates long-term prediction errors through a discount mechanism, supplemented by an N-step receding horizon optimization control framework to reduce error accumulation. Thereby, the multi-objective control framework maintains stability while achieving significantly improved control accuracy and robustness.

Meanwhile, the centroid displacement vector $x_{\mathrm{c}}$ is obtained by online acquisition of displacement data using displacement sensors in practical engineering applications, followed by real-time calculation based on rigid-body attitude monitoring methods [42]. This approach thereby eliminates potential errors associated with iterative computation.

2.2. Model for Shaft Alignment Response Under Coupled Hull–Raft Deformation

2.2.1. Shaft Alignment Response Model with Hull Deformation Effect

When hull deformation is considered, the mechanical representation based on a rigid hull foundation in Figure 2 should be revised to an elastic foundation model, as shown in Figure 3. The correspondence between Figure 3 and Figure 1 is similar to that between Figure 2 and Figure 1, with the key distinction that Figure 3 accounts for the influence of hull deformation. As illustrated in Figure 3, deformation of the hull foundation induces corresponding displacements $x_i^{Hull}$ at the lower endpoints of the air spring isolators, which connect to the hull foundation. For clarity in subsequent analysis, the direction of hull deformation is defined as follows: upward convex deformation is assigned a positive value, while downward concave deformation is assigned a negative value.

When positive deformation occurs at the hull foundation beneath the air spring isolator connection point, the isolator experiences compressive shortening. The total isolator deformation ${\delta }_i$ comprises two superimposed components:

• Hull deformation displacement $x_i^{Hull}$;
• Raft pose transformation displacement $G_i{x}_{\mathrm{c}}$, generated during the raft’s alignment adjustment.

This relationship satisfies as follows:

$${\delta }_i=G_i{x}_{\mathrm{c}}+x_i^{Hull}$$

(5)

Therefore, the equations of motion for the FR-ASMS are established as follows:

$${\displaystyle \sum _{i=1}^{n_k}{G}_i^T{K}_i(G_i{x}_{\mathrm{c}}+x_i^{Hull})}=F$$

(6)

It should be noted that it is difficult to obtain precise hull deformation measurements at the mounting position of every air spring isolator in engineering practice. To address this limitation, paired laser displacement sensors detect real-time shaft alignment offsets $x_{\mathrm{c}}^{Hull}$ induced by hull deformation [13]. Consequently, the influence quantity $x_i^{Hull}$ exerted by hull deformation on each air spring isolator is determined through displacement transformation relationships, formulated as:

$$x_i^{Hull}=G_i{x}_{\mathrm{c}}^{Hull}$$

(7)

where $x_{\mathrm{c}}^{Hull}={[\Delta x,\Delta y,\Delta z,\Delta \theta ,\Delta \varphi ,\Delta \phi ]}^T$ ($\Delta y$ does not contribute to alignment evaluation and can be ignored [80]) can be calculated as [13]:

$$\left\{\begin{array}{l}\Delta x=\Delta x_B+b\left({\alpha }_2-{\alpha }_1\right)\\ \Delta z=\Delta z_B-b\left({\beta }_2-{\beta }_1 \right)\\ \Delta \theta ={\displaystyle \frac{\Delta x_B-\Delta x_A}{b}}\\ \Delta \phi ={\displaystyle \frac{\Delta z_A-\Delta z_B}{b}}\end{array}\right.$$

(8)

where parameters ${\alpha }_1$ and ${\alpha }_2$ represent the horizontal angularities of the FR-ASMS before and after hull deformation, ${\beta }_1$ and ${\beta }_2$ denote the vertical angularities under corresponding conditions. These angular measurements are obtained through real-time monitoring using eddy-current displacement sensors. Here, numerical 1 and 2 denote pre-deformation and post-deformation states, respectively. The distance $b$ is the center-to-center distance between laser displacement sensors mounted on the drive shaft and the MPU’s output shaft. Shaft alignment variation components $\Delta x_A$, $\Delta z_A$, $\Delta x_B$ and $\Delta x_B$ are defined as follows [13]:

$$\begin{array}{l}\left[\begin{array}{c}\Delta x_A\\ \Delta z_A\end{array}\right]=\left[\begin{array}{cc}C{\omega }_y^{B2}& S{\omega }_y^{B2}\\ -S{\omega }_y^{B2}& C{\omega }_y^{B2}\end{array}\right]\left[\begin{array}{c}{X}_{A2}\\ Z_{A2}\end{array}\right]-\left[\begin{array}{cc}C{\omega }_y^{B1}& S{\omega }_y^{B1}\\ -S{\omega }_y^{B1}& C{\omega }_y^{B1}\end{array}\right]\left[\begin{array}{c}{X}_{A1}\\ Z_{A1}\end{array}\right]\\ \left[\begin{array}{c}\Delta x_B\\ \Delta z_B\end{array}\right]=\left[\begin{array}{cc}-C{\omega }_y^{A2}& -S{\omega }_y^{A2}\\ S{\omega }_y^{A2}& -C{\omega }_y^{A2}\end{array}\right]\left[\begin{array}{c}{X}_{B2}\\ Z_{B2}\end{array}\right]-\left[\begin{array}{cc}-C{\omega }_y^{A1}& -S{\omega }_y^{A1}\\ S{\omega }_y^{A1}& -C{\omega }_y^{A1}\end{array}\right]\left[\begin{array}{c}{X}_{B1}\\ Z_{B1}\end{array}\right]\end{array}$$

(9)

where ${\omega }_y^{A1}$, and ${\omega }_y^{A2}$. denote the self-rotational angles of the MPU’s output shaft before and after hull deformation, ${\omega }_y^{B1}$ and ${\omega }_y^{B2}$ represent the corresponding angles for the drive shaft. The coordinates $(X_{A1},Z_{A1})$ and $(X_{A2},Z_{A2})$ correspond to laser displacement sensor image points on the MPU’s output shaft before and after hull deformation, with $(X_{B1},Z_{B1})$ and $(X_{B2},Z_{B2})$ similarly defined for the drive shaft. All parameters are acquired by laser displacement sensors mounted at shaft ends.

When the hull foundation is considered a rigid-body structure (i.e., deformation effects are neglected), the displacement at the lower endpoint of air spring isolators becomes $x_i^{Hull}=0$. Consequently, the shaft alignment response model degenerates to the conventional rigid-body model.

2.2.2. Shaft Alignment Response Model with Raft Deformation Effect

Implementing elastic support boundary constraints fundamentally modifies stiffness parameters for key degrees of freedom [81]. For FR-ASMS, this requires superimposing the elastic support stiffness matrix $K_{spring}$ onto the inherent structural stiffness matrix $K_{plate}$ of the raft structure. The resulting response equation is characterized by:

$$(K_{plate}+K_{spring})x=F$$

(10)

where $x$ is the nodal displacement vector and $F$ is external load vector.

However, Equation (10) exhibits significantly increased computational complexity due to its high degrees of freedom, which cannot meet real-time prediction and control requirements. To enhance computational efficiency, model simplification is imperative. As shown in Figure 4, deformation of the raft induces displacements $x_{\mathrm{i}}^S$ at the upper endpoints of air spring isolators, which connect to the floating raft. The raft deformation direction convention is as follows: locally convex upward deformation is positive, while concave downward deformation is negative.

When positive deformation occurs at the upper connection points of air spring isolators on the floating raft, isolators experience tensile elongation that increases their height. The total isolator deformation ${\delta }_i$ comprises two superimposed components:

• Raft elastic deformation component $x_i^S$, directly induced by the raft deformation;
• Rigid-body displacement component $G_i{x}_{\mathrm{c}}$, generated during the raft’s alignment adjustment.

This relationship satisfies:

$${\delta }_i=G_i{x}_{\mathrm{c}}-x_i^S$$

(11)

Therefore, the equations of motion for the FR-ASMS are established as follows:

$${\displaystyle \sum _{i=1}^{n_k}{G}_i^T{K}_i(G_i{x}_{\mathrm{c}}-x_i^S)}=F$$

(12)

where the structural displacement $x_{total}=G_i{x}_{\mathrm{c}}-x_i^S$ can be calculated as follows [10]:

$$x_{total}={\Phi }_{n_k\times n}{\left({\left[\begin{array}{c}{w}_{\epsilon }{\Psi }_{M\times n}\\ w_d{\Phi }_{K\times n}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}{\Psi }_{M\times n}\\ {\Phi }_{K\times n}\end{array}\right]\right)}^{-1}{\left[\begin{array}{c}{w}_{\epsilon }{\Psi }_{M\times n}\\ w_d{\Phi }_{K\times n}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}{S}_{M\times 1}\\ W_{K\times 1}\end{array}\right]$$

(13)

where ${\Phi }_{n_k\times n}$ denotes the displacement mode matrix at are spring isolator locations, $S_{M\times 1}$ represents strain sensor measurements, ${\Psi }_{M\times n}$ corresponds to the strain mode matrix, $M$ is the number of strain sensors, and $n$ signifies the modal order, $W_{K\times 1}$ indicates displacement sensor measurements, ${\Phi }_{K\times n}$ corresponds to the displacement mode matrix, and $K$ represents the number of limited displacement sensors.

It should be emphasized that the centroid displacement vector $x_{\mathrm{c}}$ and elastic deformation components $x_i^S$ cannot be directly separated from the total displacement $x_{total}$. Based on least-squares fitting algorithm [82,83,84], the centroid displacement vector $x_{\mathrm{c}}$ can be calculated as follows:

$$x_{\mathrm{c}}={(G^{T}G)}^{-1}{G}^{T}B$$

(14)

where matrix $B$ represents tri-axial displacement components at discrete points (i.e., x-, y-, and z-directional displacements of $x_{total}$), and matrix $G={\left[G_1^T,\cdots ,G_i^T,\cdots \right]}^T$ denotes the transformation matrix composed of initial position coordinates, specifically:

$$G=\left[\begin{array}{cccccc}1& 0& 0& 0& a_1^z& -a_1^y\\ 0& 1& 0& -a_1^z& 0& a_1^x\\ 0& 0& 1& a_1^y& -a_1^x& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 1& 0& 0& 0& a_i^z& -a_i^y\\ 0& 1& 0& -a_i^z& 0& a_i^x\\ 0& 0& 1& a_i^y& -a_i^x& 0\end{array}\right]$$

(15)

where $(a_i^x,a_i^y,a_i^z)$ denotes the initial coordinates of point #i.

Therefore, the elastic deformation component $x_i^S$ can be acquired through a vector decoupling operation between the calculated rigid-body displacement component $G_i{x}_{\mathrm{c}}$ and the total structural displacement $x_{total}$, specifically:

$$x_i^S=G_i{x}_{\mathrm{c}}-x_{total}$$

(16)

2.2.3. Shaft Alignment Response Model with Coupled Hull–Raft Deformation

Building upon the independent analyses of hull and raft deformations, two critical displacement effects emerge:

• Hull deformation induces a reference offset $G_i{x}_{\mathrm{c}}^{Hull}$ at the lower isolator connection points;
• Conversely, raft deformation generates displacement deviations $x_i^S$ from the rigid-body plane at upper connection points.

Consequently, under simultaneous deformation conditions as illustrated in Figure 5, the total isolator deformation ${\delta }_i$ collectively constitutes three coupled components:

$${\delta }_i=G_i{x}_{\mathrm{c}}+G_i{x}_{\mathrm{c}}^{Hull}-x_i^S$$

(17)

Based on the air spring isolator stiffness matrix $K_i$ and the system’s geometric configuration $G_i$, the equations of motion for the FR-ASMS are established as follows:

$${\displaystyle \sum _{i=1}^{n_k}{G}_i^T{K}_i(G_i{x}_{\mathrm{c}}+G_i{x}_{\mathrm{c}}^{Hull}-x_i^S)}=F$$

(18)

3. Multi-Objective Control Algorithm Considering Coupled Hull–Raft Deformation

3.1. Formulation of the Multi-Objective Optimization Model

3.1.1. Multi-Objective Optimization Model

When accounting for the influence of coupled hull–raft deformation, the construction of the multi-objective control optimization model must simultaneously satisfy three core functionalities:

(1)

Precise adjustment of alignment components (specifically shaft alignment attitude control),

(2)

Dynamic compensation for hull deformation,

(3)

Autonomous suppression of raft deformation.

Consequently, establishing this multi-objective control optimization model necessitates meeting two key requirements:

(1)

Optimizing the control response model by updating the state transition function based on the shaft alignment response model under coupled hull–raft deformation,

(2)

Introducing a suppression of raft deformation as an additional control objective.

Therefore, this multi-objective optimization problem is formalized as follows: identify the optimal set of control strategies $U=\left(\{t_1,\Delta p_{t_1}\},\{t_2,\Delta p_{t_2}\}\cdots ,\{t_j,\Delta p_{t_j}\}\cdots \right)$, such that the control action $\{t_j,\Delta p_{t_j}\}$ applied at each control step #j satisfies all constraints and optimization objectives specified in Equation (19).

$$\begin{array}{c}\begin{array}{cc}\underset{\{t_j,\Delta p_{t_j}\}}{\min }& \left\{\begin{array}{l}{J}_1={(x_c-\Delta x)}^2\\ J_2={(z_c-\Delta z)}^2\\ J_3={(\alpha -\Delta \theta )}^2\\ J_4={(\beta -\Delta \phi )}^2\\ J_5={(\chi -({\omega }_y^{A2}-{\omega }_y^{A1}))}^2\\ J_6={\displaystyle \sum {(x_i^{Sz})}^2}\\ J_7={\displaystyle \sum {(\Delta p_{t_j})}^2}\end{array}\right.\end{array}\\ \begin{array}{cc}s.t.& \left\{\begin{array}{l}{g}_1:P_{\min }\le P_i\le P_{\max }\\ g_2:{\displaystyle \sum _{i=1}^{n_k}{G}_i^T{K}_i(G_i{x}_{\mathrm{c}}+G_i{x}_{\mathrm{c}}^{Hull}-x_i^S)}=F\\ g_3:{‖\Delta P‖}_0=1\end{array}\right.\end{array}\end{array}$$

(19)

where $t_j$ denotes the sequential operation index for air spring inflation/deflation cycles and $\Delta p_{t_j}$ represents the inflation/deflation control input magnitude. The state variables, decision variables, objective function, and constraints in Equation (19) are described as follows.

3.1.2. State Variables

The state variables comprise two categories: directly measured and computationally derived quantities, with their formal definitions detailed in

Table 1. Directly measured state variables.

Variable Description			Symbol
air spring pressure vector			$P={[p_1,p_2,\cdots ,p_{n_k}]}^T$
discrete measurements
	x-axial displacement		$U_{L\times 1}={[u_1,u_2,\cdots ,u_L]}^T$
	z-axial displacement		$W_{K\times 1}={[w_1,w_2,\cdots ,w_K]}^T$
	y-axial normal strain		$S_{M\times 1}={[{\epsilon }_{y_1},{\epsilon }_{y_2},\cdots ,{\epsilon }_{y_M}]}^T$
laser displacement sensor coordinates
	pre-deformation
		MPU’s output shaft	$(X_{A1},Z_{A1})$
		drive shaft	$(X_{B1},Z_{B1})$
	post-deformation
		MPU’s output shaft	$(X_{A2},Z_{A2})$
		drive shaft	$(X_{B2},Z_{B2})$
laser displacement sensor self-rotations
	pre-deformation
		MPU’s output shaft	${\omega }_y^{A1}$
		drive shaft	${\omega }_y^{B1}$
	post-deformation
		MPU’s output shaft	${\omega }_y^{A2}$
		drive shaft	${\omega }_y^{B2}$

and

Table 2. Computed state variables.

Variable Description			Symbol
raft misalignment
	pre-deformation
		horizontal angularity	${\alpha }_1$
		vertical angularity	${\beta }_1$
	post-deformation
		horizontal angularity	${\alpha }_2$
		vertical angularity	${\beta }_2$
raft’s centroid displacement vector			$x_{\mathrm{c}}$
shaft alignment variation by hull deformation			$x_{\mathrm{c}}^{Hull}$
elastic deformation of raft			$x_i^S$

, respectively.

3.1.3. Decision Variables

Given the physical constraints of the ASMS’s actuators and system stability requirements, only a single air spring isolator is permitted to perform inflation/deflation operations during each control step. Consequently, two decision variables must be determined per control cycle #j:

(1)

Discrete decision variable: air spring isolator identification number $t_j(1,\cdots ,n_k)$;

(2)

Continuous decision variable: pressure adjustment magnitude $\Delta p_{t_j}$.

3.1.4. Objective Function

(1)

Control Objectives $J_1$–$J_2$: shaft alignment assurance

To satisfy the precision requirements for shaft alignment positioning, horizontal offset $x_c$ and vertical offset $z_c$ must be maintained within threshold limits:

• Horizontal offset: $J_1=\min {(x_c-\Delta x)}^2$
• Vertical offset: $J_2=\min {(z_c-\Delta z)}^2$

(2)

Control Objectives $J_3$–$J_5$: overturning moment suppression

Constraining shaft-induced overturning moments requires adherence to attitude angle tolerances:

• Horizontal angularity: $J_3=\min {(\alpha -\Delta \theta )}^2$
• Vertical angularity: $J_4=\min {(\beta -\Delta \phi )}^2$
• Self-rotation: $J_5=\min {(\chi -({\omega }_y^{A2}-{\omega }_y^{A1}))}^2$

(3)

Control Objective $J_6$: raft deformation suppression

Given that air spring load control can only regulate localized deformation near mounting positions, this objective minimizes the squared deformation sum at all air spring installation points: $J_6=\min {\displaystyle \sum {(x_i^{Sz})}^2}$.

(4)

Control Objective $J_7$: pneumatic efficiency optimization

To enhance system endurance, this objective minimizes the total squared pressure adjustment magnitude per control action: $J_7=\min {\displaystyle \sum {(\Delta p_{t_j})}^2}$.

It should be noted that control objectives $J_1$–$J_4$ represent critical alignment performance thresholds requiring strict satisfaction, designated as hard control objectives. Conversely, objectives $J_5$–$J_7$ constitute performance optimization targets aimed at enhancing operational efficiency and system dynamics, classified as satisfactory control objectives.

Furthermore, distinct dynamic response characteristics emerge from specific structural configurations of FR-ASMS. Consequently, beyond the foundational control objectives established herein, targeted supplementary control objectives may be incorporated. For instance, it can effectively suppress raft bending deformation by implementing vertical displacement control objectives in slender FR-ASMS.

3.1.5. Constraints

(1)

Constraint $g_1$: pressure boundary constraint

Post-adjustment air spring isolator pressures must satisfy operational limits: $P_{\min }\le P_i\le P_{\max },\forall i\in \{1,2,\cdots ,n_k\}$.

(2)

Constraint $g_2$: dynamic state constraint

System state transitions must comply with the governing equation as Equation (18).

(3)

Constraint $g_3$: single-air spring isolator activation constraint

Only one air spring isolator may be activated per control step, enforced through the L0-norm of the pressure adjustment vector: ${‖\Delta P‖}_0=1$.

3.2. Multi-Objective Model Solving and Algorithm Design

Equation (18) reveals a fundamental distinction in the influence mechanisms of hull deformation and raft deformation on the FR-ASMS.

(1)

Hull deformation. Hull deformation is essentially induced by external environmental loads, such as changes in diving depth. Its disturbance to the system manifests as imposing a static offset onto the desired alignment attitude. This offset is primarily determined by navigation condition parameters (e.g., diving depth) and is independent of the internal pressure state of the air spring isolators. Within a single control cycle, it can be treated as a constant disturbance. Given its nature as an external and static disturbance, the proposed control method retains applicability through multi-objective satisfactory optimization [68]. The core concept involves modifying the alignment control objective to counteract the offset induced by hull deformation, effectively implementing compensation control for the offset. Experiments documented in [75] have systematically validated the effectiveness and applicability of this approach.

(2)

Raft deformation. Given a defined mass distribution of raft-mounted equipment, the raft deformation is primarily regulated by actively adjusting the load distribution of air spring isolators. This deformation exhibits the following characteristics.

First, pressure regulation of a single air spring isolator primarily induces significant localized deformation in the raft region corresponding to its load-bearing area. This effect exhibits spatial attenuation, meaning the deformation increment becomes negligible in areas distant from the regulated point.

Second, the vertical stiffness of the FR-ASMS is significantly lower than its lateral and longitudinal stiffness. Furthermore, pressure control mainly adjusts the vertical load-bearing of the air spring isolators. Consequently, the raft deformation profile can be reasonably simplified to primarily consist of the normal deformation component perpendicular to its installation base plane.

Third, to precisely induce the required deformation profile, real-time monitoring of the raft deformation state is necessary with feedback used to guide the optimized adjustment of air spring isolator pressures.

Based on these characteristics, controlling raft deformation necessitates establishing a dynamic closed-loop system incorporating real-time deformation feedback. The control law must account for the direct effect of pressure adjustments on local deformation and the weak inter-coupling between individual air spring isolator regulations. It must also actively shape the target raft deformation profile by optimizing the air spring isolator pressure distribution (hence the load distribution) to compensate for system transmission errors.

3.2.1. Prediction of Raft Centroid Displacement and Elastic Deformation

To achieve real-time shaft attitude control and raft deformation compensation in closed-loop systems, a rapid prediction method for both centroid displacement (alignment attitude) and raft deformation must be established. This method supports the solution of optimal control sequences in multi-objective optimization by pre-evaluating system responses to given control actions.

(1)

Prediction of Raft Centroid Displacement

The raft centroid displacement is predicted according to the following procedure. An integrated strategy combining numerical simulation and experimental calibration is employed to establish the sensitivity matrix $A_{(L+K)\times n_k}$ for air spring isolator pressure-displacement responses at the displacement monitoring points. Calibrated with experimental measurements, this matrix serves as a surrogate model in multi-objective optimization to predict displacement responses under varying pressure distributions. This predicted displacement is then processed using the least squares method to determine the FR-ASMS’s centroid displacement [42]. The sensitivity matrix is defined as:

$$A_{(L+K)\times n_k}=\left[\begin{array}{ccc}{\displaystyle \frac{\partial u_1}{\partial p_1}}& \dots & {\displaystyle \frac{\partial u_1}{\partial p_{n_k}}}\\ ⋮& ⋮& ⋮\\ {\displaystyle \frac{\partial u_L}{\partial p_1}}& {\displaystyle \frac{\partial u_L}{\partial p_j}}& {\displaystyle \frac{\partial u_L}{\partial p_{n_k}}}\\ {\displaystyle \frac{\partial w_{L+1}}{\partial p_1}}& {\displaystyle \frac{\partial w_{L+1}}{\partial p_j}}& {\displaystyle \frac{\partial w_{L+1}}{\partial p_{n_k}}}\\ ⋮& ⋮& ⋮\\ {\displaystyle \frac{\partial w_{L+K}}{\partial p_1}}& \dots & {\displaystyle \frac{\partial w_{L+K}}{\partial p_{n_k}}}\end{array}\right]$$

(20)

where $u_i$ denotes the measured displacement of the i-th horizontal displacement sensor, $w_{L+j}$ represents the measured displacement of the j-th vertical displacement sensor, $n_k$ is defined as the total number of air spring isolators (including both horizontal and vertical isolators).

Therefore, based on the air spring isolator pressure distribution $P_{n_k\times 1}$, the displacement responses $X_{L+K}$ corresponding to each horizontal and vertical displacement sensor can be computed according to Equation (21).

$$X_{L+K}=A_{(L+K)\times n_k}{P}_{n_k\times 1}+X_{L+K}^{(0)}$$

(21)

where $X_{L+K}={[u_1,\cdots ,u_L,w_{L+1},\cdots ,w_{L+K}]}^T$ denotes the displacement response vector, $X_{L+K}^{(0)}$ represents the initial displacement vector of all sensors under zero-pressure conditions.

Thus, the centroid displacement is decoupled and fitted using the least squares method as follows [80]:

$${\widehat{x}}_{\mathrm{c}}={\left(H_1^T{H}_1\right)}^{-1}{H}_1^T{X}_{L+K}$$

(22)

where $H_1$ is the transformation matrix constructed from the position coordinates of all sensors, specifically:

$$H_1=\left[\begin{array}{cccccc}0& 0& 1& a_1^y& -a_1^x& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 1& a_L^y& -a_L^x& 0\\ 1& 0& 0& 0& a_{L+1}^z& -a_{L+1}^y\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 1& 0& 0& 0& a_{L+K}^z& -a_{L+K}^y\end{array}\right]$$

(23)

Since $H_1^T{H}_1$ is singular, stable estimates of ${\widehat{x}}_c$ require regularization via pseudo-inverse or ridge regression techniques. Furthermore, as the motion component $y_0$ along the raft axis (y-direction) does not contribute to alignment evaluation [80], we construct the rank-reduced matrix $H_2$ by removing the second column from $H_1$. This matrix modification transforms Equation (22) into Equation (24).

$${\widehat{x}}_{c2}=H_2^{+}{X}_{L+K}$$

(24)

where ${\widehat{x}}_{c2}={\left[\begin{array}{ccccc}{x}_0& z_0& \alpha & \beta & \chi \end{array}\right]}^T$ denotes a dimension-reduced vector created by removing the second element from state vector $x_{\mathrm{c}}$.

Consequently, the predictive computational method for the RF-ASMS’s centroid displacement is given by Equation (25).

$${\widehat{x}}_{c2}=H_2^{+}\left(A_{(L+K)\times n_k}{P}_{n_k\times 1}+X_{L+K}^{(0)}\right)$$

(25)

(2)

Prediction of Raft Deformation

Therefore, the raft deformation at each vertical air spring isolator is determined based on the relationship in Equation (16), specifically:

$$x^{Sz}=\left[\begin{array}{c}\begin{array}{ccc}1& a_{V_1}^y& -a_{V_1}^x\end{array}\\ ⋮\\ \begin{array}{ccc}1& a_{V{n}_c}^y& -a_{V{n}_c}^x\end{array}\end{array}\right]\left[\begin{array}{c}{z}_0\\ \alpha \\ \beta \end{array}\right]-x_{n_c}^z$$

(26)

where $(a_{V_i}^x,a_{V_i}^y,a_{V_i}^z)$ denotes the initial spatial coordinates of the i-th vertical air spring isolator, $n_c$ represents the total number of vertical air spring isolators in the FR-ASMS, $x_{n_c}^z={\left[w_{V_1}^z,\cdots ,w_{V{n}_c}^z\right]}^T$ is the vertical displacement at the air spring isolator location. $x_{n_c}^z$ can be theoretically predicted using Equation (27).

$$x_{n_c}^z=B_{n_c\times n_k}{P}_{n_k\times 1}+x_{n_c}^{z0}$$

(27)

where $x_{total}^{z0}$ denotes the initial vertical displacement at each air spring isolator location under zero-pressure conditions, and $B_{n_c\times n_k}$ represents the sensitivity matrix relating air spring isolator internal pressure to vertical height response of vertical displacement at each air spring isolator location, which can be obtained via the computational method specified in Equation (20).

3.2.2. N-Step Receding Horizon Optimal Control Framework

While the multi-objective optimal control model presented in Section 3.1 comprehensively characterizes the performance specifications of the control trajectory, two critical constraints emerge in practical implementation:

(1)

The total length of the control trajectory is unknown a priori, and

(2)

The high dimensionality of control variables along the entire trajectory leads to computationally intensive optimization.

To address these limitations, this subsection establishes an N-step receding horizon optimization control framework structured around four core mechanisms:

(1)

Trajectory Segmentation. The complete control sequence $U=\left(u^1,\cdots ,u^k,\cdots ,u^s\right)$ (total steps s) is partitioned into consecutive N-length segments. The control subsequence for each segment is defined as $u^k=\left(\{t_1^k,\Delta p_{t_1^k}\},\{t_2^k,\Delta p_{t_2^k}\}\cdots ,\{t_N^k,\Delta p_{t_N^k}\}\right)$, where k denotes the current rolling window index.

(2)

Dynamic Error Compensation. To mitigate un-modeled dynamics, system states are updated in real-time following each control execution using acquired operational data from onboard sensors, ensuring accurate state propagation.

(3)

Limited-Horizon Prediction Handling. Performance assessment using each N-step subsequence remains confined to finite prediction horizons, thus only the initial control input $\{t_1^k,\Delta p_{t_1^k}\}$ is implemented in control law decisions.

(4)

Reinforcement-Learning Inspired Optimization. A finite-horizon cumulative reward function incorporating a discount factor $\gamma$ is formulated. Given that near-term responses may be influenced by model prediction inaccuracies, $\gamma$ is set to 0.3 to attenuate disturbances from propagated errors over the prediction horizon.

Therefore, Equation (28) provides the complete description of the N-step receding horizon optimization control framework.

$$\begin{array}{c}\begin{array}{cc}\underset{u^k}{\min }& \left\{\begin{array}{l}{J}_1={\displaystyle \sum _{j=1}^N{\gamma }^{j-1}{(x_{cj}-\Delta x)}^2}\\ J_2={\displaystyle \sum _{j=1}^N{\gamma }^{j-1}{(z_{cj}-\Delta z)}^2}\\ J_3={\displaystyle \sum _{j=1}^N{\gamma }^{j-1}{({\alpha }_j-\Delta \theta )}^2}\\ J_4={\displaystyle \sum _{j=1}^N{\gamma }^{j-1}{({\beta }_j-\Delta \phi )}^2}\\ J_5={\displaystyle \sum _{j=1}^N{\gamma }^{j-1}{({\chi }_j-({\omega }_y^{A2}-{\omega }_y^{A1}))}^2}\\ J_6={\displaystyle \sum _{j=1}^N\left({\gamma }^{j-1}{\displaystyle \sum _l^{n_k}{(x_l^{Sz})}^2}\right)}\\ J_7={\displaystyle \sum _{j=1}^N{\gamma }^{j-1}{(\Delta p_{t_j^k})}^2}\end{array}\right.\end{array}\\ \begin{array}{cc}s.t.& \left\{\begin{array}{l}{g}_1:P_{\min }\le P_i\le P_{\max }\\ g_2:x^{Sz}=f_1(A_{n_c\times n_k}{P}_{n_k\times 1}),x_c=f_2(B_{(L+K)\times n_k}{P}_{n_k\times 1})\\ g_3:{‖\Delta P‖}_0=1\end{array}\right.\end{array}\end{array}$$

(28)

3.2.3. Hybrid-Encoded Optimization Algorithm Design Based on NSGA-II

This work employs the NSGA-II algorithm to solve the N-Step receding horizon optimal control, implemented through the following procedure.

Step 1: Acquiring Current State. Directly measured state variables were obtained for all air spring isolator pressures, displacements, and strains at discrete points, and the self-rotations and coordinate values of imaging points from laser displacement sensors. These measurements were subsequently processed to derive the rigid-body displacement of the floating raft, hull deformation, and raft elastic deformation.

Step 2: Generating the Initial Population. A dual-mode encoding mechanism was employed to implement hybrid encoding for the discrete decision variables (air spring isolator identification numbers) and continuous decision variables (air pressure adjustment amounts). This process initialized the control sequence $u^k$ for subsequent optimization.

Step 3: Evaluating the Population. The multi-objective function values and constraint violations were calculated for each individual in the population. These results were then used to assess the performance and constraint satisfaction of the control sequences.

Step 4: Generating a New Population. A new population was generated through the following procedure.

First, non-dominated sorting and crowding distance calculations were performed on the current population. All solutions were then sorted based on their Pareto rank and crowding distance. Elite individuals were selected based on this ordering.

Second, parent individuals were chosen using a binary tournament selection method incorporating rank and crowding distance criteria. These selected parents underwent crossover and mutation operations to produce an offspring population. To ensure the uniqueness of air spring isolator identification numbers during crossover and mutation within the air spring isolator identification segment, tournament selection was specifically applied to this discrete variable. Additionally, air spring isolators were randomly switched with a probability $p_d$. For the continuous air pressure adjustments, a Gaussian perturbation was employed for mutation: $\Delta p_{t_j}~\aleph (0,{\sigma }^2)$ where the variance $\sigma$ was adaptively decreased over time according to $\sigma ={\sigma }_0{e}^{-t/T}$.

Third, the parent and offspring populations were merged. The combined population was then evaluated using the procedure described in Step 3. Subsequently, non-dominated sorting and crowding distance calculations were applied to the merged population. Environmental selection was finally performed on this sorted, merged population to generate the new population for the next generation.

Step 5: Updating the Pareto Front and Checking Convergence. The Pareto front was updated with the current non-dominated solutions. The hypervolume $H{V}^{(k)}$ was calculated for the solution set in the objective space relative to a predefined reference point. Convergence was assessed by evaluating whether the condition $\left|H{V}^{(j)}-H{V}^{(j-1)}\right|/H{V}^{(j)}<{10}^{-3}$ was satisfied. If this convergence criterion was not met, the process iteratively executed Steps 3, 4, and 5 until convergence was achieved.

4. Experimental Verification

Due to limitations in the existing test platform’s capability to simultaneously simulate coupled hull–raft deformation, the experimental design encompassed two distinct tests:

(1)

A compensatory control test addressing hull deformation effects was conducted on an MPU-ASMS prototype,

(2)

An adaptive control test investigating raft deformation effects was performed on a SC-FR-ASMS prototype.

4.1. Compensatory Control Test for Hull Deformation

4.1.1. Test Rig of MPU-ASMS

As the hull deformation-induced shaft misalignment monitoring technique had been experimentally validated in Ref. [13], this study omitted redundant verification. As depicted in Figure 6, the hull deformation compensation control experiments were conducted on an MPU-ASMS prototype featuring a 10-ton MPU.

Twelve JYQN-type air springs were arranged symmetrically in a diagonal configuration between the frame and mounting base with about 9.8 kN rated load capacity. Displacement monitoring employed seven non-contact eddy-current sensors: three for horizontal and four for vertical measurements.

For deep sea environments, the characteristic environmental loads acting on the MPU-ASMS undergo significant transformation: the dominant dynamic loads in shallow sea environments (e.g., wave actions and cargo loads) are replaced by extreme static loads (e.g., substantial hydrostatic pressure). To address the pronounced hull deformation induced by hydrostatic pressure, the system’s control architecture has been evolutionarily optimized. A comparative schematic of structure and working principle of MPU-ASMS is presented in Figure 7.

Figure 7a illustrates the control workflow for dynamic load compensation in shallow sea environments. In contrast, Figure 7b demonstrates the augmented system architecture incorporating a Hull Deformation Monitoring Module to address hydrostatic pressure-induced hull deformation in deep sea environments. The real-time data from this module serves as feedforward signals to the Control Execution Module, enabling active compensation for shafting misalignment caused by hull deformation. Although the hardware design of the MPU-ASMS (e.g., high-load/long-stroke air spring isolators) requires corresponding modifications for deep-sea applications, its intelligent control system maintains performance in deep sea environments through integrated advanced sensing and real-time compensation mechanisms.

The integrated control system comprised three functionally distinct modules, as detailed below:

(1)

State Monitoring Module. This module continuously monitored real-time operational parameters of the MPU-ASMS. Key measurements included the internal pressure of each pneumatic isolator, displacements (both horizontal and vertical), inclination angles, and draft depth.

(2)

Hull Deformation Monitoring Module. Triggered either by changes in draft depth surpassing predefined thresholds or at scheduled intervals, this module activated the opposing laser displacement sensors. Upon activation, it acquired the spatial coordinates (X, ~, Z) of the designated imaging points and the corresponding self-rotation angles from each sensor.

(3)

Control Execution Module. This module processed the data streams from both the State Monitoring and Hull Deformation Monitoring Modules. Based on the assessed operating conditions and the current device status, it generated control commands. These commands dynamically regulated the air spring isolators by executing inflation or deflation adjustments through dedicated control valves.

Preliminary tests confirmed significant structural rigidity in the assembly, with raft deformation demonstrating negligible magnitudes. Physical constraints additionally prevented strain sensor installation, precluding direct structural deformation monitoring. Consequently, the control objective $J_6$ was excluded from this experimental campaign.

4.1.2. Test Conditions and Procedure

Four test conditions were investigated:

(1)

Case #1: +0.5 mm horizontal offset.

(2)

Case #2: −0.5 mm horizontal offset.

(3)

Case #3: +0.5 mm vertical offset.

(4)

Case #4: −0.5 mm vertical offset.

The experimental procedure comprised the following steps:

Step 1: Establish Initial Reference State. With the MPU-ASMS under automatic control, adjustments were made until the shaft alignment met specific criteria: horizontal and vertical offsets within ±0.3 mm, and angularity (horizontal and vertical) and self-rotation angles within ±0.5 mm/m. This state was maintained for ≥10 s without any air spring isolator inflation/deflation activity and recorded as the initial reference state.

Step 2: Apply Induced Offset. Automatic control of the MPU-ASMS was disabled. The specified offset change for the current test condition was manually applied as an input to the MPU-ASMS simulating hull deformation via the Hull Deformation Monitoring Module. This induced offset was held steady for 10–30 s to ensure data stability and recorded as the applied offset state.

Step 3: Engage Active Control & Record Final State. Automatic control of the MPU-ASMS was reactivated. The system then autonomously adjusted the MPU-ASMS position to re-achieve the target criteria: horizontal and vertical offsets within ±0.3 mm, and angularity (horizontal and vertical) and self-rotation angles within ±0.5 mm/m. This actively controlled state was maintained for ≥10 s without air spring isolator adjustment and recorded as the automatic control state.

Step 4: Reset for Next Condition. The offset applied in Step 2 was removed. The system state was returned to the initial reference state by repeating Step 1, preparing for the subsequent test condition.

4.1.3. Experimental Results and Analysis of MPU-ASMS

The experimental outcomes for the four test conditions are presented in Figure 8, Figure 9, Figure 10 and Figure 11.

Based on the experimental results presented above, the following core conclusions can be drawn:

(1)

Validation of control strategy effectiveness. The proposed control method demonstrated significant effectiveness under all four typical operating conditions. Both the horizontal and vertical offsets converged rapidly to within the preset threshold (±0.3 mm). Simultaneously, the horizontal and vertical angularities converged quickly within the tolerance range of ±0.5 mm/m, fully satisfying the stringent requirements of the hard control objectives $J_1$–$J_4$ in the multi-objective optimization model.

(2)

Validation of rotational stability. Although minor fluctuations in self-rotation were observed in Case #1 and Case #3, their magnitude consistently remained within the strict bounds of ±0.5 mm/m. This performance aligns with the design expectations for a satisfactory control objective $J_5$ in the multi-objective optimization model, confirming the system’s control precision in maintaining rotational stability.

(3)

The similarity between Figure 8 (Case #1, +0.5 mm horizontal offset) and Figure 10 (Case #3, +0.5 mm vertical offset) arises because both cases involve a positive offset perturbation. While the direction of the offset differs (horizontal vs. vertical), the nature of the perturbation (positive) is the same. Similarly, the similarity between Figure 9 (Case #2, −0.5 mm horizontal offset) and Figure 11 (Case #4, −0.5 mm vertical offset) stems from their shared negative offset perturbation. Conversely, the reverse correlation observed between the positive offset group (Figure 8 and Figure 10) and the negative offset group (Figure 9 and Figure 11) is a direct consequence of the antisymmetric input perturbations (+0.5 mm vs. −0.5 mm) applied to a linear system. In such systems, antisymmetric inputs naturally produce antisymmetric outputs, which manifests as the observed reverse correlation in the results.

Periodic, ineffective inflation/deflation actions (such as opposing actions applied sequentially to a single air spring isolator) were not observed during the tests. This indicates that all control commands directly served the optimization of the target parameters, thereby verifying the achievement of the satisfactory control objective $J_6$.

In summary, the control strategy demonstrated reliable dynamic response capabilities and effective multi-objective coordination performance under hull deformation conditions.

By analyzing the angularity characteristics in Figure 8b, Figure 9b, Figure 10b and Figure 11b, the following core conclusions can be drawn:

(1)

The horizontal and vertical angularities remained stable under actively applied horizontal/vertical offset.

(2)

The control approach inherently produces an additional tilt component during the automatic control state, because the control mechanism achieves shaft alignment correction through localized air spring pressure adjustments rather than lifting or lowering the entire raft structure. Consequently, the adjustment of the offsets is inevitably accompanied by dynamic changes in angularity, indicating a strong correlation between these parameters.

4.2. Adaptive Control Test for Raft Deformation

4.2.1. Test Rig of SC-FR-ASMS

To validate the control algorithm’s capability in suppressing raft deformation, an adaptive control experiment was conducted on an SC-FR-ASMS as shown in Figure 12.

This system comprised 17 vertical load-bearing air springs and 6 horizontal stabilizing air springs with a rated load capacity of approximately 9.8 kN. The total weight of the SC-FR-ASMS was approximately 18 tons. Floating raft equipment mass distribution comprises water-lubricated bearing (about 3.0 tons), thrust bearing (about 2.5 tons), propulsion motor (about 0.9 tons), and torque loading device (about 0.7 tons).

A strain monitoring network consisting of 34 high-precision fiber grating strain sensors was installed at the raft base, including 17 longitudinal sensors along each side (Y-axis). The displacement monitoring system integrated 7 vertical and 2 horizontal sensors. Vertical measurement points were distributed at three port-side locations, three starboard-side locations, and the central propulsion output shaft end. Horizontal sensors were installed on the starboard.

Given the slender configuration of the floating raft with a high aspect-ratio, this subsection introduces a hard control objective for vertical displacement differential between port and starboard sides to enhance attitude stability and deformation suppression capabilities. The constraint threshold is set at ±1.0 mm, implemented by adding the optimization objective $J_{8i}=\begin{array}{cc}\min & {(w_{L+i})}^2\end{array}(i=1,\cdots ,6)$ to Equation (28).

The alignment attitude tolerance thresholds remained unchanged from Section 4.1: horizontal and vertical offsets within ±0.3 mm, and angularity and rotation angles within ±0.5 mm/m.

4.2.2. Experimental Results and Analysis of SC-FR-ASMS

Figure 13 illustrates the dynamic response of hard constraint objectives (four shaft alignment parameters and six vertical displacements) and a satisfactory control objective (self-rotation) during the system’s transition from a low-pressure initial state (air spring pressure ≤ 0.2 MPa) to steady-state operational alignment.

Based on experimental results, the following key conclusions are drawn:

(1)

Validation of Control Strategy Effectiveness. Consistent with findings in Section 4.1.3, both alignment offsets and angularities rapidly converged within preset thresholds. Concurrently, the six vertical displacements were strictly constrained within ±1.0 mm. This fully satisfies all hard constraint objectives $J_1$–$J_4$ and the newly added vertical displacement hard constraint objectives $J_{81}$–$J_{86}$ in the multi-objective optimization model.

(2)

Verification of Torsional Stability. The self-rotation angle exhibited an initial peak of −1.2 mm/m, attributed to the mass imbalance from the starboard-mounted MPU. Through closed-loop control, it converged within the ±0.5 mm/m threshold. This meets the design expectations for satisfactory control objective $J_5$, confirming the torsional stability control capability of the SC-FR-ASMS.

(3)

Analysis of Offset Dynamics. A −9.3 mm initial vertical offset was observed under low-pressure conditions owing to gravitational settling. A transient overshoot (+0.6 mm) occurred during control, resulting from excessive air spring inflation rates. This deviation was corrected after closed-loop pressure feedback control adjustments. During active vertical offset correction, a transient horizontal offset exceeding the threshold was observed (peak value: −0.6 mm). This phenomenon originated from the multi-objective control priority arbitration mechanism: when vertical displacement deviation exceeded critical levels, the system concentrated actuator resources on vertical correction, resulting in temporary horizontal control degradation. Following activation of the closed-loop feedback switching strategy, horizontal offset was restored within the ±0.3 mm threshold range.

(4)

Angular Deviation Performance. Both horizontal and vertical angularity were maintained within ±0.2 mm/m throughout testing, 60% below the design thresholds proving the robust disturbance rejection capabilities of the SC-FR-ASMS against angular perturbations.

(5)

Discrepancy Analysis. The discrepancies observed in Figure 13a–c stem from the distinct physical quantities they represent. Figure 13a illustrates the offset displacement, calculated via Equation (24), which characterizes the global deviation of the device. Under the low-pressure conditions, the device undergoes subsidence, leading to a negative increase in vertical offset (consistent with the displacement trend of w1 in Figure 13c). Upon pressure adjustment, this value gradually converges toward zero. Figure 13b presents the angularities, defined as the ratio of relative displacements. Despite the overall reduction in vertical displacement under the low-pressure conditions, the vertical angularity remains minimal due to synchronous variation and the slender longitudinal configuration of the device. Meanwhile, the horizontal angularity exhibits negligible fluctuations, as the horizontal offset itself undergoes minor changes (aligned with the trend in Figure 13a). As discussed in point 2, the larger self-rotation observed in the low-pressure conditions has been analyzed. Figure 13c depicts the vertical displacement variations at key locations, reflecting local deformation characteristics.

Figure 14 illustrates the variation in satisfactory control objectives (raft deformation and pneumatic energy efficiency).

Based on experimental results, the following key conclusions are drawn:

(1)

Dynamic control characteristics of the raft deformation. Figure 14a demonstrates a progressive attenuation trend in the cumulative raft deformation. The observed transient increases during control originated from the multi-objective priority decision-making mechanism: to prioritize satisfaction of hard constraints (e.g., alignment tolerance objectives $J_1$–$J_4$ and vertical displacement tolerance objectives $J_{81}$–$J_{86}$), targeted inflation of specific air springs was required, inducing localized transient deformation. These perturbations were subsequently attenuated through closed-loop feedback control, thereby verifying the active deformation suppression capability of the SC-FR-ASMS.

(2)

Evolution of energy consumption in pneumatic control. Figure 14b shows continuous increase in the compressed air consumption throughout the control process. Comparative analysis of Figure 13a,c reveals two distinct control phases: 0–190 s is a vertical offset regulation phase, and 191–250 s is a vertical displacement regulation phase. Pneumatic energy consumption exhibits strong phase-dependent characteristics. During the vertical offset regulation phase, the energy consumption increases rapidly. This is attributed to the requirement for coordinated inflation across all air spring isolators to achieve the necessary overall lift of the raft for initial vertical offset compensation. In the subsequent vertical displacement regulation phase, the rate of increase in energy consumption slows significantly. This reduction is due to the localized pressure adjustments applied only to individual air spring isolators, which are necessary for correcting the local pose attitude of the raft.

5. Discussion

The experimental results demonstrate that the proposed multi-objective control algorithm delivers outstanding performance in both simulated hull deformation (MPU-ASMS prototype) and raft deformation (SC-FR-ASMS prototype) scenarios. The compensation and adaptive control strategies successfully regulate shaft alignment parameters (offset and angularity) within predetermined thresholds, confirming the solution’s effectiveness in addressing critical challenges of deep-sea vibration isolation systems.

For hull deformation compensation, the predictive capability of the Hull Deformation Monitoring Module enables proactive compensation before disturbance impacts system performance, which is fundamental for rapid convergence. The observed self-rotation fluctuations and coupled angular dynamics originate from the inherent multi-objective optimization characteristics of the system. Under constrained actuator (air spring) resources, the control architecture deliberately permits minor variations in soft-target parameters (e.g., self-rotation) to strictly maintain hard-constraint performance (offset tolerance).

For raft deformation adaptation, the multi-objective coordination mechanism dynamically allocates control resources, leading to temporary horizontal performance degradation during critical vertical deviation correction. Vertical offset overshoot results from pneumatic system nonlinearities (e.g., inflation rate) and controller parameter overshoot. Phase-dependent pneumatic energy consumption reflects intelligent strategy switching: rapid consumption during global elevation (all air springs active) versus slow consumption during local attitude adjustment (selective actuation).

Unlike shallow-water systems targeting wave loads [34], this work solves the new challenge of deep-sea hydrostatic-induced quasi-static deformation. The key innovation lies in the original integration of real-time hull deformation monitoring with feedforward compensation mechanisms, which effectively extends the operational boundaries of intelligent vibration isolation technology.

In contrast to existing fragmented approaches to shaft alignment attitude control, the integrated multi-objective optimization framework proposed in this study demonstrates superior comprehensive performance by simultaneously coordinating four typically conflicting objectives: (1) shaft alignment, (2) hull deformation compensation, (3) raft deformation suppression, and (4) energy efficiency optimization.

While the laboratory prototype demonstrates promising results, the linearized approximation of nonlinear response characteristics in the control model may contribute to the observed transient overshoot. Future investigations will implement adaptive robust control frameworks integrated with machine learning algorithms, which dynamically identify and update the sensitivity matrix based on real-time operational data, thereby significantly enhancing control accuracy under multi-pressure operating conditions.

6. Conclusions

This study proposes an intelligent alignment control technique addressing MPU-FR-ASMS under coupled hull–raft deformation to meet the stringent requirements of deep-diving and highly integrated deep-sea. The main contributions and findings are summarized as follows:

(1)

The deformation mechanism of hull–raft coupling was elucidated, and a control response model incorporating this effect was established. This model quantifies the deformation of air spring isolators as a superposition of three components: alignment transformation displacement, hull deformation, and raft deformation. It accurately describes the dynamic relationship between shaft alignment and these deformations.

(2)

A multi-objective optimization model was constructed to simultaneously optimize precise shaft alignment adjustment, hull deformation compensation, and raft deformation suppression. To overcome the challenges of unknown control sequence length and high-dimensional variables, an innovative N-step receding horizon optimal control framework was implemented. By integrating the NSGA-II algorithm with hybrid coding techniques, real-time prediction and optimization of control actions were achieved.

(3)

The effectiveness of the proposed control model was rigorously validated through independent tests on two distinct prototypes. In the test of the MPU-ASMS, all four critical alignment attitudes converged within specified thresholds, confirming the model’s efficacy in compensating hull deformation. In the test of the SC-FR-ASMS, all four alignment attitudes were similarly satisfied. Furthermore, six additional raft vertical displacement differentials met their thresholds, and overall raft deformation magnitude progressively decreased, demonstrating the model’s capability in raft deformation suppression. Collectively, these results validate the dual efficacy of the control model in both hull deformation compensation and raft deformation inhibition.

This study provides a key technological pathway for the perception and active control of hull–raft deformation in deep-sea submersible propulsion systems, laying a critical foundation for ensuring the operational safety, stability, and vibration control of large-depth vehicles. The established control framework demonstrates significant extensibility and can be adapted to other scenarios involving complex coupled deformations, such as ship propulsion systems and offshore platforms.

Notably, accumulated prediction errors stemming from the fixed sensitivity matrix were identified as a limitation. Future work will focus on integrating machine learning algorithms to dynamically identify the sensitivity matrix, thereby enhancing control accuracy under multi-pressure operating conditions.