Surrogate Model of Hydraulic Actuator for Active Motion Compensation Hydraulic Crane

Abstract

Offshore cranes equipped with active motion compensation (AMC) systems play a vital role in marine engineering tasks such as offshore wind turbine maintenance, subsea operations, and dynamic load positioning under wave-induced disturbances. These systems rely on complex hydraulic actuators whose strongly nonlinear dynamics—often described by differential-algebraic equations (DAEs)—impose significant computational burdens, particularly in real-time applications like hardware-in-the-loop (HIL) simulation, digital twins, and model predictive control. To address this bottleneck, we propose a neural network-based surrogate model that approximates the actuator dynamics with high accuracy and low computational cost. By approximately reducing the original DAE model, we obtain a lower-dimensional ordinary differential equations (ODEs) representation, which serves as the foundation for training. The surrogate model includes three hidden layers, demonstrating strong fitting capabilities for the highly nonlinear characteristics of hydraulic systems. Bayesian regularization is adopted to train the surrogate model, effectively preventing overfitting. Simulation experiments verify that the surrogate model reduces the solving time by 95.33%, and the absolute pressure errors for chambers $p_1$ and $p_2$ are controlled within 0.1001 MPa and 0.0093 MPa, respectively. This efficient and scalable surrogate modeling framework possesses significant potential for integrating high-fidelity hydraulic actuator models into real-time digital and control systems for offshore applications.

1. Introduction

Offshore hydraulic cranes equipped with active motion compensation (AMC) systems represent critical components in offshore engineering operations [1]. These systems effectively mitigate wave-induced disturbances and play pivotal roles in the construction and maintenance of large-scale marine facilities, including oil drilling platforms [2] and offshore wind turbines [3,4]. However, most of the existing equipment predominantly employs single-degree-of-freedom (SDOF) active heave compensation (AHC) technology [5]. The transition to AMC systems imposes heightened demands on modeling, simulation, and control strategies, particularly regarding the real-time simulation and control of multi-axis hydraulic systems. Concurrently, the presence of counterbalance valves in offshore hydraulic crane circuits [6,7] introduces complex dynamic characteristics [8] and transforms system equations into differential-algebraic equations (DAEs), thereby exacerbating computational burdens [9]. These factors collectively result in significant computational overhead for simulating hydraulic actuator circuits containing counterbalance valves, presenting challenges for real-time-critical applications, such as hardware-in-the-loop (HIL) simulation, digital twins, and model predictive control.

First-principle modeling approaches typically derive differential or algebraic equations directly from hydraulic component characteristics and circuit configurations [10,11,12]. Solving such equations often demands substantial computational resources, rendering them impractical for real-time applications. To address simulation efficiency, the existing research has employed model-order reduction and surrogate modeling techniques [13,14,15]. Chu et al. [13] adopted a Bond Graph (BG) methodology to characterize hydraulic system behaviors in offshore cranes through energy flow analysis of physical systems. A. G. Agúndez et al. [14] proposed a linearization method for hydraulic-driven multibody systems by linearizing motion equations and incorporating hydraulic subsystem equations.

Surrogate methods significantly enhance computational efficiency by establishing surrogate models that approximate the original problem. The commonly used approaches include response surface methodology (RSM) and Kriging [16]. RSM aims to fit input–output relationships through polynomial models, while Kriging employs Gaussian regression processes to predict responses at unknown points based on spatial correlation. Both methods can incorporate optimal parameters through Design of Experiments (DOE) techniques [17,18]. DOE is utilized to scientifically plan experimental schemes and efficiently analyze the effects of multiple factors on system outputs, with the objective of maximizing information acquisition while minimizing experimental runs. The Factorial Designs approach within DOE, encompassing full or fractional factorial designs, enables efficient screening of critical variables. Central Composite Design (CCD) and Box–Behnken Design (BBD) are primarily applied for parameter optimization in RSM. The accuracy, precision, and reliability of RSM models are typically evaluated using key metrics such as Mean Absolute Error (MAE), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), R-squared, and Adjusted R-squared [19].

Neural networks demonstrate superior potential for modeling strongly nonlinear systems, such as counterbalance-valve-integrated (CBV-integrated) hydraulic actuator circuits, with radial basis function (RBF) neural networks being commonly employed [20,21]. Wang et al. [20] compared three centrifugal pump surrogate models (response surface model, Kriging, and RBF neural network), revealing the superior predictive accuracy of RBF neural network. Qingtong et al. [21] developed an RBF-based surrogate model for water hydraulic high-speed on/off valves in optimization frameworks. However, pressure oscillations in CBV-integrated hydraulic circuits generate steep system gradients that challenge RBF methodologies [22]. This study proposes hyperbolic tangent (tanh) activation functions [23] to address this limitation. Network architecture optimization remains crucial. Single-hidden-layer networks exhibit insufficient nonlinear fitting capacity, while deep neural networks introduce unnecessary computational overhead. We therefore develop a three-hidden-layer tanh-based neural network surrogate model that balances enhanced nonlinear approximation capability with computational efficiency. In the field of hydraulics, Jose et al. [15] implemented piecewise linearization via response surface methodology to simplify excavator arm hydraulic systems. However, conventional model reduction methods exhibit system-specific limitations, while traditional surrogate approaches like response surface and Kriging methods demonstrate inadequate capability in handling high-dimensional nonlinear systems [24].

We establish the first-principle model (DAEs) for CBV-integrated hydraulic actuators. Latin hypercube sampling (LHS) generates input samples, which are processed through the first-principle model to obtain output datasets. The input–output pairs are partitioned into three mutually exclusive sets for network training, testing, and validation. Bayesian regularization prevents overfitting during training [25]. The simulation results demonstrate that the proposed surrogate model achieves 95.33% computational time reduction while maintaining absolute pressure errors within 0.1001 MPa compared to the original first-principle model. This research applies the artificial neural network surrogate model to the proxy modeling of an active motion compensation hydraulic crane, providing a novel solution for such applications. Compared to first-principle modeling, the artificial neural network surrogate model significantly reduces computation time and addresses a gap in proxy modeling for active motion compensation hydraulic cranes. The remainder of this paper is organized as follows: Section 1 presents the research background; Section 2 establishes first-principle models for CBV-integrated hydraulic actuator systems; Section 3 details the neural network-based surrogate modeling methodology; Section 4 validates the model performance through AMC crane simulations; and Section 5 summarizes the content of the paper.

2. The First-Principle Model

2.1. Overview of the Active Motion Compensation Crane

The hydraulic crane under investigation is a knuckle boom crane employing active motion compensation (AMC) technology. It is capable of compensating for tri-axial motion, providing highly precise load positioning for fixed or floating offshore installations in operations such as offshore wind power, drilling supply, and maintenance. Its three-dimensional model and the hydraulic schematic diagram of a single telescopic cylinder are illustrated in Figure 1.

The model comprises a base 1, a turntable 2, hydraulic cylinders 3, 6, booms 4, 7, and a winch 5, among other components. It is designed to lift heavy loads stably on a sea surface affected by wave disturbances. The hydraulic circuit of its actuator is illustrated in Figure 1b, incorporating a counterbalance valve (CBV) 8, a proportional directional valve 2, and a hydraulic cylinder 10. The CBV is a three-way metering element that integrates three typical valve functions into a single component and is connected to the actuator port [26]. As part of the hydraulic actuator, it manages gravitational loads and ensures that the load does not descend due to gravity in the event of power loss [27,28,29].

Hydraulic fluid is supplied from the pressure source $p_s$, passes through a pressure-compensated valve 1, and is regulated by the proportional directional valve 2 in terms of flow direction and magnitude. It then traverses the counterbalance valve formed by components 5 and 8 before entering the hydraulic cylinder 10. The relief valve 3 provides overload protection, while check valves 4 and 6 prevent air ingress by maintaining fluid supply during system shutdown. Manual shutoff valves 7 and 9 control the on/off state of hydraulic lines.

Considering the proportional directional valve–CBV–hydraulic cylinder circuit, the hydraulic circuit model under study is obtained and will be modeled in subsequent sections.

2.2. First-Principle Modeling of the CBV-Integrated Hydraulic Actuator

The hydraulic schematic diagram of the circuit under study is shown in Figure 2.

The hydraulic circuit, as depicted in Figure 2, includes CBV for motion in both directions. For modeling purposes, unidirectional motion is typically analyzed. In the left schematic, components 2 and 3 collectively constitute the counterbalance valve, which is equivalently represented as component 2 in the right diagram during unidirectional motion. Component 1 denotes the simplified proportional directional valve. The governing dynamic equation for the hydraulic cylinder is provided by [30]

$$M\ddot{x}+C\dot{x}=A_1{p}_1-A_2{p}_2-F_{\mathrm{L}}$$

(1)

where M signifies the equivalent mass of the moving parts (comprising the mass of the cylinder piston and the load), C is the viscous damping coefficient of the cylinder, $A_1$ and $A_2$ denote the effective piston areas for the cylinder’s inlet (chamber 1) and outlet (chamber 2), respectively, and $F_L$ is the external load force acting on the cylinder.

The spool dynamics of the proportional valve are considered fast enough to be neglected. The control input to the system is denoted by u. The volumetric flow rate into the cylinder’s inlet chamber is $q_1$, and the flow rate from the outlet chamber is $q_2$. The term $k_{\mathrm{q}}$ represents the flow coefficient of the proportional valve. Based on standard orifice flow equations for the proportional valve ports [31], these flows are determined as

$$\begin{array}{cc}\hfill q_1& =k_{\mathrm{q}}u\sqrt{p_{\mathrm{s}}-p_1}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill q_2& =k_{\mathrm{q}}u\sqrt{p_{\mathrm{r}}}\hfill \end{array}$$

(3)

The displacement of the CBV spool is denoted by $x_{\mathrm{c}}$, with $x_{\mathrm{c}}>0$ indicating an open-valve state. The flow coefficient for the CBV is $k_{\mathrm{qc}}$, and its effective flow area, which is a function of spool displacement, is given by $a_{\mathrm{c}}\left(x_{\mathrm{c}}\right)$. The flow rate $q_2$ (also passing through the CBV from chamber 2) is characterized by

$$q_2=k_{\mathrm{qc}}{a}_{\mathrm{c}}\left(x_{\mathrm{c}}\right)\sqrt{p_2-p_{\mathrm{r}}}$$

(4)

Combining Equations (3) and (4), we have

$$0=k_{\mathrm{q}}^2{u}^2{p}_{\mathrm{r}}-k_{\mathrm{qc}}^2{a}_{\mathrm{c}}^2\left(x_{\mathrm{c}}\right)\left(p_2-p_{\mathrm{r}}\right)$$

(5)

The mechanical dynamics of the CBV spool are modeled as a second-order system, encompassing mass, spring, and frictional effects:

$$m_{\mathrm{c}}{\ddot{x}}_{\mathrm{c}}=-k_{\mathrm{c}}\left(x_{\mathrm{c}}+x_{\mathrm{c}0}\right)+f_{\mathrm{c}}\left({\dot{x}}_{\mathrm{c}}\right)+A_{\mathrm{c}}\left(\gamma p_1+p_2\right)$$

(6)

where $m_{\mathrm{c}}$ is the CBV spool’s mass, $k_{\mathrm{c}}$ is its spring stiffness, and $x_{\mathrm{c}0}$ denotes the initial pre-compression of the spring. The term $f_{\mathrm{c}}\left({\dot{x}}_{\mathrm{c}}\right)$ represents the friction force acting on the spool, which will be further specified (e.g., as Coulomb friction). $A_{\mathrm{c}}$ is the effective pressure control area of the spool, and $\gamma$ indicates the pilot ratio of the CBV.

Considering the fluid compressibility within the cylinder chambers and neglecting any internal or external leakage, the continuity equations for the inlet and outlet chamber flows are expressed as

$$q_1={\displaystyle \frac{V_1}{\beta \mathrm{e}}}{\dot{p}}_1+{\dot{V}}_1$$

(7)

$$q_2=-{\displaystyle \frac{V_2}{\beta \mathrm{e}}}{\dot{p}}_2-{\dot{V}}_2$$

(8)

where $V_1$ and $V_2$ are the instantaneous fluid volumes in the inlet and outlet chambers, respectively. These volumes change linearly with the cylinder displacement x, such that their rates of change are ${\dot{V}}_1=A_1\dot{x}$ and ${\dot{V}}_2=-A_2\dot{x}$.

The complete first-principle model is derived by assembling the hydraulic cylinder dynamics (1), the pressure evolution in each chamber (obtained by substituting flow expressions (2) and (3) into continuity Equations (7) and (8)), the CBV spool’s equation of motion (6), and the algebraic constraint for flow through the CBV path (5). This results in the following system of DAEs:

$$\begin{array}{cc}\hfill m_{\mathrm{c}}{\ddot{x}}_{\mathrm{c}}& =-k_{\mathrm{c}}\left(x_{\mathrm{c}}+x_{\mathrm{c}0}\right)+f_{\mathrm{c}}\left({\dot{x}}_{\mathrm{c}}\right)+A_{\mathrm{c}}\left(\gamma p_1+p_2\right)\hfill \\ \hfill M\ddot{x}& =-C\dot{x}+A_1{p}_1-A_2{p}_2-F_{\mathrm{L}}\hfill \\ \hfill {\dot{p}}_1& =-{\displaystyle \frac{A_1\beta \mathrm{e}}{V_1}}\dot{x}+{\displaystyle \frac{\beta \mathrm{e}{k}_{\mathrm{q}}}{V_1}}\sqrt{p_{\mathrm{s}}-p_1}u\hfill \\ \hfill {\dot{p}}_2& ={\displaystyle \frac{A_2\beta \mathrm{e}}{V_2}}\dot{x}-{\displaystyle \frac{\beta \mathrm{e}{k}_{\mathrm{q}}}{V_2}}\sqrt{p_{\mathrm{r}}}u\hfill \\ \hfill 0& =k_{\mathrm{q}}^2{u}^2{p}_{\mathrm{r}}-k_{\mathrm{qc}}^2{a}_{\mathrm{c}}^2\left(x_{\mathrm{c}}\right)\left(p_2-p_{\mathrm{r}}\right)\hfill \end{array}$$

(9)

3. Surrogate Model-Based Rapid Dynamics Solution

3.1. Framework of the Dynamic Surrogate Model

The primary challenge in solving CBV-integrated hydraulic actuators stems from the inability to explicitly express pressure $p_{\mathrm{r}}$ as a function of coefficients $k_{\mathrm{q}},k_{\mathrm{qc}}$, motion state $x_{\mathrm{c}}$, and pressure $p_2$, precluding its formulation as ODEs. To address this limitation, we introduce a surrogate model $\mathit{h}$.

$$\ddot{\mathit{x}}\approx \mathit{h}\left(x,\dot{x},p_1,p_2,u\right),\mathit{h}:{\mathbb{R}}^{4f}\mapsto {\mathbb{R}}^{2f}$$

(10)

Functioning as an ODE approximation for the dynamic system (9), the surrogate model $\mathit{h}$ is trained to map the system state $\mathit{x}$ and input $\mathit{u}$ to the state derivative $\dot{\mathit{x}}$. The training process involves minimizing the mean squared error (MSE) between the surrogate model predictions and the actual state derivatives obtained from the original DAEs. This dynamic surrogate modeling approach circumvents the computationally intensive solution of DAEs, thereby enhancing numerical solution efficiency at the cost of acceptable approximation errors. Figure 3 illustrates the comparative workflow between surrogate modeling and conventional dynamic equation solving.

Let $\mathit{x}=\left({\dot{x}}_{\mathrm{c}},x_{\mathrm{c}},\dot{x},x,p_1,p_2,u\right)$, where $\mathit{x}\in \mathcal{X}$ denotes the reasonable ranges of the spool displacement and velocity of the equilibrium valve, the operating speed of the valve-controlled hydraulic cylinder, pressure, and control signals. To ensure uniform distribution of initial sampling points within the design space, Latin hypercube sampling (LHS) is employed to generate a set of randomized initial values [32]. These randomly obtained initial values ${\mathit{x}}_0^i\in \mathcal{X}$ are then substituted into the dynamic system described by Equation (9) for precise computation, yielding large datasets of numerical results (${\dot{x}}_{\mathrm{c}}^i,{\ddot{x}}_{\mathrm{c}}^i,{\dot{x}}^i,{\ddot{x}}^i,{\dot{p}}_1^i,{\dot{p}}_2^i$), $i=$$1,2,\dots ,n$. These datasets are used to train and validate the parameters $\omega$ of the surrogate model. As the training error progressively decreases, a continuous nonlinear function $\mathit{h}$ is obtained, which serves as the surrogate model for the problem.

3.2. Three-Hidden-Layer Neural Network-Based Surrogate Model

We adopt a three-hidden-layer neural network as the surrogate model, which constitutes a linear combination of nonlinear functions. Equation (11) presents its general form for a single hidden layer.

$$h_k\left(\mathit{x}\right)=\sum _{i=1}^m{\mu }_i{\psi }_i\left({\mathit{w}}_i\mathit{x}+b_i\right)+\beta$$

(11)

Here, we select the hyperbolic tangent function (tanh) as the nonlinear activation function, i.e., ${\psi }_i\left(z\right)=2/(1+exp(-2z))-1$[23]. The architecture of the three-hidden-layer neural network is as follows: the input $\mathit{x}$ is first linearly transformed by weights $\mathit{w}$ into an m-dimensional feature space. Subsequently, m nonlinear functions individually process these features. After three successive hidden-layer mappings, the final output is expressed as a linear combination of nonlinear functions ${\psi }_i$ plus a bias term $\beta$, as illustrated in Figure 4.

Here, parameters ${\mathit{w}}_i,b_i,\mu$, and $\beta$ are all subject to optimization. The training set comprises n ordered input–output pairs ( $\left[x^i,{\dot{x}}^i,p_1^i,p_2^i\right],\left[{\dot{x}}^i,{\ddot{x}}^i,{\dot{p}}_1^i,{\dot{p}}_2^i\right]$ ), $i=1,2,\dots ,n$, obtained by computing randomly generated initial values through the dynamic system (9). Employing the same methodology, the validation and test sets are independently generated, ensuring mutual exclusivity among all three datasets. Specifically, the validation set serves to monitor training performance during the optimization process, while the test set is reserved for post-training evaluation of the surrogate model’s accuracy.

For the n data points in the training set, the surrogate model yields n corresponding outputs. The loss function is constructed as the sum of squared errors between all predicted and reference outputs.

$$L\left({\mathit{w}}_i,b_i,\mu ,\beta \right)=\sum _{j=1}^n{\left[\sum _{i=1}^m{\mu }_i{\psi }_i\left({\mathit{w}}_i{\mathit{x}}^j+b_i\right)+\beta -\left[x^j,{\dot{x}}^j,p_1,p_2\right]\right]}^2$$

(12)

The training process can be formally expressed as the following unconstrained optimization problem:

$$\underset{{\mathit{w}}_i,b_i,\mu ,\beta }{min}L\left({\mathit{w}}_i,b_i,\mu ,\beta \right)$$

(13)

Given that the loss function L possesses well-defined first- and second-order derivatives with respect to ${\mathit{w}}_i,b_i,\mu$, and $\beta$, we employ Bayesian regularization (Equation (13)) to solve this optimization problem, which effectively mitigates overfitting during training [25]. After each iteration, the validation set is utilized to evaluate the training progress via the loss function (Equation (12)). The optimization terminates when the loss function fails to decrease after multiple consecutive iterations, marking the completion of surrogate model training. As a global approximation of the original dynamic system within the specified parameter bounds, the trained surrogate model requires no retraining unless the parameters of the original dynamic system are modified.

4. Simulation Verification of Dynamic Surrogate Model

Using the modeling methodology described in Section 2 and Section 3, we established a forward dynamic model and trained the dynamic surrogate model for an active motion compensation hydraulic crane lifting system. Comparative analysis between first-principle modeling approaches and the proposed method demonstrates the superior performance and effectiveness of our approach.

4.1. Active Motion Compensation Hydraulic Crane Lifting System

As critical equipment in offshore wind farm operations, the active motion compensation hydraulic crane was simulated with a CBV-integrated hydraulic actuator circuit under typical constant gravity load conditions. The relationship between hydraulic cylinder displacement and load force was obtained through multibody dynamics simulation, as shown in the following figure (Figure 5):

Through piecewise cubic interpolation, we derived the corresponding parameters for Equation (9), which were subsequently substituted back into the original equation. The remaining parameters in Equation (9) are listed in

Table 1. Parameters of active motion compensation hydraulic crane lifting system.

Model Parameters (Units)	Value
M (kg)	200
$m_{\mathrm{c}}$ ($\mathrm{kg}$)	0.025
C ($\mathrm{N}·\mathrm{s}/\mathrm{mm}$)	1
$c_{\mathrm{c}}$ ($\mathrm{N}·\mathrm{s}/\mathrm{mm}$)	0.1
$A_1$ (${\mathrm{mm}}^2$)	80,424.8
$A_2$ (${\mathrm{mm}}^2$)	49,008.8
$k_{\mathrm{c}}$ ($\mathrm{N}/\mathrm{mm}$)	23.94
$k_q$ (${\mathrm{mm}}^3/(\mathrm{s}·\sqrt{\mathrm{MPa}})$)	$4.057\times {10}^5$
$k_{qc}$ (${\mathrm{mm}}^3/(\mathrm{s}·\sqrt{\mathrm{MPa}})$)	$35.355\times {10}^5$
$p_s$ (MPa)	30
$\gamma$	4.5
$p_{\mathrm{cm}}$ (MPa)	1.7
$f_{\mathrm{c}}$ ($\mathrm{N}$)	5
$\beta$ (MPa)	690
$V_{1d}$ (${\mathrm{mm}}^3$)	20
$V_{2d}$ (${\mathrm{mm}}^3$)	20

.

Based on these parameters, the first-principle model of the hydraulic lifting system was established, represented as a set of DAEs (9).

4.2. Training and Evaluation of Dynamic Surrogate Model

4.2.1. Dynamic Surrogate Model Training

Considering the characteristics of valve-controlled hydraulic cylinders, we determined the value ranges for each input of the dynamic equations as shown in

Table 2. Input value ranges for the dynamic equations.

Parameters	Minimum Value	Maximum Value
$x\left(\mathrm{mm}\right)$	0	2000
$\dot{x}(\mathrm{mm}/\mathrm{s})$	$-1000$	1000
$p_1\left(\mathrm{MPa}\right)$	$7.5$	$9.5$
$p_2\left(\mathrm{MPa}\right)$	$-1$	1

:

Latin hypercube sampling was employed to generate $2.5\times {10}^7$ initial values within the specified parameter ranges. These initial conditions were subsequently processed using Newton’s method to solve the dynamic Equation (9), yielding a comprehensive dataset containing $n=2.5\times {10}^7$ entries $(\dot{x},\ddot{x},{\dot{p}}_1,{\dot{p}}_2)$. The dataset was rigorously partitioned into mutually exclusive subsets: a training set ($2\times {10}^7$ samples), validation set ($2.5\times {10}^6$ samples), and test set ($2.5\times {10}^6$ samples), ensuring no data leakage between subsets.

Following the methodology detailed in Section 3.2, we implemented a three-hidden-layer neural network architecture with neuron configurations of [25, 10, 20] in successive layers. The mathematical representation of each hidden layer follows:

$$\mathit{h}\left(x\right)=\sum _{i=1}^m{\mu }_i\left({\displaystyle \frac{2}{1+exp(-2(w_{i}x+b_i))}}-1\right)+\beta$$

(14)

where $m=25,10,20$ denotes the neuron count per layer. This configuration yields 333 trainable parameters in Equation (14). The unconstrained optimization problem in Equation (13) was solved using Bayesian regularization. Training termination criteria were established when the validation loss failed to decrease for 100 consecutive iterations.

4.2.2. Surrogate Model Performance Evaluation

Model accuracy was quantified through test set error analysis, essentially evaluating the computational precision of the surrogate model’s outputs. To ensure unbiased evaluation, the test set maintained strict non-overlapping with both training and validation datasets. As the surrogate model operates strictly in interpolation mode, all subsets share identical input domains where $\mathit{x}\in \mathcal{X}$.

Let the test set contain $n_E$ data samples $\left([x^i,{\dot{x}}^i,p_1^i,p_2^i],[{\dot{x}}^i,{\ddot{x}}^i,{\dot{p}}_1^i,{\dot{p}}_2^i]\right),i=1,2,\dots ,n_E$. The absolute error of the surrogate model for the k-th degree of freedom is defined as

$${ϵ}_k=\sqrt{{\displaystyle \frac{1}{n_E}}{\sum }_{i=1}^{n_E}{\left[h_k\left(x^i\right)-[{\dot{x}}^i,{\ddot{x}}^i,{\dot{p}}_1^i,{\dot{p}}_2^i]\right]}^2}$$

(15)

Additionally, training iterations and computational duration serve as critical metrics for evaluating the surrogate model development process.

The system model based on the dynamic surrogate model is governed by four ODEs:

$$\left\{\begin{array}{cc}\hfill m_{\mathrm{c}}{\ddot{x}}_{\mathrm{c}}& =-k_{\mathrm{c}}(x_{\mathrm{c}}+x_{\mathrm{c}0})+f_{\mathrm{c}}\left({\dot{x}}_{\mathrm{c}}\right)+A_{\mathrm{c}}(\gamma p_1+p_2)\hfill \\ \hfill M\ddot{x}& =-C\dot{x}+A_1{p}_1-A_2{p}_2-F_{\mathrm{L}}\hfill \\ \hfill ({\dot{p}}_1,{\dot{p}}_2)& =h(x,\dot{x},p_1,p_2,u)\hfill \end{array}\right.$$

(16)

Under identical software configurations, fixed-step Runge–Kutta integration was employed to solve both the first-principle and surrogate models. This approach ensures consistent numerical treatment for fair performance comparison.

4.3. Solution Results and Comparative Analysis

The training of the surrogate model and numerical simulations of the first-principle model were conducted under identical computational conditions. The hardware platform consisted of an AMD 4800H 2.9GHz processor. The equipment was purchased in Changsha from the Chinese manufacturer ASUSTeK Computer Inc. (ASUS). Using Equation (15), the normalized absolute error of the surrogate model was calculated on the test set, with training iterations and total duration statistically summarized in

Table 3. Training performance of surrogate model.

Normalized Absolute Error	Training Iterations	Total Training Duration
$2.4\times {10}^{-9}$	1537	20 h 11 min

.

For both the DAEs and the surrogate model, identical input sequences $u_t$ (where t denotes time indices) were applied. Fixed-step numerical integration ($\mathrm{\Delta }t$) was implemented to compute one-step predictions. Output trajectories were compared to evaluate model accuracy, while computational efficiency was assessed through average step-solving time.

A proportional valve input profile was designed as the system excitation (Figure 6).

4.3.1. Evaluation of Computational Accuracy

Using the aforementioned valve input, fourth-order Runge–Kutta simulations were performed on both the first-principle model and surrogate model. Comparative results for cylinder displacement trajectories, velocity profiles, and pressure dynamics are shown in Figure 7.

The first-principle model solutions were treated as ground truth for error quantification. The absolute errors of the dynamic surrogate model relative to reference values are visualized in Figure 8.

Quantitative error metrics are summarized in

Table 4. Absolute errors of dynamic surrogate model.

Category	${\mathit{p}}_1\left(\mathbf{MPa}\right)$	${\mathit{p}}_2\left(\mathbf{MPa}\right)$
error absolute value	0.1001	0.0093

, demonstrating the surrogate model’s high computational fidelity.

It can be seen that the dynamic surrogate model can ensure a relatively high calculation accuracy.

4.3.2. Evaluation of Computational Accuracy Under Perturbed Load

Gaussian random disturbances were added to the load, yielding the new relationship between hydraulic cylinder perturbed load $F_L\ast$ and displacement x, as shown in Figure 9.

The proportional valve input curve remained identical to Figure 6. Solutions were obtained for both the first-principle model and the surrogate model under disturbed load conditions, with the computational results presented in Figure 10 and Figure 11.

The first-principle model solutions were treated as ground truth for error quantification. The absolute errors of the dynamic surrogate model relative to reference values are visualized in Figure 11.

Quantitative error metrics are summarized in

Table 5. Absolute errors of dynamic surrogate model under perturbed load.

Category	${\mathit{p}}_1\left(\mathbf{MPa}\right)$	${\mathit{p}}_2\left(\mathbf{MPa}\right)$
error absolute value	0.0402	0.0099

, demonstrating the surrogate model’s high computational fidelity under perturbed load.

It can be observed that the proposed method maintains high precision even when subjected to perturbed load forces. The perturbing force exhibits minimal impact on the proposed artificial neural network surrogate model.

4.3.3. Evaluation of Computational Efficiency and Memory Requirements

Under identical hardware/software configurations, both models were solved using the fourth-order Runge–Kutta method. The average step computation time, derived from the total duration divided by the simulation steps, is compared in

Table 6. Average single-step computation time comparison.

Model Type	Computation Time (ms)
First-Principle Model	4.3103
Surrogate Model	0.2011

.

Under identical hardware/software configurations, the memory consumption of both models is compared in

Table 7. Memory consumption comparison.

Model Type	Memory Consumption (MB)
First-Principle Model	334.35
Surrogate Model	206.09

.

The surrogate model achieved a remarkable 95.33% reduction in computation time and 128.26 MB memory consumption compared to the conventional methods. These results confirm that direct solving of dynamic equations becomes computationally prohibitive for real-time applications on consumer-grade hardware, whereas the proposed approach significantly reduces computational overhead while maintaining small step sizes and ensuring real-time performance.

5. Conclusions

To address the rapid computation challenge for CBV-integrated hydraulic actuators, this study establishes a first-principle model for such systems and generates a set of inputs via LHS. The input set is substituted into the first-principle model to obtain accurate outputs, which are then partitioned into mutually exclusive training, testing, and validation datasets. A neural network surrogate model with three hidden layers is designed and trained. By testing diverse inputs in a fourth-order fixed-step Runge–Kutta solver, the performance of the first-principle model and that of the surrogate model are compared. The results show that the trained surrogate model reduces computation time by 95.33% while maintaining low error relative to the first-principle model, significantly conserving computational resources. Applying this method to AMC hydraulic cranes can advance real-time control, optimal design, and digital twin implementation, demonstrating substantial engineering and scientific value.