First, two coordinate systems – the moving and fixed frames – are established. The origin O of the fixed coordinate system O-xyz is located at the center of the base platform's joints. The origin C of the moving-coordinate system C-xyz is located at the center of the moving platform's joints. When the moving platform is at its home position, the directions of the x, y, and z axes for both coordinate systems are identical, and the z axis of the fixed coordinate system passes through point C. The orientation of each axis is shown in Fig. 1.

As shown in Fig. 1, points B1–B6 are the six joints on the upper platform, used for installation and positioning at the upper-platform connections, and C is the center of the upper joint circle. Points A1–A6 are the six joints used for installation and positioning at the lower-platform connections, and O is the center of the lower joint circle. RB represents the radius of the upper platform's joint circle, and RA represents the radius of the lower platform's joint circle, while H represents the initial height of the platform. The angle between a pair of adjacent joints on the upper platform is α, and the angle between a pair of adjacent joints on the lower platform is β.

For the heavy-duty compensation parallel platform, five key configuration parameters affecting its kinematic characteristics are selected to form a five-dimensional decision vector x: 1x=[RB,RA,α,β,H]T Each element of the vector corresponds to a physical dimension of the mechanism, and the optimization algorithm will search for the optimal solution within a five-dimensional space. Table 1 presents the value ranges of the structural design variables for the heavy-duty wave compensation platform.

The decision space of the optimization algorithm is constrained within a five-dimensional hyper-rectangular region: 2Ω={x∈R5∣xi,min≤xi≤xi,max, 3varmin=[2000,3.0,10,10,4.0]Tvarmax=[3000,4.5,30,30,7.0]T, where varmin and varmax denote the lower and upper bounds of the variables, respectively. These limits are not assigned arbitrarily; rather, they are constrained by practical engineering considerations, such as the available deck space of the main vessel, servo-cylinder selection catalogs, and the anti-interference clearance of the joints. Through the parametric description of the geometric configuration of the heavy-duty compensation parallel platform, a design space encompassing the radial dimensions, angular distributions, and spatial height – which characterize the topological features of the mechanism – is established. By setting reasonable physical boundary constraints and proportional constraints, the search process is ensured to be conducted within a set of configurations that satisfy practical engineering requirements.

Considering the structural parameters of the heavy-duty compensation parallel platform and practical engineering requirements, the maximum actuator stroke, peak driving force, maximum dexterity, and kinematic isotropy are selected as the four objective functions for the optimization analysis.

The accuracy of the surrogate model is highly dependent on the distribution quality of the training samples within the design space. In this study, all 2500 datasets were numerically generated using a customized physical simulation program developed in MATLAB. Specifically, the input parameters were generated via a random sampling function within the structural design boundaries. The corresponding multi-physics output responses were subsequently calculated using MATLAB's mathematical solvers: the actuator stroke was solved via spatial inverse kinematics, the peak driving force was derived using the virtual work principle via the force Jacobian matrix, and the dexterity and isotropy indices were quantified using SVD. To address the issue of magnitude disparity among different performance objectives of the heavy-duty compensation parallel platform – for instance, the actuation stroke objective f1 is at the 10⁰ order of magnitude, whereas the peak driving-force objective f2 is at the 10⁵ order of magnitude – Z-score standardization is applied to the output objective vectors. This ensures that the neural network can evenly capture the features of various physical domains during the training process: 17y^i,j=yi,j-μjσj, where yi,j is the true value of the jth objective for the ith sample, and μj and are σj the sample mean and standard deviation of this objective, respectively.

A deep neural network model with a four-layer structure is established, with its overall architecture defined as 5-20-15-4. The input layer corresponds to five structural decision variables, and the two hidden layers contain 20 and 15 neurons, respectively, employing sigmoid activation functions to capture highly nonlinear mapping relationships; the output layer corresponds to four standardized performance metrics. Through the nonlinear transformation of the dual hidden layers, high-speed prediction of four performance objectives – actuator stroke, driving force, dexterity, and isotropy – is ultimately achieved at the output layer.

The number of hidden layers and neurons in the artificial neural network was determined through a systematic grid search and trial-and-error approach, aiming to balance predictive accuracy and computational efficiency. Initially, empirical formulas were utilized to define the approximate range of the number of neurons. Subsequently, various network models containing one to three hidden layers with different neuron combinations were trained and evaluated. The evaluation results indicated that a single hidden layer lacked the capacity to accurately map the highly nonlinear multi-physics responses of the 6-DOF platform, whereas a three-hidden-layer architecture was highly prone to overfitting and consumed excessive training time. Through performance comparison on the validation set, the dual-hidden-layer architecture containing 20 and 15 neurons, respectively, demonstrated the most robust generalization capability. Consequently, a deep neural network model with an overall architecture defined as 5-20-15-4 was ultimately established.

The training process employs the Levenberg–Marquardt algorithm, which features a second-order convergence rate. Its weight update formula is given by the following: 18wk+1=wk-[JtTJt+μI]-1JtTe, where Jt is the Jacobian matrix of the error with respect to the weights, μ is the damping factor, and e is the training error.

The goodness of fit is employed to independently evaluate the four performance objectives. The calculation formula is as follows: 19R2=1-∑i=1nyi-y^i2∑i=1nyi-y¯i2, where yi is the true value from the physical simulation, and y^i is the predicted value from the neural network.

The 2500 sets of sampled data are divided into a training set and a testing set at a ratio of 8:2. To eliminate potential data error risks and ensure the training reliability of the ANN, it is necessary to clarify that these 2500 datasets are not empirical measurements susceptible to observation noise but exact numerical solutions strictly generated from closed-form analytical equations such as spatial inverse kinematics and the virtual work principle. Furthermore, the numerical solver was systematically pre-verified under typical boundary conditions prior to the bulk generation to guarantee the physical credibility of the datasets. The regression analysis results of the surrogate model on the independent testing set are shown in Fig. 6, where the horizontal axis represents the true values from the physical simulation, and the vertical axis represents the predicted outputs of the neural network surrogate model. The average R2 values of all objectives reach over 0.98, demonstrating that the constructed ANN surrogate model possesses high fitting accuracy across the entire variable space, thereby satisfying the performance evaluation requirements for the subsequent NSGA-III optimization.

Furthermore, a 0.5 % level of Gaussian white noise was introduced into the training data. This operation is designed to simulate the stochastic response of the heavy-duty compensation parallel platform under uncertainties such as manufacturing errors, joint clearances, and sensor measurement noise. The experimental results demonstrate that the R2 values of the noise-injected surrogate model on the independent testing set remain between 0.987 and 0.999.

The diversity maintenance mechanism of NSGA-III is based on a reference point network. A systematic projection method is employed to construct hyperplane reference points within the objective space. For a task with M objectives and p divisions, the total number of reference points N_ref follows the combinatorial relationship below: 20Nref=M+p-1p. With the number of divisions set to p=12, a total of 455 uniformly distributed reference vectors are generated within the four-dimensional objective space. Given the significant order-of-magnitude disparities among the physical metrics of the heavy-duty compensation parallel platform – for instance, the actuation stroke metric f1 is at the 10⁰ scale, whereas the peak driving-force metric f2 is at the 10⁵ scale – the algorithm performs a normalization process in each evolutionary generation to eliminate scale bias. First, the ideal point z∗=(z1∗,z2∗,…,zM∗) is determined based on the minimum values of the population. Subsequently, the extreme points are identified by minimizing a scalarizing function to construct a hyperplane. The j-th objective function value fi,j of individual i is then mapped into the normalized interval: 21fi,j′=fi,j-zj∗aj-zj∗, where aj is the intercept of the hyperplane with the jth objective axis. This process ensures that the search process takes place within a unit simplex, thereby assigning equal evolutionary weights to all performance metrics.

The algorithm employs real-coded simulated binary crossover and polynomial mutation to generate superior offspring configurations. Before calculating the overall violation degree, two specific geometric interference criteria are explicitly defined: the dynamic actuator length li must satisfy the physical stroke limits, i.e., lmin≤li≤lmax, and the spatial deflection angle θi must not exceed the maximum allowable cone angle of the Hooke's joints, i.e., θi≤ θmax. Exceedances of these physical boundaries are directly transformed into specific interference constraint functions gk(x). Furthermore, to address the complex interference and singularity constraints of the platform, a handling logic based on feasibility rules is introduced. The overall constraint violation degree CV(x) of individual x is defined as the weighted sum of individual constraint violations: 22CV(x)=∑k=1mmax(0,gk(x)). During the parent selection phase, the algorithm adheres to the following rules: if both individuals are feasible, they are selected based on their non-domination ranks; if infeasible individuals are involved, the one with the smaller CV value is preferentially retained.

During the environmental selection phase, NSGA-III maintains the population distribution by calculating the perpendicular distance between individuals and reference lines, which replaces the crowding distance used in traditional algorithms. For an individual s and a reference line w in the normalized space, their perpendicular distance d(s,w) is expressed as follows: 23d(s,w)=(f′(s)-z∗)-(f′(s)-z∗)Twww. Based on a niche counting mechanism, the algorithm preferentially retains solutions around the reference points that have the fewest associated individuals. This dynamic screening strategy forces the population to be uniformly distributed across the four-dimensional Pareto front, thereby excavating potential optimal configurations.

Figure 8 illustrates the sensitivity correlation characteristics between the structural parameters and the objective functions.

Figure 8 reveals the sensitivity patterns of each structural parameter on the system performance. The global sensitivity indices follow the order of RB>H>RA>β>α. Among them, the upper-platform radius RB is the most critical variable; increasing RB can reduce the peak load and improve kinematic dexterity. The initial height H dominates the actuation stroke; an excessive height will lead to a sharp increase in the telescoping span and deteriorate the isotropy. Appropriately increasing the base platform radius RA helps mitigate stroke redundancy.

Taking the structural parameter RB as a vital characteristic parameter for further analysis, variations in the value of RB can simultaneously influence three optimization objectives. Within different parameter intervals, the interactive relationships among these optimization objectives may vary, as illustrated in Fig. 9.

As shown in Fig. 9, increasing RB can significantly optimize the force mapping to evenly distribute the load; as it increases from 2140 to 3110 mm, the peak driving force decreases from 117.47 to 87.44 kN. In terms of kinematics, RB enhances the mechanism dexterity with a correlation coefficient of r=0.843. Meanwhile, it exhibits a strong negative correlation with the condition number (r=-0.774), which drops substantially from approximately 22 to 14.28.

Based on the sensitivity correlations between the design variables and the performance objectives, the 455 non-dominated solutions obtained via optimization are compared against the baseline of initial structural parameters. This comparison aims to screen out the optimized solutions that exhibit improvements across three to four optimization objectives. Figure 10 illustrates the performance enhancements of the 91 selected optimal solutions relative to the initial structural baseline.

The analysis indicates that the improvements in dexterity and peak driving force are the most prominent. Across the entire solution set, dexterity increases by 1 % to 20 %, with solution 1 reaching 18.88 %. The peak driving force decreases by 8 % to 14 %, and, notably, solution 3 drops from 103.11 to 90.97 kN. Constrained by the relatively optimal initial baseline, the stroke reduction rate remains stable between 1 % and 5 %.

In this section, the traditional NSGA-II algorithm is selected as the classical baseline for comparison. Rather than treating it as a competing meta-optimizer, NSGA-II is strictly employed as a control group to demonstrate the exact limitations of the traditional crowding-distance mechanism in a four-objective space, thereby highlighting the necessity and performance leap brought by the reference-point mechanism of NSGA-III. Three core evaluation metrics in the multi-objective optimization field are introduced to conduct a quantitative comparative analysis of the optimized solution sets from the traditional NSGA-II algorithm and the enhanced multi-objective genetic algorithm NSGA-III. The spacing metric SP is utilized to measure the distribution uniformity of the non-dominated solution set. Its calculation formula is as follows: 32SP=1n-1∑i=1nd‾-di2, where n is the total number of obtained non-dominated solutions, and di is the minimum Euclidean distance from the ith solution to the other solutions in the set, expressed as 33di=minj∈{1,2,…,n},j≠i∑k=1M∥fki-fkj∥2, 34d‾=1n∑i=1ndi, where M is the number of objective functions, and d‾ is the average of all minimum distances di. Numerically, the SP metric reflects the standard deviation of individual spacing on the Pareto front. When SP = 0, it indicates that all non-dominated solutions are arranged at completely equal intervals in the objective space, reaching an ideal uniform state. A smaller SP value signifies that the distribution of the solution set tends to be more uniform, and the algorithm's ability to capture the performance front is more stable, without obvious performance gaps or individual crowding phenomena.

The maximum spread metric MS is used to evaluate the algorithm's exploration capability regarding the extreme boundaries of the objective space. Its formula is defined as the Euclidean distance between the two furthest solutions in the normalized objective space: 35MS=max⁡i,j∈{1,2,…,N}∑m=1Mfm(i)-fm(j)2, where M is the objective dimension, and fm(i) is the normalized value of the ith solution on the mth objective. A larger MS value indicates that the algorithm has a wider coverage range over the extreme parameter boundaries.

The comprehensive hypervolume indicator HV is an important metric for evaluating the comprehensive performance of multi-objective optimization algorithms, taking into account both convergence and diversity. Let the reference point be r=[r1,r2,…,rM]T. HV is defined as the hyper-geometric volume enclosed by the Pareto solution set and the reference point in the objective space: 36HV=Volume⋃i=1Nf1i,r1×…×fmi,rm. A larger HV value indicates that the solution set generated by the algorithm is closer to the true perfect front and covers a broader trade-off space, reflecting superior overall multi-objective performance. After applying range normalization to the optimization results of NSGA-II and NSGA-III, the statistical results of the above three metrics are calculated, and these are presented in Table 2:

The analysis shows that its SP value drops to 0.0497, representing a decrease of 37.4 %, and the HV value increases by 26.2 %. Although the MS value of NSGA-III is slightly lower than that of NSGA-II, it eliminates the artificial inflation of MS caused by isolated extreme solutions in the traditional algorithm. NSGA-III generates an internally highly continuous solution set, which holds greater significance for engineering guidance. The comparison of structural parameters before and after optimization is shown in Table 3.

Through comparison, it is evident that the improved NSGA-III algorithm benefits from the introduction of a dynamic penalty mechanism for physical interference. Its optimal scheme not only reasonably expands the base support span of the platform – with RA increasing to 3881 mm and RB increasing to 3060 mm, coupled with a height of 5291 mm to enhance overall stiffness and force performance – but also stably maintains α and β at 19.82 and 19.04°. This result closely aligns with the initial design of 20°, ensuring a safe installation clearance for the hinges. The comparison of the four optimization metrics between NSGA-II and NSGA-III is presented in Table 4.

Table 4 indicates that the comprehensive performance of the improved NSGA-III is superior to that of NSGA-II. By relying on mechanisms such as dynamic normalization to escape local optima, it successfully reduces the peak driving force to 90.97 kN, increases the dexterity to 0.4460, and optimizes the actuation stroke to 3567 mm. Although its condition number of 14.26 superficially appears to be inferior to the 11.14 of the latter, this is actually the optimal solution derived from a multi-dimensional trade-off after strictly avoiding the boundaries of mechanical interference and introducing the penalty mechanism, rather than a degradation in performance.

A proof-of-concept prototype of the wave compensation parallel platform was developed. This prototype employs DC linear actuators as the underlying actuation units, and its core geometric parameters are designed as follows: the radius of the upper platform hinge point circle RB is 180 mm, the radius of the lower platform hinge point circle RA is 260 mm, and the initial neutral posture height H is 370 mm. According to the optimization results, the angles between the hinge points of the upper and lower platforms, αand β, are 19.82 and 19.04°, respectively. Based on this prototype, dynamic tests of extreme postures were conducted. The upper platform executed roll-and-pitch extreme posture commands with an amplitude of ±15°, respectively, as shown in Fig. 13.

To comprehensively evaluate the reliability of the optimized configuration, the extreme posture experiment was consecutively repeated 50 times. Furthermore, a quantitative comparison between the optimized theoretical results and the physical experimental data was performed. The measurement results indicated that the relative error between the theoretical target angles and the actual executed angles of the prototype remained strictly within 1 % across all test cycles. The experimental results demonstrate that the six straight legs of the platform operate smoothly throughout the 50 consecutive cycles, with no physical collisions or interference occurring between the kinematic pairs and the connecting links.