Integrating Physics-Based and Data-Driven Approaches for Accurate Bending Prediction in Soft Pneumatic Actuators

Abstract

Soft pneumatic actuators (SPAs) are gaining attention in the field of soft robotics due to their lightweight, highly flexible, and safer interaction while operated under an unstructured environment. They are easy to fabricate, produce high output force, and are relatively very inexpensive compared to other soft actuators. However, accurate prediction of their nonlinear bending behavior is one of the main challenges, which is mainly due to the complex material properties and high deformation patterns. Therefore, this study focused on a hybrid approach that accurately captures the bending behavior of a single-chambered SPAs. This approach integrates physics-based modeling (finite element analysis (FEA) and analytical modeling) with a data-driven (polynomial regression modeling) approach to analyze the bending of single-chambered SPAs. Initially, four different hyperelastic material models (Neo-Hookean, Yeoh, Arruda–Boyce, and Ogden) were tested using FEA to analyze how material selection affects the SPA response. It is found that the Arruda–Boyce model generates the highest bending of 101° at 30 kPa pressure, while the other models consistently underestimated deformation at higher pressures. Further, an enhanced mathematical or analytical model was developed using Euler and Timoshenko beam theory with certain assumptions, such as neutral axis shifting, chamber ballooning, and shear deformation. These assumptions significantly improve the prediction accuracy and generate a bending angle of 99°at 30 kPa, which closely matches FEA bending. Further, a polynomial regression-based machine learning (ML) model was trained using analytical or mathematical bending data for faster output prediction. This data-driven approach achieves very high accuracy in the validation range, with an average absolute percentage deviation of only 0.002%. Additionally, comparison with the analytical results showed a mean absolute error (MAE) of 0.00180°, root mean squared error (RMSE) of 0.00205°, and coefficient of determination (R²) value of 0.999999808. Overall, integrating physics-based modeling with a data-driven approach provides a reliable and scalable method for SPA design. It provides practical information on material selection, analytical correction, and ML modeling, which will reduce the need for time-consuming prototyping. Finally, this hybrid approach can help to accelerate the development of soft robotic grippers, rehabilitation tools, and other bio-inspired actuation systems.

1. Introduction

Soft pneumatic actuators (SPAs) are of high importance in soft robotics as they can mimic the flexibility of biological muscles, adapt well to changing environments, and allow safe interaction with complex or unstructured surroundings [1,2]. However, accurate prediction of their mechanical response under varying internal pressure remains challenging due to the nonlinear characteristics of elastomeric materials [3]. To address this challenge, finite element analysis (FEA) has been widely employed, which enables the simulation of deformation and stress distribution through hyperelastic material models (such as Neo-Hookean, Mooney–Rivlin, Yeoh, Ogden, and Arruda–Boyce) [4,5]. Several studies comparing different hyperelastic materials (for example, Ecoflex OO-30 and Ecoflex OO-50) highlight how material selection significantly affects bending behavior under pneumatic actuation [6,7]. Similarly, studies using Dragon Skin 30, Elastosil M4601, and Smooth-Sil 950 demonstrate that prediction accuracy is closely related to the selected constitutive model [8]. More recently, FEA-calibrated hyperelastic models have been integrated into inverse kinematic formulations, which extends their use towards real-time robotic control [9]. Although FEA provides highly detailed and reliable results, its computational intensity still limits its practicality in areas that demand rapid design iterations or real-time operations [10].

To address the limitations of FEA, researchers have introduced a range of analytical models aimed at delivering faster and more interpretable predictions of SPA performance. This includes, for example, simplified formulations using Neo-Hookean hyperelasticity for pleated and bellows actuators [11], quasi-static “gray-box” models for estimating bending displacement and blocking force [12], and more advanced approaches based on minimum potential energy principles and continuum rod theory to capture three-dimensional deformations [13]. Other studies have established direct relationships between chamber pressurization, bending angle, and tip force by incorporating hyperelastic membrane theory along with the effects of inter-chamber contact moments [14,15]. While these analytical strategies achieve a practical balance between speed and accuracy, their dependence on simplified assumptions can limit their general applicability. Simultaneously, machine learning (ML) has emerged as a promising alternative for SPA modeling. Unlike conventional methods, ML depends on data-driven learning to capture nonlinear responses without the need to explicitly define constitutive material laws [16]. For example, nonlinear ML models trained on FEA datasets have successfully predicted the actuation of 4D-printed SPAs, which enables faster design and optimization cycles [17,18]. More recently, physics-guided deep learning (PGDL) frameworks have embedded governing equations within neural network structures, which produces surrogate or substitute models that combine physical accuracy with data efficiency [19]. Other studies have employed ML techniques to optimize SPA geometry and predict deformation patterns under varying loads, which bridges the gap between simulation and real-world performance [20,21]. Collectively, these studies highlight that ML-based approaches (especially when combined with physical laws) can achieve near-real-time prediction accuracy, which make them promising tools for design optimization, adaptive control, and the rapid prototyping of SPAs.

Thus, motivated by the need for versatile and computationally efficient modeling strategies, this study presents a detailed comparison of three approaches, FEA, analytical modeling, and a data-driven ML technique, for a single-chambered SPA assumed to be made up of Ecoflex OO-50. The work systematically examines how different constitutive models influence bending behavior and stress distribution under identical boundary conditions. In the first stage, FEA simulations are carried out using the Neo-Hookean, Yeoh, Ogden, and Arruda–Boyce hyperelastic models. Their predicted bending angles and stress fields are compared to identify the model that offers the best balance between bending and stress distribution. In the second stage, an analytical model of SPA bending is developed under the same loading and boundary conditions. The predictions from these models are first compared with results from established hyperelastic FEA models to assess their accuracy. In the final stage, a data-driven approach is introduced using a polynomial regression model. This ML model is trained on the bending predictions obtained from the analytical model, and its predicted results are then evaluated against the analytical results to check for consistency and accuracy in deflection. Finally, by combining these three approaches (FEA, analytical modeling, and data-driven prediction), this study aimed to improve computational efficiency, deliver reliable performance predictions, and facilitate the optimized design of SPAs.

2. Materials and Methods

2.1. Structural Design

In this work, the single-chambered SPA was modeled using Ecoflex OO-50, a silicone-based and platinum-cured elastomer commonly employed in soft robotics due to its favorable mechanical characteristics [22]. The choice of Ecoflex OO-50 was motivated by its balanced properties, including a moderate Shore hardness (OO-50), high elongation at break of about 980%, a relatively low tensile modulus of ~200 kPa, and the ability to recover reliably under repeated actuation cycles [23]. Compared with other materials such as Dragon Skin or Ecoflex OO-30, it provides an effective compromise between flexibility and load-bearing strength. This enables the SPA to achieve large deformations while maintaining its structural stability under pneumatic pressure. Furthermore, its properties (such as biocompatibility, ease of casting, and wide commercial availability) make it particularly suitable for the rapid prototyping and experimental evaluation of SPAs [24]. Previous studies have also shown that Ecoflex OO-50 performs reliably in finite element simulations using hyperelastic models like Yeoh and Arruda–Boyce, further supporting its suitability for this application. These material models have improved fit to experimental stress–strain curves compared to simpler models like Neo-Hookean, particularly in the large deformation regimes encountered during actuator bending. Therefore, this study simulates a single-chambered SPA using Yeoh, Arruda–Boyce, and Neo-Hookean to evaluate which model develops larger deformation or bending. Figure 1 shows the structural design of the single chambered SPA.

The SPA was designed as a single-chambered rectangular body with the following dimensions: a 100 mm length, 14 mm width, and 16 mm height. It includes an internal hollow cavity for pneumatic inflation. This simple geometry was purposely chosen to isolate and analyze the fundamental bending behavior of the SPA under uniform pressurization. Upon pressurization, compressed air is introduced into the internal chamber, which results in an internal force acting radially outward on all chamber walls. However, due to the difference in wall thickness (stiffness), the SPA starts to bend in one direction. This asymmetrical deformation causes the SPA to bend toward its fixed base, closely resembling the natural curvature produced by biological muscles. Such design ensures consistent and repeatable bending in a single plane, which makes it particularly suitable for applications like robotic fingers, soft grippers, and soft locomotion robots. The single-chambered configuration also simplifies fabrication and computational modeling, which provides a clearer understanding of how material properties and internal pressure affect bending behavior. Additionally, it allows for a more controlled comparison of different hyperelastic material models within the same structure and reduces the impact of geometric variations.

2.2. Finite Element Analysis

To accurately capture the nonlinear bending behavior of the single-chambered SPA, FEA simulation was carried out. The simulations utilized four widely recognized hyperelastic material models, namely Neo-Hookean, Yeoh, Arruda–Boyce, and Ogden. These models were selected due to their extensive use in representing silicone-based elastomers and their abilities to account for large deformations [25,26]. They are particularly effective under the uniaxial and biaxial stretching conditions, typically experienced by soft actuators. Specifically, the Neo-Hookean model is a fundamental hyperelastic model that assumes the material is incompressible and isotropic, which makes it well-suited for moderate deformation cases. Its strain energy density function is given by [27],

$$W={\displaystyle \frac{\mu }{2}}(\overline{I_1}-3)$$

(1)

where μ is the shear modulus and ($\overline{I_1}$) is the first deviatoric strain invariant. However, the Neo-Hookean model underestimates the material stiffness when subjected to higher strains. Furthermore, the Yeoh model is an extension of the polynomial model, which is designed to capture nonlinear behavior more accurately over large strains. Its strain energy density function depends only on the first invariant of the deformation tensor [28], which is given by:

$$W=C_{10}\left(\overline{I_1}-3\right)+{C_{20}\left(\overline{I_1}-3\right)}^2+{C_{30}\left(\overline{I_1}-3\right)}^3$$

(2)

This model works well for capturing the behavior of materials such as Ecoflex under uniaxial stretching and compression. It offers a good balance between accurate curve fitting and computational efficiency. In addition, the Arruda–Boyce model provides a more physically accurate representation of material behavior under large strains. This model is based on the statistical mechanics of polymer chains and is especially suitable for rubber-like materials subjected to multi-axial loads. The strain energy density function is given by [29]:

$$W= \mu [{\displaystyle \frac{1}{2}}\left(\overline{I_1}-3\right)+{\displaystyle \frac{1}{20}}\left({\stackrel{̿}{I}}_1^2-9\right)+{\displaystyle \frac{11}{1050}}\left({\stackrel{̿}{I}}_1^3-27\right)+···]$$

(3)

This model introduces limiting chain extensibility through the parameter λ_m, which provides enhanced accuracy when modeling large deformation responses in elastomeric actuators. Finally, the Ogden model is considered as one of the most versatile hyperelastic models, and describes the strain energy as a function of the principal stretch ratios. This makes it especially suitable for elastomers undergoing complex deformations. Its strain energy density function is given by [30]:

$$W={\sum }_{i=1}^N{\displaystyle \frac{{\mu }_i}{{\alpha }_i}}({\lambda }_1^{{\alpha }_i}+{\lambda }_2^{{\alpha }_i}+{\lambda }_3^{{\alpha }_i}-3)$$

(4)

where ${\mu }_i$ and ${\alpha }_i$ are material cosntants, λ₁, λ₂, and λ₃ are the principal stretch ratios, and N is the number of terms (usually 1 or 2 in practice). The Ogden model is particularly effective in fitting experimental data across multiple loading conditions, including tension, compression, and shear.

In this study, the SPA was modeled as a cantilever structure (Figure 2b), with one end fully fixed (zero displacement and rotation constraints) to replicate the mechanical anchoring of the base in practical application. The opposite end was free, and internal pressure was applied uniformly to the inner hollow chamber of the SPA. This loading condition creates an internal expansion force, resulting in asymmetric deformation due to the stiffness difference and thereby inducing bending. Finally, the SPA bending angle was measured manually using the configuration shown in Figure 2c. Additionally, the mesh was constructed using higher-order tetrahedral elements (Figure 2a) to ensure accuracy in capturing large deformations and stress concentrations, especially at the chamber corners and near the fixed boundary. The FEA was conducted under quasi-static conditions to focus purely on the material and geometric response without dynamic effects.

2.3. Analytical Modeling

To accurately predict the bending of the single-chambered SPA under pneumatic pressure, a mathematical model was formulated based on the Euler Bernoulli Beam concept [31]. It will provide a correlation between the input actuating pressure and output bending angle. The Euler theory states that plane sections remain plane even after the deformation and stay perpendicular to the neutral axis (N.A). This is specifically useful for long and slender actuators, as it provides closed-form expressions for curvature and angular deflection. The Euler theory serves as an initial approximation to estimate the bending angle, and the generalized form is given by [32]:

$${\varnothing }_{bending}= {\displaystyle \frac{ML}{EI}}$$

(5)

where M is the bending moment, L is the effective length, E is the elastic modulus, and I is the second moment of area of the hollow chamber. However, short structures are subjected to shear deformation, which is neglected by the Euler–Bernoulli theory. This is where the Timoshenko theory becomes relevant. It extends the Euler–Bernoulli concept by including a shear deformation term, which becomes significant in short actuators [33]. Thus, the shear strain term (shown in Figure 3a) is given by

$${\varnothing }_{shear}={\displaystyle \frac{K_t\times V\times L}{G\times A_S^{eff}}}$$

(6)

where V is shear force (given by V = Pressure × Chamber area), $K_t$ is the shear correction factor, G is the modulus of rigidity, and $A_S^{eff}$ is an effective shear area, which is given by $A_S^{eff}$ = t_w × h × n_w. In this expression, $t_w$ is wall thickness, h is height, and $n_w$ is the number of walls in the SPA. Here, the shear deformation in SPAs is far greater than what classical beam theory suggests. For example, the Timoshenko shear model gives relatively small values, but the actual measurements show much larger strains. This happens because of stress concentration at the edges of the chamber, non-uniform material behavior (the elastomer stretches more at thin regions), and geometric inconsistencies. Additionally, digital image correlation (DIC) tests clearly show that the shear at the wall edges can be 50 to 100 times higher than the theoretical predictions. This is mainly because of local stretching that is not considered in beam-type analytical models. DIC is a non-contact optical method of displacement measurement that requires only two digital images for the evaluation of displacements. In contrast to point measurement techniques, DIC can provide the complete in-plane displacement over a selected finite area of observation. In DIC, the correlation coefficient (C) measures how closely a region (subset) of the deformed image matches with its corresponding region in the undeformed reference image. When a speckle pattern is applied to a specimen surface and imaged before and after deformation, each small subset of pixels has a unique grayscale intensity distribution, denoted as F(x,y). After deformation, this subset moves to a new position (x*,y*) where the grayscale intensity distribution becomes F* (x*,y*). To determine the displacement field, DIC searches for the new subset position in the deformed image that produces the maximum correlation coefficient, meaning it best resembles the reference subset. Mathematically, the correlation coefficient (C) is calculated as

$$C={\displaystyle \frac{〈F{F}^{*}〉-〈F〉〈F^{*}〉}{\sqrt{\left[〈{\left(F^{*}-〈F^{*}〉\right)}^2〉\right] \left[〈{\left(F^{*}-〈F^{*}〉\right)}^2〉\right]}}}$$

(7)

where ⟨ ⟩ denotes the mean intensity within the subset. The value of C ranges between −1 and +1, where C = 1 indicates perfect correlation, which indicates a perfect linear relationship between the pixel intensities of the two images. In simpler terms, every pixel in the deformed subset has the same relative intensity pattern as in the original, which means DIC can track the displacement with absolute accuracy. Meanwhile, C = 0 indicates no similarity between the reference and deformed subsets. In practice, DIC analysis typically accepts correlation coefficients above 0.5–0.7 as reliable, with values below this range indicating significant decorrelation due to excessive deformation, lighting variation, or speckle loss. For easier interpretation in this study, the correlation coefficient was expressed in percentage form (e.g., C = 0.5 corresponds to 50%), representing the degree of similarity between corresponding subsets. A higher correlation percentage thus indicates stronger pixel-level consistency and more accurate strain measurement, which directly influence the determination of displacement gradients, strain fields, and related amplification parameters such as the shear amplification factor. Therefore, an amplification factor of 50 is considered for shear strain term [34]. The amplified shear strain term is given as

$${\varnothing }_{shear}=50\left({\displaystyle \frac{k_t\times V\times L}{G\times A_S^{eff}}}\right)$$

(8)

By adding Equations (5) and (8), the modified bending equation becomes

$$\varnothing =\varnothing + {\varnothing }_{shear}={\displaystyle \frac{ML}{EI}}+50\left({\displaystyle \frac{k_t\times V\times L}{G\times A_S^{eff}}}\right)$$

(9)

However, the shear correction factor ($k_t$) is a multiplier used in Timoshenko Beam theory [35], since the shear stresses in the beam section are not uniformly distributed. Therefore, as per the Timoshenko beam theory, $k_t$ is given by

$$k_t={\displaystyle \frac{12+11\nu }{10+10\nu }}$$

(10)

where v is Poisson’s ratio of the SPA material. Generally, the value of the shear correction factor ($k_t$) varies between 1.16 and 1.2 for SPAs with a wall thickness greater than 2 mm [36]. Here, the considered SPA material is Ecoflex OO-50 and its Poisson’s ratio is around 0.499 [37]. Thus, the ($k_t$) becomes 1.2 for our study. Furthermore, SPA walls are subjected to not only axial deformation but also radial expansion during inflation (shown in Figure 3c). This radial expansion is referred to as a ballooning effect in the SPA, which significantly alters the SPA geometry as well as the bending response. This ballooning effect is particularly observed in actuators made up of hyperelastic materials (such as Ecoflex, Dragon skin, and Elastosil), where the walls are thin and highly compliant. Thus, the effective or expanded height (h) of the chamber under actuating pressure (P) is given by [38]:

$$h=h_o(1+\alpha P)$$

(11)

where $h_o$ is the initial unpressurized height of the chamber, and α is the ballooning coefficient or radial expansion coefficient (empirical or derived from material testing). The value of α for Ecoflex OO-50 is 0.8 × 10⁻⁶ [37].

Another important consideration in modelling a soft actuator is the shifting of the neural layer or axis. In traditional beam bending, the neutral axis lies at the top surface of the lower layer and experiences zero strain during deformation. However, the distribution of pressure and material properties across chamber leads to asymmetric strain, which shifts the neutral axis slightly downward from its original position (shown in Figure 3b). This shift changes the lever arms of the internal pressure forces, which makes it necessary to redefine the distances from the new neutral axis to the chambers. This new distance (d_c) is given by [38]:

$$d_c={\displaystyle \frac{h}{k_t}}$$

(12)

where h is height of the SPA and $k_t$ is the shear correction factor. Thus, the moment generated due to the deformation of chamber ($M_C$) is given by

$$M_C=(P_C\times A_C\times d_C)$$

(13)

Finally, using Equation (12) in (8), the bending angle of the single-chambered SPA is given by

$$\varnothing ={\displaystyle \frac{\left(M_C\right)\mathrm{L}}{EI}}+50\left({\displaystyle \frac{K_t\times V\times L}{G\times A_S^{eff}}}\right)={\displaystyle \frac{(\mathrm{P}\mathrm{c}\times \mathrm{A}\mathrm{c}\times \mathrm{d}\mathrm{c})\mathrm{L}}{EI}}+50\left({\displaystyle \frac{K_t\times V\times L}{G\times A_S^{eff}}}\right) \mathrm{radians}$$

(14)

2.4. Data-Driven Modeling

Among the number of various ML models, the polynomial regression model was chosen for this analysis because of its capability to cope with non-linear functions within small datasets. This model enhances linear regression by including higher-order terms of the independent variable, which enables it to accurately represent the non-linear deformation characteristics of SPAs resulting from hyperelastic properties of materials and significant geometric deformations. The general formulation for a polynomial regression of degree n is expressed as follows [39,40]:

$$y={\beta }_o+{\beta }_{1}x+{\beta }_2{x}^2+{\beta }_3{x}^3+\cdots +{\beta }_n{x}^n+ϵ$$

(15)

where y represents bending angle, x represents applied pressure, β_i represents polynomial coefficients, and ϵ represents the error term. The selection of the polynomial order was determined through a process of iterative experimentation, which aimed to strike a balance between accuracy and model complexity to eliminate the risk of overfitting. In addition, hyperparameter optimization was conducted via a grid search methodology, complemented by the implementation of a k-fold cross-validation strategy (k = 5) to ensure a robust assessment of performance. This methodology guaranteed that every subset of the dataset was utilized for both training and validation purposes, thereby minimizing bias and variance in performance evaluation.

The bending results generated from analytical prediction have been considered for the training and evaluation of the model. The ML model covered an actuating pressure ranging from 1 to 25 kPa as the training dataset, by keeping all geometric parameters as constant. This was done to focus particularly on the effect of actuator pressure on SPA deformation. Further, to test its ability to extrapolate, the model was then employed to predict bending angles for extended pressure, from 26 to 30 kPa. Figure 4 describes the workflow used for the data-driven modeling approach. During feature selection, the applied pressure (P) was chosen as the primary input feature and the bending angle (θ) as the output feature. This selection was made because pressure directly affects the SPA deformation, while other parameters (such as geometry and material properties) were kept constant to clearly capture the nonlinear relationship between pressure and bending. Therefore, a sensitivity analysis was not conducted in this study, as the influence of additional features was outside the scope of the present study. However, we acknowledge that incorporating a multi-feature sensitivity analysis (such as, varying wall thickness, chamber length, or material stiffness) could further enhance the robustness of the model and will be considered in future work. Additionally, the performance of the polynomial regression model was evaluated using several statistical metrics, such as mean absolute error (MAE), root mean square error (RMSE), and the coefficient of determination (R²). The MAE measures the average deviation of predictions, RMSE captures the impact of larger errors, and R² quantifies how well the model explains variations in bending angles. Collectively, these metrics show that the trained polynomial regression model provides reliable and generalized predictions, which offers a computationally efficient alternative to repeated analytical and FEA simulations.

3. Results

This section covers the key results from the analysis performed under the methods and analysis section of the paper.

3.1. Finite Element Analysis (FEA)

The FEA simulation was carried out using Abaqus 6.14, which is a widely employed multi-domain finite element software. The single-chambered SPA is assumed to be made of Ecoflex-OO50, which is selected for its flexibility and suitability for soft robotic applications. The SPA was analyzed under cantilever beam conditions, where one edge was constrained from any motion while the remaining body could freely deform under varying actuating pressure. The hollow chamber of the SPA was subjected to a maximum actuating pressure of 30 kPa and the resulting bending angles were measured as per the configuration shown in Figure 2c. Further, the bending angles and maximum stress experienced by four different hyperelastic material models at pneumatic pressures of 5 and 30 kPa (Neo-Hookean (Figure 5), Yeoh (Figure 6), Arruda–Boyce (Figure 7), and Ogden (Figure 8)) were compared with each other, as shown in Figure 9. However, to accurately define the bending response of Ecoflex-OO50 across these models, material constants from the literature were considered [18,19,20,21]. For the Neo-Hookean model, a single material constant of C10 = 22.4 kPa was considered. The Yeoh model consists of higher-order terms to capture larger deformations; thus, the material constants are C10 = 21.9 kPa, C20 = 0.069 kPa, and C30 = 0.0167 kPa. The Arruda–Boyce model is based on molecular chain network theory, where the shear modulus was defined as μ = 36.1 kPa and the limiting network stretch parameter was set to λ_m = 2.42. Finally, the Ogden model is known for its flexibility in fitting complex deformation behaviors. Therefore, the material constants are ${\mu }_1$ = −37.1 kPa, ${\mu }_2$ = 23.1 kPa, ${\mu }_3$ = 70.2 kPa, ${\alpha }_1$ = 1.63, ${\alpha }_2$ = 3.36, and ${\alpha }_3$ = −2.92.

3.2. Analytical Modeling

In the analytical modeling, both Euler–Bernoulli Beam theory and Timoshenko Beam theory were considered to capture the combined effects of bending moments caused by pressure forces as well as shear deformations. Certain important assumptions were also included, such as the shear deformation, shifting of the neutral layer, and ballooning effect that occurs under high pressurization of the pneumatic actuator. Normally, the neutral layer passes through the centroid of the actuator body; however, due to ballooning, it shifts downward toward the bottom layer of the actuator. Furthermore, the estimated shear deformation is significantly smaller than the actual deformation observed through digital image correlation (DIC), as per the Timoshenko Beam theory. To account for this difference, a shear amplification factor was introduced, typically ranging between 50 and 100. In addition, the shear correction factor (k_t) was used as a multiplier in the Timoshenko model, since shear stresses within the beam cross-section are not uniformly distributed. Therefore, by applying this analytical model with considered assumptions, the bending response of the SPA was evaluated across a pressure range of 1–30 kPa (shown in Figure 10), yielding the bending deformation values for the given loading conditions.

Moreover, among all four hyperelastic constitutive models applied in the FEA, the Arruda–Boyce model produced the highest bending angle (shown in Figure 9a). For this reason, its results were further compared with the analytical bending outcomes to evaluate whether both modeling approaches yield consistent results (shown in Figure 11a). At lower pressures, such as 5 kPa, the absolute percentage deviation was found to be as high as 173.5%. However, as the pressure increased, the deviation gradually reduced, reaching a minimum of 2.1% at 30 kPa (shown in Figure 11b). The absolute percentage deviation was determined using the following relation:

$$Absolute Percent Deviation= {\displaystyle \frac{\left|{FEA}_{bending}-{Analytical}_{bending}\right|}{{FEA}_{bending}}}\times 100$$

(16)

3.3. Data Driven Model

The polynomial regression model was built using data from the analytical bending predictions, which cover pressures between 1 and 30 kPa, and the geometric parameters of the single chambered SPA design. To check how well the model could generalize, the data was split into two parts, 1–25 kPa for training and 26–30 kPa for validation. During hyperparameter tuning, polynomial degrees from 2 to 5 were tested, and the cubic model (degree 3) was found to give the best balance between accuracy and complexity. After training (shown as blue points in Figure 12), the model was used to predict bending in the 26–30 kPa range, with the extrapolated results shown as red points in Figure 12. However, the bending pressure curve (shown in Figure 12) is linear because the dataset chosen for the validation of the polynomial regression ML model was based on the results obtained from the analytical model. In analytical modeling, it is assumed that the bending response is approximately proportional to the applied pressure and that, within this pressure range, the nonlinear effects of shear deformation, ballooning, and material strain-stiffening remain moderate. This causes the SPA to behave almost linearly from 1–25 kPa, and the extrapolated region (26–30 kPa) also follows this near-linear trend because it lies just beyond the training range and does not yet reach the highly nonlinear strain-stiffening zone of the material.

The ML-predicted bending values closely matched the analytical estimations (shown in

Table 1. Comparison between polynomial regression and analytically predicted bending results (26–30kPa) with absolute percentage deviation.

S. No.	Pressure (kPa)	Bending Angle (Degrees)		Absolute Percentage Deviation (%)
		Analytical	ML Model
1	26	85.61	85.607	0.004
2	27	88.91	88.913	0.003
3	28	92.22	92.221	0.001
4	29	95.53	95.529	0.001
5	30	98.84	98.839	0.001

), with an average absolute percentage deviation of 0.002%. This demonstrates the reliability of the polynomial regression model for extrapolation beyond the training range. For example, at 28 kPa, the ML model predicted a bending angle of 92.221°, while the analytical approach generated 92.220°. This agreement suggests that the polynomial regression model can serve as a fastest yet accurate predictive tool for SPA bending behavior when experimental or simulation or analytical measurements are limited. Additionally, the accuracy of the ML-based predictions was quantitatively evaluated using the mean absolute error (MAE), root mean squared error (RMSE), and the coefficient of determination (R²) between the predicted and analytical bending angles for pressures ranging from 26–30 kPa. These values can be evaluated using the relation

$$MAE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left|y_i-\widehat{y_i}\right|$$

(17)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}}{\sum }_{i=1}^n{(y_i-\widehat{y_i})}^2$$

(18)

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{(y_i-\widehat{y_i})}^2}{{\sum }_{i=1}^n{(y_i-\overline{y_i})}^2}}$$

(19)

where $y_i$ is actual or analytical bending value, ($\widehat{y_i}$) is the ML-predicted bending value, ($\overline{y_i}$) is the mean of the actual or analytical bending values, and n = number of samples. The results generated an MAE of 0.00180°, RMSE of 0.00205°, and an R² value of 0.999999808, which indicates an almost perfect agreement between the ML-predicted and analytical values. This minimum deviation confirms the robustness and reliability of the proposed predictive model in predicting the bending behavior of the single-chambered SPA with high precision.

4. Discussion

This study demonstrates how combining FEA, analytical modeling, and ML provides a rigorous framework for examining the bending response of single-chamber SPAs. Each technique offers unique advantages; for example, FEA provided high-fidelity predictions of bending deformation under different constitutive models, analytical modeling incorporated essential physical corrections for practical usability, and ML introduced a fast surrogate for real-time predictive applications. Collectively, these methods create a holistic understanding of SPA mechanics that is both computationally efficient and scientifically robust. FEA revealed that the selection of the hyperelastic model substantially influences the predicted bending angle. At 30 kPa, the Arruda–Boyce model developed the largest bending of 101°, whereas the Neo-Hookean model predicted only 84°, underestimating deformation at higher strains. The Yeoh and Ogden models predicted intermediate values of 93° and 96°, respectively. These variations underscore the fact that simple models, while computationally less expensive, may fail to capture strain-stiffening behavior at large deformations. The superior performance of the Arruda–Boyce formulation reflects its molecular chain-based representation, which accounts for the limited extensibility of elastomer networks [22,23]. These results confirm that constitutive model selection plays a decisive role in determining actuator reliability and should therefore guide the design process in soft robotics. Furthermore, the analytical model presented here extends classical beam formulations by including neutral axis shifting, ballooning-induced geometric changes, and shear amplification. Without these corrections, the Euler–Bernoulli beam approximation underestimated bending by nearly 15–20% at low pressures. The incorporation of Timoshenko shear deformation (amplified by an empirical factor) improved agreement with FEA at higher pressures, while ballooning effects significantly influenced chamber curvature. For example, the analytical prediction of 99° bending at 30 kPa closely matched the Arruda–Boyce FEA result of 101°, which validates the necessity of these modifications. Such improvements highlight the potential of refined beam theories to provide accurate yet computationally light alternatives for SPA design [31,32].

Moreover, the ML-based polynomial regression model replicated analytical predictions with remarkable accuracy. Across the validation range (26–30 kPa), the average absolute percentage deviation was just 0.002%, which effectively reproduces bending angles with negligible error. This demonstrates that even relatively simple regression methods can function as highly efficient surrogate models, making them suitable for rapid optimization or embedded real-time control. The success of this data-driven approach aligns with recent studies advocating for hybrid strategies that couple physics-based principles with AI for accelerating the design and deployment of soft robots [37,38]. Importantly, extending training with experimental data could transform ML into a predictive substitute for both FEA and analytical solvers in future applications. The integration of these approaches carries several important implications for soft robotics research. The comparative FEA results provide a benchmark for selecting constitutive models depending on the application requirements and balancing accuracy against computational resources. The modified analytical model offers a middle ground between theoretical simplicity and predictive reliability, which makes it particularly useful during the early stages of SPA design. Meanwhile, ML delivers the speed required for real-time control, an essential feature for adaptive robotic systems such as wearable rehabilitation devices and delicate food-handling grippers [20,22]. Overall, this hybrid framework reduces trial-and-error prototyping, reduces material waste, and opens new avenues for more systematic SPA design. However, the study has certain limitations despite its contributions. The FEA simulation depends mainly on material constants obtained from the literature rather than direct experimental characterization. This may introduce discrepancies in absolute bending predictions. Further, the analytical corrections for ballooning and shear deformation employed empirical amplification factors, which may vary with actuator geometry or elastomer grade. Finally, the ML model was trained solely on analytically generated datasets, which require validation against experimental measurements to ensure robustness in real-world conditions.

5. Conclusions

This article provides a detailed evaluation of the bending behavior of a single-chambered SPA by integrating a physics-based (FEA and analytical modeling) method with a data-driven (polynomial regression ML model) approach. Initially, the FEA simulation demonstrated that the type of hyperelastic material model strongly affects the SPA performance. Specifically, the Arruda–Boyce model predicted a maximum bending of 101° at 30 kPa by capturing strain-stiffening behavior more accurately than the Neo-Hookean, Yeoh, and Ogden models. Furthermore, the developed analytical model based on extended classical beam theories consider several assumptions, such as shear deformation, the ballooning effect, and neutral layer shift. By considering these non-linear effects, the analytical bending results come in close approximation with the FEA bending results. Specifically, this is evident at higher actuating pressure values where traditional beam theories show 15 to 20% deviations from FEA. For example, the analytical model predicted 99° bending at 30 kPa, which is approximately similar to FEA bending (i.e., 101°) at the same pressure. This validates the effectiveness of the considered assumptions. Therefore, it can be concluded that the analytical approach provides a computationally efficient and physically meaningful tool for predicting SPA bending, which eliminates the gap between theoretical models and time-consuming simulations. Finally, a polynomial regression ML model has been used to highlight the potential of the data-driven approach as a reliable alternative prediction tool. The ML model effectively reproduces analytical predictions with an average absolute percentage deviation of 0.002% in the extrapolated actuating pressure range of 26–30 kPa. Furthermore, in comparison, the results generated an MAE of 0.00180°, RMSE of 0.00205°, and an R² value of 0.999999808, which shows that the bending results predicted by the data-driven ML model are in almost perfect agreement with the analytically predicted bending. This demonstrates the advantages of a hybrid framework that combines physics-based modeling with data-driven learning for real-time control, design optimization, and adaptive soft robotic applications.

Furthermore, the proposed hybrid framework can be effectively applied to the design and analysis of various soft robotic systems where controlled deformation and precise motion prediction are essential. Its adaptability makes it suitable for single and multi-chambered SPAs used in soft grippers, assistive wearable devices, and biomedical robots. This hybrid framework is also extendable to real-time control and adaptive learning environments, where quick feedback and optimization are required. Moreover, this method provides a foundation for exploring the influence of geometry, material composition, and actuation pressure on complex deformation modes, thereby supporting performance tuning and design optimization for task-specific applications. However, the applicability of the proposed hybrid framework is presently constrained by several factors. For example, the dependency on predefined material constants from the literature may introduce deviations when applied to custom-fabricated elastomers with different mechanical characteristics. Additionally, the empirical assumptions used to represent ballooning and shear deformation may limit precision under high-strain or cyclic loading conditions. Moreover, the ML model is trained solely on analytically generated data, which restricts its generalization to unmodeled behaviors, such as viscoelastic damping, hysteresis, or material fatigue. Therefore, to broaden the scope and reliability of the method, future work should include experimental calibration, dynamic loading validation, and multi-physics coupling for enhanced predictive robustness.