Figure 2 illustrates the structural configuration of the soft actuator used for model validation. The actuator is cast from silicone, and circular surface grooves are subsequently patterned via laser engraving. These grooves enable the integration of Kevlar fibers (DuPont, USA), which reinforce the structure and constrain undesired deformation^[27]. An inextensible layer is integrated into the base of the soft actuator to constrain longitudinal expansion during pressurization, thereby enabling localized control over its nonlinear mechanical behavior. Furthermore, to optimize performance, the fiber helix angle should ideally approach 90°.

Due to the inherent hyperelasticity of the silicone elastomer, the actuator exhibits pronounced nonlinear deformation behavior. Consequently, the moment-curvature equation is employed to characterize its bending response. The application of internal pressure P generates a resultant thrust force F on the actuator’s cross-section. The relationship between the input pressure and the resulting output force is governed by the actuator’s geometry.

(1) $$ dF=P(r\mathrm{cos}\psi)d(r\mathrm{sin}\psi)=Pr^2\mathrm{cos}^2\psi d\psi $$

where r is the inner radius of the soft actuator and θ is the angle between r and the x-axis. The bending moment M is derived from the resultant moment of the normal stresses σ_x acting over the cross-sectional area S. Accordingly, M is defined as:

(2) $$ M=\int _S\sigma _xydS=\int_0^{2\pi }(r(\mathrm{sin}\psi+1)+r_t)dF $$

From the fundamental principle of mechanics, the curvature k is proportional to the bending moment M, as expressed in the Euler-Bernoulli framework^[28].

(3) $$ M=kEI $$

where EI is the flexural rigidity, R = $$ \frac{1}{k} $$ represents the radius of curvature, and I = ∫_Ay²dS denotes the second moment of area. To facilitate the integration with Lagrangian dynamics, Equation (3) is reformulated in terms of the bending angle θ (see Figure 3):

(4) $$ k=\frac{1}{R}=\frac{1}{\frac{L}{\theta}}\Rightarrow \theta=\frac{ML}{EI} $$

where L is the effective length of the soft actuator.

To facilitate the formulation of the reduced-order dynamic model, three key physical assumptions are introduced and justified. The first assumption is that the volume of the elastomer remains invariant throughout the deformation process. This is physically grounded in the near-incompressibility inherent to the silicone material used for the soft actuator, which is characterized by a Poisson’s ratio of ν ≈ 0.5.

The second assumption is that the deformation of the soft actuator is restricted to a single bending plane. This constraint is kinematically enforced by the structural design; specifically, the embedded Kevlar fibers and the inextensible bottom layer effectively suppress radial expansion and out-of-plane twisting.

The third assumption posits that the soft actuator undergoes constant-curvature deformation. While this geometric approximation is highly effective and physically valid at low-to-moderate actuation pressures, it presents inherent limitations under extreme conditions. Specifically, as the input pressure approaches 90 kPa, the silicone elastomer enters a highly nonlinear hyperelastic regime. Under high-pressure conditions, unmodeled localized phenomena, such as the ballooning effect and minor fiber slippage, may arise, causing the actuator profile to deviate from the ideal constant-curvature assumption. This modeling mismatch constitutes a major source of the increased prediction error observed at elevated pressures. Consequently, the proposed model achieves its highest predictive accuracy within the low-to-moderate pressure operating range.

To construct the Lagrangian dynamic model^[29] for the soft actuator, it is essential to consider both kinetic energy T and potential energy V. The primary forms of these energies are as follows:

(5) $$ \ell =\mathbf{T}-\mathbf{V}=\mathbf{T}-(S_E+G_P) $$

Where ℓ represents the Lagrangian of the system. S_E and G_P represent strain energy and gravitational potential energy, respectively. By applying the Lagrangian function to the soft actuator, Equation (6) can be derived^[26].

(6) $$ \frac{d}{dt}\frac{\partial \ell }{\partial \dot{q}_i}-\frac{\partial \ell }{\partial q_i}=Q_i\Leftrightarrow \frac{d}{dt}\frac{\partial \ell }{\partial \dot{\theta}}-\frac{\partial \ell }{\partial \theta}=Q_i $$

Where Q_i denotes a non-conservative generalized force related to P. Based on the principle of virtual work, the generalized force Q_i is derived as:

(7) $$ \left\{\begin{matrix} Q_i=P\frac{\partial V_f(\theta )}{\partial \theta}\\ V_f(\theta )=V_t(\theta )-V_e \end{matrix}\right. $$

where V_f(θ) represents the volume of the internal channels of the soft actuator, V_t(θ) is the total volume of soft actuators, which can be obtained from Equation (8).

(8) $$ \begin{array}{l} V_{t}(\theta)=\int_{0}^{\theta} r_{o} \cos \psi d(\sin \psi) \int_{0}^{2 \pi}\left(R+r_{o}+r_{o} \sin \psi\right) d \alpha \\ =\int_{0}^{2 \pi} \int_{0}^{\theta} r_{o}^{2} \cos ^{2} \psi\left(R+r_{o}+r_{o} \sin \psi\right) d \alpha d \psi \\ =\int_{0}^{2 \pi} \int_{0}^{\theta} r_{o}^{2} \cos ^{2} \psi\left(R+r_{o}\right)+r_{o}^{2} \cos ^{2} \psi\left(r_{o} \sin \psi\right) d \alpha d \psi \\ =\int_{0}^{2 \pi} \int_{0}^{\theta} r_{o}^{2} \cos ^{2} \psi\left(R+r_{o}\right)+r_{o}^{3} \cos ^{2} \psi(\sin \psi) d \alpha d \psi \\ =\int_{0}^{2 \pi} r_{o}^{2} \cos ^{2} \psi\left(L+r_{o} \theta\right)+\theta r_{o}^{3} \cos ^{2} \psi(\sin \psi) d \psi \\ =\pi \theta r_{o}^{3}+\pi r_{o}^{2} L \end{array} $$

Under the assumption of silicone incompressibility, the elastomer volume V_e remains constant throughout the deformation process and can be expressed by Equation (9).

(9) $$ V_e=(\pi r_o^2-\pi r_i^2)L=\pi L(r_o^2-r_i^2) $$

The strain energy S_E of soft actuator is determined by integrating the strain energy density U₀ over the elastomer volume V_e. For the silicone elastomer, U₀ is defined using the Neo-Hookean model as 0.5G(I₁ - 3), where G is the shear modulus of the soft actuators, I₁ is the first invariant of the Cauchy–Green strain tensor. Given the radial constraints imposed by the fiber reinforcement, we assume λ₂ = 1 and λ₃ = $$ \frac{1}{\lambda_1} $$.

The strain energy of soft actuator S_E can be calculated by the following expression:

(10) $$ \begin{array}{l} S_{E}=\int_{V_{e}} \frac{1}{2} G\left(I_{1}-3\right) d V \\ =\int_{V_{e}} \frac{1}{2} G\left(\lambda_{1}^{2}+\frac{1}{\lambda_{1}^{2}}-2\right) d V \\ =\int_{0}^{r_{t}} \int_{0}^{2 \pi} \frac{G}{2}\left(\lambda_{1}^{2}+\frac{1}{\lambda_{1}^{2}}-2\right)(r+\tau) L d \psi d \tau \end{array} $$

Accordingly, the axial strain λ₁ is determined by the bending angle θ and the associated geometric parameters:

(11) $$ \lambda _1=\left ( \frac{((r+\tau)\mathrm{sin}\psi +r)\theta} {R\theta } \right ) =1+\frac{((r+\tau )\mathrm{sin}\psi +r)\theta }{L} $$

To accurately model the gravitational potential energy in underwater environments, the coupled effects of gravity and buoyancy must be considered. As both the surrounding medium and internal driving fluid are water, the net body force arises from the density difference between the silicone elastomer and the surrounding fluid. Under the assumption of constant actuator volume during deformation, the effective weight is expressed in terms of this density difference, as given in Equation (12).

(12) $$ \left\{\begin{matrix} m_sg=\rho _sV_eg\\ F_f=\rho_wV_eg \end{matrix}\right. \Rightarrow mg=\rho_sV_eg-\rho_wV_eg=(\rho_s-\rho_w)V_eg $$

Where m_s, ρ_s represent the mass and density of the soft actuator, g represents gravitational acceleration, F_f and ρ_w represent the buoyancy force and the density of water, respectively. Thus, the calculation method for gravitational potential energy G_P is as follows:

(13) $$ G_p=-\frac{mg}{L}\int_{0}^{\theta }(R-R\mathrm{cos}\alpha)Rd\alpha =-\frac{mgL}{\theta ^2}\int_0^{\theta}(1-\mathrm{cos}\alpha )d\alpha $$

The tip position of the soft actuator is given by Equation (14).

(14) $$ \left\{\begin{matrix} \begin{aligned} x(s)&=R\mathrm{sin}\theta (s)=R\mathrm{sin}\frac{s}{R} \\ y(s)&=-R(1-\mathrm{cos}\theta (s))=R\left (1-\mathrm{cos}\frac{s}{R} \right ) \end{aligned} \end{matrix}\right. $$

where s denotes the arc length along the actuator’s centerline, and θ(s) represents the bending angle associated with segment s, as illustrated in Figure 3. Thus, the derived formula for kinetic energy is as follows:

(15) $$ \begin{aligned} \mathbf{T} &=\frac{1}{2}mv^2 \\ &=\frac{m}{2L}\int_{0}^{L} \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 ds \\ &=\int_{0}^{L} \left( \frac{dx}{d\theta} \frac{d\theta}{dt} \right)^2 + \frac{m}{2L} \left( \frac{dy}{d\theta} \frac{d\theta}{dt} \right)^2 ds \\ &=mL^2\dot{\theta}^{\,2} \left( \frac{1}{6\theta^2} +\frac{1}{\theta^4}(1+\cos\theta) -\frac{2}{\theta^5}\sin\theta \right) \end{aligned} $$

Although all necessary parameters for the dynamic model have been defined, the inherent strong nonlinearity of these equations makes them unsuitable for high-frequency, real-time implementation.

To improve the computational efficiency of the dynamic model, a Taylor-series-based numerical approximation is employed. The nonlinear terms in the governing equations are truncated at the fifth order to balance modeling accuracy and computational cost. While higher-order terms offer marginal precision gains, they introduce disproportionate calculation burdens that would preclude high-frequency execution. Conversely, the fifth-order truncation effectively preserves the dominant nonlinearities of the system while ensuring the low-latency response required for real-time control applications. The detailed implementation is as follows:

(16) $$ \left\{\begin{matrix} \begin{aligned} (1+ax)^{-2}&=1-2ax+3(ax)^2-4(ax)^3+\cdots \\ \mathrm{cos}x&=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots \\ \mathrm{sin}x&=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots \end{aligned} \end{matrix}\right. $$

By applying the aforementioned Taylor series expansion, the kinetic energy T, gravitational potential energy G_P, and strain energy S_E are approximated and simplified as follows:

(17) $$ \left\{ \begin{aligned} \mathbf{T} &\approx \left( \frac{1}{40} -\frac{\theta^{2}}{1008} \right) mL^{2} \left(\frac{d\theta}{dt}\right)^{2} \\ G_{P} &= -\frac{mgL}{\theta^{2}} \int_{0}^{\theta}(1-\cos\alpha)\,d\alpha = -\frac{mgL}{\theta^{2}} (\theta-\sin\theta) \approx -mgL \left( \frac{\theta}{6} -\frac{\theta^{3}}{40} +\frac{\theta^{5}}{5040} \right) \\[4pt] S_{E} &\approx \frac{2G\theta^{2}}{L^{2}} \left( \frac{5\pi}{4}r_{o}^{4} -\frac{\pi}{4}r_{i}^{4} -\pi r_{o}^{2}r_{i}^{2} \right)\\ \frac{d}{dt} \frac{\partial\ell }{\partial \dot{\theta}} &\approx \left( \frac{1}{20} -\frac{\theta^{2}}{504} \right) mL^{2}\frac{d^{2}\theta}{d^{2}t} -\frac{\theta}{252} mL^{2} \left( \frac{d\theta}{dt} \right)^{2} \end{aligned} \right. $$

The truncation order for the Taylor series expansion was selected to balance mathematical convergence with computational efficiency. Given that the exact nonlinear integrals governing the actuator’s system energies lack closed-form analytical solutions suitable for real-time control, the model leverages the rapid convergence of the Taylor series to maintain high-fidelity approximations. At the maximum experimental bending angle (θ_max at 90 kPa), the coefficients of the retained terms (e.g., $$ \frac{1}{5040} $$ for θ⁵) indicate that subsequent higher-order terms contribute negligibly to the total system energy. Retaining higher-order terms would exponentially increase the computational burden of the ODE45 numerical solver without meaningfully improving precision.

Next, to obtain the numerical solution for Equation (6), we need to establish the boundary conditions.

Boundary Condition 1: As shown in Figure 3, the initial segment of the soft actuator is fixed, so its curvature is zero, θ = 0.

Boundary Condition 2: Assuming no external point load is applied to the distal tip of the soft actuator, the endpoint curvature is primarily governed by the internal pressure-induced moment. Consequently, the curvature at the free end can be directly determined from the input pressure P via the moment-curvature relationship expressed in Equation (4).

By combining the moment–curvature equation with the Lagrangian dynamics formulation, Equation (6) is solved numerically using a shooting method. The resulting nonlinear system is implemented in MATLAB using the built-in functions fsolve and ode45.

The specific computational framework is as follows:Initial Guess: To initiate the iterative solving process, a strategically selected initial guess is provided to the numerical solver to ensure robust convergence.Numerical Integration via ODE45: The governing differential equations are numerically integrated using the ode45 solver, starting from the estimated initial parameters across the spatial domain.Iterative Boundary Matching: The fsolve algorithm is used to iteratively update the unknown initial conditions. The iterations proceed until the integrated terminal states satisfy the prescribed boundary conditions, minimizing the boundary residuals.

Before validating the proposed model, it is necessary to calibrate the material parameters used to fabricate the soft actuator. Previous studies indicate that the material parameters EI of soft actuators are nonlinear, and their values change with variations in input pressure^[30,31]. To obtain accurate values for EI, we collected shape and position data of the soft actuators with pressures ranging from 10 to 110 kPa. To facilitate parameter estimation, and given the annular geometry of the soft actuator’s cross-section, the second moment of area I is defined as π$$ \left ( \frac{r_o^4-r_i^4}{4} \right ) $$. Finally, by matching the model’s simulation data with the experimental results of the soft actuator, we can derive the fitting formula for how the parameter E of the soft actuator changes with input pressure P.

To calibrate the model under realistic experimental conditions, an input pressure range of 10-110 kPa was defined. The resulting trajectory fitting is illustrated in Figure 4, where M-Trajectory and T-Trajectory denote the predicted deformation and the actual observed data, respectively. Based on these fitting results, the empirical formula characterizing the Young’s modulus E of the actuator is formulated as shown in Equation (18).

(18) $$ E=1.795*10^{-19}*P^5-6.36*10^{-14}*P^4+8.535*10^{-9}*P^3-5.566*10^{-4}*P^2+15.97*P+6.562*10^5 $$

where the unit of Young’s modulus E is MPa, the unit of P is Pa.

While physically motivated hyperelastic constitutive models capture the fundamental mechanics of silicone elastomers, their integration into dynamic ODE solvers significantly increases computational cost. As a result, real-time control becomes infeasible. To maximize computational efficiency, an empirical lumped-parameter algebraic mapping was utilized [Equation (18)]. This 5th-order polynomial was calibrated over the 10-110 kPa range, achieving a high goodness-of-fit with an R-squared of 0.9848.

To verify the accuracy of the model presented in this study, we selected dynamic data with input pressures ranging from 20 to 90 kPa, as shown in Figure 5. The input form of P is as follows:

(19) $$ P=A\mathrm{sin}\left ( \frac{2\pi }{T}t \right ) $$

where A represents the amplitude of the pressure P, has a range of values between 0 and 90 kPa in the actual experiment, as shown in Figure 5A-D. The pressure is measured using a hydraulic sensor, with data recorded every 0.5 s. T represents the period, and t represents the time. The input pressure control frequency ranges between 0.1 and 0.2 Hz.

The model validation experiment was performed for input pressures ranging from 0 to 20 kPa. Figure 6A and B display the simulated and actual movement trajectories of the soft actuator’s end in the x and y directions, respectively. Figure 6C and D present the error rates in these directions, with the experimental data denoted by Exp_Data and the model simulation data denoted by M_Data. As observed in Figure 6A and B, there is a high degree of overlap between the simulated and experimental trajectories, indicating good agreement between the model and actual performance. To objectively evaluate the effectiveness of the model, we define the error rate E_Rate calculation formula as follows:

(20) $$ E_{Rate}=\frac{||M_{Date}-Exp_{Data}||}{L}\times 100\% $$

The error rates calculated using Equation (20) are shown in Figure 6C and D.

The proposed model demonstrates a maximum relative error rate of less than 1.17% in the x-direction and less than 3.96% in the y-direction, normalized by the total length of the soft actuator. Furthermore, the model achieved a mean computational time of 0.0094 s per step, effectively fulfilling the stringent low-latency requirements for real-time implementation.

In the second validation experiment, the dynamic response of the soft actuator was further evaluated by increasing the pressure amplitude to 50 kPa and adjusting the input frequency. As illustrated in Figure 7A and B, the proposed model exhibits robust tracking fidelity across both the x and y directions. Quantitative results in Figure 7C and D reveal that the peak relative errors are constrained within 4.55% and 5.15% respectively. These findings confirm that the model maintains consistent performance even under elevated input pressures. Furthermore, the mean computational latency per iteration under these conditions was recorded at 0.0127 s.

In the third experiment, the actuation pressure range was extended to 80 kPa, while the pressure control frequency was also appropriately adjusted. Figure 8A and B illustrates the trajectory tracking performance of the soft actuator tip under the above experimental conditions. These figures show the effectiveness of the model in tracking the desired trajectories over the extended pressure range. Furthermore, Figure 8C and D illustrate that the proposed model restricts the peak relative prediction errors for the soft actuator’s end trajectory to under 7.08% in the x-direction and 5.99% in the y-direction. Under the experimental condition, the average time required to solve each step is 0.0122 s.

In the fourth validation scenario, the input pressure reached 90 kPa to assess the model’s robustness under large hyperelastic deformations. The corresponding trajectory tracking results are depicted in Figure 9A and B. Specifically, Figure 9C and D demonstrate that the proposed model maintains a peak relative tracking error of less than 9.23% in the x-direction and 5.88% in the y-direction. Despite the presence of unmodeled local deformations at this pressure level, the results indicate that the model can accurately predict the actuator’s spatial trajectory with acceptable error levels. Furthermore, the mean computational latency remained low at 0.015 s per iteration.

In addition to visual comparisons and peak errors, the mean absolute error (MAE) and root mean square error (RMSE) were calculated for both the x and y trajectories across all four validation experiments (20, 50, 80, and 90 kPa). These statistical metrics are summarized in Table 1, providing a clear and objective assessment of the model’s tracking reliability.

As indicated by the statistical metrics, the model exhibits excellent tracking accuracy and repeatability under low to moderate pressures (0-80 kPa). However, a noticeable increase in the prediction error is observed at the extreme pressure of 90 kPa, where the peak error reaches 9.23%. This deviation stems from the inherent physical limitations of the assumed kinematic geometry. At lower pressures, the soft actuator undergoes quasi-linear deformation, adhering strictly to the constant-curvature assumption. Conversely, as the internal pressure approaches 90 kPa, the silicone elastomer enters a highly nonlinear hyperelastic regime. Extreme pressurization induces unmodeled local deformations, such as slight radial expansion despite the fiber constraints, as well as visco-elastic hysteresis. These phenomena alter the local stiffness and curvature, causing the actuator’s actual shape to deviate from an ideal uniform curvature. Recognizing this physical boundary is crucial, as it defines the optimal and reliable operating range (0-80 kPa) for the proposed physics-informed reduced-order model. Although direct quantitative comparison of tracking errors with established approaches such as FEM or Cosserat rod models is often hindered by differences in experimental configurations, their computational trade-offs are widely acknowledged in the literature. Purely numerical methodologies, particularly FEM-based frameworks, offer exceptional theoretical fidelity by capturing high-dimensional degrees of freedom; however, their execution times typically range from seconds to minutes per step, confining them to offline design rather than real-time control. Similarly, Cosserat rod models effectively resolve 3D continuum dynamics. However, solving the underlying partial differential equations introduces latency on the order of tens to hundreds of milliseconds, limiting their suitability for high-frequency control loops. Furthermore, even analytical Lagrangian frameworks can encounter computational bottlenecks when integrated with non-linear environmental forces, such as buoyancy and hydrodynamic drag.

In contrast, the physics-informed reduced-order model proposed in this study is specifically optimized to break this computational bottleneck. By utilizing Taylor series expansion to reduce the model’s order, the proposed method achieves an extremely efficient execution time of approximately 0.015 s per step. Although this profound simplification results in a bounded peak error of 9.23% under extreme hyperelastic deformations (90 kPa), it successfully fulfills the stringent low-latency requirements of dynamic closed-loop systems. This establishes a highly pragmatic trade-off, prioritizing extreme computational efficiency while maintaining an operationally acceptable tracking accuracy for soft actuators in complex environments.