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Abstract

Soft pneumatic actuators (SPAs) typically offer a fixed trajectory, resulting in one specific tip motion for a given range of pressure. When multiple trajectories are needed, these actuators require re-fabrication with altered structural designs, with different lengths, chamber sizes, and wall thicknesses etc. Passive modular variable stiffness SPAs present a significant advantage by enabling the realization of many distinct trajectories without structural redesign. Although various mathematical modeling techniques are widely used to predict their tip motion by treating it a kinematics problem, solving the inverse problem in the presence of modular strain-limiting layer (SLL) configurations is challenging. It is essential to determine the configuration of such a slender actuator in the form of a robot manipulator to deploy it for a specific function and application without re-fabricating them, by simply varying the SLL per the configuration required for a particular tip-point trajectory. To this aim, this paper introduces a hybrid methodology (based on feed-forward neural network) and a convolutional neural network-based method to predict the required SLL configuration for a particular tip trajectory of the SPA. This methodology is generic enough to apply to such actuators to predict their configuration as per their specific tip point trajectory in Cartesian space. The results presented for a slender SPA have demonstrated that the proposed method has predicted its configurations for a range of applications typified by an endoscope prototype, a soft robotic gripping application, and a system mimicking human finger movement with an average error of 1.65%. This study offers a versatile methodology for “function and application specific” SPAs or robot manipulators without re-fabricating them, by strategically combining SLL and machine learning-based prediction to generate a specific trajectory.

Keywords

soft robotics soft robotic materials and designs soft sensors and actuators zig-zag soft pneumatic actuators variable-stiffness strain-limiting layers stiffness modulation machine learning techniques in stiffness modulation

Introduction

Soft robotics presents a compelling alternative to conventional rigid robotic systems, especially for applications requiring interaction with soft objects and environments Gunawardane et al. (2019); Xavier et al. (2022); Ye et al. (2024). Particularly when engaging with exceedingly soft materials, rigid robots must exercise extreme caution in force control due to substantial contact stiffness mismatches between the robot and the object Kwon et al. (2023). Conversely, soft robots leverage compliant materials characterized by flexibility and adaptability, thereby demanding less computational resources, and facilitating a smoother and safer interaction with soft objects Hines et al. (2017); Robertson et al. (2021); Tawk and Alici (2021); Xavier et al. (2022). This inherent softness and flexibility empowers soft actuators to be used as a slender manipulator or a combination of such actuators to undertake sophisticated tasks such as grasping, in-hand manipulation, whole-arm grasping, navigating clutter, endoscope steering, organ retraction, and beyond Graule et al. (2022b). Simplifying design complexity and obviating the need for disparate actuators for each task necessitates the development of soft actuators with a universal design, enabling swift adaptation or interchangeability for diverse tasks Gunawardane et al. (2024); Kan et al. (2022). To realize this objective, it is imperative to engineer actuators capable of swiftly realizing various trajectories with minimal alterations to their structural design Gunawardane et al. (2024, 2022); Peters et al. (2024); Rogatinsky et al. (2022); Zhang et al. (2023b).

Considering the requirement of achieving multiple trajectories, several methods have been developed for designing soft pneumatic actuators (SPAs), specifically for both balloon and membrane type SPAs Cappello et al. (2018); Kim et al. (2019); Rogatinsky et al. (2023); Yang and Asbeck (2020). Typically, physical parameters of SPAs with passive strain-limiting layers (SLLs), e.g. length, chamber design, and SLL properties need to be readjusted to alter their tip trajectories Gunawardane et al. (2024); Hu et al. (2018); Moseley et al. (2016); Wang et al. (2023a). The SLL essentially controls excessive strains to maintain the SPA’s tip trajectory along a predefined path while compensating gravitational loading Gunawardane et al. (2024, 2022); Zhang et al. (2023b). When the stiffness of the SPA remains constant, the actuator cannot modify its tip trajectory without re-designing and re-fabricating, which is a significant limitation in passive SLL designs. However, when balloon-type SPAs are used with braided strings, the trajectory can be varied by adjusting the braid angles Connolly et al. (2015); Guan et al. (2020) without making significant structural modifications. However, it is very challenging to repeatedly rewind the SLL of the SPA with different angles, without leading to substantial operational disruptions and downtime.

To achieve on-demand trajectory control for SPAs, modular passive SLL designs were introduced Gunawardane et al. (2025a and 2025b); Zhang et al. (2023). In these designs, the SLL is separated from the main actuator and constructed in a modular fashion using press-fittable modules Gunawardane et al. (2025a and 2025b) or bands with variable stiffness Zhang et al. (2023b). The effective stiffness of the SLL varies based on its physical properties, configuration, and materials, enabling on-demand trajectory control. While first-principle-based modeling techniques are frequently employed to design SPAs, they face challenges in determining the specific SLL configuration needed to achieve a desired trajectory or spatial point in these SPAs Gunawardane et al. (2025a and 2025b). Although these models can predict the flexural rigidity and effective stiffness of the SLL, identifying the precise formal configuration required to realize an arbitrary trajectory remains difficult Gunawardane et al. (2025a and 2025b).

Static and dynamic modeling techniques have been employed to design, model, and control SPAs. Continuum mechanics models, such as the Cosserat rod, Kirchhoff, and Euler-Bernoulli models, have been utilized alongside geometric models like the piecewise continuous curvature (PCC) model to represent the behaviour of SPAs Boyer et al. (2020); Della Santina et al. (2021); Gilbert and Godage (2019); Gazzola et al. (2018); Godage et al. (2015); Graule et al. (2022b); Tang et al. (2021); Wang et al. (2023a); Xiao et al. (2023). Furthermore, the Simulation Open Framework Architecture (SOFA) serves as an example of computationally efficient finite element methods used to design, optimize, and control SPAs and soft robots Courtecuisse et al. (2012); Godage et al. (2015); Thalman and Lee (2020); Xavier et al. (2023). While these advanced models are effective, they face challenges when applied to multi-material and complex structures, such as SPAs with multi-material reconfigurable SLLs. For instance, the Zig-zag modular passive variable stiffness SPAs can achieve over 100 different trajectories depending on the material type and SLL configuration Gunawardane et al. (2025a and and 2025b). Consequently, employing conventional modeling techniques to develop accurate models identify the formal configuration for these SPAs is difficult. This challenge is particularly pronounced in inverse problems, where constructing a model to determine the specific configuration required to realize a particular trajectory is notably complex.

Machine learning (ML) techniques, such as feed-forward neural networks (FFNN) Sapai et al. (2023); Yao et al. (2024), reinforcement learning Graule et al. (2022a); Zhang et al. (2023c); Yao et al. (2023) convolutional neural networks (CNNs) Xiao et al. (2023), graph neural network encoders Gazzola et al. (2018), and physics-informed recurrent neural networks Mosser et al. (2023), have recently been widely used for structural design Di Lecce et al. (2022); Graule et al. (2022c); Hussain et al. (2020); Mosser et al. (2023); Wang et al. (2023a); Yao et al. (2024), optimization Yao et al. (2024); Li et al. (2023), control Lu et al. (2023); Wang et al. (2021); Zhang et al. (2023c), learning complex dynamics Sapai et al. (2023), and proprioception Wang et al. (2023b); Zhang et al. (2023a) of SPAs and soft robots. Several efforts have been made to utilize ML in designing SPAs that conform to specific trajectories. Examples include the Soft Pneumatic Actuator Design Framework (SPADA), which employs simulation data with an artificial-neural-network (ANN) and a genetic algorithm Yao et al. (2024), a CNN model combined with a Bezier curve-based genetic algorithm Mosser et al. (2023), and an ANN-based method for the automatic design of inflatable soft membranes Forte et al. (2022). Additionally, a metaheuristic algorithm has been used for inverse material identification for predefined movements Di Lecce et al. (2022). While these methods effectively identify or optimize SPAs for a particular single trajectory with a non-modifiable single SLL, further advancements are necessary to develop techniques that can predict formal SLL configurations for specific trajectories for SPAs with on-demand trajectory control.

This paper investigates the examination of the curvature index and flexural rigidity to identify configurations of reconfigurable SLLs for zig-zag SPAs and its trajectory variations. Within this framework, the paper proposes a new hybrid approach based on a feed-forward neural network (FFNN) and a CNN-based approach to predict SLL configurations for SPAs. The present paper contributes to 1). Examine the variation in curvature of the variable stiffness SPA utilizing the curvature index and the Euler-Bernoulli beam theory, 2). Establishment of a new FFNN model with a polynomial model aimed at predicting SLL configurations for a particular trajectory in SPA’s working envelope, 3). Develop a CNN-based model to identify formal SLL configurations for a trajectory in SPA’s working envelope, and 4). Compare the ability of predicting SLL configurations using FFNN and CNN methods and demonstrating their use in several practical applications such as robotic gripping, endoscopic camera control, and mimicking of human finger movements.

The remainder of this article is structured as follows: The next section presents the methodology, encompassing mathematical modeling, ML model development, fabrication techniques, control mechanisms, and practical applications. This is followed by a section detailing the results, which includes the outcomes derived from the models and identifies various configurations. The final section provides the conclusions and discusses potential avenues for future research.

Methodology

Zig-zag SPA design

The zig-zag SPA design served as the primary structural component responsible for generating motion in the system Gunawardane et al. (2024, 2025a, 2025b). This design was developed based on membrane-type mold cast SPA design. Structurally, the zig-zag SPA resembles a spring; when pressurized, it unfolds and fully expands, resulting in pure linear motion in the absence of a secondary load. The SPA features a uniform cross-sectional area and exhibits a 30% extension at 350 kPa with a 100° bending angle for a continuously 3D-printed SLL, and it achieves a blocking force of 9.34 N during rectilinear motion Gunawardane et al. (2024). The design parameters and physical characteristics of the zig-zag SPA are detailed in Table 1.

Table 1.

Parameters and corresponding physical values of the zig-zag SPA. The zig-zag SPA features a uniform cross-section and is fabricated using TPU NinjaFlex material with a Flashforge Dreamer NX 3D printer.

Symbol	Definition	Value
L	Length of the SPA	0.134 m
E	Elastic modulus	Varying
h	Height of the SPA	0.012 m
b	Width of the SPA	0.016 m
P	Input pressure	0-350 kPa
θ	Bending angle	0-100°
α	Angle of the zig-zag	45°
A	Cross-section area (Internal)	–
q	Force on the beam	P sin α
q∗	Shear force	P cos α
x	Linear displacement	0-0.4 m
A _c	Cross-section area (Chamber)	–

The bending motion and on-demand trajectory control of the SPA are achieved by introducing different SLL materials and configurations. As illustrated in Figure 1, ePLA+ (enhance polylactic acid plus) were developed (3 mm ePLA+ parts are glued outside SPA) on both sides of the SPA to allow for the press-fitting of various SLL module. These modules were fabricated using materials such as Dragon Skin and EcoFlex, and they were developed in different sizes: module one at 122 mm, module two at 53 mm, module three at 33 mm, and module six at 8 mm. The modules also vary in material stiffness, as detailed in Table 2. Adjusting the sizes and stiffness on both sides of the SPA alters the effective stiffness of the SLL, thereby enabling on-demand control of the SPA’s tip trajectory.

Figure 1.

3D-Printedzig-zag SPA with Press-fittable Modular SLL Design, (a) the 3D-printed zig-zag SPA and its mold-cast SLL modules of various sizes. Blue snaps were developed on the SPA to accommodate press-fittable SLL modules in different configurations. (b) the zig-zag SPA with SLLs on both sides: the left side has DS20, and the right side has E30, resulting in a bending motion (Configuration: 2_2D20-2_2E30).

Table 2.

Different press-fittable SLL module materials and their 100% elastic modulus. “D” denotes Dragon Skin, “E” denotes EcoFlex, and “Mat.” refers to the material type; for example, E20 corresponds to EcoFlex 20.

Mat	10	20	30	50
E	x	55[E20]	68[E30]	83[E50]
D	152[D10]	338[D20]	593[D30]	x

Each side of the SPA is designed to accommodate the press-fitting of diverse modules, featuring various combinations (as showin in Figure 1) and materials (as summarized in Table 2). The following convention delineates the configuration of each side: every pair of snap locks is regarded as one unit (see Figure 1), and press-fittable SLL modules are assigned numbers 1, 2, 3, and six corresponding to their respective lengths (see Figure 1). In this naming convention, each numerical value signifies the type of SLL part in each side, and the subscript indicates the quantity of that part on each side of the SPA, along with specifying the material used. Absence of an SLL is denoted as 0, and the subscript designates the number of empty snaps. This notation commences from the top and progresses towards the bottom of the SPA for each side. For instance, if the left side of the SPA features one SLL part 1 [D10], and the right side incorporates two SLL parts 3 [E10] and one SLL part 6 [E10], this configuration would be represented as [1_1D10-3_2E106_1E10].

The versatility of the SPA is heightened by the utilization of different materials and their varied combinations on different sides. This allows for a substantial number of stiffness variations in the SLL, resulting in diverse tip trajectories in the SPA. Considering the extensive range of possibilities in material combinations, this study focuses on two materials, D10-D30 and EF30-D30, representing a significance difference in the SLL, for both simulations and experimental validations. However, any combination of materials listed in Table 2, or similar polymer materials within this range of elasticity, can be used as the SLL material.

Fabrication of SPA

The hollow compartments of the SPA, specifically the zig-zag SPA without the SLL, were produced using a Flashforge Dreamer NX 3D-printer. The printing process employed NinjaFlex Snow 3DNF0017505 from NINJATEK. Standard temperature conditions suitable for the respective material were maintained during the 3D-printing process, and the printer was configured for Thermoplastic Polyurethane (TPU) by applying the special settings specified in Gunawardane et al. (2024, 2025a, 2025b). Slicing and transferring the prints to the printer were executed using Simplify 3D and FlashPrint software. For the formation of Polylactic Acid (PLA) components utilized in the molds for the SLL modules, standard 3D-printing settings for enhanced PLA were employed. The primary material chosen for manufacturing the rigid components (molds) was eSUN ePLA+.

Mathematical background

Curvature index

The curvature index (I) is defined as the ratio of the lengths of the two sides of the SPA after press-fitting a specific SLL configuration and pressurizing the SPA to its maximum pressure (see Figure 2). This index can be used to study the variation in trajectory and the change in length (δL) on both sides of the SPA, thereby controlling the formation of the tip trajectory’s curvature.

\hat{I} = \frac{L_{1}}{L_{2}}

(1)

δ L = L - \frac{L_{1} + L_{2}}{2}

(2)

where

\hat{I} =

curvature index, L₁ = length of the left side of the SPA after applying the pressure, L₂ = length of the right side after applying the pressure, L = original length of the SPA, δL = average extension of the SPA after pressurizing.

Figure 2.

Geometric parameters and structural assumptions for the curvature index and modeling. (a) definition of L₁ and L₂ for curvature index calculation. (b) Cantilever structure assumption used for deriving the Euler-Bernoulli-based model. The relationship between pressure and angle was derived using the equation M = PAe in conjunction with the Euler-Bernoulli beam theory, as described in Alici et al. (2018).

Euler-Bernoulli model

The static modeling of bending motions in the square zig-zag SPA is developed through the utilization of the Euler-Bernoulli beam theorem Alici et al. (2018); Gunawardane et al. (2025a, 2025b); Hu et al. (2018). In this model, the length of the beam represents the fabricated SPA, and its cross-section is assumed to be square (refer to Table 1 for model parameters, definitions, and values). The Euler-Bernoulli model (derived as in Alici et al. (2018)) assumes that the actuator experiences only bending motion (see Figure 2), with the pressure exerted on each surface acting as a uniformly distributed force on the beam. The bending of the beam can be represented as: (considering small angles ∂y/∂x = θ),

θ (p) = \frac{A^{2} e L}{A_{c} E^{2} I} P^{2} + \frac{A e L}{E I} P

(3)

assuming P² ≈ 0 because E² ≫ 0 :

θ (P) = \frac{A e L}{E I} P

(4)

Upon scrutiny of the static model, it becomes evident that the flexural rigidity (EI) stands out as the primary parameter influencing the trajectory variation of the SPA. Subsequently, the experimental data obtained from the SPAs were fitted to the model equation, allowing for an exploration of the variation in EI. This approach provides an approximate means to determine the configuration necessary for the SPA to replicate a desired trajectory. By utilizing the model equation for a newly desired trajectory, it becomes feasible to identify the corresponding EI value. This emphasizes that EI can serve as a parameter for predicting new trajectories. Given a desired trajectory, one can calculate the corresponding EI and select an appropriate configuration. However, a comprehensive list of all available configurations and their associated EI values should be accessible to the user. This method presents a practical approach for attaining diverse trajectories in SPAs with specific motion profiles.

Finite element model

The computer-aided design (CAD) models for the SPA, were created using SolidWorks 2021 and imported to Abaqus for finite element analysis (FEA). An iterative simulation approach is used, where the FEAs were executed to ensure that the SPA is configured to the desired movements. Parameters such as chamber size and thickness are calculated iteratively to achieve the required SPA’s tip trajectory. This comprehensive simulation approach is integral to the design refinement process and ensures the SPA’s functionality aligns with the intended specifications. In the simulation, the primary structure, namely the TPU zig-zag SPA, was modeled utilizing a hyperelastic Ogden model Gunawardane et al. (2024, 2025a and 2025b) with N = 5 (the strain energy function consists of five terms, representing a fifth-order Ogden model. This configuration provided the best fit for the material data when using Abaqus fitting), employing the parameters detailed in Table 3. The strain energy function for the model can be written as,

W = \sum_{n}^{i = 1} \frac{μ_{i}}{α_{i}} [λ_{1}^{α_{i}} + λ_{2}^{α_{i}} + λ_{3}^{α_{i}} - 3]

(5)

Table 3.

The parameters of the Ogden N = 5 hyper-elastic model & Yeoh model utilized in the Abaqus simulations Graule et al. (2022a), Yao et al. (2024). Note: The parameters not mentioned are zero.

Material	Parameters
TPU	μ₁ = 903.005, μ₂ = −723.556, μ₃ = 264.028, μ₄ = −669.432, μ₅ = 236.657
TPU	α₁ = 3.719, α₂ = 5.239, α₃ = 6.187, α₄ = 2.264, α₅ = 1.415
D10	C₁₀ = 2.78 × 10⁻², C₂₀ = 1.31 × 10⁻³, C₃₀ = −7.55 × 10⁻⁶
D20	C₁₀ = 4.69 × 10⁻², C₂₀ = 2.4 × 10⁻³, C₃₀ = −1.86 × 10⁻⁵
D30	C₁₀ = 6.38 × 10⁻², C₂₀ = 6.22 × 10⁻³, C₃₀ = −1.24 × 10⁻⁴
E10	C₁₀ = 1.73 × 10⁻³, C₂₀ = 1.11 × 10⁻⁴, C₃₀ = −2.2 × 10⁻⁷
E30	C₁₀ = 9.31 × 10⁻⁴, C₂₀ = 3.45 × 10⁻⁴, C₃₀ = −1.88 × 10⁻⁶
E50	C₁₀ = 9.7 × 10⁻³, C₂₀ = 2.1 × 10⁻⁴, C₃₀ = −4.36 × 10⁻⁷

The principal Cauchy stresses can be represented as,

σ_{uniax} = \sum_{n}^{i = 1} 2 μ_{i} [λ^{(α_{i - 1})} - λ^{(\frac{1}{2} α_{i + 1})}]

(6)

In a similar manner, the Yeoh model Yao et al. (2024) was selected for the SLL modules employing the parameters detailed in Table 1. The strain energy function for the model can be written as,

W = \sum_{i = 1}^{n} C_{i} {(\bar{I_{1}} - 3)}^{i},

(7)

where

\bar{I_{1}} = λ_{1}^{2} + λ_{2}^{2} + λ_{3}^{2}

is the first invariant of the right Cauchy-Green deformation tensor, and C_i are material constants.

The principal Cauchy stress under uniaxial tension is given by

σ_{uniax} = 2 (λ^{2} - \frac{1}{λ}) \sum_{i = 1}^{n} i C_{i} {(I_{1} - 3)}^{i - 1},

(8)

λ_i principle stretch ratio and I₁ is the first invariant of the deformation tensor. C and μ are material constants, with μ being associated with the stiffness in the Ogden model. α is the nonlinear exponent in the Ogden model that controls the degree of non-linearity.

Identification of formal Configuration using machine learning

The Euler-Bernoulli model and the Finite Element Model are effective methods for predicting trajectory based on known actuator specifications, similar to forward kinematics in robotics. Conversely, addressing the inverse problem requires developing a methodology to determine the SLL configuration necessary to reach a specific point within the actuator’s operational range, akin to inverse kinematics. Moreover, implementing a mathematical model to predict SLL configurations for such cases presents significant challenges. For example, the Euler-Bernoulli model can be used to obtain the flexural rigidity of the configuration for a particular trajectory, but mapping them to configurations is very tedious to model mathematically. Consequently, ML models have been developed to predict the necessary configurations essential for achieving specific points or trajectories within the actuator’s working envelope.

Hybrid model-based method

A hybrid method based on FFNN and a polynomial model was developed to predict the SLL configuration that meets a specific point/ tip trajectory within the working envelope of the SPA. This FFNN-based model utilized the bending angle θ and the pressure input (P), along with the SLL configuration of the SPA. The bending angle is defined as the angle between the resting position and the pressurized position of the straight line joining the center of the base to the center of the tip of the SPA (see Figure 3). To accurately represent the trajectory of the SPA, five auxiliary variables (θ_i, i = 1, 2, 3, 4, 5) were introduced at equal intervals along the system (see Figure 3). A polynomial model was then used to relate the tip trajectory to the auxiliary angles, establishing a relationship between each intermediate angle and the final tip angle, θ₆,

θ_{i} = β_{0} + β_{1} θ_{N} + β_{2} θ_{N}^{2} + \dots . + β_{n} θ_{N}^{n} + ϵ

(9)

where θ_i is the dependent variable, θ_N is the independent variable (N = 6), β_n is the model coefficient, n is the degree of the polynomial, and ϵ is the term representing the error.

Figure 3.

The final tip angle and the intermediate angles of the polynomial model. m_i is represented by (x_i, y_i) in the Cartesian plane, and θ_i is defined as $θ_{i} = \tan^{- 1} (y_{i} / x_{i})$ .

The model’s accuracy was determined using the coefficient of determination R².

R^{2} = 1 - \frac{\sum_{i} {(y_{i} - f_{i})}^{2}}{\sum_{i} (y_{i} - \bar{y})}

(10)

Where y_i is the data set, f_i is the fitted value, and

\bar{y}

is the mean of the observed data,

\bar{y} = 1 / k \sum_{i = 1}^{k} y_{i}

. θ_i, i = 1, 2..6 along with P were used as inputs to the FFNN model. The FFNN model was as follows,

{\hat{y}}_{i} = g (W_{0, i}^{k - 1} + \sum_{j = 1}^{n} f (z_{j}^{k - 1} w_{j, i}^{k - 1}))

(11)

Z_{k, i} = w_{0, i}^{k} + \sum_{j = 1}^{n_{k - 1}} g (z_{k - 1}, j), w_{j, i}^{k}

(12)

where

\hat{y}

is the predicted output, Z is the pre-activation output, W weights, g is the activation function in the final layer, and f is the general activation function. The hidden layers of the model were activated using LeakyReLu Ian Goodfellow and Courville (2016) function as follows,

f (x) = m a x (0.01 x, x)

(13)

The output layer of the model was activated using Softmax activation as follows,

S o f t m a x {(Z)}_{i} = \frac{e^{Z_{i}}}{\sum_{j = 1}^{K} e^{Z_{j}}}

(14)

i = 1, …, K, Z is the input vector of length K, Z_i is the ith element of Z, e is the base of the natural logarithm, K is the number of classes. Categorical cross entropy loss function was used to train the model as follows,

L = - \sum_{i = 1}^{p} y_{i} \log {\hat{y}}_{i}

(15)

p = number of outputs, y_i = true class distribution,

{\hat{y}}_{i}

= predicted class distribution

The FFNN model was developed with seven input neurons, representing the bending angle of the tip, five auxiliary angles, and the pressure input. It included two hidden layers, comprising 512 and 100 neurons, respectively, and an output layer with 102 neurons corresponding to the SLL configurations. To prevent overfitting, a dropout rate of 0.2 was applied during the training process. The final integration of the FFNN model and the polynomial-based input is illustrated in Figure 4. The target column, ”SLL Configurations,” was label encoded, assigning a unique number to each category within the variable to facilitate algorithmic interpretation. One-hot encoding was applied to convert the 102 SLL configurations into numerical values ranging from 1 to 102. After shuffling, the dataset was split into training and validation sets, with 80% allocated to training and 20% to validation. Both sets were then standardized using the following equation,

z = \frac{x - μ}{σ}

(16)

where

μ = 1 / k \sum_{i = 1}^{k} (x_{i})

and

σ = \sqrt{1 / k \sum_{i = 1}^{k} {(x_{i} - μ)}^{2}}

; μ = mean, σ = standard deviation.

Figure 4.

Hybrid model for identifying the optimal SPA configuration. The FFNN model is used to identify the optimal configuration of the SPA for a specific trajectory (set of points corresponding to input pressure values) within its working envelope. For an arbitrary trajectory, pressure and angle values are employed to generate auxiliary angles using an auxiliary variable-generating polynomial model. These angles, along with the pressure inputs, are then fed into the FFNN to predict the SLL configuration.

Convolutional neural network method

To identify a unique configuration that satisfies a specific trajectory within the actuator’s working envelope, a CNN-based method was implemented (see Figure 5). This method utilizes images depicting the relationship between pressure and angles across various configurations. The CNN model consists of a sequence of layers, including convolutional layers interspersed with max-pooling layers, followed by a fully connected network, and culminating in an output layer with SoftMax activation. Each convolutional layer applies filters to the input, and the operation performed by a convolutional layer can be expressed as follows,

a_{ijk}^{(l)} = σ (\sum_{m} \sum_{n} \sum_{c} ω_{mnc}^{(l)} x_{(i + m) (j + n) c} + b_{k}^{(l)})

(17)

where

a_{ijk}^{(l)}

is the activation of the k-th filter at position (i, j) in layer l,

ω_{mnc}^{(l)}

represents the weights of the k-th filter at position (m, n) across channel c, x is the input matrix,

b_{k}^{(l)}

is the bias term, and σ is the activation function, typically ReLU (max(0, x)).

Figure 5.

CNN model for identifying optimal SLL configurations for the SPA. For a given trajectory, spatial coordinate values are converted into angles. These angles are then input into the CNN as a pressure versus angle curve to predict the SLL configuration.

The pooling layers were used to reduce the spatial dimensions (height and width) of the input volume for the next convolutional layer. Max pooling, was used in this network as follows,

a_{ijk}^{(l)} = \max_{p ϵ P_{i j}} a_{pjk}^{(l - k)}

(18)

where P_ij is the pooling region (i, j).

The neural network at the end of a CNN model is a fully connected layer, as illustrated in Figure 5. The operation in this section of the CNN model is as follows:

y_{i} = σ (\sum W_{i j} x_{j} + b_{i})

(19)

where W represents the weights, x is the input from the previous layer, b is the bias, and y is the output. The final layer is responsible in classification task and therefore SoftMax activation is used.

The pressure (P) versus angle (θ) response curves were transformed into images of size 224 × 224 × 1 and fed into the input layer. The network architecture comprised several convolutional layers. The initial convolutional layer applied 32 filters of size 3 × 3 with ReLU activation. This was succeeded by a max-pooling layer employing a 2 × 2 filter, reducing spatial dimensions by half. Subsequent convolutional layers increased filter depth from 64 to 128, each followed by a max-pooling layer, progressively reducing dimensionality and enhancing feature abstraction. Following three sets of convolutional and max-pooling layers, the network flattened the 3D output into a 1-D vector for input into the fully connected layers. A dense layer with 128 neurons and ReLU activation was employed, integrating a dropout rate of 0.4 to mitigate over-fitting within the fully connected layer. The output layer comprised a dense layer with a softmax activation function, containing 102 neurons to correspond with the SLL configurations.

Control & experimental setup

The movements of the SPA were generated using pressurized air from the SPEEDAIRE compressor, regulated by the AURORA 1.4 MPa air regulator, and controlled by a FESTO proportional pressure regulator (VPPM-6L-L-1G18-0L10H-V1P-S1C1) (experiments were carried out in a pressure range of 0 to 340 kPa). The SPA movements were recorded with a Microsoft LifeCam HD 3000. Video analysis and processing were performed using the Tracker Video Analysis and Modeling Tool for Physics. The operation of the pressure regulator was managed by a power supply (GWINSTEK GPS-3303) and a function generator (HEWLET PACKARD 33,120A), both connected to the valve. All applications and test setups were interfaced with a PC via Arduino and MATLAB for control purposes. Figure 6 provides an illustration of the experimental setup.

Figure 6.

Experimental Setup for SPA Characterization. The experimental setup for characterizing the SPA includes a function generator used to control a pressure regulator, delivering pressure signals to the SPAs ranging from 0 to 340 kPa with a sinusoidal signal at a frequency between 50 and 300 mHz. The motion of the SPA tip was recorded by the camera, and the videos were processed frame by frame to track the tip position at varying pressure values.

Data processing and computation

The code was developed using Google Colab and Jupyter Notebook with Python. The Scikit-learn and TensorFlow libraries were utilized to process the data and develop the FFNN and CNN models. The code and recorded data are accessible here https://github.com/HiroshanGunawardane/ML_SelectiveStiffness-. A summary of the captured videos can be found in Video Attachment.

Application demonstration

Three primary applications were tested to demonstrate the versatility of the modular SLL across various configurations. In these applications, predefined or assumed trajectories were used in the ML models to predict the configurations required to achieve the desired tip trajectory of the SPA.

Endoscope camera control

In endoscopic applications, optical scanning utilizing bending fibers may lead to image distortion because the bending fiber forms a non-vertical angle with the imaging plane Moallemi et al. (2020). To demonstrate the SPA’s controllable bending curvature to achieve a vertical angle with respect to a plane, a prototype of an endoscope was fabricated, as depicted in Figure 7. A small camera was securely embedded at the tip of the SPA, and image distortion was systematically measured for various configurations to quantify the impact of the passive variable stiffness SLL on its design. While a conventional SPA typically exhibits pure bending or extension, configurations like 2_1D100₃-0₂3_1D106_2D10 enable the SPA to move perpendicular to its plane of bending. This capability has the potential to significantly reduce image distortion, proving highly effective for applications requiring clear imaging, such as in minimally invasive surgeries. The experiment aimed to assess image distortion utilized an A4 sheet with equally spaced small black circles. In the absence of image distortion, the captured image should depict a perfect circle with r₁ = r₂, and the ratio r₁/r₂ should equal one. Conversely, if the circles are distorted, the circle will appear in an oval shape, with r₁ ≠ r₂, and the ratio r₁/r₂ unequal to one. Distortion measurement in the experiment involved assessing the ratio between r₁ and r₂ for various configurations that tend to induce more parallel motion to the plane of scanning. A summary of the endoscopic camera operation can be found in Video Attachment.

Figure 7.

Endoscopic camera application. Prototype design of the endoscope featuring a small camera at the tip and the calculation of image distortion from the endoscope camera. Circular images, where r1/r2 = 1, are considered non-distorted, while oval-shaped images, where r1/r2 ≠ 1, are considered distorted. This was tested over the pressure range of 0 to 400 kPa, with the final P > 0 configuration corresponding to the maximum pressure.

Robotic gripper

A soft robotic gripper was developed using various combinations of SPAs configured with two, three, and four actuators. This gripper was attached as an end-effector to the Kinova Gen two six-degree-of-freedom robotic manipulator for experimentation. The SPAs were secured in a 3D-printed holder to ensure proper positioning and then fixed to the end of the robot’s final link, serving as the soft gripping end-effector. The required tip trajectory of each actuator was determined based on the size and shape of the object to be gripped. The CNN-based model was then employed to identify the necessary configuration for gripping the object. The gripper’s performance was evaluated based on its ability to effectively grip standard objects using different numbers of actuators and SLL configurations. Gripping tests were conducted with a variety of objects, including a dessert cup, a wine glass, a slinky, a table tennis ball, a sponge, and a dice with different words used for playing games (see Figure 8). A summary of the robotics gripping operation can be found in Video Attachment.

Figure 8.

Various SLL configurations and their effectiveness in grasping different objects when integrated into a robotic end-effector. The end-effector was mounted on a Kinova Gen two serial link robotic manipulator for experimentation. Objects of various shapes and sizes were used for the gripping tasks. The size of the gripper and the SPA trajectory for each gripping task were predefined, and the ML model was used to predict the required SLL configuration.

Mimicking human finger

The human finger consists of three primary joints and three segments, functioning similarly to a serial link mechanism. The motion of the fingertip varies according to the angular changes at each joint, producing different trajectories. The capacity to adjust these angles creates a workspace, as illustrated in Figure 9. For this experiment, only the planar motion of the finger was considered. The smallest and largest possible trajectories of the human finger were recorded (limiting the motion into a single quadrant), and the potential to generate SLL configurations using different ML models to replicate similar motions was examined. A summary of the mimicking human finger motion operation can be found in Video Attachment.

Figure 9.

Recording of human fingertip motion. Recordings of human fingertip motion to analyze its ability to vary the curvature index and to mimic similar motion in the zig-zag SPA with on-demand trajectory control. The finger’s motion was constrained to one quadrant, and both the smallest and largest possible trajectories were considered to examine the working envelope. The angle is measured between the finger extension axis (y-axis) and the line connecting the base to the tip at a specific point. The fingertip position was tracked through video processing of a marker placed at the location of the red dot.

Results

Curvature index and flexural rigidity study

Figure 10(a) illustrates the variations in curvature index and flexural rigidity across all configurations of the SPA. The curvature index varies between one and 0.65, with a maximum bending angle of approximately 60° observed in the trajectories (the curvature index is calculated using equation (1)). Figure 10(b) depicts the net extension of the SPA for different configurations, highlighting the variation in net extension and its contribution to different trajectories. Figure 10(c) summarizes the variation in flexural rigidity for different SLL configurations, while Figure 10(d) presents the relationship between flexural rigidity and the maximum bending angle of each trajectory (flexural rigidity is calculated by fitting the trajectory data into equation(4)). Additionally, Figure 10(d) indicates that a decrease in flexural rigidity corresponds to an increase in the maximum bending angle of the SPA. In general, flexural rigidity decreases at higher angles. To achieve larger angles, the actuator must be soft, which necessitates a lower flexural rigidity. In such cases, bending becomes more prominent than extension in the motion. However, at smaller bending angles, higher flexural rigidity results in smaller angles, while lower flexural rigidities make extension more dominant than bending, although still producing small angles. Overall, Figure 10 summarizes the distinct trajectories of all 102 SPA configurations under a pressure input of 0-340 kPa.

Figure 10.

(a) Curvature index angle versus maximum bending angle for different configurations. (b) extension of the SPA for different SLL configurations. Behavior of the SPA under various SLL configurations. δL = L − L₁ + L₂/2 (c) variation in flexural rigidity for different SLL configurations. (d) flexural rigidity versus maximum bending angle.

Hybrid model results

To determine the optimal number of auxiliary variables for the hybrid model, we evaluated the accuracy of the FFNN with varying auxiliary variable inputs, as presented in Table 4. An increase in the number of auxiliary angles resulted in higher FFNN accuracy; however, it also substantially extended the training time. After comparing different configurations, five auxiliary angles were selected, balancing both accuracy and computational efficiency.

Table 4.

Number of auxiliary angle inputs to the FFNN model and its performance metrics. Computation time represents the duration required to stabilize training accuracy across epochs (Training was conducted over a total of 50 epochs). For each test, the size of the training dataset is 3345 times the number of auxiliary angles, while the size of the validation dataset is 836 times the number of auxiliary angles.

No. of Aux. Angle	Accuracy (%)	Computation time (s)	Time cost to reach 1% accuracy
0	35.65	149.04	4.18
1	79.31	86.8	1.09
2	93.3	114.19	1.22
3	94.38	121.57	1.28
4	92.94	93.27	1
5	91.63	77.98	0.85

The accuracy of the polynomial model in predicting auxiliary variables was evaluated. 20 percent of the dataset was designated as a validation set to assess the regression model. Through a trial-and-error approach, a 4^th order polynomial was selected to predict the angles, avoiding overfitting. After training the model, the evaluation process indicated an accuracy of 94% for both the training and validation sets, with an average standard deviation of 8° across all angles. The average RMSE for the training set was 0.89° with a standard deviation of 0.38°, and the R² value ranged from 80.82% to 99.92%, indicating a good fit. For the validation set, the average RMSE was 0.91° with a standard deviation of 0.4°, and the R² value ranged from 80.42% to 99.92%, also indicating a good fit. These results demonstrate that the polynomial regression model has a strong predictive capability for the auxiliary angles (see Table 5).

Table 5.

Prediction accuracy and root-mean-square error of the polynomial model to predict θ₁ to θ₅.

Aux. Angle	Accuracy (%)		RMSE °
Aux. Angle	Training set	Validation set	Training set	Validation set
θ ₁	80.82	80.72	1.19	1.18
θ ₂	93.65	94.05	1.30	1.36
θ ₃	98.73	98.68	0.96	0.99
θ ₄	99.76	99.75	0.57	0.58
θ ₅	99.92	99.92	0.42	0.41

The FFNN model was trained using both simulation and experimental data, with the confusion matrix of the trained model shown in Figure 11(a) and 11(b). The simulation data resulted in an accuracy of 92.%, with a precision of 0.92, recall of 0.93, and an F1-score of 0.93. Similarly, the experimental data achieved an accuracy of 94.43%, precision of 0.94, recall of 0.91, and an F1-score of 0.93 (see Table 6). To analyze the robustness of the predictions, errors of 1%, 2%, and 5% were artificially added to the trajectory, resulting in a gradual decrease in accuracy as shown in Figure 11(c). The model can handle errors of approximately 2% while maintaining an accuracy of around 70%, but accuracy drops to between 15% and 40% when the error increases to 5%. Four arbitrary trajectories were generated using existing trajectories (assuming Traj. One is higher than Traj. 2) with the formula: Traj. New = (Traj. 1 + Traj. 2)/2 + (Traj. 1 - Traj. 2)/4. These trajectories were then inputted into the FFNN model, and the prediction accuracies were recorded. The prediction accuracy for these trajectories was 92.33% for the simulation results and 87.38% for the experimental results.

Figure 11.

Results from the FFNN Model. (a) confusion matrix for the training of simulated data. (b) confusion matrix for the training of the experimental data. (c) robustness of the FFNN model, tested by adding errors of 1%, 2%, and 5% to the original data and evaluating prediction accuracy.

CNN model results

The CNN model was trained using both simulation and experimental data, with the confusion matrix of the trained model shown in Figure 12(a) and 12(b). The simulation data resulted in an accuracy of 90.04%, with a precision of 0.85, recall of 0.9, and an F1-score of 0.86. Similarly, the experimental data achieved an accuracy of 100%, precision of 1, recall of 1, and an F1-score of 1 (see Table 6). To analyze the robustness of the predictions, errors of 1%, 2%, and 5% were artificially added to the trajectory, resulting in a gradual decrease in accuracy as shown in Figure 12(c). The model can handle errors of approximately 2% while maintaining an accuracy of around 70%–80%, but accuracy drops to between 30% and 50% when the error increases to 5%. Four arbitrary trajectories were generated using existing trajectories (assuming Traj. One is higher than Traj. 2) with the formula: Traj. New = (Traj. 1 + Traj. 2)/2 + (Traj. 1 - Traj. 2)/4. These trajectories were then inputted into the CNN model, and the prediction accuracies were recorded. The prediction accuracy for these trajectories was 95.24% for the simulation results and 99.96% for the experimental results.

Figure 12.

Results from the CNN Model. (a) confusion matrix for the training of simulated data. (b) confusion matrix for the training of the experimental data. (c) robustness of the CNN model, tested by adding errors of 1%, 2%, and 5% to the original data and evaluating prediction accuracy.

Table 6.

Performance metrics for FFNN and CNN models under experimental and simulation conditions. Accuracy is in percentages.

Model	Type	Accu	Precision	Recall	F1
FFNN	Exp	94.43	0.94	0.93	0.93
FFNN	Sim	92.09	0.92	0.91	0.91
CNN	Exp	97.06	0.96	0.97	0.96
CNN	Sim	90.04	0.85	0.90	0.86

Application demonstrations

Various robotic gripping tasks were executed using different objects, employing distinctive SLL configurations of the SPA. The bending-type SLL configurations demonstrated their capability to securely grip diverse objects of varying sizes and shapes. Leveraging the intricate shapes achievable with this SPA, a multitude of unique gripping tasks were successfully accomplished. For these tasks, the required trajectory of the SPA was determined by observing the workspace, and ML models were used to predict the SLL configurations. Video Attachment provides a comprehensive overview of the diverse and intricate gripping configurations attainable through these SPAs, along with several examples of gripping tasks.

The endoscope prototype was tested using a similar approach to the gripping experiment. First, using the front camera, the required tip trajectory to move the endoscopic camera was predicted. Then, the ML model was used to predict the SLL configuration. One of the main challenges with the typical use of endoscope cameras is image distortion. Traditional bending SPAs and linear SPAs cannot move in complex trajectories, specifically failing to move the endoscope camera parallel to the perpendicular plane to the actuators’ plane. Table 7 and Video Attachment demonstrate the current SPA’s ability to move its trajectory in such motions, significantly reducing image distortion.

Table 7.

Outcome of the endoscopy prototype experiment. The camera’s image distortion was computed and assessed across various configurations to examine their respective contribution in mitigating image distortion when motion along the x axis is predominant. Configuration C1 to C5 are as follows; ”2_1D100₃ − 0₂3_1D106_2D10”, ”1_1D10 − 3_3D10”, ”0₆ − 0₂2_2D103_1D10”.

Config	Distortion	% Rel. motion in y w.r.t. x
Linear	11.5	72.36
Bending	12.71	173.43
Config.1	121.21	42.54
Config.2	6.89	32.52
Config.3	8.47	6.33

The recorded human finger movements, specifically along the x-axis and y-axis, were scaled up by a factor of 1.27 to match the size of the SPA. Subsequently, the elapsed time of the finger’s movement was mapped to pressure using a 1:1 ratio. These parameters were then input into the FFNN and CNN models to generate the corresponding SLL configurations, aiming to replicate the same motion within the SPA. The minimum and maximum trajectory movements of the human finger were recorded, and the respective SLL configurations were predicted. These SLL configurations were then applied to the SPA, and the resulting trajectory was recorded using the camera. Table 8 summarizes the predictions for the smallest and largest configurations using each model. The FFNN model predicted 0 − 2_2D20 for the smallest trajectory and 0 − 2_2D30 for the largest trajectory, with likelihoods of 52.24 and 29.70, respectively. Similarly, the CNN model predicted 0 − 2_2D30 for the smallest trajectory and 0 − 1_1D20 for the largest trajectory, with likelihoods of 99.99 and 40.26, respectively. As illustrated in Figure 13, the pressure versus angle curves for these trajectories indicate that the FFNN model closely predicted similar SPA configurations for the smallest trajectory, while the CNN model closely predicted similar SPA configurations for the largest trajectory.

Table 8.

Different finger trajectories and SLL configuration predictions were analyzed using FFNN and CNN models. The likelihood represents the highest likelihood value produced by each machine learning model for its respective configuration.

Finger Traj	FFNN prediction	FFNN likelihood	R-square	MAE
SmallestTraj	two D20 [2] right side	52.24	0.6	8.31
LargestTraj	two D30 [2] right side	29.70	0.87	5.24
	CNN prediction	CNN likelihood	R-square	MAE
SmallestTraj	two D30 [2] right side	99.99	0.88	4.34
LargestTraj	one D20 [2] right side	40.26	0.92	3.93

Figure 13.

The human finger movements were mimicked using various SLL configurations within the SPA, focusing on pressure versus angle curves. The pressure versus angle relationships of the finger and the respective configurations were analyzed, yielding R² values as follows: Largest 0.979, Smallest 0.909, 0 − 2_2D30 0.939, 0 − 2_2D20 0.926, and 0 − 1_1D20 0.942. The largest and smallest curves represent the experimental fingertip trajectories, obtained directly from video recordings. The remaining curves correspond to model predictions. Specifically, 0–2_2D30 denotes two [2] Dragon Skin 30 actuators, 0–2_2D20 refers to two [2] Dragon Skin 20 actuators, and 0–1_1D20 represents one [1] Dragon Skin 20 actuator.

Conclusions and future work

Based on the analysis conducted using the curvature index model and the Euler-Bernoulli model, as summarized in Figure 10, the trajectory of this particular SPA design can exhibit significant variation depending on the material type and the configuration of the SLL. To address the challenge of identifying the desired trajectory for specific applications, ML techniques were employed. Both the FFNN model, utilizing a hybrid approach, and the CNN model have demonstrated the capability to accurately predict the SLL.

The FFNN model utilizes five auxiliary variables generated by a fourth-order polynomial model to capture the full variation of the actuator’s motion when pressurized. In contrast, the CNN model employs a different approach, using a 2D image of the pressure angle variation to predict the SLL configurations. The FFNN model is particularly effective for predicting SLL configurations when the required trajectory is partially known. On the other hand, the CNN model is better suited for situations where the pressure angle variation curve is fully known.

One of the key challenges in the current method is that it has only been developed for planar trajectories that predominantly involve pure bending. However, these SPAs can generate complex trajectories, such as highly complex S-shaped paths, and have demonstrated their ability to produce multi-degree-of-freedom movements Gunawardane et al. (2025a and 2025b). For applications such as endoscopic camera control, it would be beneficial to explore these capabilities further to enable S-shaped behaviors and 3D trajectory control. It is anticipated that the CNN model will be extended to incorporate these features in the future.

Supplemental Material

Footnotes

ORCID iDs

Palpolage Don Shehan Hiroshan Gunawardane

Gursel Alici

Funding

The authors sincerely acknowledge the generous support provided by the funding sources for this research: The Friedman Award for Scholars in Health at the University of British Columbia and the Natural Sciences and Engineering Research Council of Canada (NSERC) CREATE grant (565429-2022).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Supplemental Material

Supplemental material for this article is available online.

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Supplementary Material

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0.00 MB

Configuration identification of on-demand variable stiffness strain-limiting layers in zig-zag soft pneumatic actuators using deep learning methods

Abstract

Keywords

Introduction

Methodology

Zig-zag SPA design

Fabrication of SPA

Mathematical background

Curvature index

Euler-Bernoulli model

Finite element model

Identification of formal Configuration using machine learning

Hybrid model-based method

Convolutional neural network method

Control & experimental setup

Data processing and computation

Application demonstration

Endoscope camera control

Robotic gripper

Mimicking human finger

Results

Curvature index and flexural rigidity study

Hybrid model results

CNN model results

Application demonstrations

Conclusions and future work

Supplemental Material

Footnotes

ORCID iDs

Funding

Declaration of conflicting interests

Supplemental Material

References

Supplementary Material