1 Introduction

For decades, researchers have endeavored to design prosthetic hands. Notable examples of such efforts include the DLR Hand [1], UB Hand III and IV [2,3,4], DEXMART hand [5], BRL/IIT/Pisa Softhand [6], and Hannes hand [7]. The intricate nature of the human hand poses significant challenges in replicating it in robotic prostheses [8]. One such challenge arises when developing the outer skin to cover these prostheses, aiming to closely mimic the capabilities and appearance of natural hands.

The covering skin, often termed as soft pads for robotic prostheses, has been extensively investigated in the literature for four primary objectives [9]: (a) manipulation through soft fingers; (b) modeling soft contacts; (c) embedding appropriate tactile sensors; and (d) designing soft fingers; The first research category concentrates on grasping and manipulating objects using soft robotic fingers [10]. The second category encompasses studies aimed at developing mathematical models for soft contacts, with key models discussed in [11,12,13,14,15,16,17]. The third category aims to embed appropriate tactile sensors on soft fingers [18,19,20,21,22]. The last category encompasses research centered on developing optimized soft finger pads and exploring the mechanical behavior of different materials for such soft pads [23,24,25,26,27,28,29,30].

Given the pivotal role of soft pads’ mechanical properties in improving the performance and acceptance of robotic prostheses, it is essential to intensify research efforts aimed at optimizing these components. This research direction closely aligns with the investigation of compliant surfaces and their impacts for the safety, acceptance, and functionality of humanoid robots and prosthetic devices for users [24,25,26].

Regarding safety, employing a soft pad can mitigate damage in accidental collisions with objects. This is because the impact forces are dispersed more effectively over larger contact areas. Unlike traditional machines, which are designed to be fast and stiff to achieve precise position control, their speed and power make them unsafe for interaction with humans. To create mechatronic systems that are more human-friendly, structures need to be designed to be more compliant compared to traditional technologies. In this regard, the soft pad acts as a passive device that reduces potential damage and injuries in the event of accidental impacts [24].

In terms of user acceptance, a soft pad is crucial for robots that interact with humans, such as rehabilitation devices or humanoid robots. An artificial skin with appropriate stiffness, texture, color, and shape can diminish the perception of the robot as a mere machine, particularly when interacting with specific demographics like the elderly or children [25].

Fig. 1
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(a) Sample homogeneous (top) and differentiated (bottom) soft pads; and (b) Human fingertip

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Regarding functionality, the presence of surface compliance can significantly influence the robot’s performance during force/position-controlled tasks, akin to the function of human fingers or feet covered by soft tissues. Concerning local compliance in anthropomorphic hands, soft pads enable shape adaptation, conformability, and provide large contact areas, thereby reducing mechanical stresses and contact pressure, while also allowing torque exertion perpendicular to the contact plane [26,27,28, 31]. Furthermore, they facilitate energy dissipation during shocks or vibrations, extend the duration of contact under a constant load, and improve the sustainable normal torque for a given surface friction by enhancing pressure distribution [16].

Most studies on robotic soft pads have employed a thick layer of uniform material shaped around a rigid core (see Fig. 1(a), the model at the top). The shape of the pad and the rigid core are selected based on the scenario depicted in Fig. 1 (b) to enhance realism in applications involving human fingers. However, human fingers exhibit complex nonlinear mechanical characteristics [32], attributed to the presence of bones and nails that restrict the deformation of soft tissues. The stiffness of a human finger increases nonlinearly as the soft tissue flattens, while the contact area varies with grasping force, affecting the overall finger stiffness [33]. Regrettably, these nonlinearities cannot be accurately replicated by uniform soft pads. To tackle this limitation arising from the discrepancy between homogeneous soft material layers on robotic limbs and biological models, the adoption of soft pads with a differentiated structure has been proposed (see Fig. 1(a), the model at the bottom). Differentiated structure entails the use of solid material to partition the pad’s total thickness into layers with distinct structural designs, such as a continuous outer skin layer coupled with an inner layer containing voids. The objective is to adjust the pad’s compliance and resulting power law, specifically within the constraints of the material and allowable pad thickness, to enhance compliance compared to a non-structured pad. Various internal layer structures have been described in the literature to achieve this objective [24,25,26,27,28, 34,35,36]. While this approach presents a promising avenue for varying the compliance of soft pads, it necessitates flexibility in adjusting the compliance for diverse applications and purposes. Using the human finger as an example, the distinct regions within a single finger (proximal, medial, and distal phalanges) exhibit varying compliances. Therefore, to closely replicate the mechanical characteristics of the human hand, it is of high importance to develop a methodology for optimizing the shape of differentiated structures in soft pads to accommodate the compliance of different parts within the hand. This could also be advantageous for other designers employing soft pads, enabling them to modulate the overall compliance of a given soft pad to tailor its properties for specific applications.

This paper introduces five distinct differentiated structures aimed at adjusting the compliance of a given soft pad to match the desired force-displacement curve. Following this, nonlinear FEA models are developed for the soft pads, considering compressive contact loads and incorporating the aforementioned rigid core. Shape optimizations are subsequently applied through FEA to enable the generation of force-displacement curves with specific shapes, aiming to replicate the compliance of a human finger. The main contributions of this article are as follows:

  • Introduction of five differentiated structures to vary the compliance of soft pads, addressing the limitation of homogeneous ones.

  • Development of detailed FEA models considering nonlinearities arising from geometry, which experiences large deformations, material properties due to the use of rubber-like materials, and contact, necessitating an iterative procedure to solve boundary conditions on the contact pair.

  • Utilization of a computer-aided engineering (CAE) framework [37] incorporating FEA to optimize the differentiated structures for a specific target force-displacement curve.

  • Comparison of the optimized differentiated structures in terms of their alignment with the force-displacement curve of a human finger.

  • Introduction of a differentiated structure for soft pads, a novel optimized design whose compliance closely matches that of a human finger.

The remainder of this paper is organized as follows: Section 2 presents the design and optimization of soft pads by introducing five distinct differentiated structures and their corresponding material selection criteria and FEA models. Additionally, the optimization problem to mimic human finger’s compliance properties has been formulated in this section. In Section 3, the results of the optimizations and effectiveness of the method have are discussed. Finally, Section 4 concludes the research.

2 Soft pads’ design and optimization

Reducing material hardness, while maintaining a specific layer thickness, enhances compliance but may pose challenges in surface tribology and reliability. Conversely, reducing thickness, for a given material, decreases pad compliance, resulting in reduced contact area under a given load and compromised contact robustness. For robotic limb designers aiming to create slender, bio-mimetic limbs without altering the overall size of the internal rigid core, this presents a significant challenge, particularly evident in designing humanoid robot hands’ fingers. Striking a balance between material properties and layer thickness is a common approach. Alternatively, designing pads with a differentiated structure, incorporating layers of different materials or using a single material with varied layer designs, holds promise. This manuscript focuses on designing differentiated layers using a single material, easily manufacturable via 3D printers. This facilitates relatively simple, fast, and cost-effective production of complex-shaped items. The objective is to adjust pad compliance and consequently alter contact areas for a given load while maintaining consistent dimensions for both the inner rigid core and the outer pad surface.

Fig. 2
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Proposed differentiated soft pads (a) Pad I; (b) Pad II; (c) Pad III; (d) Pad IV; and (e) Pad V

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2.1 Proposed differentiated soft pads

The main focus of this paper is on designing soft pads suitable for application on the fingers of a robotic prosthetic hand, resembling the size of a human hand. To ensure comparability with previous studies, the test items were designed with dimensions identical to those utilized in [17]. Featuring a soft pad geometry, an external continuous layer covering an intermediate layer containing voids, allowing for the adjustment of its compliance based on the design and distribution of these voids. These voids may contain air or be filled with incompressible viscous liquids, offering additional effects of potential interest. Illustrated in Fig. 2 are five fundamental design axisymmetric patterns that have been developed:

  • Pad I features circumferential ribs with varying thickness along its length, where the thickness of the bottom side of each rib is divided by a constant value “c” at the top. The ribs are inclined at a 75-degree angle about the inner arc, with their top sides constrained to be tangential to the rigid core. Additionally, the ribs are equally spaced from each other at a distance of 0.2 mm (Fig. 2(a)).

  • Pad II incorporates bulb-shaped voids, where adjusting the radius of the arcs determines the size of the voids, yielding either larger or smaller voids. These voids are evenly spaced angularly from each other at an angle of \(\theta \)=25 degrees (Fig. 2(b)).

  • Pad III features voids with a length of 2.5 mm and arcs with a length of \(\theta _{\text {2}}\). Additionally, they are spaced apart from each other at an angle of \(\theta _{\text {1}}\) (Fig. 2(c)).

  • Pad IV is characterized by evenly spaced hemispherical protrusions, each with a radius of “r”. These protrusions are distanced from each other by an angle of \(\theta \). It is important to note that their upper sides are constrained to be tangential to the rigid core (Fig. 2(d)).

  • Pad V features circumferential ribs with a constant thickness throughout its length (unlike pad I). These ribs are inclined at a 45-degree angle about the inner arc, with their upper sides constrained to be tangential to the rigid core. Moreover, the ribs are evenly spaced from each other at a distance of 2 mm (Fig. 2(e)).

In Fig. 2, the gray elements represent the rigid core and the rigid wall. In all designs, the radius of the rigid core is set to 7 mm, and the thickness of the soft pads is maintained at 3 mm.

2.2 Material selection

This study aimed to design differentiated structures using a single material, leveraging additive manufacturing techniques for the efficient production of complex-shaped items. The ideal material for covering robotic prosthetic hands should possess essential characteristics like flexibility, defor-mability, compactness, and durability. Similar to the nonlinear mechanical behavior observed in the human hand due to the presence of bone and nail, leading to self-hardening, soft pads exhibit a comparable behavior owing to their rigid core. Consequently, the stiffness of these structures is heavily influenced by both material properties, especially hardness, and the design of the intermediate layers. The recent availability of elastic materials compatible with additive manufacturing has facilitated the affordable and swift production of complex-shaped objects. However, a limitation arises from the absence of constitutive models accurately describing the mechanical behavior of these materials. In this research, Tango Plus Fullcure 930, a commonly used polymeric resin in additive manufacturing, was chosen for its experimentally determined mechanical properties and available hyperelastic constitutive models [25]. With a hardness similar to that of the human thumb, approximately 25 Shore A, this material was selected for designing soft pads. It is worth noting that hyperelastic constitutive laws, such as Ogden models for incompressible media, are essential for describing the quasi-static behavior of the material, as the Young modulus cannot be defined, while the Poisson’s ratio is set at \(\nu \) = 0.5. Detailed parameters of the hyperelastic model are provided in Table 1.

Table 1 Hyperelastic parameters of Tango Plus Fullcure 930 [25]
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Regarding the choice of material for the rigid core and rigid wall, a linear elastic material called PLA (Polylactic Acid) from the COMSOL Multiphysics material library has been chosen. A Poisson’s ratio of 0.35 has been assigned to this material as it is not defined automatically within the software.

2.3 FEA model

As extensively discussed in [38], a comprehensive understanding of a soft pad involves investigation of its various properties and behaviors. At the forefront is how the pad responds to normal contact loads when interacting with a rigid object, influencing the relationship between load and contact penetration. At this initial stage of analysis, emphasis is placed on the static behavior to refine the soft pads’ ability to mimic the compliance of a human finger. The objective is to demonstrate that by adjusting the inner layer geometry once the material and overall pad thickness have been determined, the pad’s static compliance can be controlled. In the simulated scenario, the pad is subjected to pressure against the rigid core and wall, while measurements are taken of the resulting displacement and contact force.

Fig. 3
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Meshed models and the resultant Von Mises stress field for (a) Pad I; (b) Pad II; (c) Pad III; (d) Pad IV; (e) Pad V

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The chosen simulation software for this analysis is COMSOL Multiphysics 6.1. Geometries have been designed to accommodate a 2D axisymmetric model instead of 3D ones, resulting in lower computational costs. This decision is crucial as the subsequent optimization steps will involve running multiple iterations for each shape, making models with reduced computational costs essential in the initial design stage. It’s worth noting that the geometries need to be developed within the software to define parameters for optimization. The simulations are conducted using the “Solid Mechanics” module in COMSOL. A fixed boundary condition is set for the bottom side of the rigid wall, while a prescribed displacement is applied to the top of the rigid core. Contacts have been established between the rigid core and the soft pad, as well as between the soft pad and the rigid wall.

Regarding FEA models of soft pads with differentiated structure designs, it’s important to note that the layer is not fixed to the inner surface of the rigid core, allowing for potential sliding. This boundary condition is selected to emulate the actual operating conditions of soft pads during their application in robotic hands. While the internal structure of soft pads is not physically bound to the hand’s endoskeleton, conducting FEA simulations under this assumption is valuable for qualitatively exploring various shape trends.

In these simulations, a high friction coefficient of \(\mu = 0.4\) is applied between the soft pad and the rigid core, while a lower coefficient of \(\mu = 0.1\) is used between the soft pad and the rigid wall. It’s worth noting that while the friction coefficient between the soft pad and the rigid wall has been experimentally determined in [25], the coefficient for the soft pad and the rigid core’s contact remains undetermined. Consequently, achieving experimental validation in future research for the friction between the soft pad with differentiated structural design and the rigid core is crucial for simulating more realistic friction laws beyond Coulomb friction.

Similar to previous studies [39, 40], our simulations employ a simple Coulomb friction model, chosen for its ease of use compared to other friction models in the literature (e.g., [31]). Despite the simplicity of this frictional model, comparisons between numerical and experimental results have shown promising alignment, as demonstrated in [25]. According to the Coulomb model, the tangential stress (\(\tau \)) is given by \(\tau = \mu \cdot \sigma + \delta \), where \(\sigma \) represents the normal stress, \(\delta \) denotes the initial cohesion, and \(\mu \) is the friction coefficient. Although the software allows for analyzing gradual displacements and variations in relative reaction forces, the simulations remain static in nature.

For all models, the “Free Quad” mesh type is employed. This two-dimensional element, shaped as a quadrilateral, comprises 4 nodes and exhibits excellent performance in simulating significant deformations. The mesh sizes for the soft pads and rigid components (rigid core and rigid wall) are configured as “Extremely Fine” and “Extra Fine,” respectively. Subsequently, a stationary solver with a relative tolerance of 0.1 is utilized for the analysis. The meshed models and the resultant Von Mises stress field for the pads are presented in Fig. 3(a)–(e).

The trends of reaction forces, represented by the normal load versus pad flattening for all the pads, are depicted in Fig. 4. It is evident from Fig. 4 that all the pads exhibit a self-hardening behavior, which is further augmented by the inclusion of the rigid core. Additionally, as anticipated, the stiffness of the specimens is significantly influenced by both the material properties, particularly hardness, and the design of the intermediate layer. Given a constant pad thickness, a proper selection of such design can improve the self-hardening characteristics of the pads.

2.4 Optimization problem

The primary objective of this paper is to replicate the compliance of the human finger by fine-tuning the shape parameters of the proposed patterns for soft pads. The compliance of the human finger has been extensively studied and documented in existing literature. Of particular interest in this study is the stiffness of the distal phalanx, a crucial aspect measured in [26]. The setup used in the experiment, illustrated in Fig. 5, consists of a custom-made mono-axial force sensor and a linear electric motor (Linmot P01-23\(\times \)80) with an integrated high-resolution position transducer (with a resolution of 1\(\mu \)m). This system offers precise force and position control with customizable movement patterns, while data is automatically collected by a PC. During the test, a rigid indenter is applied to the human finger at very low speeds to examine its quasi-static properties. To reduce friction at the contact point, petroleum jelly is applied to both the indenter and the finger before the test. Indenting the human distal fingertip with the indenter and measuring the resulting force and displacement allowed for the plotting of its stiffness curve, which is the focus of the following optimization.

Fig. 4
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Initial results of finite element analysis before optimization

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Fig. 5
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Experimental setup for measuring the stiffness of the human fingertip [26]

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The optimization setup is established within COMSOL Multiphysics by importing data points representing the force-displacement curve of the human distal phalanx into the parameter estimation module. To ensure that the simulation results closely match the experimental data, a least squares fit optimization method is employed. The objective function for this optimization can be formulated as follows:

$$\begin{aligned} F(\theta ) = \sum _{i=1}^{n} (y_i^{\text {exp}} - y_i^{sim}(\theta ))^2 \end{aligned}$$
(1)
$$\begin{aligned} \text {Subjectto: } \theta _i^{\text {min}} \le \theta _i \le \theta _i^{\text {max}} \text {for } i = 1, 2, ..., m \end{aligned}$$
(2)

where:

  • \(\theta \) represents the parameters of the model;

  • n is the number of data points;

  • \(y_i^{\text {exp}}\) is the experimental data at the ith data point;

  • and \(y_i^{sim}(\theta )\) is the simulated result at the ith data point using the model parameters.

  • m is the total number of parameters.

The parameter estimation method is set to BOBYQA, chosen due to its requirement for mesh adjustments with each iteration, with an optimality tolerance set to 0.01. Additionally, the maximum number of model evaluations is limited to 1000. The parameter to be optimized in each pad, along with its corresponding upper and lower bounds for all the proposed patterns, is presented in Table 2. Moreover, the shapes of the pads in their initial state, along with their respective upper and lower bounds, are shown in Fig. 6

3 Results and discussion

The optimization was executed in COMSOL Multiphysics with minimal computational expense, employing 2D axisymmetric models. The results of the optimization trials to align the shapes of the output curves of the pads with human finger compliance are illustrated in Fig. 7(a)–(e). The solver autonomously determines the number of trials by seeking a solution that best aligns with the target curve. The optimal parameter values (pertaining to the aforementioned variables) for the pads are detailed in Table 3.

Table 2 Optimization parameters and their corresponding initial values as well as upper and lower bounds
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Fig. 6
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Shapes of the pads in their initial state, along with their respective upper and lower bounds

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Fig. 7
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Optimization trials vs. human finger’s compliance: (a) Pad I; (b) Pad II; (c) Pad III; (d) Pad IV; (e) Pad V

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The optimum compliance for each pad is plotted in Fig. 8, showcasing how closely they align with the compliance of a human finger. Notably, Pad I closely captures the compliance of a human finger, exhibiting a linear response until approximately 0.6 mm, followed by nonlinear behavior and self-hardening, akin to the human finger’s behavior. Additionally, Pad V demonstrates some resemblance to the target compliance. This suggests that introducing multiple variables for the solver to optimize could yield results closer to the desired curve.

Optimizing differentiated structures for soft pads to attain a targeted force-displacement profile emerges as a highly efficient approach. Specifically, this method holds promise for designing prosthetic hands, allowing for the generation of outer surfaces that closely mimic those of natural hands. Beyond mechanical advantages, this could significantly enhance the acceptance of prosthetics among users and society overall.

When aiming to replicate the sensation of human soft tissues, it’s essential to consider the varied compliance across different anatomical regions, such as the proximal, medial, and distal phalanges of the human finger. In this study, our focus is solely on the hemispherical part of the distal phalanx, establishing a methodology for optimizing distinct structures. Analyzing the contact behavior and compliance of each sub-region through experimental or numerical means can help determine the most suitable morphology and material composition for prosthetic hand design. Once the desired compliance of an individual component is established, the compliance of the pads can be optimized to match that of the specific area of interest for effective coverage.

Table 3 Optimal parameter values for the pads, taking into account the compliance of the human finger
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Fig. 8
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Optimum compliance of the pads compared to human finger compliance

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In future research, expanding models to incorporate additional variables will provide the solver with greater flexibility to accurately capture the behavior of the target compliance. Furthermore, it is imperative to validate experimentally the performance of the optimized pads, ensuring that they indeed exhibit the desired characteristics. Additionally, annotating the fabrication process with specific comments on these soft pads could provide valuable insights into the resulting structural properties and performance of the soft pads. This approach could offer a deeper understanding of how different fabrication parameters or techniques influence the final product, leading to potential improvements in design and functionality.

4 Conclusions

In this study, efforts have been made to establish an optimization framework for designing differentiated structures capable of producing soft pads with a specified force-displacement curve (specific compliance). The primary objective has been to emulate the compliance characteristics of the human finger through FEA optimization. To achieve this goal, five distinct patterns for differentiated structures, along with their corresponding FEA models, have been introduced and thoroughly discussed. Furthermore, careful consideration has been given to the selection of appropriate materials for this purpose. Subsequently, by formulating an appropriate optimization function, the shapes of these patterns have been fine-tuned to match the compliance of the human finger. The findings suggest that this method holds promise as an efficient approach for tailoring soft pads to meet specific compliance requirements. Notably, the target in this paper has been to replicate the compliance of the human finger, and it has been demonstrated that one of the proposed and optimized pads (Pad I) closely approximates this target through the proposed methodology. In the future, it is necessary to develop experimental setups to validate the outcomes of the study, ensuring their practical applicability.