The field of robotics has witnessed a paradigm shift with the advent of soft robotics, moving away from rigid, pre-programmed movements towards adaptable, compliant, and safe interactions with the environment. A critical component in this evolution is the end effector, the interface between a robotic system and the object of manipulation. Traditional rigid grippers, while precise and powerful, often struggle with fragile, irregularly shaped, or variable objects. This has spurred significant research into soft grippers, often inspired by biological systems like octopus arms, starfish, or elephant trunks, utilizing materials such as silicone elastomers, shape memory alloys, and electroactive polymers. Among various actuation methods—including hydraulic, thermal, and electrical—pneumatic actuation remains highly popular due to its cleanliness, availability, and rapid response. However, many state-of-the-art soft end effectors are multi-fingered structures that grasp objects from the outside. This external grasping paradigm can present challenges when handling containers with liquids or objects with significantly varying outer contours, as uneven actuation pressure across fingers may lead to unstable grips and spillage.
To address these limitations, we propose a fundamentally different grasping strategy: internal radial expansion. This work details the design, comprehensive theoretical modeling, finite element analysis, and experimental validation of a novel pneumatic, air-supported soft end effector capable of gripping objects from within, such as containers of various calibers. This internal support mechanism provides a stable, conformal, and distributed holding force, minimizing localized stress on the object and enabling the secure handling of vessels containing liquids. Our approach centers on a single, radially-expanding soft actuator, simplifying control compared to multi-digit systems while achieving robust and adaptable grasping.
Design Principle and Structural Configuration
The proposed soft end effector consists of two primary subsystems: the soft pneumatic actuator and the mechanical interface/connection assembly.
The core innovation lies in the design of the soft actuator. It is conceived as a cylindrical silicone body with an internal cylindrical air chamber. A key design feature is the strategic variation in wall thickness. The top and bottom walls of the air chamber are significantly thicker, forming strain-limiting layers. In contrast, the circumferential side wall is deliberately made thin. This asymmetry in material distribution dictates the deformation behavior upon pressurization. When pressurized air is introduced into the internal chamber, the thinner sidewall offers less resistance to expansion compared to the constrained top and bottom. Consequently, the actuator undergoes preferential radial expansion, inflating outwards like a balloon, while its axial (height) deformation is minimized. The final inflated shape approximates an oblate spheroid or a barrel shape, providing a large, continuous contact surface ideal for internal support.
The geometric parameters defining the actuator are crucial for its performance. The table below summarizes the nominal design dimensions:
| Parameter | Symbol | Value (mm) |
|---|---|---|
| Outer Diameter | \(D\) | 30.0 |
| Side Wall Thickness | \(d_1\) | 1.0 |
| Air Chamber Diameter | \(d_2\) | 28.0 |
| Total Height | \(H\) | 63.0 |
| Top/Bottom Wall Thickness | \(h_1\) | 30.0 |
| Air Chamber Height | \(h_2\) | 28.0 |
The second subsystem is the connection assembly, which serves to securely attach the soft actuator to a robotic arm or a fixed platform. Its design ensures a robust and leak-proof interface. The internal bore of the connector features a helical groove and an inverted conical section with distributed fan-shaped slots at the top. During the assembly process, uncured silicone is applied between the actuator stem and this connector. The silicone flows into these grooves and slots, and upon curing, it creates a strong mechanical interlock, preventing detachment even under significant axial loads during gripping operations. This design is critical for the reliability of the overall end effector.
Material Constitutive Modeling
Silicone elastomers, the material of choice for our soft actuator, exhibit hyperelastic behavior. Their stress-strain relationship is nonlinear, reversible, and characterized by large deformations. Accurately modeling this behavior is essential for predicting the actuator’s performance. Numerous constitutive models exist, falling broadly into two categories: those based on the statistical thermodynamics of molecular chains (e.g., Arruda-Boyce, Treloar) and those based on phenomenological strain energy functions (e.g., Neo-Hookean, Mooney-Rivlin, Yeoh).
For this application involving large, multi-axial deformations, the Yeoh model is particularly advantageous. It is known for its good fit over a large strain range with a relatively simple form, and its parameters can be determined from standard uniaxial tensile tests. The Yeoh strain energy density function \(W\) is expressed as a power series in terms of the first invariant of the Cauchy-Green deformation tensor, \(I_1\). A second-order model often provides sufficient accuracy:
$$W = C_{10}(I_1 – 3) + C_{20}(I_1 – 3)^2$$
where \(C_{10}\) and \(C_{20}\) are material constants specific to the silicone rubber used (e.g., Ecoflex series or Dragon Skin). The first invariant \(I_1\) is defined as:
$$I_1 = \lambda_1^2 + \lambda_2^2 + \lambda_3^2$$
Here, \(\lambda_1, \lambda_2, \lambda_3\) represent the principal stretch ratios along the material’s principal axes. For an incompressible material like silicone, the volume is conserved during deformation:
$$\lambda_1 \lambda_2 \lambda_3 = 1$$
In our simplified analytical model for the radially expanding section, we assume plane strain conditions in the axial direction (thick top/bottom walls restrict axial stretch), hence \(\lambda_3 = 1\). For the circumferential expansion, we define the circumferential stretch as \(\lambda_c = \lambda_1\). Due to incompressibility and axisymmetry, the radial stretch is \(\lambda_r = \lambda_2 = 1 / \lambda_c\). Therefore, the first invariant simplifies to:
$$I_1 = \lambda_c^2 + \frac{1}{\lambda_c^2} + 1$$
And the strain energy density for our deformation state becomes:
$$W = C_{10} \left( \lambda_c^2 + \frac{1}{\lambda_c^2} – 2 \right) + C_{20} \left( \lambda_c^2 + \frac{1}{\lambda_c^2} – 2 \right)^2$$
Theoretical Model for Actuator Expansion
To establish a predictive relationship between the applied pneumatic pressure \(p\) and the resulting radial expansion (quantified by the outer diameter \(L\)), we develop a mechanics model based on energy principles. We simplify the actuator’s inflated geometry. The actively deforming section (the thin sidewall) is modeled as a cylindrical segment that, upon pressurization, expands into a toroidal/spherical segment.
Let the initial inner radius of the air chamber be \(r_1 = d_2/2\) and the initial outer radius of the sidewall be \(r_2 = D/2\). The initial height of the deforming sidewall is \(h = h_2\). The initial volume of the silicone in the sidewall is:
$$V_r = \pi (r_2^2 – r_1^2) h$$
The initial volume of the air chamber is:
$$V_a = \pi r_1^2 h$$
Upon pressurization, the sidewall bulges outward. We approximate the deformed air chamber as a cylindrical center portion (of reduced height \(h’\)) capped by two spherical segments. Let the bend angle of the sidewall be \(\theta\), measured from the vertical. The radius of the approximating sphere forming the bulge is \(R = r_1 / \cos(\theta/2)\). The new height of the cylindrical center portion is \(h’ = 2 r_1 \tan(\theta/2)\). The total volume of the deformed chamber \(V’_a\) is the sum of the central cylinder and the two spherical caps:
$$V’_a = \pi r_1^2 h’ + 2 \cdot \frac{2\pi}{3} R^3 (1 – \cos(\theta/2))$$
A more compact form is:
$$V’_a = \pi r_1^2 h’ + \frac{4}{3} \pi R^3 \left(\frac{\theta}{\pi}\right)$$
where the term \(\theta/\pi\) approximates the volume fraction of the sphere. The increase in chamber volume due to deformation is \(\Delta V_a = V’_a – V_a\).
We apply the principle of virtual work, equating the work done by the internal pressure to the change in the stored elastic energy within the silicone material. Assuming no other external forces and neglecting inertia, the balance is:
$$p \delta(\Delta V_a) = V_r \delta W$$
where \(\delta\) denotes a virtual variation. Relating the stretch \(\lambda_c\) to the bend angle \(\theta\) through geometric considerations gives \(\lambda_c = \theta / \sin \theta\) for the circumferential fiber at the midline of the sidewall. Substituting the expression for \(W(\lambda_c)\) and the geometric relations for \(\Delta V_a(\theta)\) and \(V_r\) into the virtual work equation, and differentiating with respect to \(\theta\), yields the nonlinear analytical relationship between pressure \(p\) and bend angle \(\theta\):
$$p(\theta) = \frac{ 3\cos^3(\theta/2) \cdot \pi (r_1^2 – r_2^2) h \cdot \frac{dW}{d\theta} }{ 6\pi r_1^2 \cos(\theta/2) + 4 r_1^3 + 12 \theta \tan(\theta/2) r_1^3 }$$
The term \(\frac{dW}{d\theta}\) is derived using the chain rule: \(\frac{dW}{d\theta} = \frac{dW}{d\lambda_c} \cdot \frac{d\lambda_c}{d\theta}\). Finally, the expanded outer diameter \(L\) (the key performance metric for gripping) is related to the bend angle \(\theta\) by:
$$L(\theta) = 2 [R + r_1] = 2r_1 \left(1 + \frac{1}{\cos(\theta/2)}\right) = \frac{2r_1 (1 + \cos(\theta/2))}{\cos(\theta/2)}$$
Thus, the coupled equations \(p(\theta)\) and \(L(\theta)\) form the complete theoretical model, allowing us to predict the required pressure to achieve a desired gripping diameter \(L\) for the soft end effector.
Finite Element Analysis (FEA) Simulation
To validate the theoretical model and gain deeper insight into the stress distribution and deformation geometry, we performed nonlinear finite element analysis using Abaqus/Standard. The soft actuator was modeled as a 3D deformable body with hyperelastic material properties defined by the Yeoh model (using typical coefficients for a soft silicone, e.g., \(C_{10} \approx 0.05\) MPa, \(C_{20} \approx 0.005\) MPa). The thick top and bottom sections were included, with the bottom face fixed in all degrees of freedom. The internal cavity surface was subjected to a uniformly distributed pressure load. An analytical rigid surface was used to model the connection assembly interface. The model employed hybrid elements (C3D8H) to handle the incompressibility of the material.
The simulation was conducted for increasing pressure levels from 0 to 0.05 MPa in increments of 0.01 MPa. The results confirmed the design intent: the actuator expanded predominantly in the radial direction with minimal axial elongation. The von Mises stress distribution showed higher stresses concentrated in the thin sidewall during inflation, while the thick sections remained relatively unstressed, acting as effective constraints.
The key output from FEA was the maximum outer diameter \(L\) at each pressure step. The table below compares the FEA-predicted diameters with those from the simplified theoretical model (using appropriate material constants) and later with experimental results:
| Gauge Pressure \(p\) (MPa) | Theoretical \(L\) (mm) | FEA \(L\) (mm) | Experimental \(L\) (mm) |
|---|---|---|---|
| 0.00 | 30.00 | 30.00 | 30.00 |
| 0.01 | 32.5 | 31.86 | 33.44 |
| 0.02 | 34.8 | 33.92 | 35.67 |
| 0.03 | 36.9 | 35.93 | 37.72 |
| 0.04 | 38.8 | 37.81 | 39.66 |
| 0.05 | 40.6 | 39.59 | 41.45 |
The FEA results lie between the theoretical and experimental values. The theoretical model tends to slightly overpredict the expansion due to its simplified geometry and idealized constraints, while FEA accounts for full 3D effects and material model more accurately. The experimental results show the largest expansion, which can be attributed to factors like slight variations in material properties from the assumed values, manufacturing imperfections (wall thickness variations), and the viscoelastic nature of real silicone not captured in the quasi-static Yeoh model. Nevertheless, the consistent trend across all three datasets validates the fundamental working principle and the order-of-magnitude accuracy of the models. This agreement confirms the end effector‘s predictable behavior.
Fabrication and Experimental Characterization
The soft actuator was fabricated using a multi-part, 3D-printed mold and silicone casting. The mold was designed in three main sections: a top plate, a middle section defining the sidewall and chamber geometry, and a bottom plate. A two-part platinum-cure silicone rubber (e.g., Ecoflex 00-30 or similar, Shore hardness ~A35) was mixed, degassed, and poured into the assembled mold. After curing, the actuator was demolded. The connection assembly was 3D-printed separately. The actuator was then bonded to the connector using the same silicone rubber as an adhesive, ensuring it flowed into the designed grooves and slots to create a permanent, high-strength mechanical bond, completing the assembly of the soft end effector.
A test platform was established, comprising an air supply (compressor or regulated gas cylinder), a precision electronic pressure regulator, solenoid valves for control, pressure sensors, and a data acquisition system. The end effector was mounted vertically. For the expansion characterization, the internal pressure was increased in controlled steps. At each pressure setpoint, high-resolution images were taken, and the maximum outer diameter \(L\) was measured using digital image processing techniques. The results, as shown in the previous table, demonstrated a smooth, repeatable, and monotonic increase in diameter with pressure.
The core functionality test involved gripping various objects. The end effector was inserted into the opening of different containers. Upon pressurization, it expanded radially, creating a firm frictional grip against the container’s inner wall. Successful grasping was demonstrated for a wide range of objects:
- Cylindrical containers (cups, beakers) of diameters from 35 mm to 50 mm.
- Square and rectangular cross-section boxes.
- Objects with irregular inner contours (e.g., the cavity of a headphone case).
In all cases, the grip was stable and distributed. For cylindrical containers, the axisymmetric expansion provided a perfectly conformal contact. For non-cylindrical objects, the soft material passively adapted to the shape, showcasing the inherent compliance and adaptability of this soft end effector design. The internal support mechanism proved particularly advantageous for gripping lightweight containers that could easily tip if grasped externally from one side.
Discussion and Comparative Analysis
The internally-actuated, air-supported end effector presents a compelling alternative to traditional external grippers. The table below highlights its key advantages and trade-offs compared to common multi-fingered soft grippers and universal jamming grippers.
| Feature | Proposed Internal Support End Effector | Multi-fingered Soft Gripper | Universal Jamming Gripper |
|---|---|---|---|
| Grasping Principle | Radial expansion from inside | Enveloping or pinching from outside | Molding and jamming around object |
| Contact Force Distribution | Uniform, outward radial pressure | Can be uneven across fingers | Conforms well, force depends on jammed state |
| Stability for Liquid Containers | Excellent (centralized, symmetric support) | Potentially unstable (asymmetric grip) | Good, but may require precise positioning |
| Adaptability to Opening Size | High (continuous diameter adjustment) | Limited by finger length and motion range | High, but requires object to be partially submerged in granules |
| Control Complexity | Low (single pressure input) | Higher (multiple chambers/actuators) | Medium (pressure for molding, vacuum for jamming) |
| Object Geometry Requirement | Requires an opening/ cavity | Requires accessible external surfaces | No inherent requirement, versatile |
The theoretical model, while simplified, provides a valuable first-order design tool. Discrepancies with FEA and experiment point to areas for model refinement: incorporating the exact deformed geometry rather than the spherical approximation, accounting for the strain-stiffening effect at higher stretches (perhaps using a 3rd-order Yeoh term), and including the energy stored in the bending of the thick top/bottom plates. The FEA serves as a powerful intermediate tool for detailed design optimization—for instance, exploring how the ratio of sidewall to endcap thickness, or the initial chamber aspect ratio, affects the expansion range and blocking force.
The performance of this end effector can be quantified further by measuring the axial extraction force (the force required to pull a container off the inflated actuator) as a function of internal pressure and container material (coefficient of friction). This force \(F_{extract}\) is related to the normal pressure \(p_{contact}\) and the contact area \(A_{contact}\) by:
$$F_{extract} \propto \mu \cdot p_{contact} \cdot A_{contact}$$
where \(\mu\) is the coefficient of friction. Since \(p_{contact}\) is approximately equal to the internal actuation pressure \(p\) (for a thin wall), and \(A_{contact} \approx \pi L h_{contact}\), the holding force scales linearly with input pressure and expanded diameter. This predictable scaling is a significant advantage for controlled manipulation tasks.
Potential Applications and Future Development
The unique gripping strategy of this soft end effector opens up numerous application avenues beyond simple container handling:
- Food and Beverage Industry: Handling glass jars, plastic bottles, or paper cups without marring the exterior surface, ideal for packaging or serving robots.
- Biomedical and Laboratory Automation: Grasping sample vials, test tubes, or beakers, minimizing the risk of breakage and contamination. The gentle, compliant touch is also suitable for handling biological tissues or delicate organoids in culture dishes.
- Logistics and Warehousing: Picking and placing boxes or totes from within, which can be more reliable than external suction cups or grippers on damaged or uneven boxes.
- Home Assistance and Service Robotics: A robot could use such an end effector to pick up a mug by its handle opening or stabilize a moving object by inserting the effector into a cavity.
- Space and Extreme Environments: The simplicity and lack of intricate moving parts make it robust. It could be used in space stations to handle experiment containers or tools with standardized ports.
Future work will focus on several enhancements:
1. Material and Structure: Integrating fiber reinforcements in the sidewall to tailor the expansion profile (e.g., limit axial expansion further or create non-axisymmetric shapes).
2. Sensing: Embedding flexible sensors (capacitive, resistive) into the actuator wall to provide real-time feedback on contact and grip diameter, enabling closed-loop control.
3. Multi-Material Printing: Using direct 3D printing of soft materials to create actuators with graded stiffness, eliminating the need for complex molding.
4. Advanced Modeling: Developing a fully coupled fluid-structure interaction (FSI) model to capture the dynamics of inflation and the effects of air compressibility for high-speed operations.
5. Modular Systems: Creating an array of smaller internally-actuating units on a single platform for grasping multiple objects or objects with multiple internal cavities simultaneously.
Conclusion
In this work, we have presented the complete design cycle—from conceptualization and analytical modeling to simulation, fabrication, and experimental validation—of a novel pneumatically actuated soft end effector based on the principle of internal radial support. The core design utilizes a thickness-gradient in a cylindrical silicone actuator to achieve predictable and substantial radial expansion under pneumatic pressure. We derived a nonlinear analytical model based on hyperelastic material theory and the principle of virtual work, establishing the relationship between input pressure and expanded diameter. This model was corroborated by trends observed in detailed finite element analysis and physical experiments, confirming its utility as a design guide. The fabricated prototype demonstrated successful and stable grasping of containers with a wide variety of opening sizes and shapes, validating the effectiveness of the internal support strategy. This end effector offers a compelling alternative to external grippers, particularly for handling vessels, objects with cavities, and scenarios requiring a stable, symmetric grip. Its simplicity, compliance, and predictable behavior make it a promising tool for advancing the capabilities of soft robotic manipulation in diverse industrial, service, and research settings. The development of this end effector underscores the potential of innovative kinematic and structural strategies in soft robotics to solve specific manipulation challenges that are difficult for traditional rigid or even conventional soft grippers to address.