End-effectors, the vital interface between robotic systems and their environment, have long been a focal point of automation research. Traditional, rigid grippers, while precise and powerful, often struggle with delicate, irregularly shaped, or fragile objects. The burgeoning field of soft robotics offers a compelling alternative, leveraging compliant materials to create end-effectors capable of adaptive, gentle, and versatile manipulation. Inspired by biological systems like elephant trunks, starfish, and octopus arms, these soft end-effectors have demonstrated remarkable potential. However, a prevalent design paradigm—the multi-fingered soft gripper—faces inherent challenges when tasked with stably grasping containers of varying diameters, particularly those holding liquids. The independent actuation of fingers can lead to unbalanced gripping forces, and the external enveloping grasp may not conform well to objects with significant contour variations. To address these limitations, we propose a novel gripping strategy: an air-supported, internal expansion end-effector. This work details the design, modeling, simulation, and experimental validation of this new class of soft end-effector, demonstrating its ability to securely hold containers from the inside out.
The core innovation of our approach lies in its departure from external multi-finger grasping. Instead of attempting to conform to an object’s exterior, our end-effector is designed to be inserted into a hollow object, such as a cup or bottle, and then actuated to expand radially. This internal expansion creates a distributed, high-friction contact along the inner wall of the object, enabling a stable and balanced grip that is largely independent of the object’s external shape. The design is elegantly simple, consisting of two main components: the soft actuator body and a rigid connection fixture.
The soft actuator is the heart of the end-effector. It is fabricated as a cylindrical bellows-like structure from a hyperelastic silicone elastomer. Its geometry is carefully tailored to promote controlled radial expansion upon pressurization. The actuator features a central air chamber. The sidewalls of this chamber are relatively thin, making them the primary deformation zone. In contrast, the top and bottom walls of the cylinder are significantly thicker, acting as strain-limiting layers that constrain axial elongation. Consequently, when pressurized air is introduced into the chamber, the compliant sidewalls bulge outward symmetrically, causing the entire actuator to expand primarily in its radial direction, transforming its cross-section from a circle into a larger, near-spherical shape. The key structural parameters are summarized in Table 1.
| Parameter | Value (mm) |
|---|---|
| Initial Diameter (D) | 30.0 |
| Sidewall Thickness (d₁) | 1.0 |
| Air Chamber Diameter (d₂) | 28.0 |
| Total Height (H) | 63.0 |
| Top/Bottom Wall Thickness (h₁) | 30.0 |
| Air Chamber Height (h₂) | 28.0 |
The connection fixture serves as the mechanical interface between the soft silicone actuator and the robotic arm or test platform. It is a rigid component designed to securely anchor the soft actuator while providing a sealed passage for the pneumatic supply line. Its interior features a helical groove and an inverted conical section with radial slots. During the assembly process, uncured silicone is used to bond the actuator to this fixture. The grooves and slots allow the liquid silicone to flow into and fill all voids, creating a large interfacial contact area and a robust, leak-proof mechanical connection crucial for reliable performance of the end-effector.
To predict and understand the behavior of this soft end-effector, developing a mathematical model linking the input pressure to the output radial expansion is essential. This begins with characterizing the silicone material. Silicone is a hyperelastic, nearly incompressible material, meaning its stress-strain relationship is nonlinear and its volume remains constant during deformation. Among the many constitutive models available, the Yeoh model is well-suited for large-strain analysis and offers a relatively simple form. Its strain energy density function \(W\) is expressed as:
$$W = C_{10}(I_1 – 3) + C_{20}(I_1 – 3)^2$$
where \(C_{10}\) and \(C_{20}\) are material constants determined experimentally, and \(I_1\) is the first invariant of the Cauchy-Green deformation tensor. For our axisymmetric expansion case, we can define principal stretch ratios. Assuming incompressibility (\(\lambda_1 \lambda_2 \lambda_3 = 1\)) and no deformation in the axial (height) direction of the strain-limiting layers (\(\lambda_3 = 1\)), we have \(\lambda_1 = \lambda\) and \(\lambda_2 = 1/\lambda\), where \(\lambda\) is the circumferential stretch. The first invariant becomes \(I_1 = \lambda^2 + \lambda^{-2} + 1\).
To model the actuator’s mechanics, we simplify its geometry post-inflation. The expanded air chamber is approximated as a cylindrical midsection with two spherical end caps. Let the original inner radius be \(r_1\) and the original outer radius be \(r_2\). The central angle subtended by each spherical cap is denoted by \(\theta\). Using geometric relations, the radius of the spherical cap is \(R = r_1 / \cos(\theta/2)\), and the new height of the cylindrical section is \(h’ = 2r_1 \tan(\theta/2)\). The relationship between the circumferential stretch \(\lambda\) and the bending angle \(\theta\) is given by \(\lambda = \theta / \sin\theta\).
The volume of silicone \(V_r\) remains constant. The change in the internal air chamber volume \(\Delta V_a\) upon inflation is calculated from the geometry. Applying the principle of virtual work, the work done by the internal air pressure \(p\) during an incremental volume change equals the change in the elastic strain energy stored in the silicone material:
$$p \, d(\Delta V_a) = V_r \, dW$$
Differentiating this expression with respect to the deformation angle \(\theta\) yields the nonlinear analytical relationship between the input pressure \(p\) and the geometric parameter \(\theta\):
$$p = \frac{3\cos^3\left(\frac{\theta}{2}\right) \pi (r_1^2 – r_2^2) h \frac{dW}{d\theta}}{6\pi r_1^2 \cos\left(\frac{\theta}{2}\right) + 4r_1^3 + 12\theta \tan\left(\frac{\theta}{2}\right) r_1^3}$$
where \(h\) is the original height of the deformable sidewall. The final expanded outer diameter \(L\) of the end-effector, which is critical for gripping, is related to \(\theta\) by:
$$L = \frac{2r_1 \left[1 + \cos\left(\frac{\theta}{2}\right)\right]}{\cos\left(\frac{\theta}{2}\right)}$$
Thus, by selecting a desired expanded diameter \(L\) for a target container, the required driving pressure \(p\) can be determined theoretically using this model.
Prior to physical fabrication, we performed Finite Element Analysis (FEA) using Abaqus/CAE to simulate the actuator’s deformation. A model of the silicone actuator was created, and the hyperelastic material properties were defined using a Yeoh model with constants derived from standardized silicone tests. A pressure load was applied to the internal surfaces of the air chamber. The simulation results confirmed the design intent: under pressure, the sidewalls expanded radially outward with minimal axial change, forming a near-spherical contact profile ideal for internal grasping. The displacement contours clearly showed symmetric inflation. We simulated pressure increments of 0.01 MPa up to 0.05 MPa. The resulting maximum radial diameters from the FEA are listed in Table 2, indicating a significant expansion capability suitable for a range of container sizes.
| Pressure, p (MPa) | Expanded Diameter, L (mm) – FEA |
|---|---|
| 0.00 | 30.00 |
| 0.01 | 31.86 |
| 0.02 | 33.92 |
| 0.03 | 35.93 |
| 0.04 | 37.81 |
| 0.05 | 39.59 |
Following the simulation, we proceeded to fabricate the physical soft end-effector. The process began with 3D-printing a multi-part mold designed in CAD software. A two-part, platinum-cure silicone elastomer (with a Shore hardness of approximately A35) was mixed and degassed, then poured into the assembled mold. After curing, the demolded soft actuator was bonded into the 3D-printed connection fixture using additional uncured silicone, ensuring a monolithic and airtight assembly. The final prototype was then integrated into a test platform consisting of a support frame, an air pump, precision pressure regulators, solenoid valves, and tubing.
The experimental investigation started with characterizing the expansion performance. The end-effector was pressurized in 0.01 MPa increments from 0 to 0.05 MPa. At each steady-state pressure, the maximum outer diameter was measured manually using calipers. The average results from multiple trials are presented in Table 3.
| Pressure, p (MPa) | Expanded Diameter, L (mm) – Experiment |
|---|---|
| 0.00 | 30.00 |
| 0.01 | 33.44 |
| 0.02 | 35.67 |
| 0.03 | 37.72 |
| 0.04 | 39.66 |
| 0.05 | 41.45 |
The data from the theoretical model, FEA simulation, and physical experiment were compiled and plotted for comparison. All three curves showed a consistent monotonic trend: the diameter of the end-effector increases with applied pressure. The experimental results aligned closely with the theoretical prediction, validating the derived mathematical model based on the Yeoh constitutive law and virtual work principle. Some deviation between the FEA curve and the other two was observed, which can be attributed to slight differences in the assigned material parameters in the simulation and minor imperfections in the fabricated prototype’s geometry compared to the ideal CAD model. Nevertheless, the overall congruence confirms the model’s utility for predicting the behavior of this air-supported end-effector.
The ultimate test of any end-effector is its grasping performance. We conducted a series of gripping demonstrations with containers of various inner diameters and shapes. The soft end-effector, mounted on a simple vertical stage, was lowered into a container. A low pressure (typically between 0.02 MPa and 0.04 MPa, depending on the container size) was then applied. The radial expansion caused the silicone to press firmly against the inner wall, generating sufficient frictional force to lift the container reliably. Successful grasps were demonstrated on cylindrical cups and bottles with different mouth openings. Notably, the end-effector also effectively gripped objects with non-circular openings, such as a square packaging box and a rectangular headphone case, showcasing its adaptability to shape variations. This internal gripping method provides a stable, enveloping contact that is inherently balanced, eliminating the tilting or squeezing concerns associated with external multi-finger grippers, especially when handling vessels containing liquid.
In conclusion, we have successfully designed, modeled, and validated a novel air-supported soft end-effector based on a principle of internal radial expansion. The key contributions include: (1) A new gripping paradigm for hollow objects that offers inherent stability and balanced force distribution. (2) A practical and manufacturable design comprising a custom-molded silicone actuator and a connection fixture. (3) A nonlinear analytical model derived from hyperelasticity theory that accurately relates input air pressure to output expansion diameter, providing a valuable tool for design and control. (4) Experimental verification showing that the prototype can effectively grip containers of different calibers and even some non-circular shapes. This end-effector presents a compelling solution for applications in warehousing, food handling, and assembly, where securely grasping vessels from the inside is advantageous or necessary. Future work will focus on integrating sensor feedback for pressure control, optimizing the geometry for higher load capacity, and exploring multi-chamber designs for more complex manipulation tasks.