A novel rigid-soft gripper for safe and reliable object handling

Abstract

This research paper proposes a novel robotic gripper that combines rigid and soft components. The challenge of developing robotic grippers that can safely handle a wide variety of objects and remain stable, especially under harsh vibration, is addressed. The gripper uses a main rigid structure for maintaining the stiffness of the robot hand and soft pads under air pressurization for handling the softly and flexibly the objects. The proposed design capitalizes on the advantages of both rigid and soft grippers, by using rigid clamps to achieve robust gripping movements. Simultaneously, exploiting the soft pads’ deformation to generate gripping force, this integration not only has high reliability but also guarantees safety in gripping objects. The novel concept, design, and fabrication of the gripper are introduced. Then, the gripper’s ability was validated through mathematical modeling, analysis, and experiments. In particular, the geometrical kinematics of the rigid mechanism and deformation of the soft element is mathematical modeled and numerical simulation. Subsequently, the effective gripping size and contact force are analyzed. Experiments are conducted to verify the accuracy of the mathematical model and simulations, as well as to assess the capabilities of the proposed gripper design. The gripping experiment validates that the gripper can securely grasp a diverse range of objects while ensuring stability during subsequent movements, even under intense high-frequency vibration conditions. This gripper design can be applied for handling a wide range of objects with varying sizes, masses, fragility, and placement positions, specially when the gripping process necessitates both safe interaction and high reliability.

1 Introduction

Robotic manipulation encompasses a range of actions performed by robots in their environment, including grasping objects, packing orders into boxes, transporting items from one location to another, etc. [1, 2]. Of all these actions, grasping or gripping is the most prevalent and fundamental task. Grasping or gripping can be defined as the physical act of securely holding onto an object. In essence, gripping refers to the act of moving an object from one place to another, commonly known as the “pick and place” action. Gripping is a crucial aspect of robotic manipulation, and as a result, grippers are one of the most commonly used end-effectors in robotics. The development of the gripper is associated with innovative research achievements in the robotic field and has been demonstrated by various research achievements. Presently, state-of-the-art robotic gripper designs focus on several critical goals, notably emphasizing flexibility and adaptability to accommodate a wide range of object shapes and sizes. These grippers aim for precision, integrating intelligent sensors and control systems to precisely adjust gripping positions and necessary contact forces. Moreover, improving picking efficiency involves factors like object mass and picking speed as significant objectives. Equally important is safety, ensuring the protection of both users and objects during the picking processes [3, 4].

The current existing gripper designs typically align with three main directions: traditional grippers utilizing rigid components, soft grippers constructed from soft materials, and hybrid grippers combining rigid and soft components.

Typically, traditional grippers are made up of rigid links, joints, sensing elements, and control systems [5, 6]. Moreover, to increase the rigidity of the robot arm, hybrid serial-parallel linkages or closed loop mechanisms were added to the main structure of the robot [7, 8]. These grippers have several advantages, such as enhancing picking efficiency, reliability, and accuracy. Nonetheless, their lack of flexibility due to the limited movement range of the rigid links and joints makes them unsuitable for handling various types of gripping objects, especially deformable or fragile ones. In order to manipulate delicate objects without causing damage, a sophisticated controller and tactile sensing system are required to monitor the precise contact force. Numerous efforts have been made to enhance the design and performance of the conventional gripper to address these issues.

For instance, a parallel gripper with an optical mirroring mechanism based on the Periscope principle is presented in [9], allowing for vision-based sensing to detect slipping during grasping. This gripper has proven to have high gripping reliability with various shapes and weights of gripping objects without slipping. In [10], the authors propose a method to estimate the remaining useful life of a two-jaw parallel gripper. The design of a generalized gripper capable of multiple grasping modes for various sheet parts was introduced in [11]. This gripper integrates link-type fingers, sucker-type fingers, and two sets of compact force transmission mechanisms with large mechanical gain to adapt to different grasping modes. Additionally, the researchers in [12] presented a general method to improve the design of a customizable robot gripper. This method involved optimizing both the chosen parameters of the gripper mechanism and the parameters of the finger geometry by utilizing task-specific finger designs acquired through dynamic simulation. Several methods to control and optimize the grasping force were presented in [13,14,15]. However, these grippers have intricate structures, sensor systems, and advanced control algorithms, and are only suitable for gripping rigid objects.

Soft robotics emerged as a new area of the robotic field a few decades ago, utilizing novel soft materials like silicone elastomers, urethane, and hydrogels...to enable new robot functions [16, 17]. By leveraging the softness and deformability of these materials, soft robots are highly compliant and can easily adapt to their environment with minimal risk of damage, a capability that rigid robots lack. In the soft robotic research field, the development of soft grippers has many achievements. Soft grippers normally rely on the deformation of soft materials to generate a grasping form, which allows them to maintain a large contact area and generate significant frictional force and moment without the need for excessive contact force. This enables safe interaction with grasping objects. There have been various designs of soft robotic and soft-fingered hands described in the literature [18,19,20], with most exploiting the high response, energy-efficient, and omnidirectional characteristics of pneumatic/hydraulic actuators. Soft grippers based on such actuating mechanisms typically consist of serial chambers with thin walls designed to generate the gripping shape under pressurization activation. However, due to the deformable properties of soft materials, soft robotic grippers are only suitable for handling lightweight objects with predetermined placement positions. The gripping stability and robustness of soft grippers are not as reliable as conventional grippers.

Several studies have focused on enhancing the loading capacity of soft grippers. For instance, one such study [21] proposes an untethered soft pneumatic gripper that integrates a silicone elastomer structure with pneumatic artificial muscle to achieve a high payload. Similarly, another study [22] presents a universal method for equipping the tips of soft bending actuators with bioinspired force booster fingers, which significantly enhances the payload of multi-finger soft grippers while retaining their flexibility. In order to improve the effectiveness of manipulating objects with varying sizes, shapes, and postures, the authors of [23] introduced a soft gripper with four fingers that can accommodate two different gripping sizes and four distinct gripping modes. The fingers of this gripper are designed to mimic the functionality of the fiber-reinforced bending actuator. However, these grippers have a complex design, are expensive, and are only suitable for gripping objects with predetermined postures. In robotic systems, grippers are mounted on the robot arm to manipulate various objects of different weights and sizes, including delicate objects that are easily damaged, such as glass and fruits, and objects in different postures such as standing or lying down. Thus, robotic grippers need to be flexible, diverse, and reliable without damaging objects.

One of the solutions that have been utilized to enhance the flexibility, reliability, and handling capacity of soft grippers involves employing hybrid grippers, which combine rigid and soft components. In a study by the author [24, 25], a soft gripper with rigid elements was introduced. This gripper consisted of three fingers that operated based on the principle of pneumatic activation, utilizing serial chambers. Rigid parts were incorporated within the soft chambers to augment the fingertip force and actuation speed. Similarly, in another study [26], the authors proposed a hybrid gripper design inspired by the human metacarpophalangeal joint. This design incorporated a combination of a pneumatic synergistic alignment joint, constrained by a rigid universal joint and a rigid link. Although these hybrid grippers exhibit complex structures and require intricate control, they possess a fingered gripper structure, rendering them unstable under specific working conditions such as vibration or disorder instability. As the application of robots continues to advance in the current industrial revolution 4.0, it becomes necessary for robots and their mounted grippers to operate in various environments. For instance, a gripper can be mounted on a mobile robot, which can traverse rough and undulating terrain after completing the gripping task, resulting in significant vibrations for both the robot and the gripper. Consequently, the gripper must ensure safety by preventing deformation or breakage, while also maintaining high stability to prevent slipping or dropping of objects.

To address these requirements, we propose a gripper design that exploits the benefits of both rigid and soft grippers by combining rigid parts and soft elements. The gripping motion is achieved through the operation of rigid parts by using cylinder actuators, while the soft pads are inflated under air pressurization to directly contact the object and generate gripping forces. The objectives of the proposed gripper involve efficiently grasping diverse objects, including rigid, fragile, and deformable items, situated in various orientations such as standing or lying down, while emphasizing high reliability and stability, especially in scenarios involving robust vibrations-like integrating on mobile robots moving on rugged terrain.

In the realm of robotics gripper designs, there has been considerable attention given to the combining of soft pads with rigid clamps. For instance, in [27], the authors conducted a finite analysis into the viscosity alterations of magnetorheological fluid within a jaw gripper cushion. Then, they introduced a gripper concept featuring thermoplastic polyurethane cushion-finger pads filled with MR fluid. In a other publication [28], a novel robotic gripper with soft surfaces and underactuated joints was introduced. The soft surface comprised a deformable rubber bag filled with incompressible fluid and a microgripper inside the fluid. This innovative design enabled three distinct modes of operation: parallel gripping, pinching, and enveloping. Furthermore, in another article [29], a circular shell gripper was proposed, comprising a rigid external shell encompassing four internal air chambers of soft material. This gripper design facilitated versatile handling postures and adept twisting manipulations.

In this research, the novelty of this design is based on utilizing two curvature rigid clamps designed with hollow cavities along their profiles. These hollow spaces house soft pads that can be inflated using compressed air. This gripper design ensures that the rigid clamps move easily in the initial stages of the gripping process when the gripper is securing the object. Subsequently, the soft pads deform and inflate, establishing direct contact with the object to generate the necessary force for secure gripping. This design also allows for the manipulation of diverse objects, irrespective of size or orientation, whether they are positioned upright or lying. Unlike existing grippers [27, 28] employing pre-curved soft pads with dome shape attached to rigid clamps, which impose limitations on object size and convenient pickup positions. Moreover, in comparison with the circular shell design of the gripper in [29], the curvature rigid clamp is designed with comb teeth. This feature facilitates the gripping of challenging objects, such as heavy cylindrical items lying on the ground. In addition, the integration of teeth on the curved rigid clamps, combined with the gripping force generated by the soft pads, enhances safety when handling delicate or easily deformable objects. It also effectively prevents object slipping or accidental displacement even under intense vibration conditions during the movement process after completing gripping manipulation. This is an invaluable attribute, especially when these grippers are mounted on mobile robot systems operating in varying terrains and encountering substantial vibrations during movement. Consequently, the design of the proposed gripper facilitates the improvement of the robot system’s reliability and performance.

The rest of this paper is organized as follows: In Sect. 2, we describe the design and fabrication of the gripper. In Sect. 3, the mathematical modeling of the gripper’s gripping is proposed. Finally, the design of the controlling system and validated experiments of the gripper with experimental results, discussion, and conclusions is presented.

2 Design and fabrication of the gripper

2.1 Design descriptions

The proposed gripper design targets specific objectives: Firstly, it could effortlessly attach to robot arms, facilitating the efficient picking of objects within the robotics system. Secondly, the movement of the rigid clamps and activation of the soft pads will be air-driven, optimizing compressed air usage. The compressed air pressure level remains below 70 kPa to maintain a compact compressed air supply system, allowing seamless integration with the actual robot system without compromising movement efficiency. The gripper is designed to handle objects within a diameter range of 15 mm to less than 70 mm and with a mass not exceeding 1 kg. It accommodates objects varying in hardness, from delicate items such as eggs and fruits to considerably harder objects. Moreover, the gripper is capable of manipulating objects placed in both standing and lying positions. After picking up the object, the gripper can withstand significant vibrations during movement without falling or damaging the object.

To fulfill these main objectives, the proposed gripper design is detailed as follows:

The design of the gripper has six main parts, including two rigid clamps, soft pads, rotational joints, a connected body, two pneumatic cylinder actuators, and a rotational base for mounting the gripper into the robot arm. The design of the gripper is depicted in Fig. 1, where the rigid clamp has a hollow cavity to embed the soft pad. The rigid clamps connect to the connected body and the rotational joints by bolt joints. The opening and closing motions of the gripper are operated through the translational motion of two pneumatic cylinder actuators. The soft pad is a closed-soft chamber, with a thin wall thickness of 2 mm. The soft chamber has a small tube to input air pressure for generating inflated deformation of the soft chamber’s outer surface, which contacts the gripping objects and generates the gripping force.

Fig. 1

The design of gripper

The detailed designs of those parts are described as follows:

The rigid clamp includes two similar parts of the left and right clamp, which have a semicircular cross-section with an inside hollow cavity to hold the soft pad. The end of the rigid clamps has a comb teeth design for knitting together during the gripping process to increase the gripping ability with a small diameter range of objects. Additionally, this design effectively prevents object slippage even when the gripper undergoes intense vibrations during subsequent movements after completing the gripping manipulation. The curvature radius of the rigid clamps and the dimension of comb teeth are determined based on mathematical modeling calculations, presented in Sect. 3.1. In the cavity of the rigid clamp, there is a hole of a diameter of 5 mm to connect the air tube of the soft pad. The rigid clamps have been designed with a protruding part on the outer surface for connecting with the piston through the rotational joint. The remaining dimensions of the rigid clamps are configured to facilitate seamless assembly with other elements, as depicted in Fig. 2.

Fig. 2

Design of rigid clamps

The soft pad comprises a thin-walled hollow chamber that has a thickness of 2 mm and a circular cross-section with an outside diameter of 42 mm and an inside diameter of 30 mm. The outer and inner surface of the soft chamber comes into contact with the inside hollow cavity of the rigid clamp and the gripping objects, respectively. In the center of the outer surface of the soft part, there is a hole with a diameter of $\phi = 5$ mm to enable air pressure to enter through the air tube. The soft pad is fabricated from Silicone RTV 225, a material with a hardness of 25 A, and is designed to withstand air pressure within the range of 30 kPa to 70 kPa. The diameter and thickness dimensions of the soft pad are determined through calculations, outlined in Sect. 3.2, considering its deformation and the resultant contact force necessary to accommodate the specified size range and mass of objects that the gripper can handle. Figure 3 depicts a detailed design of the soft pad.

Fig. 3

The design of soft pad

The connection parts include a rigid-connected body, rotational joints, and a rotational base, whereas the rigid-connected body is designed for connection between the rigid clamp, pneumatic cylinder actuator, and rotational base. It has a rectangular box shape with a length of 127 mm. There are four 8-mm-diameter holes on the lower side to connect the rigid body to the rigid clamps, four 5-mm-diameter holes on the upper side to connect to the two pneumatic cylinders, and a single 8-mm-diameter hole in the middle side to connect to the rotational base. The design of the rigid-connected body can be observed in Fig. 4.

Fig. 4

The design of rigid-connected body

The rotational joint is designed with a rectangular cross section, and it has three holes with a diameter of 5 mm to connect the rigid clamp and the pneumatic cylinder. The remaining dimensions are shown in Fig. 5.

Fig. 5

The design of rotational joint

The rotational base is designed to connect the gripper to the robot arm for implementing the gripping tasks. The design and dimensions of this base are shown in Fig. 6.

Fig. 6

The design of rotational base

The pneumatic cylinder actuator is a dual directions type of CDJ210*5*35 with a cylindrical cross-section. The outer diameter of the cylinder is 10 mm, and the piston diameter is 4 mm. The working stroke of the cylinder is 35 mm. The length of the cylinder in the initial position is 110 mm. The schematic of the cylinder actuator is shown in Fig. 7.

Fig. 7

The schematic of cylinder actuator

All parts of the gripper are assembled with commercially standard screws. The completed assembly of the gripper is shown in Fig. 1.

The dimensions of the rigid-connected body, rotational joints, rotational base, and pneumatic cylinder actuator are designed and selected to facilitate the straightforward assembly of the gripper components and their integration with the robot arm. Moreover, these dimensions ensure the gripper maintains a compact size suitable for handling objects within the target size range. It is important to note that the dimensions of the components in this proposed gripper design can be adaptable and customizable to accommodate different size ranges and object masses.

2.2 Fabrication of gripper

The soft pad is made from the soft material of silicone RV-25 by using the modeling method, where the molds of the soft pad are fabricated by 3D printing, using the Zortrax 200 machine. This mold is divided into three parts: The first part and the second part (Fig. 8a,b) are used to make the main part and form the air cavity of the soft pad. The second mold part (Fig. 8c) is used to create the sealing cap for the soft pad.

Fig. 8

The mold to fabricate soft pad

The fabrication process of the soft pad can be summarized as:

• (1) Using 3D printing technology to fabricate the molds for the fabrication of soft pads (Fig. 8).

  Fig. 9

  

  Fabrication process by molding method of soft pad

• (2) Mixing the A-B component of the liquid silicone rubber RTV 225 with a ratio of 1:1, then poured the mixed liquid silicon rubber into the first mold part and cured at room temperature after sixteen hours. (It may be shortened using a microwave oven.) After the curing process was completed, the first mold part is removed to get the main part of the soft pad (Fig. 9).

• (3) After finishing the second stage, the soft pad still has an opened end, which is closed with the same mixed silicone rubber by molding method, using the second mold part. The curing condition is the same as the curing process of the soft pad’s main part. Finally, the second mold is removed to complete the fabrication of the soft pad.

The rigid parts of the gripper including the rigid clamps, the rigid-connected body, the rotational joints, and the rotational base are designed by 3D software—Solidworks then is fabricated by 3D printing technology, using the 200 Zortrax machines.

Next, the soft pad is attached to the hollow cavity of the rigid clamps by special adhesive glue. Finally, the pressure cylinder actuator is assembled to the rigid clams and rigid-connected body for completing the gripper as shown in Fig. 1.

The entire process of manufacturing a complete gripper, involving the production of rigid components via 3D printing, fabricating soft pads by molding method, and assembling all components, typically takes approximately 48 h.

3 Mathematical modeling for the gripper design

3.1 Geometrical kinematics of the rigid mechanism of the gripper

The gripping size range of the proposed gripper design is primarily dependent on the geometric kinematics of the rigid mechanism and the deformation of the soft pad. Thus, to investigate the relationship between the stroke of the cylinder actuator and the geometric kinematics of the rigid clamp, a geometrical kinematics model is introduced.

Figure 10 shows the schematic model of the proposed gripper, where (1) is a cylinder, (2) is a piston, (3) is a rigid clamp, and a rigid base is denoted by (0). The piston and rigid clamps are connected by two rotational joints, while the cylinder is connected to the piston and rigid base by translational and rotational joints, respectively (Fig. 10).

Fig. 10

The schematic principle of the gripper

The gripper’s motion is assessed through the calculation of the number of free degrees of the system as follows:

The number of free degrees of the gripper is determined as:

$$\begin{aligned} w = 3n - 2p \end{aligned}$$

(1)

where n is the number of dynamic links and p is the number of joints.

Thus, we have:

$$\begin{aligned} w = 3 \times 3 - 2 \times 4 = 1 \end{aligned}$$

(2)

The gripper’s dynamic structure possesses a single free degree, meaning that the gripper mechanism converts the piston’s reciprocating motion into rotational movement of the rigid clamp to execute closing actions, and conversely for opening motion.

The motion of the gripper can be described as:

The closed state of the gripper is at position $O_1$ of the center point. In such a state, the piston’s head is at $C_1$, its corresponding position relative to the connected-rigid body is determined by the angle $\alpha$, and the piston’s length is $L_1$. To implement the gripping task, the rigid clamps rotate around the rotary joints at point A, through the reciprocating motion of the piston. In the opened state of the gripper, the center of the rigid clamp is at position $O_2$, corresponding to the position of the piston at point $C_2,$ and the piston length is then $L_2$. The reciprocating displacement of the piston $\Delta L$ can be determined as:

$$\begin{aligned} \Delta L = {L_1} - {L_2} \end{aligned}$$

(3)

To analyze the relationship between piston displacement $\Delta L$ and the gripper rotation angle $\varphi$, we note that:

$$\begin{aligned} A{C_1} = \sqrt{{b^2} + {L_1}^2 - 2b.{L_1}.\cos \alpha } = A{C_2} = \sqrt{{b^2} + {L_2}^2 - 2b.{L_2}.\cos (\alpha + \varphi )} \end{aligned}$$

(4)

Thus, we have:

$$\begin{aligned} {b^2} + {L_1}^2 - 2b.{L_1}.\cos \alpha = {b^2} + {L_2}^2 - 2b.{L_2}.\cos (\alpha + \varphi ) \end{aligned}$$

(5)

Then:

$$\begin{aligned} 2b.{L_2}.\cos (\alpha + \varphi ) = {L_2}^2 - {L_1}^2 + 2b.{L_1}.\cos \alpha = \Delta L.(2{L_1} - \Delta L) + 2b.{L_1}.\cos \alpha \end{aligned}$$

(6)

Finally, we have the relationship:

$$\begin{aligned} \varphi = arc\left( {\frac{{\Delta L.(2{L_1} - \Delta L) + 2b.{L_1}.\cos \alpha }}{{2b.{L_2}}}} \right) - \alpha \end{aligned}$$

(7)

The closed state of the gripper is at position $O_1$ of the center point. In such a state, the width of the gripper is:

$$\begin{aligned} {D_1} = a + 2r-{L_c} \end{aligned}$$

(8)

where a is the distance between two center points of two rigid clamps at the closed state and $L_c$ is the length of the comb teeth. At the opened state of the gripper, the width of the gripper is estimated as:

$$\begin{aligned} D_{\max }^2 = 2r + a + 2r\sin \varphi \end{aligned}$$

(9)

where $\varphi$ is the rotational angle of the gripper, determined as in equation (7). The design parameters of the gripper are: $L_1 = 145$ mm, $L_2 = 110$ mm, $\Delta L = 35$ mm, $\alpha =33^\circ$; $b= 36$ mm, $r = 30$ mm, $a = 2$ mm, $L_c = 10$ mm. Thus, based on Eqs. (8), and (9) the width of the gripper at the closed state $D_1$ and opened state $D_{\max }^2$ can be calculated as:

$$\begin{aligned} \begin{array}{l} D_1 = 65 mm\\ D_{\max }^2= 108mm \end{array} \end{aligned}$$

(10)

3.2 Deformation modeling and analysis of the soft pads

Because, the proposed gripper operates on a principle where rigid clamps initially secure the object, and subsequently, the soft pads deform and inflate using compressed air to make direct contact with the object, generating the gripping force. Therefore, the gripper’s capacity to handle objects, considering their size and mass, depends not only on the dimensions of the rigid clamps but also significantly on the deformation of the soft pad. Accurately predicting the soft pad’s deformation enables the calculation of the compressed air pressure needed for each specific dimension of the gripping object, ensuring an appropriate gripping force suited to the mass of each object. Thus, in this section, we propose a model to estimate the deformation of the soft pad under pressure actuation (Fig. 11).

Fig. 11

The schematic model of gripper’s motion

Firstly, to predict the deformation of the soft pad, the numerical simulation is conducted by using the commercial-numerical simulation software Abaqus, which can conduct a non-linear dynamic simulation. Compared to the design concept of the gripper, the simulation model is constructed, shown in Fig. 12. The model consists of two parts: a soft pad and a rigid clamp, and their sizes are identical to their design, presented in section 2.1. The soft pad was modeled as a deformable solid with the hyper-elastic material model. The Yeoh model with the parameters of $C_{10} = 0.1$, $C_{20} = 0.02$, $C_{30} = 0.0002$, $D_1 = D_2 = D_3 = 0$ [30] was used in the simulations. The rigid clamps were modeled as linear material with a Young modulus of 2000 MPa. The rigid clamp and the soft pad were merged to limit the relative motions at the boundaries among them. The developed finite element analysis (FEA) model uses the linear hexahedral elements of type C3D8R with a global mesh seed of 1.5 mm, a total of 11956 nodal points, and 9670 elements was used as shown in Fig. 12. The deformation of the soft pad was simulated by using static analysis, where the merged structure is bound to fix five boundary surfaces and the surface of the soft pad’s skin is free. The effect of air pressure actuation was modeled by uniformly distributed pressure on the internal surfaces of the soft pad’s cavity. The setting of the analysis step includes the nonlinear effects of large displacements. The range of pressure values was varied from 30 kPa to 70 kPa to investigate its values’ effect on the deformation of the soft finger.

Fig. 12

Simulated results of soft pad’s deformation with various pressure values

The deformation of the soft pad with various pressure values is shown in Fig. 12. Under air pressure activation, the soft pad is inflated to generate the deformable part, named a Wrinkle [31, 32]. The simulated results show that under the linear increase in the pressure values, the rise of the wrinkle’s height is nonlinear, due to the nonlinear physical properties of the soft material and nonlinear geometrical properties of large deformation.

Secondly, we attempt to estimate the wrinkle shape by a mathematical model that will be used in the contact model of the soft pad with gripping objects.

Under pressure activation, the soft pad is inflated to generate the wrinkle (as depicted in Fig. 13). The skin of the soft pad is deformed according to three directions: axial, radial, and circumferential direction.

Fig. 13

A schematic deformation of the soft pad

For estimating the wrinkle shape, two assumptions are given as follows:

• (1) The shape of the wrinkle is a round cylinder with an arc cross-section as in Fig. 14.

• (2) The thickness of the skin of the soft pad is constant, and thus, the strain in the radial direction is negligible.

Fig. 14

A schematic of calculation of wrinkle’s height

Let ${\lambda _1},{\lambda _2},{\lambda _3}$ be the strain in axial, radial, and circumferential directions, respectively.

Due to the fabricated material of the soft pad being silicone rubber, that is, the incompressibility of elastomer material, we have:

$$\begin{aligned} {\lambda _1}.{\lambda _2}.{\lambda _3} = 1 \end{aligned}$$

(11)

According to the non-Hookean model, the expression of the free energy of deformation W (strain energy) can be calculated as:

$$\begin{aligned} W = \frac{G}{2}\left( {{\lambda _1}^2 + {\lambda _2}^2 + {\lambda _3}^2} -3\right) \end{aligned}$$

(12)

where G is the initial shear modulus of the material, and G is calculated as:

$$\begin{aligned} G = \frac{E}{{2(1 + \upsilon )}} \end{aligned}$$

(13)

With E, $\upsilon$ is the Young modulus and the Poisson coefficient of the material, for the elastomer material $\upsilon =0.5$, thus $G = E/3$, the Young modulus $E = 550$ Pa [30].

From assumption 2), and combining with the incompressibility of elastomer material, we have:

$$\begin{aligned} {\lambda _1} = \lambda ,{\lambda _2} = 1,{\lambda _3} = \frac{1}{\lambda } \end{aligned}$$

(14)

Let c be the semi-width of the soft pad, and $h, R_d$ be the height and radius of the wrinkle. The radius of the inner curvature of the soft pad’s cavity is $R_1$, which is determined in the design of the gripper.

The radius of the wrinkle can be calculated from the below equation:

$$\begin{aligned} H = {R_1} - \sqrt{R_1^2 - {c^2}} + {R_d} - \sqrt{R_d^2 - {c^2}} \end{aligned}$$

(15)

Then, based on the conservation energy, we assume that the total supported energy of pressurization is transferred to the deformation energy of the soft pad’s skin.

The deformation energy of the skin layer ${W_{ed}}$ and supported energy of pressurization ${W_{es}}$ are computed as:

$$\begin{aligned} \begin{array}{l} {W_{ed}} = {V_k}.W\\ {W_{es}} = {V_c}.p \end{array} \end{aligned}$$

(16)

where $V_k$ and $V_c$ are the volume of the wrinkle and the increased volume of the soft pad’s cavity, respectively, while p is the pressure value.

A balanced equation of energy can be written as:

$$\begin{aligned} {W_{ed}} = {W_{es}} \end{aligned}$$

(17)

Let D be the length of the wrinkle and also be the length of the deformation skin-covered section of the soft pad. We get:

$$\begin{aligned} \begin{array}{l} {V_k} = {S_k}.D\\ {V_c} = {S_c}.D \end{array} \end{aligned}$$

(18)

where $S_k$ and $S_c$ are the cross-section area of the wrinkle and the increased cavity of the soft pad, respectively, and can be calculated as follows:

$$\begin{aligned} \begin{array}{l} {S_k} = \frac{{2\theta }}{{2\pi }}\pi \left( {{t^2} + 2{R_d}t} \right) = \theta \left( {{t^2} + 2{R_d}t} \right) \\ {S_c} = \theta R_d^2 + \Phi R_1^2 - c\left( {{R_d}\cos \theta + {R_1}\cos \Phi } \right) \end{array} \end{aligned}$$

(19)

where $\theta = \arcsin \left( {\frac{c}{{{R_d}}}} \right) ;\Phi = \arcsin \left( {\frac{c}{{{R_1}}}} \right)$ and t is the thickness of the skin layer.

The principal stretch (stretch in the axial direction) can be expressed as: $\lambda = l/{l_0}$, where l is the elongated length of deformed skin (the length of the wrinkle’s arc cross section). Given assumption 2) that the thickness of the skin is constant, the principal stretch can be estimated as:

$$\begin{aligned} \lambda = \frac{l}{{{l_0}}} = \frac{{{S_k}}}{{{S_0}}} = \frac{{\theta \left( {{t^2} + 2{R_d}t} \right) }}{{\Phi \left( {{t^2} + 2{R_1}t} \right) }} \end{aligned}$$

(20)

Put Eqs. (18) and (19) into equation (17), the balance energy equation can be written as:

$$\begin{aligned} {S_k}.D.W = {S_c}.D.p \end{aligned}$$

(21)

Then, we have:

$$\begin{aligned} \theta \left( {{t^2} + 2{R_d}t} \right) \frac{G}{2}\left( {{\lambda ^2} + \frac{1}{{{\lambda ^2}}} - 2} \right) = \left( {\theta R_d^2 + \Phi R_1^2 - c\left( {{R_d}\cos \theta + {R_1}\cos \Phi } \right) } \right) p \end{aligned}$$

(22)

where $\lambda$ is determined from Eq. (20). Solving Eq. (22) by using the function “fsolve”¹ of the MATLAB program, the radius of the wrinkle $R_d$ is computed.

After estimating the wrinkle’s radius, its height can be calculated as Eq. (15), and these values are summarized in the following table.

Table 1 Calculated values of wrinkle radius and wrinkle height with various pressurization

By estimating the radius and height of the wrinkle, the wrinkle’s shape with different activated pressure values can be assessed and is depicted in Fig. 15.

Fig. 15

The estimation of the wrinkle’s shape under different pressurization

From Table 1, it can be seen that with the range of pressure values from 30 kPa to 70 kPa the maximum and minimum values of the wrinkle’s height can be determined as: $H_{\min } =17.5$ mm and $H_{\max } =27.9$ mm with the pressure value of 30 kPa and 70 kPa, respectively. Then, by combining these values with the detailed analysis of rigid clamp motion provided in Sect. 3.1, the minimum and maximum diameters of objects that the gripper can grasp can be determined using the following equation:

$$\begin{aligned} \begin{array}{l} D_{min}=D_1- 2H_{max}\\ D_{max}=D_{\max }^2- 2H_{min} \end{array} \end{aligned}$$

(23)

Utilizing Eqs. (23) and (10) in Sect. 3.1, the minimum and maximum diameters of objects that the gripper can grasp within the pressure range from 30 kPa to 70 kPa can be determined as: $D_{\min }=9.2$ mm, and $D_{\max }=73$ mm. These values satisfy predetermined requirements for the size of objects that the gripper design can handle.

Note that the diameter range of the grasped object can be adjusted by altering the dimensions of the rigid clamps, soft pad, and pressure values.

3.3 Modeling the contact force

In the OXY coordinate, the shape of the wrinkle and the shape of the object can be described by circular equations as follows:

$$\begin{aligned} \begin{array}{l} {\left( {x - {x_0}} \right) ^2} + {\left( {y - {y_0}} \right) ^2} = {R_d}^2\\ {\left( {x - {x_1}} \right) ^2} + {\left( {y - {y_1}} \right) ^2} = {R_b}^2 \end{array} \end{aligned}$$

(24)

where $x_0, y_0$ and $x_1, y_1$ are the coordinate of the central point’s wrinkle and the object, respectively.

Fig. 16

A schematic to estimate contact force

The gripping capability of the gripper can be assessed by estimating the contact force between the soft pad and gripping object, which is calculated through the contact area S and pressure value p as the following equation:

$$\begin{aligned} {F_{\textrm{contact}}} = p.S \end{aligned}$$

(25)

To estimate the contact area S, we used some assumptions: (1) The wrinkle’s shape has a semicircular cross-section, and its profile has been described by Eq. (24). (2) The profile of the gripping objects is also semicircular with the radius $R_b$. The schematic contact of the wrinkle and the object is depicted in Fig. 16.

Based on the geometrical relationship as in Fig. 16, the contact area S is calculated as follows:

$$\begin{aligned} S = \frac{{\alpha {R_d}^2}}{2} - \frac{1}{2}{R_d}^2\sin \Omega + \frac{{\alpha {R_b}^2}}{2} - \frac{1}{2}{R_b}^2\sin \psi \end{aligned}$$

(26)

with

$$\begin{aligned} \begin{array}{l} \Omega = 2\arccos \frac{{{d^2} + {R_b}^2 - {R_d}^2}}{{2d{R_b}}}\\ \psi = 2\arccos \frac{{{d^2} + {R_d}^2 - {R_b}^2}}{{2d{R_d}}}\\ d = \sqrt{{{({x_0} - {x_1})}^2} + {{({y_0} - {y_1})}^2}} \end{array} \end{aligned}$$

(27)

where $(x_0,y_0)$ and $(x_1,y_1)$ are the center’s coordinate of the wrinkle and gripping object, respectively. $R_d$ and $R_b$ are the radius of the wrinkle and gripping object, respectively.

So from Eq. (25), combined with the calculated results of the wrinkle shape and the motion of the rigid clamps, the contact force between the wrinkle and gripping objects can be estimated, thus the gripping capability of the gripper is assessed, and we can control the air pressure value to obtain the contact force for the specific weight of the certain gripping object.

4 Validations

4.1 Validation of wrinkle’s height’ calculation

To validate the effectiveness of the estimation of the wrinkle’s height by mathematical calculation and numerical simulation, the experiment was conducted to evaluate the deformation of the soft pad. In this experiment, the soft pad was activated by the air pressure through the pump, and the pressure’s value is controlled by the air regulator. Under the activation of the compressed air, the soft pad is deformed, generating a wrinkle. The height value of the generated wrinkle is measured by using the calipers. The wrinkle’s height is estimated by subtracting the thickness of the rigid clamp from the measured value. The experimental setup used for this measurement is illustrated in Fig. 17.

Fig. 17

Experimental setup to measure the wrinkle’s height

Figure 18 presents a comparison of the results of the wrinkle height obtained through calculation, simulation, and experiment. The comparison indicates that the height of the generated wrinkle increases in the order of theoretical calculation, numerical solution, and experimental measurement. This can be attributed to the assumption of using linear materials in the theoretical model, which neglects the nonlinearity property, as well as the difference in coefficients of the hyper-elastic material used in the simulation. Nonetheless, the maximum error between the theoretical and experimental results is less than $12\%$, which can validate the effectiveness of the theoretical calculations.

Fig. 18

Comparison of calculation, simulation, and experimental results of wrinkle height

4.2 Validation of contact force’s calculation

To validate the effectiveness of the proposed contact force model, experiments were conducted using the Universal Micro Material Tester (UMT) system (CETR-US),² as illustrated in Fig. 19. Spherical objects of varying radii were fabricated using 3D printing rapid prototyping technology to serve as the upper models in the UMT system. These spheres were attached to the vertical sliding motor through a suspension. This motor can provide precise movement capabilities with a position resolution of $10^{-3}$ mm. The 3D-printed spheres were designed with threaded holes for easy assembly with the suspension. A 2-dimensional force sensor, with a measured range of 0.2 N to 20 N and resolution of $10^{-3}$ N, was mounted on the suspension to measure the contact force.

The lower sample is the rigid clamp with the soft pad within its cavity, firmly placed on a horizontal sliding table. The inflation of the soft pad was achieved through a compressed air pressure system, regulated by a pneumatic adjustment system consisting of solenoid valves and pressure gauges. The pressure values were set to equal those used in the theoretical model. The movement of the vertical sliding motor was automated and controlled by using a computer with the UMT system software.

Fig. 19

Experiment to validate theoretical model of contact force

The experimental procedure is as follows: Initially, the soft pad was inflated with the specific pressure values. Subsequently, the spherical object is moved down to contact with the surface of the inflated soft pad and then pressed onto this surface. The software determined the pressing depth, which corresponded to the contact depth value d between the soft pad and the object, as calculated in Eq. (27) of the theoretical model. (d is depicted in Fig. 16.) The contact force between the object and the soft pad was displayed on the system’s software and recorded. Each experiment was repeated five times, and the resulting contact force values were averaged across these trials to derive the final values of the contact force.

Fig. 20

Comparison of experimental and theoretical values of contact force

The comparison between the experimental contact force measurements with error bars and the corresponding theoretical calculation results is shown in Fig. 20. It can be seen that for the case of gripping spherical objects with various radii, when the pressure value increases, the contact force initially increases at a faster rate but eventually increases more slowly. This suggests that high contact forces can be achieved without the need for excessively high pressure. Furthermore, the maximum error between the theoretical and experimental results is within $13\%$, validating the effectiveness of the contact force’s theoretical model.

4.3 Validation of the gripper’s performance

Next, to validate the gripping ability of the proposed gripper, real gripping experiments were carried out. Firstly to conduct the gripping experiment, the pneumatic control system was established, and its circuit diagram is presented in Fig. 21. The system operates by controlling the flow of compressed air and can be described as follows:

Fig. 21

Circuit diagram of the gripper’s pneumatic controlling system

The compressed air produced by the pneumatic pump is split into two channels using two solenoid valves, namely the solenoid reversing valve 5/3 and the solenoid reversing valve 5/2. The first channel, which passes through the solenoid reversing valve 5/3, is utilized to control the soft pad’s deformation. When the valve is in the middle position (with no signal sent to Y1 or Y2), the airflow is blocked. However, if Y1 is activated, the compressed air from the source pump will be directed to the single-acting cylinder through the reversing valve and the throttle valve for inflating the soft pad. During the reverse phase, if Y2 is triggered, the air from the single-acting cylinder will be released through the throttle valve to return the soft pad to its initial shape.

The second channel, which passes through the solenoid reversing valve 5/2, is used to control the motion of the double-acting cylinder. If the Y3 valve is in the initial position, the double-acting cylinder will move forward under the influence of the airflow from the source through the throttle valve. However, if Y3 is activated, the double-acting cylinder will move backward under the pressure of compressed air.

Fig. 22

The setup of gripping experiments

The next step involves connecting the gripper to a MITSUBISHI RV-12SD robot arm with 6 degrees of freedom, allowing for movement at varying velocities during the gripping experiments, as depicted in Fig. 22. The motion trajectory and speed of the robot arm are managed using commercial software. An Arduino controller circuit is utilized to operate the pneumatic controlling system, which includes the double-acting cylinder, solenoid reversing valves, and compressed air source. During the experiment, a wide variety of objects with different shapes, sizes, weights, and positions were utilized for pick-and-place manipulation.

The operation of components in the gripping system for ensuring effective and reliable gripping manipulation can be described as follows:

Initially, based on the object’s orientation, including standing or lying down, the robot arm adjusts the gripper’s position accordingly, either vertically or horizontally. Subsequently, the robot arm positions the gripper to enclose the object. At this stage, the two rigid clamps remain in an open state and are then actuated to close using pneumatic cylinder actuators. These actuators are controlled by opening corresponding valves that supply compressed air to initiate their movement. Once the rigid clamps are closed, the soft pads undergo deformation and inflation by activating pneumatic valves that supply air to them. This inflation process generates a gripping force, securely grasping the object. The pressure levels for activating the soft pads are customized to suit each object’s size and weight and are regulated by a compressed air supply system, encompassing pressure gauges and solenoid valves.

After gripping the object, the robot arm executes predefined movements along a specific trajectory to locate the grasped object to its release position. During the object release stage, the supply of compressed air to both the cylinder actuators and soft pads is interrupted simultaneously, allowing the object to be released from the gripper. In the operation of the system, the opening or closing process of solenoid valves to regulate the supply or disconnection of compressed air is governed by the pneumatic control circuit.

The experimental results, as shown in Fig. 23, verify the successful gripping performance of the proposed robotic gripper on a variety of objects. The gripper’s design with two rigid clamps allows for easy handling of objects placed in various positions, such as horizontal cylinders, which can be challenging for soft grippers. In addition, the gripper maintains flexibility through the soft contact of the inflated soft pad, ensuring the protection of fragile objects.

The experimental parameters including the shape, size, weight, and properties of the gripping objects are summarized in Table 2.

Fig. 23

Operation of the gripper with various objects

Table 2 Summarized data of the gripping objects

In evaluating the grasping efficiency, the gripper’s complete cycle time, from positioning to grasping an object to the release position, depends on the distance traveled. In the experimental setup, this distance was set at 1 m, resulting in a total duration of about 45 s, in which the time taken for the gripper to pick up and elevate the object is notably brief, taking less than 10 s.

Our gripper’s design also specifically targets the goal of maintaining objects even when subjected to significant vibrations during movement after completing the gripping process, which can arise from factors such as large swings of the robot arm or the mobility of robots, especially on rough surfaces. Therefore, in order to assess this ability of our proposed gripper, the next section will focus on testing the gripper’s ability to maintain stable objects under harsh vibrations. The vibration experiment was carried out by using the LDS electrodynamic vibration system.³ This system is designed for conducting mechanical shock and vibration testing, using a variety of excitation modes, including sinusoidal, random, and transient. It can accommodate a wide range of payloads and applies a force rating. The experiment used sinusoidal excitation with variable acceleration, where $A = {A_0} \sin(\omega .t)$, with the frequency and amplitude automatically set up by the commercial control software of the LDS system. The gripper was secured to the shaker of the LDS system via a mounting rod, fabricated with the Zortrax 2000 3D printer, and an egg was used as the object being gripped in the vibration experiment. The vibration testing setup is shown in Fig. 24.

Fig. 24

Vibration experiment setup

During vibration testing, strain gauges were embedded in the soft pad’s skin to monitor the gripper’s contact with the grasped objects. The contact state is assessed through the signals from strain-gauge sensors, collected by using the LMS system, which can simultaneously collect signals from 16 channels at high frequency.⁴ The collected data were sent to a computer through SIMCENTER TESTLAB, a commercial software that displays the data in real time and saves it for analysis. The object’s stability is evaluated based on the slipping phenomenon, which is detected by tracking changes in the strain-gauge signal. The deformation amplitude at the strain gauge’s integrated position on the soft pad is represented by the signal obtained through the LMS system. Then, a fast Fourier transformation (FFT) is conducted on the raw deformation signal to characterize the variation in the strain-gauge signal. The FFT spectrum is presented in Figs. 25, 26 and 27.

Fig. 25

FFT spectrum with various resonant frequencies of the obtained signal

Fig. 26

FFT spectrum with frequency vibration of 15 HZ

Fig. 27

FFT spectrum with frequency vibration of 115 HZ

In the initial vibration experiment, the frequency gradually increased from 2 Hz to 190 Hz to identify the resonant frequencies in the strain-gauge signal. The resulting FFT spectrum is illustrated in Fig. 25. Based on the figure, it can be observed that the strain signal collected from the strain gauge during the vibration experiment had several resonant frequencies. The largest strain was obtained when the first resonant frequency occurred at 15 Hz. The sharpness of the resonant frequency peaks suggests that there was no significant slipping between the soft pad and the object during dynamic vibration.

Two additional vibration experiments were conducted to confirm this observation, with the LDS system oscillating at frequencies of 15 HZ and 115 HZ. The FFT spectra obtained from the strain-gauge signal show that the resonant frequencies of the received signal match the vibration frequency set by the LDS system, confirming that there is no slippage between the soft pad and the gripper during vibration. Moreover, the signal received from the strain gauge reflects the soft pad’s deformation at the gauge’s position and correlates with the contact force between the soft pad and the object. Thus, the coincidence of the gripper’s vibration frequency and the strain gauge’s signal frequency indicates a consistent contact force, particularly at resonance frequencies, during vigorous shaking. These results affirm that the proposed gripper can maintain the contact force, leading to the stability of the objects during the movement after completing the gripping process, even under adverse vibration conditions with high frequency, enabling safe interaction with grasped objects.

5 Discussion and conclusion

Developing robotic hands or grippers that can handle a wide variety of objects with safety and flexibility is a challenging task. Traditional rigid grippers are reliable but struggle with deformable or fragile objects, as they may exert excessive force and cause damage, while soft grippers have been developed to mimic the human hand’s grasping capabilities but lack stability and robustness, especially under harsh vibration. Soft robotic grippers only can handle gripping objects with lightweight and predetermined placement positions.

In this study, we propose a novel robotic gripper that combines the robustness of rigid components with the safety of soft grippers. The gripper uses rigid parts for gripping movement and soft pads that are inflated under air pressurization to generate gripping forces. The operation of the gripper is realized through cylinder actuators. Both the actuator operation and the activation of the soft pad utilize compressed air, thereby maximizing the efficiency of the compressed air system.

The gripping capability of the gripper is analyzed by the theoretical model and experimentally validated by the gripping process, using the real robot arm system. The analysis and experimental findings demonstrate that, based on the proposed design parameters, the gripper exhibits the capacity to manipulate the object’s mass up to approximately 1 kg, with diameters ranging between 15 mm and 70 mm. It is important to highlight the adaptability of the gripper’s design, allowing adjustments for handling other object sizes. This customization can be formulated by considering the interplay between the gripper’s design components—such as the dimensions of the rigid clamps, soft pads, and the resultant of the gripping area and the contact force, as presented in Sect. 3 of this article. In comparison with other existing grippers, the proposed design has the advantage of effortlessly executing pickup and release operations, accommodating a diverse array of objects with varying hardness levels, fragility, and multiple positions, including standing and lying down.

Then, vibration experiments were conducted to validate that after picking up the object, the gripper can withstand significant vibrations without causing the gripped object to slip or fall during robotic movement, aligning with a primary objective of the proposed gripper design. To address this, we affixed strain gauges onto the surface of the soft pad, that direct contact with the gripping objects. In this approach, we assume that if no relative movement occurs between the soft pad and the object, consequently when the gripper vibrates, the vibration frequency of the gripper coincides with the signal frequency of the strain gauge. Thus, the signal received from the strain gauge also indicates a consistent contact force, particularly at resonance frequencies, during vigorous shaking.

During vibration tests conducted in an LDS system, the sine wave signal is selected to induce vibration due to its simplicity and controllability. Furthermore, the sinusoidal oscillation signal mirrors a form of oscillation commonly encountered in real-world scenarios, such as when the robot moves on rough and undulating terrain. Specifically chosen significant vibration frequencies and amplitudes are deliberately employed to assess the gripper’s stability after object pickup, even under considerably strong vibration conditions. This approach validates the gripper’s durability and reliability within real-world operating scenarios when the actual oscillations have smaller amplitudes and frequencies than experimental conditions.

The gripper design presented in this study can be applied for handling a wide range of objects in various real applications. For instance, integrating grippers onto robotic arms or mobile robots within manufacturing lines facilitates the automated handling of items, ranging from production processes to packaging. In certain industries such as pharmaceuticals or chemicals, where direct human exposure to manufactured products may pose risks, grippers aid in the safe and automated manipulation of items. Moreover, grippers can find applicability in logistics and warehouse operations, with various tasks such as arrangement, or transportation of goods. Additionally, the gripper can play an important role in mobile or collaborative robots, particularly in scenarios where objects are picked up, and then transported over rugged terrain, which may generate substantial vibrations posing a risk of object displacement.

However, some primary constraints of our gripper design are the following. Similar to other soft grippers employing pneumatic actuation principles, our gripper utilizes inflating deformation of the soft pad to generate contact force and secure objects. Consequently, this design is susceptible to damage when handling sharp objects, potentially causing punctures in the soft pad and resulting in air leaks. Moreover, the gripper’s reliance on a compressed air supply system, which involves driving pneumatic cylinders and inducing deformation in the soft pad, restricts its flexibility, particularly when integrated into mobile robot systems. Additionally, maintaining air pressure necessitates energy consumption. The pneumatic pump’s operation often generates noise, which can be problematic in noise-sensitive environments where maintaining low noise levels is crucial such as medical settings or places where noise disturbances are undesirable.

When considering the manufacturing process for grippers that combine rigid clamps with soft pads, the production of rigid clamps can be executed swiftly using 3D printing technology. However, manufacturing soft pads involves the molding method, which tends to be time-consuming. The soft material requires sufficient time to solidify before the mold can be extracted. Moreover, the mold casting process is primarily manual, demanding precision and meticulousness during operations. Additionally, fixing the soft pad into the rigid clamp’s cavity necessitates specialized glue to ensure a secure connection without causing damage to the soft pad, especially when inflating the soft pad with air pressure for deformation purposes. Consequently, the forthcoming task of this research involves refining the gripper manufacturing process, aiming to reduce production time while maintaining a level of reliable accuracy.

Moreover, to achieve optimal performance for the practical implementation of the gripper and automate its performance, the integration of additional cameras and AI (Artificial Intelligence) devices into the robotic system is important. These devices accurately predict object characteristics like placement, size, volume, and weight. This information allows for precise adjustments in the gripper’s position and determines the optimal pressure required to activate the soft pads for suitable gripping objects. Besides, to optimize the gripper’s design and its performance, an approach combining dynamic simulation, mathematical modeling of the gripping process, and experimentation will be important. These issues will be the forthcoming focus of our future research.