Design of Adaptive Grippers for Fruit-Picking Robots Considering Contact Behavior

Abstract

Adaptability to unstructured objects and the avoidance of target damage are critical challenges for flexible grippers in fruit-picking robots. Most existing flexible grippers have many problems in terms of control complexity, stability and cost. This paper proposes a flexible finger design method that considers contact behavior. The new approach incorporates topological design of contact targets and introduces contact stress constraints to directly obtain a flexible finger structure with low contact stress and good adaptability. The study explores the effects of design parameters, including virtual spring stiffness, volume fraction, design domain size, and discretization, on the outcomes of the flexible finger topology optimization. Two flexible finger structures were selected for comparative analysis. The experimental results verified the effectiveness of the design method and the maximum contact stress was reduced by about 70%. An adaptive two-finger gripper was developed. This design allows the gripper to achieve damage-free grasping without additional sensors and control systems. The adaptive and contact performances of the grippers with different driving modes were analyzed. Practical grasping tests were also performed, including evaluation of adaptive performance, stability, and maximum grasping weight. The results indicate that gripper 2 with flexible finger 2 excelled in contact stress and adaptive wrapping, making it well-suited for grasping unstructured and fragile objects. This paper provides valuable insights for the design and application of flexible grippers for picking robots, offering a promising solution to enhance adaptability while minimizing target damage.

1. Introduction

Agricultural robots are developing rapidly [1,2]. The related technologies are being widely used in apples [3], cherry tomatoes [4,5], bananas [6], cucumbers [7] and other fruits and vegetables [8,9] during the production process. Robotic grippers are one of the most crucial components of robot manipulators. Adaptive flexible grippers can not only grasp fragile and unstructured objects, but also interact safely with humans and environments. These advantages make adaptive flexible grippers have great potential in the fruit picking [10].

For example, Chen et al. used a compliant structure gripper with fin effect [11] to develop a robotic apple harvesting system, which can realize adaptive picking of apples with a picking success rate of 62.8% [12]. They then designed a flexible tapered finger and, based on this, developed a soft gripper, increasing the picking success rate to 70.77% [13]. Inspired by duck feet and octopus tentacles, Cai et al. [14] proposed a pneumatically webbed soft gripper, which added four webs to the four pneumatically flexible fingers. This flexible gripper can grasp multiple targets at once while preventing them from falling out of the fingers. Cao et al. [15] reduced the control complexity by reducing the number of fingers, using three flexible fingers and a bellows actuator to achieve flexible grasping of fruits. Although these flexible grippers are more adaptive than traditional rigid manipulators, these flexible grippers do not consider the contact force conditions during the grasping process. In reality, the use of flexible grippers still does not guarantee damage-free handling of fruits if the contact force is not thoroughly considered [16,17].

Most existing research utilizes tactile sensors or smart materials in flexible fingers or fingertips to measure grasping force, enabling accurate real-time control of the gripper [18]. These sensors are predominantly based on force-sensitive resistors or resistive foil strain gauges, and typically comprise flexible thin films and electrodes [19,20]. Elfferich et al. [17] actively control flexible harvesting of blackberries by integrating force sensors into flexible fingertips. While these sensor-based force detection techniques are straightforward, they do come with certain constraints. Firstly, the bulkiness and significant weight of many conventional force sensors pose substantial challenges to their integration with grippers. For instance, organizing the sensor wiring can be particularly cumbersome, especially with a large number of sensors. Additionally, embedding or attaching external sensors may alter the stiffness and flexibility of soft grippers [21]. Secondly, these intricate sensors are often highly sensitive to environmental factors, and necessitate stringent conditions regarding temperature, humidity, cleanliness, and corrosion resistance. Consequently, these requirements lead to numerous issues in terms of reliability, stability, and cost, which limiting the application of grippers in unstructured environments.

In addition to directly combining sensors to measure force, indirect force estimation technology that utilizes the compliance of soft robots is also a common method [22,23]. The core concept of these methods involves modeling the correlation between deformations and forces, tracking the robots’ deformations, and ultimately calculating the contact forces. Thus, the challenge of force sensing can be transformed into a matter of static modeling and deformation measurement [21]. For example, Zhang et al. [24] controlled apple harvesting by establishing a grasping force sensing model of force and finger-bending angle. Ji et al. [25] proposed a variable damping impedance control strategy, which made the gripping force tracking effect smoother and improved the dynamic performance. The flexible gripper designed by Lin et al. [26] uses visual perception technology to realize tactile perception and control contact force. Visentin et al. [27] reduced cameras and designed a hemispherical soft dome gripper. Force estimation was performed by visually identifying different deformations of the pattern on the hemispherical soft dome. However, the large-sized targets can easily affect the recognition accuracy.

In general, the current flexible grippers of fruit-picking robots mainly use sensors, smart materials and force models to reduce the redundant grasping force. However, these methods can lead to issues such as complex control, long response times and high manufacturing costs. If a gripper can achieve both adaptivity and low contact stress solely through its unique topology structure, without relying on sensors or other complex technologies, the fruit grasping operations can be greatly simplified and made more cost-effective. This is important for agricultural production. For example, it is used in fruit-picking robots and automated fruit sorting and packaging, as shown in Figure 1.

However, another important issue is how to design the above-mentioned flexible adaptive grippers considering contact stress. Existing designs of flexible grippers lack clear design standards and comprehensive design methods. By taking cues from bionics, origami techniques [28,29] or drawing inspirations from the inherent characteristics of their grasping targets, various grippers have been developed, e.g., the bionic soft grippers inspired by animals [30,31] or plants [32,33]. However, current designs are highly dependent on anthropogenic factors, such as the designers’ experience, professional knowledge, and inspiration. The design of flexible grippers remains a challenging task.

Topology optimization is an effective approach for addressing the conceptual design challenges of structures [34], compliant mechanisms [35,36], soft grippers [37,38], soft robots [39], etc. This approach can automatically generate the topology, shape and size of the flexible grippers based on predefined design requirements [40,41], which is friendly for designers without extensive professional experience. Therefore, topology optimization and generative design have gradually become the main design methods for flexible grippers [42,43]. For example, Liu et al. [44] successfully developed flexible grippers with high mechanical advantage, including two-finger flexible grippers [45] and three-finger soft grippers [46,47]. In addition, they also adopted a topological method with two output ports that were in opposite directions to design multi-material flexible fingers [37]. Based on the single input and double outputs strategies, Huang et al. [48] introduced an actual contact target to design the flexible finger advantages with better performance in contact force. There are currently few studies on topology optimization of flexible fingers considering target contact stress.

This paper proposes a topology design method for flexible grippers that considers contact behavior. The design results can be directly applied to the lossless flexible gripper of fruit-picking robots. Unlike the previous solutions, our method can avoid complex and expensive sensing control systems and solve the problem of damage to targets by picking robots in terms of structural design. The contributions of this paper mainly include two aspects:

(1)

Establishing a flexible gripper topology design method that considers contact behavior, and analyzing the impact of topology design parameters on the design process and final results.

(2)

An adaptive finger structure with low contact stress is presented. This structure has superior performance (adaptability and contact performance) and versatility in applications. Based on the above-mentioned fingers, flexible grippers with different driving methods were designed. This gripper can be used in fruit grasping tasks to avoid fruit damage, such as fruit picking, sorting and packaging, etc.

The rest of this paper is as follows: In Section 2, a flexible finger topology design model considering contact behavior is established, and the influence of design parameters on the results are analyzed. In Section 3, the effectiveness of the proposed computer-aided design method is verified through FEA and experiments. The grippers with different drives are introduced, and their performance testing and analysis are carried out. The performance and characteristics during gripper gripping are compared and discussed. In Section 4, conclusions are drawn and the research results are summarized.

2. Flexible Finger Topology Optimization with Contact Constraints

This section introduces a novel strategy for flexible finger design considering contact behavior, provides a detailed description of its topology optimization design scheme, and analyzes the main factors affecting the design results of flexible fingers.

2.1. Topology Optimization Formulations

The design of flexible fingers is transformed into a topology optimization problem considering the contact behavior of the target object. The optimization model can be expressed as

$$\begin{array}{cc}\hfill max:& u_{out}={\mathbf{U}}_2^T\mathbf{K}{\mathbf{U}}_1=\sum _{i=1}^N{\left(x_i\right)}^{\rho }{\mathbf{u}}_{e2}^T{\mathbf{k}}_{\mathbf{e}}{\mathbf{u}}_{e1}\hfill \\ \hfill subjectto:& V\left(\mathbf{x}\right)/V_0=f\hfill \\ \hfill & \mathbf{K}{\mathbf{U}}_1={\mathbf{F}}_1\hfill \\ \hfill & \mathbf{K}{\mathbf{U}}_2={\mathbf{F}}_1+\mathrm{h}\mathit{l}\hfill \\ \hfill & {\mathbf{F}}_1-{\mathbf{F}}^{\mathrm{int}}(\tilde{x},U)=0\hfill \\ \hfill & \sigma <{\sigma }_{\max }\hfill \\ \hfill & \mathbf{0}<{\mathbf{x}}_{\min }⩽\mathbf{x}⩽\mathbf{1}\hfill \end{array}$$

(1)

where $\mathbf{K}$ is the global stiffness matrix of design domain; ${\mathbf{F}}_1$ is the external nodal load vector; ${\mathbf{U}}_1$ and ${\mathbf{U}}_2$ are the global displacement vectors under the actuation of ${\mathbf{F}}_1$ and ${\mathbf{F}}_1+\mathrm{h}\mathit{l}$, respectively; ${\mathbf{u}}_{e1}$ (or ${\mathbf{u}}_{e2}$) and ${\mathbf{k}}_{\mathbf{e}}$ are the element displacement vector and stiffness matrix, respectively; $\mathbf{x}$ is the vector of design variables; ${\mathbf{x}}_{min}$ is a vector of minimum relative densities; N is the number of elements used to discretize the design domain; $\rho$ is the penalization power (typically $\rho =3$); $V\left(\mathbf{x}\right)$ and $V_0$ are the material volume and design domain volume, respectively; f is the prescribed volume fraction; ${\mathbf{F}}^{\mathrm{int}}(\tilde{x},U)$ is the internal nodal load vector; and $\sigma$ and ${\sigma }_{\max }$ are the contact stress and the specified maximum stress, respectively.

The density filter is employed [49]. It transforms the design variable, ${\mathbf{x}}_j$, of element $\mathbf{j}$ to the element relative density, ${\tilde{x}}_i$, of element i as follows:

$$\begin{array}{c}\hfill \begin{array}{c}{\tilde{x}}_i=\sum _{j=1}^{N_e}{w}_{ij}{x}_j\hfill \end{array}\end{array}$$

(2)

where $w_{ij}$ is a normalized weight factor defined as follows:

$$\begin{array}{c}\hfill \begin{array}{c}{w}_{\mathrm{ij}}=\frac{H_{\mathrm{ij}}{v}_i}{{\sum }_{m=1}^{N_e}{H}_{mj}{v}_m}\hfill \end{array}\end{array}$$

(3)

where $v_i$ and $v_m$ are the volume of element i and m, respectively, $N_e$ is the number of elements in the design domain, and $H_{ij}$ is a distance factor, defined as follows:

$$\begin{array}{c}\hfill \begin{array}{c}{H}_{\mathrm{mj}}=max\left(0,r_{max}-D_{\mathrm{mj}}\right)\hfill \end{array}\end{array}$$

(4)

in which $r_{max}$ is a constant filter radius, $D_{\mathrm{mj}}$ is the center- to-center distance of elements m and j.

The FEA results of two load cases are used to perform sensitivity analysis [50]. One is the load case which, when the applied external force, is ${\mathbf{F}}_1$, and the other is the load case when the applied external force increases slightly to ${\mathbf{F}}_1+\mathrm{h}\mathit{l}$, where h is set to $0.01$ for structural design. The updated Lagrangian approach is employed in ANSYS. The sensitivity of the objective function $u_{out}$ and material volume V with respect to the element density $x_i$ is given by:

$$\begin{array}{c}\hfill \begin{array}{c}\frac{\partial u_{out}}{\partial x_i}=-\frac{1}{h}(\frac{p}{{\tilde{x}}_i}\Delta Q_{1,i}-\frac{p{\tilde{x}}_i^{p-1}}{1-{\tilde{x}}_i^p+{10}^{-10}}\Delta Q_{2,i})\hfill \end{array}\end{array}$$

(5)

$$\begin{array}{c}\hfill \begin{array}{c}\frac{\partial V}{\partial x_i}=1\hfill \end{array}\end{array}$$

(6)

where $\Delta Q_{1,i}$ is the increment of the strain energy of original element i, and $\Delta Q_{2,i}$ is the increment of the strain energy of hyperelastic element i. Here, the small constant ${10}^{-10}$ is added to the denominator of the second right-hand term to avoid division by zero when ${\tilde{x}}_i=1$. The increment of the strain energy are given by the following:

$$\begin{array}{cc}\hfill & \Delta Q_{1,i}=Q_{1,i}^{\left(2\right)}-Q_{1,i}^{\left(1\right)}\hfill \\ \hfill & \Delta Q_{2,i}=Q_{2,i}^{\left(2\right)}-Q_{2,i}^{\left(1\right)}\hfill \end{array}$$

(7)

where $Q_{1,i}^{\left(2\right)}$ and $Q_{2,i}^{\left(2\right)}$ are the strain energy of element i and the hyperelastic element i in the second load case, respectively. $Q_{1,i}^{\left(1\right)}$ and $Q_{2,i}^{\left(1\right)}$ are the strain energy of element i and the hyperelastic element i in the first load case, respectively. Using the finite difference method to represent the sensitivity of contact stress $\sigma$ with respect to element density ${\mathbf{x}}_i$ is given by:

$$\begin{array}{c}\hfill \begin{array}{c}\frac{\partial \sigma }{\partial x_i}=\frac{\sigma \left(x_i\right)-\sigma \left(x_{i-1}\right)}{x_i-x_{i-1}}\hfill \end{array}\end{array}$$

(8)

The design variable is updated by the method of moving asymptotes (MMA) [51] after sensitivity analysis.

The topology optimization is implemented in the MATLAB R2020. Ideally, the topology optimization problem for flexible fingers can be represented as shown in Figure 2. The lower end of the left side of the design domain $\mathsf{\Omega }$ is fixed, and a horizontal force $F_1$ is applied to the upper end of the left side to represent the active load. The influence of external objects is introduced at the midpoint of the bottom edge, and the lower right corner produces a vertical displacement $u_{out}$. The design for the flexible finger comprises one input port, one fixed port, and one output port, while considering the contact target. The design domain undergoes discretization utilizing a grid composed of $nelx\times nely$ four-node square elements, with each element measuring $1\times 1$ mm. The Young’s modulus E and Poisson’s ratio $\mu$ are set to ${10}^8$ Pa and 0.3, respectively. Noteworthy numerical parameters encompass the values penalty factor $p=3$, and sensitivity filter radius $r_{min}=1.8$. The magnitude of the actuating external force, $F_1$, is 10 N. To find out the suitable design parameters for the problem, the values of input stiffness $K_{in}$, output stiffness $K_{out}$ and volume fraction $V_f$ are selected within the range of $[200,650]$, $[40,120]$ and $[0.1,0.4]$, respectively.

2.2. Design Results of Flexible Fingers

Two sets of valid flexible fingers were chosen from the entire set of topological results, specifically referring to finger 1 as depicted in Figure 3a and finger 2 as illustrated in Figure 3c. For finger 1, its design considering the contact behavior of the target object, and the specific design parameter value were configured as follows: $K_{in}=500$ N/m, $K_{out}=100$ N/m and $V_f=0.25$. The topology optimization algorithm for finger 1 achieved convergence after 102 iterations. Figure 3a showcases the relationship between the number of iterations and the objective function value. Specific results for selected iterations (3rd, 20th, 80th, and 102th) are also presented. Notably, the topological configuration tends to remain relatively stable beyond the 80th iteration. In the graphical representation, the white region corresponds to elements with pseudo-densities equal to $x_{min}$, the dark region signifies elements with pseudo-densities equal to 1, while the gray region denotes elements with pseudo-densities ranging between $x_{min}$ and 1. Figure 3b shows the volume fraction and contact stress convergence histories of finger 1.

Finger 2 was the topological result obtained by introducing contact stress constraints, and all other conditions are set the same as finger 1. Figure 3c illustrates the relationship between the number of iterations and the normalized objective function value for finger 2. Specific results for some specific iterations (3rd, 25th, 100th, and 150th) are presented within the figure. The volume fraction and contact stress convergence histories for finger 2 are shown in Figure 3d. At the beginning of the iteration, a poor topology (e.g., 3rd in Figure 3) will result in a large maximum contact stress. It can be seen from the figure that for a design subject to contact stress constraints, the contact stress tends to stabilize during its iteration process. The final topologies of both finger 1 and finger 2 have a $V_f$ of 0.25. Finger 2 has a simpler internal structure and a smaller contact stress than finger 1.

2.3. Design Parameters’ Influence on Topology

The design parameters employed in topology optimization can exert significant influence on the resulting topology and performances. To comprehensively investigate the impact of design parameters on the structure and functionality of the flexible finger, the effects of $K_{in}$, $K_{out}$, $V_f$, $L_x\text{:}{L}_y$, and $nelx\times nely$ were varied and analyzed individually on the basis of a referenced set of parameters. Since finger 2 shows better grasping performance in our following discussion, the parameters of finger 2 are taken as the referenced set of design parameters.

Firstly, the spring stiffness $K_{in}$ is changed within the range of $[200,650]$ N/m, while the $K_{out}=100$ N/m, $V_f=0.25$, $L_x\text{:}{L}_y=2\text{:}1$ and $nelx\times nely=100\times 50$. A serial of topology optimizations has been carried out. Figure 4 shows part of the topology optimization results, including the convergence histories and final topologies for $K_{in}=$ 200, 350, 400, and 650 N/m. The structural flexibility is characterized in terms of MSE. According to the convergence histories, the MSE of the final structure will reduce if $K_{in}$ takes a larger value, which indicates a greater output flexibility generated by the input forces. By comparing the final topologies from Figure 4a, it becomes evident that the choice of $K_{in}$ significantly influences the optimized topology. One can see that the final topology changes significantly in structure at the input port, i.e., the top left side of the design domain.

A smaller value of $K_{in}$ will result in a suboptimal final topology, with the lower left end of the structure biased towards the bottom center of the flexible finger. This situation will lead to higher contact stress, e.g., $K_{in}=200$ N/m. That can be improved, and feasible topologies can be obtained when $K_{in}$ exceeds 350 N/m. It is also found that a large value of $K_{in}$ will increase the complexity of the final structure and the contact stress values, e.g., $K_{in}=650$ N/m. In contrast, finger 2 has smaller contact stresses and a more complete internal beam, which indicates that $K_{in}=500$ N/m can be a better virtual spring stiffness for the input port.

Secondly, the spring stiffness $K_{out}$ changes in the range of $[40,120]$ N/m, while the $K_{in}=500$ N/m, $V_f=0.25$, $L_x\text{:}{L}_y=2\text{:}1$ and $nelx\times nely=100\times 50$. Figure 5 shows some topology optimization results, including the convergence history and final topologies for $K_{out}=$ 40, 60, 80 and 120 N/m. By comparing the final topologies from Figure 5a, it can be clearly seen that all the final topologies obtained exhibit similar overall profiles with slight differences in their internal structure. According to the convergence histories, a clear inverse correlation emerges between $K_{out}$ and the flexibility of the final topologies. As the value of $K_{out}$ increases, MSE of the structure decreases, making the final topology less flexible at its output port and the maximum contact stress decrease. Despite the output flexibility decreases, a larger $K_{out}$ will result in a topology with larger payload capacity, i.e., greater stiffness to withstand the load imposed by the target. For a soft gripper, it should have sufficient flexibility to adapt the shape and size of target, but also have the capability to securely grasp the object and prevent it from slipping. To balance between the flexibility and carrying capacity, $K_{out}=100$ N/m is selected as the appropriate stiffness at the output port.

Thirdly, the volume fraction $V_f$ changes in the range of $[0.1,0.4]$, while the $K_{in}=500$ N/m, $K_{out}=100$ N/m, $L_x\text{:}{L}_y=2\text{:}1$ and $nelx\times nely=100\times 50$. Figure 6 shows some topology optimization results, including the convergence history and final topologies for $V_f=$ 0.15, 0.20, 0.30 and 0.40. Through a comparative analysis of the final topologies, it becomes evident that final topologies by different $V_f$ have analogous overall profiles, but diverging in the thickness dimensions of the beams. After $V_f$ exceeds 0.15, the hinge of the obtained final topology and the MSE change less. As the value of $V_f$ increases, MSE of the structure increases, making the final topology more flexible at its output port and greater weight. The contact stress of the structure decreases with the increase in $V_f$. In particular, $V_f=0.3$ will bring the maximum contact stress value. Considering the flexibility, contact stress and practicality of flexible fingers, $V_f=0.25$ is a suitable choice.

Then, the design domain size $L_x\text{:}{L}_y$ changes in the range of $[1\text{:}1,3.5\text{:}1]$, while the $K_{in}=500$ N/m, $K_{out}=100$ N/m, $V_f=0.25$ and $nelx\times nely=100\times 50$. A series of topology optimizations were performed for different $L_x\text{:}{L}_y$. Figure 7 shows some topology optimization results, including the convergence history and final topologies for $L_x\text{:}{L}_y=$ 1:1, 1.5:1, 2.5:1 and 3.5:1. As can be seen from all final topologies in Figure 7a, different design domain sizes ($L_x\text{:}{L}_y$) change the overall structure of the final topology. According to the convergence histories, different $L_x\text{:}{L}_y$ shows similar MSE, i.e., there is no obvious relationship between $L_x\text{:}{L}_y$ and the flexibility of the final topology in this design problem. However, both smaller $L_x\text{:}{L}_y$ and larger $L_x\text{:}{L}_y$ will make the contact stress of the final topology larger. Therefore, $L_x\text{:}{L}_y=2\text{:}1$ is a more appropriate choice.

Finally, the discretization is analyzed by varying the grid density. The $nelx\times nely$ changes in the range of $[80\times 40,200\times 100]$, while the $K_{in}=500$ N/m, $K_{out}=100$ N/m, $V_f=0.25$ and $L_x\text{:}{L}_y=2\text{:}1$. A series of topology optimizations were performed for different $nelx\times nely$. Figure 8 shows some topology optimization results, including the convergence history and final topologies for $nelx\times nely=$ $80\times 40$, $160\times 80$, $180\times 90$ and $200\times 100$. From Figure 8a, it can be clearly seen that different grid densities ($nelx\times nely$) produce different final results. When the grid density becomes larger, the time required by the topology program increases significantly. And the final structure will have many small branch structures, which will reduce the manufacturability of the flexible fingers. According to the convergence histories, the grid density has little effect on the MSE and contact stress of the topology structure. Considering the flexibility and manufacturability of the flexible fingers, $nelx\times nely=100\times 50$ is selected as a suitable grid density.

3. Numerical and Experimental Studies of the Flexible Grippers

To comprehensively evaluate the performance of the designed flexible fingers, finger 1 and finger 2 were made into flexible grippers with three different actuation modes. Figure 9 shows the three two-finger grippers, denoted as translational gripper 1, translational gripper 2, angular gripper 1, angular gripper 2, active gripper 1, and active gripper 2. The length of each finger is limited to $L_1$, and the diameter of target object is represented as D. Additionally, uniformity was maintained in terms of the input loading of the grippers and the initial position of the object with respect to the grippers. To evaluate the adaptability and performance of the flexible grippers, FEA simulations were conducted using ANSYS 2021. Experimental tests were carried out to further investigate the performances of the grippers. This comprehensive analysis can better confirm the advantages of flexible grippers that consider contact behavior and the application potential in picking robots.

3.1. Verification of Contact-Constrained Flexible Fingers

To verify the contact and deformation performance of the flexible fingers, a test device was designed (shown in Figure 10). First, all the fingers were fabricated through 3D printing, employing thermoplastic polyurethane elastomer (TPU) with a hardness of 95 Shore A. The test flexible finger was fixed on a horizontal platform. Finally, the target was fixed to the test platform through the X-axis sensor, Y-axis sensor and connectors. Five contact sensors were installed circumferentially on the right side of the target. The X-axis sensor and Y-axis sensor used in the experiment are the compression sensor, which are provided by Zhongnuo Chuanli, China, and the model is ZNLBM-IIX. Each sensor has a measuring range of 0∼500 N and an accuracy of 0.15 N. The contact sensor model is ZNLBM-X, and each sensor has a measuring range of 0∼100 N and an accuracy of 0.03 N. To simultaneously measure the grasping force exerted on the object along the X-axis and Y-axis, a connector was designed to connect the two sensors, as shown in Figure 10.

A horizontal linear stage displaced the flexible finger against the test object by a 20 mm stroke in 1 mm steps. At each step, all of the sensors record the force generated. To obtain more comprehensive contact stress data, the axial rotation platform rotated the target 10 degrees around the center of the circle, and a second test was performed on each finger to obtain contact data at additional five positions. Note that the experiment was repeated five sets of ten times and averaged to reduce errors.

The results of FEA and experimental results were compared. Figure 11a shows the FEA and experimental deformation results when the object displacement was 15 mm. It can be clearly seen from the figure that the FEA deformation results of the two fingers are consistent with the experimental results, and the flexible fingers can fit the target well. Figure 11b compares FEA data and experimental data of fingers’ contact stress, respectively. Figure 11c compares FEA data and experimental data of fingers’ resultant force, respectively. The FEA results closely matched the experimental data from the physical tests, which confirms the accuracy of the FEA results.

The maximum standard deviation for the two flexible fingers in the contact stress were 1.8% and 0.6%, respectively. The deviation is caused by the nonlinear elastic material model in FEA. It can be found from the figure that, first, the contact stress and force response of finger 1 increase approximately linearly at small loading displacements and are changing a nonlinear fashion as the object displacement continues to increase. The contact area increment between finger 1 and the object becomes smaller, and the Y-axis force on the object decreases. This results in a reduction in finger 1 contact stress and force response. Finally, the object displacement further increases, and the Y-axis force increases in reverse which causing the contact stress and force response to increase rapidly. In contrast, the contact stress and force response of finger 2 are changing a linear fashion throughout the loading phase. These results affirm the effectiveness of our design strategy and the ability of finger 2 to have less contact stress in grasping objects.

3.2. Shape and Size Adaptability

To evaluate the shape adaptability of the flexible fingers, six types of grippers were used to grasp objects of various shapes, including circles, ovals, and squares. The corners of the square objects were rounded, in order to avoid the FEA convergence problem caused by the sharp corners. A uniform input displacement or angle in the input-direction was applied to the grippers, as depicted in Figure 9. Figure 12 shows part of the FEA deformation results of the two translational grippers. The figures clearly indicate that the grippers composed of the two types of fingers exhibit great shape adaptability. Gripper 1 fits well with the surfaces of both circular and elliptical objects. Notably, gripper 2 exhibits more pronounced fingertip deformation for wrapping. Although there are gaps between the deformed fingers and the square target, gripper 2 still exhibits substantial overall adaptability in shape. These results confirm the flexible fingers’ ability of grasping objects of different shapes.

To assess the size adaptability of the flexible fingers, six types of grippers were used to grasp circular targets of different diameters, and the target-to-finger length ratios ($\frac{D}{L_1}$) changed within $[0.2,1]$. Figure 13 shows part of the size adaptability results of the translational grippers. The analysis focused on the two grippers’ performance in grasping circular targets with different sizes, whose $\frac{D}{L_1}=0.5$, $0.75$, and 1, respectively. The results indicate that all grippers exhibit effective adaptation to the targets of different sizes. Gripper 1 shows a general wrapping effect. In contrast, gripper 2 showed great wrapping for all target sizes. This difference is mainly reflected in the different deformation of the fingertips. As the size of the target object increases, the wrapping effect of the gripper 2 continues to improve, further proving the superior size adaptability of the flexible fingers.

3.3. Performance Analysis of Flexible Grippers

To comprehensively evaluate the performance of the six grippers during object grasping, the grasping performance will be quantitatively compared in terms of the fingertip deformation and mean contact stress. Taking the six objects in Figure 12 and Figure 13 as grasping targets, a series of FEA simulations about the grasping process of the six grippers were carried out. Figure 14 and Figure 15 are the corresponding performance data graph when the grippers grasping six targets, showing the impact of input loading on performance indicators. Note that all results exclude the stage when the flexible finger is not in contact with the target, and the input loading starts from contact with the target.

In Figure 14a,b, it can be observed that when the translational grippers grasp targets of different shapes, fingertip deformation and mean contact stress increase with the input displacement. For small input displacements, the growth is linear. However, for excessively large input displacements, it becomes nonlinear. Specifically, the fingertip deformation of translational gripper 1 is larger than that of gripper 2 when grasping circular targets. As the input displacement increases, the fingertip deformation of gripper 2 approaches or even exceeds that of gripper 1. Notably, throughout the entire process of the translational gripper grasping an oval target, the fingertip deformation of gripper 2 consistently surpasses that of gripper 1. Additionally, the mean contact stress of gripper 2 is consistently smaller than that of gripper 1 during the process of grasping both circular and oval targets. The indices related to the square target differ significantly due to all the grippers contacting the square target at the corners. The mean contact stress of translational gripper 2 is generally larger than that of gripper 1, but the fingertip deformation of gripper 2 consistently surpasses that of gripper 1 when grasping the square target. Figure 12c,f further reveal that the grasping effect of gripper 2 is better. These results collectively suggest that translational gripper 2 is more suitable than gripper 1 for grasping circular and oval targets. Despite the higher contact stress when grasping square targets, translational gripper 2 still demonstrates good shape adaptability.

From Figure 14c,d, it becomes evident that most fingertip deformations of the angle gripper change linearly when grasping different shaped targets. Nonlinear increases occur only when angle gripper 1 grasps the oval target at a larger input angle. Furthermore, the fingertip deformations of angle gripper 1 and gripper 2 are very close when grasping circular targets, with angle gripper 2 exhibiting smaller deformations for the other two shapes. When grasping targets of three shapes, the mean contact stress of angle gripper 2 is consistently smaller than that of gripper 1. Figure 14e,f illustrate that when the active gripper grasps objects of different shapes, fingertip deformation exhibits linear growth under small input displacement, becoming nonlinear for larger displacements. Moreover, throughout the entire grasping process, the fingertip deformation of active gripper 2 surpasses that of gripper 1. For circular and oval targets, the mean contact stress of active gripper 2 is smaller than that of gripper 1 under small displacement loading. However, when the input displacement exceeds 18 mm, the mean contact stress of gripper 1 becomes smaller. Notably, the mean contact stress of active gripper 2 is consistently smaller than that of gripper 1 when grasping square targets. In conclusion, the results indicate that the three grippers by finger 2 exhibit good shape adaptability, maintain small contact stress, and perform better when grasping targets with a certain curvature.

Figure 15 shows the performance data of six grippers when grasping circular targets of different sizes ($\frac{D}{L_1}=0.5$, 0.75 and 1). As depicted in Figure 15a, when grasping targets of varying sizes, the fingertip deformation of translational gripper 1 exhibits linear growth under small input displacements. However, for large input displacements, the fingertip deformation of translational gripper 1 changes nonlinearly. Conversely, the fingertip deformation of translational gripper 2 demonstrates linear growth when grasping targets of three different sizes. Specifically, when grasping the target with $\frac{D}{L_1}=0.5$, the fingertip deformation of translational gripper 2 is close to that of gripper 1. As the target size increases to $\frac{D}{L_1}=0.75$, it becomes smaller than that of gripper 1. For $\frac{D}{L_1}=1$, the fingertip deformation of translational gripper 2 is $50.14\%$ larger than that of gripper 1. Contrastingly, when grasping circular targets of different sizes, the mean contact stress of translational gripper 2 is consistently smaller than that of gripper 1, as shown in Figure 15b. The mean contact stress of translational gripper 2 is $55.70\%$, $53.38\%$, and $70.89\%$ smaller than that of gripper 1 when grasping targets with $\frac{D}{L_1}=0.5$, 0.75 and 1, respectively. Figure 15c illustrates that when grasping targets of different sizes, the fingertip deformations of both angle grippers show similar linear growth at smaller input angles. This phenomenon is attributed to the target contacts the angle gripper near the fixed end firstly, while the fingertip part experiences rigid displacement. As the input angle increases, this phenomenon improves, and the fingertip deformation of the angle gripper shows a nonlinear change. Additionally, when grasping a target with $\frac{D}{L_1}=0.75$, the fingertip deformation of angle gripper 2 is very close to that of gripper 1. However, as the target size increases, it surpasses the value of gripper 1 (e.g., $\frac{D}{L_1}=0.75$ and 1). Figure 15d shows that at a smaller input angle, the mean contact stress of angle gripper 2 is smaller than that of gripper 1. When the input angle exceeds $17.{44}^{\circ }$, the contact stress of angle gripper 2 becomes larger than that of gripper 1 in grasping the target with $\frac{D}{L_1}=1$. Finally, in Figure 15e,f, when grasping targets of different sizes, the fingertip deformation of active gripper 2 is greater than that of gripper 1. For $\frac{D}{L_1}=0.55$ and 0.7 targets, the fingertip deformation of the active gripper shows a linear growth trend. Moreover, the mean contact stress of active gripper 2 is much smaller than that of gripper 1. Conversely, when grasping a larger target (e.g., $\frac{D}{L_1}=1$), the fingertip deformation of the active gripper changes nonlinearly. The mean contact stress of active gripper 2 is smaller than that of gripper 1 only when the input displacement is small. When the input displacement exceeds $16.25$ mm, it becomes greater than that of gripper 1. The presented results indicate that the three grippers by fingers 2 exhibit good size adaptability and are capable of maintaining low contact stress. In particular, translational gripper 2 by flexible finger 2 significantly mitigates contact stress. Comparing the data in

Table 1. The damage stress of fruits [52,53,54].

Fruits	Stress/Mpa
Apple	0.13
Strawberry	0.093
Pomelo	0.15
Citron	0.18
Lemon	1.14
Grapefruit	0.26
Orange	0.51

, it can be seen that this characteristic makes it well-suited for applications in fruit-picking robots.

In conclusion, the comprehensive analysis presented above offers a holistic view of the grasping performance of the six grippers when handling different objects. The results show that, compared with the grippers by flexible fingers 1, the grippers by fingers 2 have similar or even larger fingertip deformation, and at the same time has a smaller mean contact stress. This means that the grippers composed of fingers 2 is more suitable for handling unstructured fragile targets, and also confirms the effectiveness of the design method proposed in this article.

3.4. Experimental Results

In this section, adaptive gripping performance, gripping stability, and maximum gripping weight of the grippers were assessed through prototype tests. The thickness of all the fingers were set to 20 mm. The fingers are mounted on a parallel gripper actuated by pressure.

Figure 16 shows the test results of translational grippers 1 and 2 in grasping objects with different shapes and size, respectively. Both grippers exhibit great shape adaptability and size adaptability, which agrees well with the simulation results. The findings indicate that relatively small fingers can effectively wrap around and grasp larger objects, highlighting the potential of the designed flexible fingers for diverse object-grasping applications. To further validate the adaptability and efficacy of the developed grippers, a variety of fruits, including oranges, apples, tangerines, and pears, were utilized as targets in grasping tests (as shown in Supplementary Movie S1). The results show that the soft gripper can successfully grasp these fruits of various shapes and sizes, highlighting the advantages of the designed flexible fingers in fruits picking. Moreover, the experimental results illustrate that translational gripper 2 with flexible finger 2 can stably grasp fruits without causing damage. The test results ($\frac{D}{L_1}=0.82$ and 1.33) show that flexible finger 2 exhibits effective grasping of large-size objects during the test. Furthermore, gripper 2 consistently demonstrates good grasping stability in these scenarios. The above results not only reaffirm the effectiveness of the proposed design method, but also substantiate the necessity of incorporating contact stress constraints in flexible finger design, highlighting its practical applicability.

As shown in Figure 17a, the payload tests were conducted to assess the maximum allowable weight that the two translational grippers can carry. Figure 17b shows the experimental results depicting the maximum load capacity of the two flexible grippers under varying input pressures. It is noteworthy that higher input pressure leads to a larger payload, and as the input pressure increases, the payload of the grippers becomes less sensitive to pressure variations. At an input pressure of 0.5 MPa, gripper 1 achieves a maximum payload of 3.2 kg, while that of gripper 2 can reach 3.52 kg. Using the selected finger dimensions and parameters, the weights of gripper 1 and gripper 2 are 87.43 g and 84.41 g, respectively. This results in payload-to-weight ratios of 36.6:1 and 41.7:1, respectively. The high payload-to-weight ratios indicate that implementing flexible grippers on fruit-picking robots has the potential to reduce power consumption. It is important to note that the maximum payload is influenced by factors such as friction and the contact region between the gripper and an object. Consequently, the estimated maximum payload may vary for different objects with distinct materials, sizes, shapes, and surface conditions. The payload test validates gripper 2’s effectiveness and stability in gripping heavy objects.

4. Conclusions

This paper proposes an adaptive finger structure with low contact stress, which has both good contact performance and adaptability. This study proposes a new method to the topology design of the above-mentioned structures and explores the effects of design parameters, including virtual spring stiffness, volume fraction, design domain size, and discretization, on the results of flexible fingers. Based on the above finger structure, flexible grippers with different driving modes are designed. These grippers maintain low contact stresses on their own and require no additional components, making them cost-effective and highly versatile. When grasping different fragile fruits, you only need to change the material and thickness of the finger to adjust its contact stress extreme value. The adaptable finger structures with low contact stress have broad potential application scenarios.

The results of this study indicate that topology optimization design parameters will affect the shape of the flexible finger, its degree of flexibility, the size of the internal beams, and the extremes of contact stress. The results of finite element and actual tests show that finger 2 has smaller contact stress and greater wrapping force than finger 1, confirming the effectiveness of the proposed design method. Based on these flexible fingers, six types of flexible grippers were developed, including translational grippers, angle grippers, and active grippers. A comparative analysis of these grippers using FEA evaluated their performance. Practical tests were conducted on the translational gripper 1 and gripper 2, including the evaluation of adaptive gripping performance, gripping stability and maximum gripping weight. The results show that the two designs of translational grippers show good adaptability in terms of shape and size. Gripper 2 with flexible finger 2 excels in contact stress and adaptive wrapping, making it well-suited for grasping unstructured and fragile objects. Overall, the proposed method provides valuable insights into the design of flexible grippers for fruit-picking robots and highlights the potential of the designed flexible grippers in agricultural harvesting applications.