Abstract

The design of grippers for the agro-industry is challenging. To be cost-effective, the picked object should be moved around fast requiring a firm grip on the fruit of different hardnesses, shapes, and sizes without causing damage. This article presents a self-adaptive flexure-based gripper design optimized for high acceleration loads. A main novelty is that it is actuated through a push–pull flexure that is loaded in tension when the gripper closes, allowing it to handle high actuation forces without the risk of buckling. To create a robust gripper that can handle relatively high loads, flexures are used that are reinforced and have a thickness variation over the length. The optimal thickness distribution of these flexures is derived analytically to facilitate the design process. The derived principles are generally applicable to flexure hinges. The resulting advanced cartwheel flexure joint, as used in this gripper, has a 2.5 times higher support stiffness and a 1.5 times higher buckling load when compared to a conventional cartwheel joint of the same size and actuation stiffness. The PP-gripper is numerically optimized for a high pull-out force, using analytical design insights as a starting point. The gripper can grip circular objects with radii between 30 and 40 mm. The pull-out force is 21.4 N, with a maximum actuation force of 100 N. Good correspondence is found between the geometric design approach, the numerically optimized design, and the results of the experimental validation.

1 Introduction

The design of grippers for the agro-industry is challenging. Fruit, being a relatively low-value product, requires fast gripping to be cost-effective [1]. Additionally, the varying sizes and weights of fruit, coupled with their susceptibility to damage from excessive pressure, necessitate grippers that are adaptable to size and shape and capable of accurate force control. A recent literature survey, published in 2020, indicates that the commercial application of robotic grippers in agro-industry is not yet prevalent [2], and harvesting fruit is mainly done manually.

Numerous studies have been conducted on robotic grippers in the agro-industry. Reference [3] provides a general review of the automation process and includes 47 papers on end-effector designs. Reference [2] also reviews end-effector designs, and Ref. [1] compares eight end-effectors. Notable examples of recent gripper designs include a three-finger gripper based on the fin-ray concept [4], a gripper with four pneumatically actuated soft robotic fingers [5], and a bistable gripper with two fingers [6].

Reference [1] lists requirements for evaluating grippers for agro-industry, mainly related to the percentage of undamaged fruit, picking time, and cost-effectiveness. References [7,8] provide performance metrics that are more technical. These metrics are mainly related to the pull-out force, support stiffness, equally distribution of forces on the object, and the range of object sizes that can be grasped.

An effective strategy for achieving the latter two metrics is underactuated gripping [8], or self-adaptive gripping, where the number of actuators is less than the number of degrees-of-freedom. In well-designed underactuated grippers, the remaining underactuated degrees-of-freedom passively adapt to the shape of the object [8], thereby reducing the complexity of contact force control. It also lowers the component count and weight and therefore renders the gripper more cost-effective. However, a poor design can lead to an unstable grasp, meaning that the gripper loses contact with the object at some phalanges, which can result in unwanted motion, unpredictable behavior, and potentially high contact forces. More advantages and disadvantages concerning underactuated gripping are discussed in Refs. [810].

Another effective design choice is to use a flexure-based gripper [6,1116], which transmits the motion through the deformation of elastic elements, rather than rolling or sliding elements. Therefore, the gripper can be made from a single part, allowing minimal assembly. As no rolling or sliding parts are in contact, the gripper is easy to clean.

Two types of flexure-based grippers can be distinguished: soft robotic grippers and lumped compliance grippers. Soft robotic grippers are flexible over the full length of the fingers, so they have an infinitely number of underactuated degrees-of-freedom. Some of the studies on these highly adaptable soft robotic grippers in the agro-industry include Refs. [4,5]. Lumped compliance grippers, on the other hand, are designed to deform only at specific hinge locations, similar to rigid link mechanisms. Reference [6] presents an example of such a gripper, suitable for picking soft objects like fruit. The advantage of lumped compliance over soft robotic grippers is that the relationship between kinematics and interaction forces with motor displacement and force is much more predictable. This improves the contact force control. Moreover, in combination with the absence of friction, wear, and play in flexure-based systems, the motion is very predictable. The predictability enables the use of a minimal number of sensors inside the gripper to estimate the contact forces without the need for separate contact force sensors. Force prediction of soft robotic grippers requires typically many more sensors, as illustrated in Refs. [17,18]. The current article focuses on lumped compliance flexure-based grippers.

The current literature on lumped compliance, flexure-based grippers features either simple notch flexures with a limited range of motion or load-bearing capacity [1113], or the gripper's dimensions significantly exceed those of the grasped object [6,14,16,19]. These larger sizes result from actuation forces applied through a linkage-based pushing mechanism on the outer side of the phalanges, increasing the gripper width.

A more compact actuation method uses a tendon or a cable to transfer the actuation force [2023]. Because a tendon is loaded with tension, it can carry a high force without the risk of buckling. However, this actuation approach involves multiple parts, and the cable is typically guided using some small holes or a sheath such that the gripper is not monolithic, compromises hygiene, and suffers from friction in contrast to flexure-based grippers. Additionally, as a tendon is only loaded on tension, multiple tendons or preload mechanisms are typically required to actuate the gripper.

In pursuit of a compact, predictable, and cleanable gripper for high-speed fruit handling, this article presents a flexure-based, underactuated gripper, actuated through a tendon-like structure, as shown in Fig. 1. This design of this so-called PP-gripper (push–pull gripper) is a result of five innovative contributions as illustrated in Fig. 2:

  • A flexure-based push–pull flexure is presented that transmits the actuation force. It combines the benefits of tendons and flexures, including a relatively slender design, zero friction, easy cleaning, and a monolithic design avoiding the need for assembly. Reinforcements enable the push–pull flexures to withstand the small compressive forces required to open the gripper, omitting the need for a preload mechanism. At the same time, it allows to transmit high tension forces required for firm grasping. An overview of the resulting gripper design is given in Sec. 2.

  • Reinforced flexures with varying thicknesses are used for optimal performance. Analytical formulas for the optimal thickness are presented to facilitate fast design optimization (Sec. 3).

  • These reinforced, variable thickness, flexures are applied to obtain a new type of cartwheel flexure joint, which is used as finger joints (Sec. 4).

  • An analytical design approach is presented to obtain the overall geometry of the gripper, aiming for a firm grip such that high accelerations are achievable. The analytical approach is followed by a numerical optimization (Sec. 5).

  • Experimental validation of the gripper's performance is conducted, including the effects of creep and contact friction (Sec. 6).

Fig. 1
PP-gripper: flexure-based tomato gripper, and current version is partly made from transparent material for visualization
Fig. 1
PP-gripper: flexure-based tomato gripper, and current version is partly made from transparent material for visualization
Close modal
Fig. 2
Overview of the main contributions of the article and its structure
Fig. 2
Overview of the main contributions of the article and its structure
Close modal

2 Conceptual Design

This section introduces the design of the PP-gripper (Sec. 2.1) followed by the concepts that are used in later sections. Section 2.2 explains the design of the push–pull flexure. An analysis of the degrees-of-freedom of the gripper is given in Sec. 2.3, which leads to several design bounds used in Sec. 5. Section 2.4 introduces the pull-loose force as a design criterion that is used in Sec. 5. Section 2.5 summarizes the design principles that are used to develop the gripper.

2.1 Design Overview.

The conceptual gripper design, shown in Fig. 3, consists of a base and two fingers, each with two phalanges. The phalanges are connected by rotational hinges and the base is fixed to the ground. These hinges will be based on cartwheel hinges as detailed in Sec. 4. The gripper is fully flexure based, allowing it to be manufactured from a single piece by 3D printing. The properties of the used material, PA 2200, are given in Table 1.

Fig. 3
Conceptual gripper design
Fig. 3
Conceptual gripper design
Close modal
Table 1

Material properties of PA 2200

PropertyValue
Flexural modulus1500 MPa
Poisson ratio0.42
Tensile strength50 MPa
Allowed stress (60% of tensile strength is used as limit)30 MPa
PropertyValue
Flexural modulus1500 MPa
Poisson ratio0.42
Tensile strength50 MPa
Allowed stress (60% of tensile strength is used as limit)30 MPa

2.2 Actuation Through Push–Pull Flexures.

Each finger is actuated by a push–pull flexure that is connected to the distal phalange and guided by three perpendicular flexures fixed to either the base or the proximal phalange, see Fig. 3. A single actuator moves in the vertical direction and drives both push–pull flexures. When closing the gripper, the push–pull flexures are loaded under tension; therefore, they can transmit a high amount of force without the risk of buckling. Additionally, with the proposed reinforcements, they can also handle the small compressive forces needed to push the gripper open against the end-stops mounted around the hinges.

2.3 Evaluation of Degrees-of-Freedom.

The gripper is underactuated as each finger contains two hinges (providing two degrees-of-freedom) and only one push–pull flexure for the actuation. This means that the gripper can adapt to the shape of the object. For effective operation, the gripper must be exactly constrained when holding an object. An overconstrained situation might result in high contact forces for minor variations of the object shape. This is undesirable for handling delicate items like fruit. Conversely, an underconstrained situation leads to unpredictable movement of the object within the gripper, limiting permissible accelerations and, consequently, operational speed.

Four assumptions are made when evaluating the degrees-of-freedom. First, the friction of the five contacts between the object and the gripper is neglected, implying that each contact only constrains the degree-of-freedom normal to the contact face. Second, the analysis is two dimensional, assuming that the neglected friction constraints motion of the object in the third dimension. Third, the object is assumed to be perfectly circular, and its rotational degree-of-freedom is neglected, as it is primarily constrained by the neglected friction. Finally, force-driven actuation is assumed, leaving the vertical degree-of-freedom of the actuator unconstraint.

Without considering contacts and push–pull flexures, the system has seven degrees-of-freedom: the two translations of the object, the two hinge rotations of each finger, and the motion of the actuator. Each of the two push–pull flexures constraints one degree-of-freedom between the actuator and a distal phalange. Combined with the five contacts, there are seven constraints in the system, matching the number of degrees-of-freedom. This indicates that the closed gripper with all contacts engaged is exactly constrained.

If one of the proximal phalanges loses contact, an undesired situation occurs in which unpredictable movement of the finger can occur. This is called closing ejection [10], see Fig. 4(a). If the base loses contact, an undesired situation occurs in which the object can easily move up and down. This is called opening ejection, see Fig. 4(b). The dimensions of the gripper should be designed such that these ejection situations will not occur. This is added as one of the design constraints in Sec. 5.1 of this paper.

Fig. 4
(a) Closing ejection and (b) opening ejection, and contacts are lost while closing the gripper
Fig. 4
(a) Closing ejection and (b) opening ejection, and contacts are lost while closing the gripper
Close modal

2.4 Design Criterion: Pull-Loose Force.

The gripper will be designed such that it has a high pull-out force. This is because a high pull-out force results in a quite tight grip, such that the gripper can accelerate fast without losing the object. However, we will not consider the pulling force on the object that is required to remove the object fully out of the gripper (which is called the pull-out force) but a force that is closely related. Namely, the pulling force that is required to lose the contact between the object and the base of the gripper. This force will be referred to as the pull-loose force. A reason for choosing this as a criterion is that losing the contact with the base already results in an undesired underconstrained situation in which the object can move uncontrollably within the gripper.

2.5 Overall Design Principles for Flexure Mechanisms.

Flexure mechanisms typically consist of multiple thick parts that are supposed to behave rigidly and thin parts, called flexures or leaf springs, that can bend (and twist) to allow motion. Using smart combinations of multiple flexures, joints can be designed that only allow motion in desired directions, for example, only rotation around a dedicated axis. Because these joints only allow specific motion, geometrically, a flexure-based design can be modeled as a rigid link mechanism.

Flexure joints have several advantages with respect to sliding or roller bearings. As there are no rolling or sliding parts in contact, they are easy to clean. As there is no need for lubrication and no wear, flexures do not need maintenance and are suitable for vacuum or dirty environments. Flexures can be often made of a single part, reducing assembly time and increasing alignment accuracy. A big advantage is that flexures behave very predictable due to the absence of friction, wear, and backlash, improving controllability and making them very suitable for precision applications.

A challenge in designing flexures is that they, on the one hand, should allow sufficient range of motion and low compliance in the motion directions, requiring the flexures to be thin. On the other hand, flexures should provide sufficient load-bearing capacity and stiffness in their supporting directions, which increases with thicker flexures. Stress is often an important constraint in the design of flexures because of the desired elastic deformation, sometimes combined with high loads in supporting directions.

The strength of flexures is limited by stress and buckling, both are addressed in the current design. Flexures can typically carry relatively high loads in tension, but loading in compression is limited due to the risk of buckling. Using reinforced flexures with varying cross sections addresses these challenges.

3 Optimal Thickness Distribution of Reinforced Flexures

To be able to open the gripper, the push–pull flexure should have high a buckling load. This is achieved by using reinforced flexures with a varying thickness. Figure 5 shows such a flexure, consisting of two compliant parts and reinforcement. A challenge is that tuning the thickness is difficult. Variation through numeric optimization can prove this difficulty as it requires many design parameters to describe the possible shapes. This section derives the optimal thickness distribution as a function of the position of the reinforcement. The result is more widely applicable to reinforced flexures with a varying thickness.

Fig. 5
Reinforced flexure design with varying cross section is based on a linearly varying bending moment. Thickness is expressed as a function of coordinate x running from zero to L.
Fig. 5
Reinforced flexure design with varying cross section is based on a linearly varying bending moment. Thickness is expressed as a function of coordinate x running from zero to L.
Close modal

3.1 Typical Flexure Requirements and Assumptions.

Flexures are typically subjected to two conflicting requirements. On the one hand, flexures should provide sufficient support, i.e., support stiffness and buckling resistance. This would result in a short as flexural part and a long as reinforcement. On the other hand, flexures should allow a certain range of motion without exceeding stress limits in the material. Therefore, the flexural parts should be sufficiently thin and long.

To satisfy these two requirements, we can obtain the optimal thickness, which is the thickness that maximizes the support stiffness for a given required range of motion, under the following assumptions:

  1. Manufacturing tolerances place a lower limit on the flexure thickness.

  2. The flexure bends only, i.e., no torsional deformation is required.

  3. The load on the flexure in the supporting directions is limited, such that the stress in the flexures is dominated by deflection, i.e., the internal bending moment. This is true for most flexure applications.

  4. The bending deformation is sufficiently small such that linear beam equations hold.

  5. There is a point in the flexure where the bending moment is zero. If this is not true, reinforcing the flexure is typically not a good option, for example, if the bending moment is constant over the flexure.

  6. The design is driven by requirements on buckling resistance, support stiffness, and stress limits, and is not dominated by a requirement on the actuation stiffness.

3.2 Design Decisions.

In the following three steps, the optimal thickness is derived from a reinforced variable-thickness flexure as shown in Fig. 5. In this figure, the lengths of both flexural parts are expressed in terms of the total length using scaling factors f1 and f3. The thickness of these parts is expressed as a function of the coordinate x by tP1 and tP3. The thickness of the reinforcement is selected to be sufficiently large, such that it does not deform significantly under load.

3.2.1 Step 1: Position of the Reinforcement.

The internal bending moment is linearly distributed as a function of x, because of assumptions 2–4. At one position, indicated by xA, the bending moment will be zero (assumption 5). The part of the flexure near this position will not be subject to much strain, so this part can be reinforced to increase support properties. Therefore, the center of the reinforcement will be at the position where the bending moment is zero. The center of the reinforcement can be expressed in terms of, currently unknown, f1 and f3:
(1)
The bending moment can be therefore expressed as follows:
(2)
where g is an unknown factor.

3.2.2 Step 2: Thickness Is Driven by Stress.

Recall the two conflicting requirements stated earlier: On the one hand, flexures should be as thick as possible for optimal support properties. On the other hand, this thickness is limited by stress under the required range of motion. This implies that the stress in optimally designed flexures becomes maximal over their full length when the flexure is in the maximally deflected state. This stress is dominated by bending (assumption 3). The highest stress as a result of bending occurs at the top and bottom surfaces of the flexure and can be derived based on the bending moment as follows [24]:
(3)
Here, w is the width of the flexure. Because the highest stress, σhighest should be a constant and w is a constant, and the thickness should be proportional to the square root of the bending moment. By substituting Eq. (2), it is derived that the thickness should be proportional to the square root of the distance to xA.
(4)

3.2.3 Step 3: Length Flexural Parts as Small as Possible.

A last insight is that typically the best flexure is obtained when the length of the deforming parts is as small as possible. This means that the thickness of the deforming parts should be small to allow for sufficient deformation. However, it was assumed that the minimal thickness of a flexure is limited (assumption 1).

According to step 2 (Eq. (4)), the lowest thickness is at the position where the bending moment in the flexural parts is minimal, which is at the positions close to the reinforcement. Therefore, the thickness of the flexures at these positions should be the minimal thickness. This means that the thickness distribution can be calculated as follows:
(5)
where LA is half of the length of the reinforcement.

3.2.4 Conclusion on the Design Decisions.

The full flexure thickness has been expressed in terms of the position and the length of the reinforcement as captured by f1 and f3. This means that only these two parameters need to be optimized, significantly simplifying the numerical optimization or the manual tuning of the thickness for optimal support stiffness.

For modeling reasons, the thickness distribution of the push–pull flexures in the gripper is chosen to be linear, using the extreme values from the aforementioned analysis.

4 Cartwheel Hinge Design

The proximal and distal hinges are cartwheel hinges with varying-thickness, reinforced flexures. This section details the choice for this flexure configuration. The general behavior of this configuration is explained in Sec. 4.1. In Secs. 4.2 and 4.3, design optimizations are performed on a single joint in order to quantify the effect of the thickness variation and reinforcements. The design optimization of the actual cartwheel flexures in the gripper is performed in Sec. 5.

4.1 General Behavior of the Cartwheel Hinge.

The cartwheel design [25,26], see Fig. 6(a), is chosen because it is relatively simple and compact and allows for a sufficient range of motion. Reinforcements are added to positions where the bending moment is close to zero. This position is relatively close to the outer parts of the hinge. As a result, the inner part of the flexures is the longest and contributes most significantly to the rotation of the flexure hinge. This can also be observed in the deformed configuration shown in Fig. 6(b). The outer part of the flexures is short and only releases small rotational deformation. However, this small rotation significantly reduces the stress on the inner part of the flexures.

Fig. 6
Cartwheel with reinforced flexures: (a) cartwheel dimensions, (b) deformed shape, and (c) buckling mode after deformation
Fig. 6
Cartwheel with reinforced flexures: (a) cartwheel dimensions, (b) deformed shape, and (c) buckling mode after deformation
Close modal

4.2 Input Cartwheel Optimization.

Four optimizations are performed in order to evaluate the effectiveness of the reinforcements and the thickness variation:

  1. No thickness variation, no reinforcements

  2. Linear thickness variation, no reinforcements

  3. No thickness variation, with reinforcements

  4. Linear thickness variation, with reinforcements

The first design parameter is the thickness of the flexure at the center, tA. For the cases with varying thickness, the thickness of the other side of this flexure is a second design parameter, tB. For the cases with a reinforcement, the normalized length f1 is an extra design parameter. Because the outer part of the reinforced flexures is very short and only functions to allow small rotation, there is no need to optimize this. It is given a constant thickness of tC=1mm and a small, normalized length of f3=0.1. The angles of α and β are 22.5 deg, L=30mm, and w=17mm. The flexure hinge is made from PA 2200, see Table 1. The hinge rotates 15 deg in each direction and should not exceed the maximum allowed stress in the material of 30 MPa. The minimal thickness of the flexures is 0.8 mm. The hinge is optimized for a low compliance in the X-direction in its deformed configuration, while constraining the rotation in the actuation direction. The cost function is defined as follows:
(6)
where Cx is the compliance in the X-direction, σhgh is the highest stress in the flexures, and σalw=30MPa is the allowable stress. The buckling loads of the resulting designs are also computed, and the relevant buckling mode is visualized in Fig. 6(c).

4.3 Results Cartwheel Optimization.

Table 2 and Fig. 7 show the results of the four optimization cases. The support compliance is improved significantly using reinforcements. This is because the total length of the flexure parts is significantly reduced. Figure 7(c) shows that the critical buckling load is significantly improved by allowing varying thickness. This is because of the increased overall thickness of the compliant parts. A disadvantage of the increased thickness is that the required actuation moment also increases (Fig. 7(d)). However, the required actuation moment with the combination of reinforced, varying-thickness flexures (case 4) is better than that of the variant without varying thickness and reinforcements (case 1). The support stiffness of case 4 is about 2.5 times higher than that of case 1, and the buckling load is about 1.5 times higher.

Fig. 7
Resulting design properties of the cartwheel hinge designs: (a) and (b) support stiffness in X- and Y-directions, respectively, evaluated after deformation in the center of compliance ((a) Kx (106 N/m) and (b) Ky (106 N/m)), (c) buckling load (kN), and (d) the required actuation moment (Nm) for 15 deg of deformation
Fig. 7
Resulting design properties of the cartwheel hinge designs: (a) and (b) support stiffness in X- and Y-directions, respectively, evaluated after deformation in the center of compliance ((a) Kx (106 N/m) and (b) Ky (106 N/m)), (c) buckling load (kN), and (d) the required actuation moment (Nm) for 15 deg of deformation
Close modal
Table 2

Cartwheel optimization: overview of input choices that define the different cases and the resulting values of the design parameters

Case numberInput choicesResulting design parameters
Varied thicknessReinforcedtA (mm)tB (mm)f1 (−)
11.18
2X1.751.00
3X0.800.23
4XX1.180.800.21
Case numberInput choicesResulting design parameters
Varied thicknessReinforcedtA (mm)tB (mm)f1 (−)
11.18
2X1.751.00
3X0.800.23
4XX1.180.800.21

In all four cases, the constraint on the maximum stress became active. In both reinforced cases, the thickness tB has the minimal allowed value, such that the length of the inner flexural part (as defined by f1) becomes as small as possible. These results agree with steps 2 and 3 in the derivation of the optimal thickness distribution for general reinforced flexures in Sec. 3.2. These results show the potential of using both reinforcement and varying thicknesses in cartwheel hinges for both improved support stiffness and improved buckling loads without the cost of additional actuation stiffness.

5 Gripper Design

The proposed push–pull flexures and the cartwheels with reinforcements and varying thicknesses are integrated into a gripper design. Before the final flexure-based design is presented, this section proposes a procedure to design and evaluate the kinematic parameters (e.g., link lengths) of such a gripper. Section 5.1 presents the design requirements, constraints, and nine design parameters that define the overall geometry of the gripper, excluding flexure dimensions. In Sec. 5.2, numerical values are chosen for seven of these parameters based on design insights. To gain more design intuition, one of the remaining parameters is varied to show its impact on the gripper performance in Sec. 5.3. The two remaining parameters are numerically optimized together with 15 design parameters that define the dimensions of the flexures, in Sec. 5.4.

5.1 Gripper Requirements.

This subsection presents the basic principles for the design of the gripper.

5.1.1 Design Aim.

The gripper will be designed for a firm grasp such that high accelerations can be achieved without losing the object. Therefore, the pull-loose force should be as high as possible. This is the force required to lose contact with the base. It is equal to the contact force at the base as friction is not considered. Therefore, the objective of the following design optimization is to maximize the contact force between the base and the object.

5.1.2 Design Boundaries.

The following constraints are used.

  1. The gripper should be able to pick all cylindrical objects with a radius between 30 and 40 mm, which is a typical size of tomatoes.

  2. The maximum actuation force on the handle of the gripper is 100 N (so, 50 N per finger).

  3. The maximum pushing force on the handle to open the gripper is 40 N (20 N per finger). However, a pushing force that is a factor two higher should be applied to make sure that the gripper is firmly pushed against the end-stops in an open configuration, such that the open gripper can accelerate without the uncontrolled movement of the fingers.

  4. The minimal thickness of the flexures is 0.8 mm, and this is limited by the 3D-SLS-printing technique.

  5. The maximum stress in the material is 30 MPa, see Table 1.

  6. The safety factor on buckling is two.

  7. The gripper should contact the object at all five contact surfaces. To make sure that all phalanges keep contact with the object (so, to avoid the closing ejection mentioned in Sec. 2.3), the contact force on the proximal phalange should be a minimal 10% of the force on the distal phalange. Note that opening ejection—also mentioned in Sec. 2.3—does not have to be constrained as the objective is to maximize the contact force at the base.

  8. In the open configuration, the distance between both fingertips should be minimal two times the largest object radius, so 80 mm, to ensure that the largest objects also fit through the opening, see Fig. 8.

  9. The angle γ (see Fig. 8) that defines the orientation of the distal phalange in the open configuration should be 25 deg. This value is chosen based on intuition: a large angle (say 80 deg) would result in a very wide gripper when it is open. A very small angle (say 0 deg) would require large rotations of the hinges from the open to the closed configuration.

Fig. 8
Open configuration
Fig. 8
Open configuration
Close modal

5.1.3 Design Parameters and Other Relevant Parameters.

Figure 9 shows the parameters that define the size and the configuration of the closed gripper. The radius of the gripped object is r. Nine free design parameters are indicated by the boxes in the figure, and a derivation of the dependent parameters is presented here. The design parameters s0, s1, s2, and s3 define the distance between the contact faces and the corresponding hinges. D0 defines the distance between the symmetry axis and the proximal hinge, and L1 and L2 define the length of the base and phalanges, respectively, which are measured parallel to the contact faces. The parameters d1 and d2 define the distance between the hinges and the corresponding part of the push–pull flexure.

Fig. 9
Gripper dimensions of the closed gripper: boxed symbols are the design parameters, and green arrows define the six forces on the gripper
Fig. 9
Gripper dimensions of the closed gripper: boxed symbols are the design parameters, and green arrows define the six forces on the gripper
Close modal

The angles ϕ1 and ϕ2 define the orientations of the phalanges, measured along their contact faces. These angles vary while opening and closing the gripper. D1 and D2 define the distance between the hinges and the contact points, parallel to the contact face. Appendix  A presents relations for these four parameters for the closed gripper as a function of the radius and the design parameters. Furthermore, ϕ1 and ϕ2 are calculated for the open configuration as a function of the design parameters. The contact forces Fc0, Fc1, and Fc2 are derived as a function of the actuation force in the push–pull flexure. The equations given in Appendix  A are similar to the equations in Refs. [11,27]. However, in Appendix  A, the width of phalanges is also considered, i.e., s0, s1, s2, and s3.

5.2 Parameter Reduction.

In this subsection, values are chosen for seven of the nine design parameters that are shown in Fig. 9. The goal is to obtain insight into how the parameters affect the performance and to simplify the numerical optimization that will follow in Sec. 5.4. Table 3 summarizes the steps and presents the used values. The used kinematic and static equations are derived in Appendix  A. This section starts with a few remarks on the simplified modeling.

Table 3

Summary of the design approach

StepDescriptionUsed values
1Choose s1, based on the desired gripper sizes1=16mm
2Set s0=s1s0=16mm
3Set d1=0.9s1+0.2rmaxd1=22.4mm
4Set s20.5s1s2=8mm
5Set s3=s2s3=8mm
6Set D00.9(rmax+s1)L0=50.4mm
7Choose L2 as a function of L1 by substituting Eqs. (11) and (13) into Eq. (7).See Sec. 5.3.
StepDescriptionUsed values
1Choose s1, based on the desired gripper sizes1=16mm
2Set s0=s1s0=16mm
3Set d1=0.9s1+0.2rmaxd1=22.4mm
4Set s20.5s1s2=8mm
5Set s3=s2s3=8mm
6Set D00.9(rmax+s1)L0=50.4mm
7Choose L2 as a function of L1 by substituting Eqs. (11) and (13) into Eq. (7).See Sec. 5.3.

5.2.1 Simplified Modeling.

The location of the push–pull flexure can be tuned by many parameters. In this article, only the distances d1 and d2 are considered. These are the most effective parameters to tune how the actuation force relates to the moment that is generated around the proximal and distal hinges: d2 defines how the actuation force FT2 relates to the moment that the push–pull flexure generates on the distal phalange around the distal hinge, see Fig. 10(a). The parameter d1 defines the relation between the actuation force FT1 and the moment on the entire finger around the proximal hinge, see Fig. 10(b).

Fig. 10
Relation between actuation force and moment applied around hinges: (a) distal phalange and (b) full phalange
Fig. 10
Relation between actuation force and moment applied around hinges: (a) distal phalange and (b) full phalange
Close modal

In the parameter reduction, we will make several assumptions for further simplification. In the numerical optimization in Sec. 5.4, assumption 2–5 will be released.

  1. Friction in the contacts between the object and the gripper is neglected.

  2. The model behaves fully symmetric on the symmetry axis that is indicated in Fig. 4.

  3. Distances d1 and d2 are assumed to be independent of the gripper configuration. (In reality, they will slightly vary, while the gripper opens or closes.)

  4. The flexure stiffness of the joints and the push–pull flexure is neglected.

  5. It is assumed that force FT in all the three longitudinal parts of the push–pull flexure is equal. This is the case if the two perpendicular parts are oriented in such a way that the angles with the longitudinal parts, θ, are equal, see Fig. 10(b). So, FT1=FT2=FT.

5.2.2 Parameter Reduction Steps

  • Step 1: The more moment can be exerted by the push–pull flexure around the hinges, and the more the force applied on the object, the higher the pull-loose force is. Therefore, the distance between the proximal hinge and the push–pull flexure, d1, should be as high as possible. However, to prevent collision between the push–pull flexure and the object, it cannot be much larger than the width of the first phalange (s1). Therefore, s1 should be as large as possible. However, a large s1 results in a very wide gripper that is less easy to move around. The parameter s1 should be therefore chosen to obtain the desired gripper size. In this article, it is set to 0.4rmax.

  • Step 2: The height of the base has relatively limited influence on the performance. This is mainly because a higher base will be compensated by a higher optimal proximal length in the optimization below, resulting in a very similar gripper. For simplicity, it is set equal to the height of the proximal phalange: s0=s1.

  • Step 3: The distance d1 is set to be 0.9s1+0.2rmax to prevent the risk of collision between the push–pull flexure and the object.

  • Step 4: The height of the distal phalange does not have a major influence on the pull-loose force. This is because this height is measured parallel to the force line of the contact force between the object and the distal phalange. Therefore, it does not influence the relation between the actuation force and the contact forces on the object, which can be seen in Eqs. (19)–(23). A very high value would make the gripper unnecessarily large. On the other hand, the height should be sufficiently larger than the distance d2 to allow enough space for the push–pull flexure. Because the distance d2 will turn out to be much lower than d1 (see Sec. 5.3), a sufficient guess is to set s2 to be half of s1.

  • Step 5: Similar to s2, the variable s3 should be sufficiently larger than the distance d2. Making s3 much larger than s2 unnecessarily increases the gripper size. The distance s3 is therefore chosen to be equal to s2.

  • Step 6: The closed angle of the proximal phalange (ϕ1closed) should be as large as possible to make sure that the contact force Fc1 does not push the object too much in the positive vertical direction, lowering the pull-loose force. Therefore, it is an advantage to set D0 relatively high (see Eqs. (11) and (12)). However, a high D0 will result in a wide gripper. For D0=rmax+s1, the proximal phalange is fully vertical. In this article, it is chosen to be 90% of this such that the proximal phalange is almost vertical in the closed configuration for the largest object.

  • Step 7: The length of the distal phalange L2 should be slightly higher than the contact distance D2 to guarantee a good contact surface at this phalange. Therefore, it is chosen to be
    (7)

Note that D2 depends on the length of the proximal phalange, L1 (see Eq. (13)), which is still to be chosen. However, in the numerical optimization, the theoretical value of D2 can be computed based on a chosen value for L1 by using Eqs. (11) and (13), by which L2 is determined from Eq. (7).

5.3 Theoretical Performance.

The two remaining parameters are L1 and d2. These will be obtained in the numerical design optimization in the next section. The goal of the current section is to specify the theoretical performance in terms of these parameters to obtain insights and validate the optimum from the numerical optimization. In Sec. 5.3.1, the distance d2 will be used to tune the ratio between contact forces on the proximal and distal phalange. Then the performance is derived as a function of the length L1, and this performance is evaluated.

5.3.1 Choice for d2.

The maximal pull-loose force, i.e., the force on the base, is obtained if contact force Fc2 is as high as possible as this force is pushing the object the most downward. The actuation force that causes a moment around the proximal hinge, see Fig. 10(b), is divided over both contact forces on the finger (see also Eq. (20)). Loosely speaking this means that the contact force Fc2 is maximal if contact force Fc1 is minimal. However, to prevent closing ejection, we required that Fc1 is at least 10% of Fc2. We will therefore set the minimum allowed value: Fc1=0.1Fc2. The distance d2 can be used to tune the ratio between both contact forces. By substituting this relation and Eq. (19) into Eq. (20), the relation for d2 is obtained as follows:
(8)

5.3.2 Choice for L1.

By substituting Eq. (8) into Eq. (23) for the smallest and largest object size, we find an expression for Fc0/FT in terms of parameter L1. The result for the current choices is shown in Fig. 11.

Fig. 11
Normalized base force and hinge rotations as a function of the normalized length of proximal phalange for current choices of step 1–6
Fig. 11
Normalized base force and hinge rotations as a function of the normalized length of proximal phalange for current choices of step 1–6
Close modal

The gripper is made using flexure-based hinges, which have a limited range of motion. Moreover, the performance of flexure joints, like the support properties, decreases drastically with the increasing range of motion, see, e.g., Ref. [28]. Therefore, the total rotation that both hinges must make from the open to the closed configuration should be limited.

The required range of motion for both hinges can be computed using the following equation:
(9)

In the aforementioned equation, the closed angles derived from Eqs. (12) and (14) can be substituted for the smallest object size. Furthermore, the open angles derived from Eqs. (17) and (18) can also be substituted. The result is shown in Fig. 11.

A larger L1 gives a better ratio Fc0/FT because the contact force Fc2 moves vertically downward. However, it also results in more required rotation of the second hinge. A compromise is to set L1=1.8rmax. The resulting design variables are as follows: L1=72mm, d2=6.0mm, and L2=31.6mm.

5.3.3 Evaluation of the Theoretical Performance.

An important input for the design is the desired range of object sizes that the gripper has to hold. The width of the proximal phalange, s1, is a trade-off between the desired pull-loose force and the gripper size. The length of the proximal phalange is a trade-off between the pull-loose force and the required hinge deflections. In the current design, the width of the open gripper is 137mm, which is measured as the distance between both distal hinges. The ratio between the pull-loose force and the actuation force is: Fc0/FT=0.19, and the total rotation of the second hinge is about 20deg.

5.4 Numerical Optimization.

The gripper is numerically optimized to obtain proper flexure dimensions and values for the remaining two geometrical design parameters. This subsection describes the boundary conditions and the result of the numerical optimization. Details of the cost function are given in Appendix  B, and detailed optimization results are given in Appendix  C. Figure 12 shows the flexure-based design of the gripper combined with the simplified geometry of the conceptual design.

Fig. 12
Flexure design: purple dotted lines indicate corresponding part sizes and hinges according to the conceptual design of Fig. 3
Fig. 12
Flexure design: purple dotted lines indicate corresponding part sizes and hinges according to the conceptual design of Fig. 3
Close modal

5.4.1 Modeling and Optimization Algorithm.

The gripper is modeled in an implementation of SPACAR [29,30] in matlab. The flexures in the cartwheel joints and the push–pull flexures are assumed to be flexible, and each flexure is modeled using two serial-connected nonlinear beam elements [31]. The design is optimized using the covariance matrix adaption evaluation strategy [32].

5.4.2 Design Parameters and Other Dimensions.

Seventeen design parameters are input for the optimization. The parameter L1 is optimized as it is difficult to derive its optimal value analytically as indicated in Sec. 5.3. Another design parameter is d2 as the factor d1/d2 has a lot of impact on the contact force distribution. Its theoretical optimal value has been derived in Sec. 5.3, see Eq. (8). However, for this derivation, the elastic forces were neglected, and the distances d1 and d2 were assumed to be independent of the gripper configuration making it impossible to use the result in the current optimization. The remaining seven geometric parameters are based on the analytical design approach in Sec. 5.2. Fifteen design parameters are used to tune the flexure dimensions. Three design parameters are used for both cartwheels (tA, tB, and f1 in Fig. 6), and their overall size and orientation are chosen based on the available space. Nine parameters are optimized to tune the thicknesses and reinforcement lengths, using the principles from Sec. 3. The lengths and orientations of the seven parts of the push–pull flexure are chosen based on the available build space. The width of the flexures in the cartwheel joints is 17 mm, and the width of the push–pull flexures is 12 mm.

5.4.3 Determining the Undeformed Configuration.

The undeformed configuration can still be chosen by the hinge angles ϕ1 and ϕ2 in this configuration. It is advantaging for the contact forces if the gripper is closed as far as possible in the undeformed configuration, such that the flexural stiffness will help to keep the gripper closed instead of counteracting this. Therefore, the undeformed configuration is selected such that the gripper can be just pushed open without exceeding the maximum actuator force for pushing, without causing buckling and without exceeding the maximum allowed stress in the flexures; see design boundaries 3, 5, and 6 as listed in Sec. 5.1 This is further detailed in Appendix  B.

5.4.4 Resulting Design Parameters.

The optimized length of the proximal phalange is L1=68mm, and the distance d2=5.7mm. Values of the other parameters are given in Appendix  C. The flexures of the proximal hinge became relatively thick: 1.02–1.7 mm, where the thickness of the distal hinge is becoming as small as possible: 0.8 mm. This is explained in Sec. 5.5.

5.5 Evaluation of the Design

5.5.1 Evaluation of the Pull-Loose Force.

Figure 13 shows the relation between the actuation force and the pull-loose force of the optimized design. For the large object, the pull-loose force is 4 N if no actuation force is applied thanks to the elastic force in the gripper, see “A” in the figure. For the small object, the graph has two slope changes, see “B” and “C.” Between these positions, the object has only contact with the base and the proximal phalanges. For five-point contact, the actuation force needs to be higher than 21 N due to the actuation stiffness of the gripper. For the maximum allowed actuation force of 100 N, the pull-loose forces of both object sizes are the same. This is the result of optimizing the gripper for the minimal value of both. The pull-loose force of the optimized design is 21.4 N.

Fig. 13
Relation between actuation force and pull-loose force of the optimal design. Point A indicates the pull-loose force without actuation force. Points B and C show the positions at which contact starts with the proximal and distal phalanges, respectively.
Fig. 13
Relation between actuation force and pull-loose force of the optimal design. Point A indicates the pull-loose force without actuation force. Points B and C show the positions at which contact starts with the proximal and distal phalanges, respectively.
Close modal

5.5.2 Comparison Between the Optimized Design and Theoretical Performance.

The optimized length of the proximal phalange is L1=68mm. So, the ratio L1/rmax=1.7, which is only slightly lower than the value of 1.8 that was chosen based on the analytical results presented in Sec. 5.3, see the graphs shown in Fig. 11. The optimized distance d2=5.7mm. This is larger than the value expected from Eq. (8), which is 4.9 mm, mainly due to the neglected extra elastic forces in the analytical approach. In order to compare the resulting pull-loose force with the theoretical performance derived in Sec. 5.3, it should be noted that the force FT1 in the push–pull flexure near the proximal hinge, see Fig. 10(b), is about 0.58 times the actuation force, as the actuation force is divided over two push–pull flexures. In the optimized design, the resulting ratio Fc0/FT=21.4/(100×0.58)=0.37, which is higher than the value of 0.31 as shown in Fig. 11(a). This is because the elastic driving force in the gripper adds extra force to the large object.

5.5.3 Performance Metrics and Boundaries.

Table 4 presents the quantitative performance metrics of the final design. Table 5 presents the active constraints in three configurations. The constraint on the minimal thickness of the flexures is active at multiple locations by design. Figure 14 shows the von Mises stress for the two extreme configurations: fully open and fully closed. As already stated in Sec. 3.1, the stress in the flexures is typically dominated by the elastic deflection due to bending rather than loads in the supporting directions. Therefore, the stress in some of the three perpendicular parts of the push–pull flexure is quite high as these are subjected to significant bending. The stress in the four longitudinal parts of the push–pull flexure is limited as these do not have to bend significantly, even though they must transmit the full actuation force. The proximal hinge becomes as stiff as possible in the optimal design, which is limited by the maximum stress when the gripper is fully open. This is because it is advantageous to have extra elastic force on this hinge. In the push–pull flexure, the stress never reaches the maximum allowed value, so this flexure can withstand the required actuation forces. The design is not limited by buckling thanks to the reinforced and varied thickness of the flexures.

Fig. 14
Von Mises stress in the tendon and hinges of one finger in the open and in the closed configurations for an object with the minimal radius of 30 mm
Fig. 14
Von Mises stress in the tendon and hinges of one finger in the open and in the closed configurations for an object with the minimal radius of 30 mm
Close modal
Table 4

Performance metrics of the numerically optimized gripper

Objective/constraintValue
Range object sizes (radius)30 mm–40 mm
Maximum actuation force100 N
Width in open configuration133 mm
Pull-loose force21.4 N
Maximum contact force on object21.4 N
Objective/constraintValue
Range object sizes (radius)30 mm–40 mm
Maximum actuation force100 N
Width in open configuration133 mm
Pull-loose force21.4 N
Maximum contact force on object21.4 N
Table 5

Active constraints and the resulting values in the three extreme configurations

ConstraintOpen configurationClosed configuration of small objectClosed configuration of large object
Maximum stress, maximum 30 MPaActive, 30 MPaNot active, 21 MPaNot active, 16 MPa
Buckling, minimum safety factor 2Almost active, 2.1Not active, 6.9Not active, 4.7
Actuation force, maximum 100 NActive, 100 NActive, 100 N
Pushing force, maximum 40 NActive, 40 N
Five-point contact, Fc1/Fc2 larger than 0.1Not active, 0.56Active, 0.1
ConstraintOpen configurationClosed configuration of small objectClosed configuration of large object
Maximum stress, maximum 30 MPaActive, 30 MPaNot active, 21 MPaNot active, 16 MPa
Buckling, minimum safety factor 2Almost active, 2.1Not active, 6.9Not active, 4.7
Actuation force, maximum 100 NActive, 100 NActive, 100 N
Pushing force, maximum 40 NActive, 40 N
Five-point contact, Fc1/Fc2 larger than 0.1Not active, 0.56Active, 0.1

6 Design Validation

In this section, the gripper design is experimentally validated by measuring the pull-loose force for a range of object sizes and actuation forces. Section 6.1 presents the measurement setup, Sec. 6.2 describes the six types of measurement that have been performed, and Sec. 6.3 shows the results of these measurements. In Sec. 6.4, the PP-gripper is compared to the existing literature.

6.1 Measurement Setup.

The measurement setup is designed to supply a measurable actuation force at the gripper and a measurable pulling force on the object. To detect pull-loose, the displacement between the gripper and the object is measured.

Figure 15 illustrates the gripper design, and STEP-files of the gripper are available in the Supplemental Material on the ASME Digital Collection. The gripper has been designed as a monolithic part, with the push–pull flexure going through the base and the proximal phalange. The version that has been used in the measurement setup is partly made from the transparent material for demonstration purposes. To avoid peak stresses, fillets with a radius of 1 mm were added to the positions where the flexural parts are connected to the reinforcements or to other parts. The length of all reinforcements was decreased by 1 mm to compensate for these fillets. A flexure-based linear guide with a handle was added, which has an actuation stiffness of approximately 600 N/m. Figure 16 shows different stages of the grasping process.

Fig. 15
Final gripper design: 3D view and 2D cross-sectional view in which the end-stops and several parts around the handle are hidden
Fig. 15
Final gripper design: 3D view and 2D cross-sectional view in which the end-stops and several parts around the handle are hidden
Close modal
Fig. 16
Grasping process. From the undeformed state (1), the gripper opens (2), after which it moves to the object (3, 4) and closes (5, 6).
Fig. 16
Grasping process. From the undeformed state (1), the gripper opens (2), after which it moves to the object (3, 4) and closes (5, 6).
Close modal

Figure 17 shows the measurement setup. The gripper was mounted upside down on the frame. Actuation force was applied using weights with a known mass, transmitted through a string guided by two pulleys. A circular object, with either a minimum object radius of 30 mm or a maximum object radius of 40 mm, was placed in the gripper. The external pull force was applied to the object using weights with a known mass. The vertical displacement between the object and the gripper was measured by a laser triangulation displacement sensor (optoNCDT ILD1402-50) that was mounted to the gripper, and the distance to a block attached to the object was measured.

Fig. 17
Measurement setup
Fig. 17
Measurement setup
Close modal

6.2 Measurements.

Six types of measurements were performed; the first three to characterize the test setup itself and the second three to validate the gripper design.

  1. The friction of the contact between the object and the gripper was measured. The smallest object was placed on the gripper. A mass of about 1 kg was added to the object to get a representative contact force. It was determined for which slope of the contact face the object just started to slip. The friction coefficient can be determined based on this angle.

  2. The effect of friction on the pulleys was measured. This was investigated under pretension by determining the difference in mass needed to exceed the friction force of the pulleys. Therefore, equal weights were attached to each end of the string that is guided by the two pulleys of the measurement setup. On one side, mass was added until the unbalance was large enough to exceed the friction force. This is performed once with a mass of about 0.5 kg on both sides and once with a mass of about 2.7 kg on both sides.

  3. The creep in the PA 2200 gripper without an object was measured. This was measured in the experimental setup by applying weights on the grip force side and measuring the resulting vertical displacement of the handle. No object was placed inside the gripper so that it could open and close freely and only the driving stiffness of the gripper was measured. First, a weight of 19.89 N was applied for several hours to eliminate any creep effects from the previous load changes. The measurement started with the 19.89 N load, and after 20 s, the load reduced to 4.98 N. The displacement of the handle was measured using the laser displacement sensor with the laser pointing at the handle.

  4. To quantify pull-loose force for the smallest object (radius of 30 mm), the object is pulled out of the gripper while measuring the displacement and force. The actuation force was held at a constant value (24.9 N and 39.8 N). The measurements are then compared to simulations that use creep and friction correction with the values obtained in experiments 1 and 3.

  5. For the largest object (radius of 40 mm) such an evaluation is not possible as the pull-loose force is equal to the pull-out force, such that the object instantaneously moves out of the hand if the pull-loose force is exceeded. Instead, the pull-loose force has been determined for actuation forces ranging from 5 to 70 N.

  6. Sixteen tomatoes were gripped to detect if the gripping resulted in visual damage to the tomatoes. An actuation force of either 34.8 N or 69.6 N was used. In half of the cases, the tomato was placed in the center of the gripper, and in the other cases, the tomato was on purpose misaligned, see Fig. 18. Two tomatoes were not gripped at all. The tomatoes were visually checked, just before the gripping, directly after the gripping, and 7 days after the gripping experiment.

Fig. 18
In experiment 6, the tomatoes are placed in the center (b) or highly misaligned (c)
Fig. 18
In experiment 6, the tomatoes are placed in the center (b) or highly misaligned (c)
Close modal

6.3 Results

6.3.1 Measurement 1: Contact Friction.

The friction coefficient between the test object and the gripper has been measured to be 0.10. This friction coefficient is used in the simulations that validate the experimental data presented in the following sections.

6.3.2 Measurement 2: Pulley Friction.

If a mass of about 0.5 kg is applied on the gripper, a difference of 9 g results in the motion of the string with weights. For a mass of about 2.7 kg, this difference is 25 g. This means that the effect of pulley friction on the contact force is less than 1.8%, and it is therefore neglected in the analysis of the results of experiments 4 and 5.

6.3.3 Measurement 3: Creep in the Gripper.

Figure 19 shows the resulting creep of the actuator handle as a function of time. The displacement after 1 h is 3.05 mm, and this is 14% higher than the displacement expected from the simulation model. To compensate for this creep, the Young's modulus of the material in the simulations that validate the experimental data has been reduced from 1500 MPa to 1320 MPa. Note that this modification in the simulation also compensates for other uncertainties in the material properties and for manufacturing tolerances on the thickness of the flexures.

Fig. 19
Creep measurement of the gripper after applying a changing actuator force at 20 s
Fig. 19
Creep measurement of the gripper after applying a changing actuator force at 20 s
Close modal

6.3.4 Measurement 4: Displacement of the Smallest Object.

Figure 20(a) shows the experimental results for the smallest object. The relation between the pulling force on the object and the displacement of the object is given, for two values of the actuation force. This relation is shown for displacing the object out and into the gripper, for which the relation differs due to the friction force direction change between the object and the gripper. Directly above the zero displacement, the object loses contact with the base of the gripper, and this is the pull-loose force. The sudden change of the slope in the graphs (between 1 and 2 mm displacement) is where the contact with both proximal phalanges is lost. The measurement with the largest error with respect to the simulation is indicated in the graph, showing that the measured forces can be predicted by the model within 91%.

Fig. 20
Results of a small object, and the largest error is 9%
Fig. 20
Results of a small object, and the largest error is 9%
Close modal

6.3.5 Measurement 5: Pulling Force on the Largest Object.

Figure 21 shows the relation between the actuation force and the pull-loose force of the largest object. The pull-loose force is slightly lower than expected from simulations, but it shows the expected behavior, and the differences are within 15%.

Fig. 21
Results of a large object, and the largest difference is 15%
Fig. 21
Results of a large object, and the largest difference is 15%
Close modal

6.3.6 Measurement 6: Fruit Damage.

Table 6 shows that small visual defects were observed on 14 of the 18 tomatoes, even before the experiment started. These defects typically slightly worsened during the 7 days after the experiment. In only one case, the gripping caused a small incision because of the gripper’s sharp edge cutting into a highly misaligned tomato. In any further version of the gripper, this defect can be probably prevented by adding small radii to all sharp edges and a soft interaction material. The first image in Fig. 22 shows this defect, and the other two images show the worst other defects. Based on these results, it can be concluded that tomatoes can be firmly gripped with the PP-gripper if they are not highly misaligned.

Fig. 22
The visual damage on the tomatoes after 7 days. Only the damage shown in (a) is caused by a highly misaligned gripper. (a) Due to gripping and (b) on the onset of the experiment.
Fig. 22
The visual damage on the tomatoes after 7 days. Only the damage shown in (a) is caused by a highly misaligned gripper. (a) Due to gripping and (b) on the onset of the experiment.
Close modal
Table 6

Number of tomatoes without damage, with damage, and with damage that is specifically due to the gripping

CaseNo damageDamage, not due to grippingDamage due to gripping
Not gripped (reference)02
Actuation force: 34.8 N, tomato in center220
Actuation force: 34.8 N, tomato misaligned220
Actuation force: 69.6 N, tomato in center040
Actuation force: 69.6 N, tomato misaligned031
CaseNo damageDamage, not due to grippingDamage due to gripping
Not gripped (reference)02
Actuation force: 34.8 N, tomato in center220
Actuation force: 34.8 N, tomato misaligned220
Actuation force: 69.6 N, tomato in center040
Actuation force: 69.6 N, tomato misaligned031

6.4 Comparison to Existing Gripers.

In this section, the PP-gripper is compared to several other flexure-based grippers from the literature, Fig. 23 visualizes these grippers. Three aspects are compared: the range of object sizes that the gripper can grasp, the width of the open gripper, and the pull-out force. As shown by the theoretical model, the PP-gripper can pick objects with a radius from 30 to 40 mm, it is 133 mm wide, and the pull-loose force is 21.4 N, based on a maximum actuation force of 100 N. The physical model that is used in the experiments is slightly wider: 140 mm. The experiments showed that the pull-out force is comparable to the pull-loose force in the theoretical model.

Fig. 23
Grippers that are used as comparison for the PP-gripper. Figures are obtained from Refs. [11,33–35]. (a) Lumped compliance gripper with notch flexures, (b) lumped compliance gripper with metal leaf springs, (c) soft-robotics fin-ray gripper, and (d) improved soft-robotics fin-ray gripper.
Fig. 23
Grippers that are used as comparison for the PP-gripper. Figures are obtained from Refs. [11,33–35]. (a) Lumped compliance gripper with notch flexures, (b) lumped compliance gripper with metal leaf springs, (c) soft-robotics fin-ray gripper, and (d) improved soft-robotics fin-ray gripper.
Close modal

The first two grippers that are used for comparison are lumped compliance grippers. Gripper (a) [11] is a gripper with notch flexures. It is about 200 mm wide and suitable for two-point grasping and four-point enveloped grasping. The latter grasping type is experimentally tested for objects with radii from 27.5 to 40 mm. Using an actuation force of 6.04 N per finger, the sum of the contact forces on one finger is 8.7 N. Gripper (b) [33] is a cable-driven gripper with metal leaf springs as joints. Based on a theoretical model, which was experimentally validated, this gripper was optimized to grasp objects with a radius of 20 mm. The width of this gripper is 85 mm, and the maximum pull-out force is 2.7 N. Compared to these two grippers, the PP-gripper can handle a significantly higher actuation force, resulting in a significantly higher pull-out force. Another advantage of the PP-gripper is that it is relatively 35% less wide.

The second two grippers used for comparison are soft-robotics grippers, based on the fin-ray lay-out, which have the disadvantage that they are less force transparent than lumped compliance grippers like the PP-gripper. Gripper (c) is the Festo fin-ray gripper with a size 80 [34]. For this gripper, data are provided for object radii from 20 to 40 mm, showing a pull-out force of 13.5 N based on an actuation force of 160 N. It is about 100 mm wide. Gripper (d) is an improved version of this fin-ray gripper [35]. It can grasp objects with a radius from 2.5 to 57.5 mm. The pull-out force is 50 N, which is measured for only one object with a radius of 42.5 mm that was placed between one finger and a human hand, see Fig. 23. When comparing these two grippers, the PP-gripper can provide a comparable pull-out force, but can grasp a smaller range of object sizes.

7 Conclusions

A compact, underactuated, force-transparent, fully flexure-based gripper has been presented. This PP-gripper is produced from a single 3D-printed part, eliminating the need for lubrication and enhancing its hygiene and suitability for dirty environments.

The actuation force is transmitted by an integrated push–pull flexure strategically positioned on the object side of the joints for a small footprint. This configuration ensures tension loading on this flexure when the gripper closes. The flexure can therefore transmit a substantial actuation force without the risk of flexure buckling, thus establishing a firm grip on the object.

Reinforced flexures with varying flexure thicknesses are incorporated to enhance performance. To keep the number of design parameters small, the optimal thickness variation is derived analytically. The derived result is also applicable for reinforced flexures in other applications.

The gripper features four novel cartwheel-style flexure hinges. By reinforcing the flexures and allowing a varying thickness, the support stiffness is increased by a factor of 2.5 and the buckling load is 1.5 times higher, compared to conventional cartwheels with the same size and comparable actuation stiffness.

A design approach is outlined to derive a gripper geometry, which maximizes the pull-loose force across a range of object sizes, such that an object is gripped firmly, and fast accelerations are achievable without losing the object. It shows that this pull-loose force mainly depends on the total gripper width and the actuation force. The length of the proximal phalanges is a trade-off between the pull-loose force and the required hinge rotations. The flexures and two of the geometrical design parameters are optimized through numerical techniques. It shows that the flexure stiffness positively affects the pull-loose force.

An optimized gripper design is presented that can grip round objects with radii ranging from 30 to 40 mm. The width of the open gripper is 133 mm. It achieves a pull-loose force of 21.4 N, with a maximum actuation force of 100 N. The gripper is experimentally validated. The creep in the material is quantified, and the correspondence between the measured pull-out force and the simulations is over 85%.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Appendix A: Geometric and Static Relations

In this appendix, three types of relations are derived. In Geometric Relations for Closed Gripper section, the joint angles ϕ1 and ϕ2 and the contact positions D1 and D2 are derived for the closed gripper. The angles ϕ1 and ϕ2 are derived for an open gripper in Geometric Relations for Open Gripper section. In Static Relations section, the static force equilibrium is derived for a closed gripper. All these equations are used to reduce the number of design parameters (Sec. 5.2) and obtain the theoretical performance of the gripper (Sec. 5.3).

Geometric Relations for Closed Gripper

Figure 24(a) shows half of the gripper. Based on the two dotted two triangles, the relations for D1 and ϕ1closed can be derived as function of the design parameters and object radius.

Fig. 24
Geometric relations of the hinge angles and the positions of the contacts for (a) the proximal phalange and (b) the distal phalange
Fig. 24
Geometric relations of the hinge angles and the positions of the contacts for (a) the proximal phalange and (b) the distal phalange
Close modal
The variable T1, which measures the distance between the center of the object and the proximal hinge, can be derived using the first triangle:
(A1)
The line with a length r+s1 goes through the contact point between the proximal phalange and the object. The distance D1 measures perpendicular to this line and can be derived using the second triangle:
(A2)
The angle ϕ1closed is the sum of the two smaller triangles:
(A3)
Figure 24(b) shows two triangles based on three lines through the center of the object, and the middle line defines the distance between the center of the object and the distal hinge. The other two lines go through the contact points. The distances L1D1 and D2 measure the distance between the distal hinge and these two lines. The distance D2 is expressed as follows:
(A4)
The angle ϕ2closed is the sum of the two smaller angles:
(A5)

Geometric Relations for Open Gripper

As stated in Sec. 5.1, the open gripper configuration should meet two requirements. In the first place, the largest object should fit between both tips, see Fig. 25. The horizontal distance between the center of the gripper and the tip of the fingers is computed as a function of both angles ϕ1open and ϕ2open:
(A6)
Fig. 25
Geometric parameters for the open configuration, in which the largest object just fits between the fingertips
Fig. 25
Geometric parameters for the open configuration, in which the largest object just fits between the fingertips
Close modal
The second requirement is that the angle γ in Fig. 25 should be 25 deg. Therefore, the sum of both open angles is expressed as follows:
(A7)
By substituting Eq. (A7) into Eq. (A6), the first angle can be derived as follows:
(A8)
The second angle is then computed based on this angle:
(A9)

Static Relations

Figure 26 shows three situations from which three equilibrium equations are derived. By considering the equilibrium of moments on the distal phalange around the distal hinge (Fig. 26(a)), a relation between the pulling force in the push–pull flexure, FT, and contact force, Fc2, can be derived:
(A10)
Fig. 26
Static relations
By considering the equilibrium of moment around the proximal hinge for a whole finger (Fig. 26(b)), a relation between the contact forces and the pulling force is obtained
(A11)
By substituting Eq. (A10) into this equation to omit Fc2, the contact force Fc1 can be expressed in terms of FT:
(A12)
By considering the vertical equilibrium on the object, see Fig. 26(c), the contact force to the base is obtained as follows:
(A13)
By substituting Eqs. (A10) and (A12) into Eq. (A13), this contact force is expressed in terms of the actuation force:
(A14)

Appendix B: Details of the Cost Function

This appendix presents an overview of the steps that are executed in the cost function of the numerical optimization in Sec. 5.4. The design parameters of the gripper are the length of the proximal phalange (L1), the distance d2, and 15 parameters that define the reinforcement lengths and the thickness of the flexures.

  1. The seven geometrical parameters that are no design parameters are computed based on the steps in Sec. 5.2, see Table 3. Because the length L1 is the input of the cost function, also the length L2 can be computed in step 7 of this approach.

  2. The hinge angles of the undeformed configuration are obtained using the following substeps:

    1. The hinge angles of the open configuration are computed using Eqs. (A8) and (A9). The hinge angles for the closed configuration with the minimal object size are computed using Eqs. (A3) and (A5).

    2. A preliminary design is defined in which the hinge angles are exactly between these open and closed values.

    3. In the first simulation, a small pushing force (0.1 N) is applied, to slightly open the gripper of this preliminary design.

    4. Using a linear extrapolation of the result, the maximum possible pushing force to open the gripper is estimated, based on the maximum stress, the buckling load, and maximum allowed pushing force.

    5. Using the results of step 2.3, it is linearly estimated how far the hinges will rotate using this maximum possible pushing force: ϕ1pushmax and ϕ2pushmax.

    6. The angles in the undeformed configuration are defined to be as close as possible to the closed configuration using the following equation:
      (B1)

  3. In a second simulation, with the new undeformed configuration, the gripper is pushed open using the approximated force from step 2.4.

  4. For the minimal and maximum object size, the following steps are applied:

    1. The gripper is closed in 50 load steps, starting from the open configuration of step 3 and ending with the maximum allowed pulling force of 50 N.

    2. The last load step that satisfies all constraints is determined: i.e., the constraints on maximum stress, buckling load, and the minimal ratio between the contact forces: Fc1/Fc2>0.1. At this load step, the contact force to the base, Fc0, is evaluated.

  5. The cost is defined as follows:

    1. If the gripper can be closed for the minimal and maximal object size (which means that the base and both phalanges make contact for both object sizes), the contact force Fc0 is computed. The cost is defined as the inverse of the lowest of both contact forces.

    2. If the gripper cannot completely close under the constraints, the cost is defined such that the optimization is guided in the right direction.

In detail, the cost function consists of a set of five functions of which the highest applicable value is the final cost:
(B2)
where
  • dmin is the minimal distance in meters between the phalanges and the object. It is determined for the smallest and the largest object, and the maximum of both cases is chosen in max(dmin).

  • Fc1c2min is the minimal contact force in Newton between the phalanges and the object. It is determined for the smallest and the largest object, and the maximum of both cases is chosen in max(Fc1c2min).

  • σmax is the maximum stress in the simulation in megapascal, and σalw is the allowed stress, i.e., 30 MPa.

  • λmin is the minimal load buckling factor in the simulations, and λalw is the allowed value of 2.

  • Fc0 is the contact force between the base and the object in Newton. It is determined for the smallest and the largest object and the minimum of both cases is chosen in min(Fc0).

Appendix C: Optimization Results

This appendix gives the detailed results of the numerical optimization. Seventeen parameters are optimized using a genetic algorithm. The optimum is reached in 463 generations of each 12 function evaluations. The total computation time is 13.5 h using 12 parallel cores on a 2.19 GHz Intel processor. Besides this, five similar optimizations have been performed with slight variations of the input, all with comparable results. This indicates that the global optimum has been found each time.

Table 7 shows the optimized values of the design parameters, including the upper and lower bounds. The thickness distribution of each of the seven parts of the push–pull tendon can be described with six parameters, as shown in Fig. 27. To limit the number of design parameters, several dimensions are optimized by the same design parameter. Some thicknesses were given the minimal allowed value. Other thicknesses are obtained using the analytical approach that is detailed in Sec. 3. The resulting dimensions are given in Table 8. These analytical derived values are obtained by substituting the right positions in Eq. (5), resulting in the following equation:
(B3)
Fig. 27
Overview of the parameters affected by the numerical optimization and some reference lengths. Each of the seven parts of the push–pull flexure is described by six parameters, and the arrows near the part numbers indicate the direction in which they are defined.
Fig. 27
Overview of the parameters affected by the numerical optimization and some reference lengths. Each of the seven parts of the push–pull flexure is described by six parameters, and the arrows near the part numbers indicate the direction in which they are defined.
Close modal
Table 7

Design parameters: optimized value and limits

DescriptionOptimized valueLower boundUpper bound
Geometric parametersL1/(rmax+s1)1.221.11.3
d1/d23.942.54.5
Push–pull flexuresf1 and f3, ii–iv0.180.10.3
f1, v0.200.20.31
f3, v0.380.20.6
f1 and f3, vi0.260.20.45
f1, vii0.250.20.45
f3, vii0.220.20.45
tA,tB,tC,tD, i1.320.81.5
tB, tC, ii–iv0.800.81.5
tA, tD, ii–iv0.800.81.5
Proximal cartwheelf10.500.10.6
tB1.020.81.5
tA/tB1.7112
Distal cartwheelf10.590.10.6
tB0.800.81
tA/tB1.0012
DescriptionOptimized valueLower boundUpper bound
Geometric parametersL1/(rmax+s1)1.221.11.3
d1/d23.942.54.5
Push–pull flexuresf1 and f3, ii–iv0.180.10.3
f1, v0.200.20.31
f3, v0.380.20.6
f1 and f3, vi0.260.20.45
f1, vii0.250.20.45
f3, vii0.220.20.45
tA,tB,tC,tD, i1.320.81.5
tB, tC, ii–iv0.800.81.5
tA, tD, ii–iv0.800.81.5
Proximal cartwheelf10.500.10.6
tB1.020.81.5
tA/tB1.7112
Distal cartwheelf10.590.10.6
tB0.800.81
tA/tB1.0012

Note: Thicknesses are given in mm, and other values are dimensionless.

Table 8

Thickness of tendons that are not explicitly optimized

ParameterPartValue (mm)Based on
tB, tCv–vii0.80Minimal value
tAv1.12Analytical
tAvi1.15Analytical
tAvii1.11Analytical
tDV1.35Analytical
tDvi1.15Analytical
tDvii1.08Analytical
ParameterPartValue (mm)Based on
tB, tCv–vii0.80Minimal value
tAv1.12Analytical
tAvi1.15Analytical
tAvii1.11Analytical
tDV1.35Analytical
tDvi1.15Analytical
tDvii1.08Analytical

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Supplementary data