Design of an Underactuated, Flexure-Based Gripper, Actuated Through a Push–Pull Flexure

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. [8–10].

Another effective design choice is to use a flexure-based gripper [6,11–16], 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 [11–13], 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 [20–23]. 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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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

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.

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.

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.

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

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.

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.

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.

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 $s3$⁠.

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.

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).

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, $FT1=FT2=FT$⁠.

• 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

  $L2=D2+0.25rmax$

  (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).

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.

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.

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.

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 $L2=31.6mm$⁠.

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 $20deg$⁠.

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.

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].

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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%.

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%.

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.

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.

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.

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%.

There are no conflicts of interest.

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

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).

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.

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:

      $ϕiundef=min(ϕiclosed,ϕiopen−ϕipushmax),i=1,2$

      (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.

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.