A Robot Gripper with Differential and Hoecken Linkages for Straight Parallel Pinch and Self-Adaptive Grasp

Abstract

Parallel pinch is an important grasp method. The end phalanx of the traditional parallel pinch and self-adaptive gripper moves in an arc trajectory, which requires the auxiliary lifting motion of the industrial manipulator, which is inconvenient to use. To solve this problem, a novel robot finger is designed and implemented—Hoecken’s finger. In this finger, the Hoecken linkage mechanism is used to realize the straight-line trajectory of the end joint, the differential mechanism set on the surface of the phalanxes is used to realize the shape self-adaptation of the first and second phalanxes, and the parallel four-bar linkage in series is used to realize the attitude keeping, thus comprehensively realizing the underactuated gripper driven by a single motor. After analyzing the grasp force and grasp motion of Hoecken’s fingers, the optimized parameters are obtained, and the Hoecken’s gripper is developed. The experimental results show that the gripper can realize the self-adaptive grasp function of straight parallel pinch, the grasp is stable, and the grasp range is large. It can be applied to more scenes that need to grasp objects.

1. Introduction

Robots are widely used in society. In the industrial field, many tasks in factories, such as welding, assembly, and painting, are completed by robots [1], because if these tasks are completed by humans, not only will a lot of time be consumed but, also, the tasks would have lower efficiency and accuracy than with robots. In other industries, such as the medical field, robots are gradually becoming indispensable due to their high precision and precise control of force application [2].

As the executive device at the end of robots, robot hands play an important role in human–computer interaction, robots, and high-risk environment operations. Robot hands can be divided into industrial grippers, dexterous hands, and soft robot hands.

The dexterous hand is the closest to the human hand. It usually refers to a robot hand that is driven independently by all joints, combined with a variety of sensor systems, and can achieve dexterous operation. It has the advantages of multiple degrees of freedom and high flexibility. For example, the Shadow hand [3], Gifu series hands [4], the coupling adaptive hand [5], the Robonaut hand developed by NASA [6], and the UTAH/MIT dexterous hand jointly developed by the University of Utah and MIT [7]. Combining them with deep neural networks is the latest application for dexterous hands [8]. Because of too much freedom in active drive, the dexterous hand not only has small grasp force but is also extremely difficult to control.

The traditional industrial gripper has a simple structure, few degrees of freedom, and good grasp stability. The traditional industrial grippers include the straight parallel pinch and self-adaptive hand designed and developed by Robotiq Company of Canada [9], the food handling gripper of Tokyo Institute of Technology [10], the industrial gripper developed by SMC Company of Italy [11], the rod array gripper developed by foreign scholars [12], the rod cluster hand developed by Tsinghua University [13], and the spherical hand [14]. When this kind of gripper is working, its end will be pinched along the arc trajectory, which makes this kind of gripper not particularly suitable for pinching objects on the parallel sheet.

The soft robot hand is a kind of robot hand that uses soft materials or uses the characteristics of the mechanism to make the executive device able to grasp freely within a certain range. Soft robot hands include the universal gripper for particle material extrusion from the University of Chicago [15] and MIT’s vacuum-driven origami soft robot hand [16]. The soft robot hand has a disadvantage: it cannot bear a very high load.

In addition to the above robot hands, some scholars have put forward the concept of an underactuated hand, that is, the number of actuators of the mechanism is less than the number of joint degrees of freedom it drives. The currently available grasping modes for underactuated manual operation include straight parallel pinch mode, coupled gripping mode, and self-adaptive grasping mode [17]. In a broad sense, all robot hands with fewer actuators than the number of joint degrees of freedom should be included in the category of underactuated robot hands, for example, the direct-driven, optimized, and light-mass hand exoskeleton [18,19] and the remote control dexterous hand designed by Malta University [20]. This kind of three-joint finger can use the rope drive mechanism to realize the use of a few motors to drive the multi-degrees-of-freedom fingers to grasp the action; Laval University of Canada earlier proposed an efficient linkage underactuated finger, the SARAH hand [21]. The finger of the hand has three phalanx and three joint degrees of freedom, which can achieve flexible and diverse mechanical grasp functions under the drive of a small number of drivers. The linkage mechanism used by this finger can provide great grasp force. Korean scholars have developed a straight parallel and self-adaptive hand with an arc-shaped sliding rod [22]. Meanwhile, some scholars have analyzed the grasping mode itself [23].

Based on the advantages and disadvantages of the above devices, the members of Kawasaki Heavy Industries of Japan designed a multi-linkage straight parallel pinch underactuated finger using the Chebyshev linkage mechanism [24]. However, due to the lack of traditional straight translation constraints, such as straight guides, the mechanism is relatively complex, and the finger has no adaptive envelope grasp function. Some scholars have proposed a straight parallel pinch and self-adaptive robot finger with Hoecken linkages and elastic joints [25], but the self-adaptive grasp range of this mechanism is limited, and the device contains the problem of driving force caused by pressure angle changes.

After comprehensive consideration of the above problems, a simple self-adaptive finger scheme of straight parallel pinch is proposed; that is, the combination of the Hoecken linkage mechanism and differential mechanism is used to realize the functions of straight parallel pinch and wide-range adaptive envelope grasp. It has the characteristics of compact structure, reliable grasp, and overcoming the motion dead zone caused by pressure angle. The feasibility of the design is verified via kinematic analysis and mechanical analysis. According to the analysis and design, a prototype is developed, and it is verified that the proposed scheme can meet the daily application.

This paper first introduces the working principle of the Hoecken finger in Section 2. The design of the Hoecken gripper is presented in Section 3. In Section 4 and Section 5, motion analysis, grasp force, and grasp range analysis of the Hoecken Finger are presented. A Hoecken gripper grasp experiment and the results are introduced in Section 6. Finally, Section 7 includes the conclusion and future work.

2. Working Principle of the Hoecken Finger

Hoecken’s finger work is divided into two stages: straight parallel pinch stage and self-adaptive grasp stage. The concepts of two grasping stages are shown in Figure 1. During movement, the Hoecken gripper first enters the straight parallel pinch stage and then transitions to the self-adaptive grasping stage.

2.1. Straight Parallel Pinch Stage

When carrying out straight parallel pinch, the motor rotates and drives the sixth shaft to rotate through the transmission mechanism, the sixth shaft drives the third linkage to rotate, and the third linkage drives the fourth linkage to move in the opposite direction of the object through the seventh shaft. The first linkage, the second linkage, the fourth linkage, the first phalanx, the roller, and the torsion spring comprehensively realize that when the second phalanx moves in the direction of the object, the second phalanx will only move in a horizontal straight line.

At this time, the roller moves in the straight-line slot, and the first linkage rotates around the first shaft. Since the first linkage, the third linkage, the fourth linkage, and the base form a parallelogram mechanism, the fourth linkage is parallel to AF. Due to the action of the torsion spring, the second link, and the first phalanx, the fourth link and the second phalanx form a parallel four-bar mechanism, so the second phalanx is parallel to the fourth link as a whole, and the fourth link is parallel to AF; that is, CE is parallel to AF, so the second phalanx maintains the same attitude relative to the base. The first linkage, the second linkage, the roller, and the base together constitute the Hoecken linkage mechanism. The third shaft moves to the object in a straight line in space, CE moves to the object in a horizontal straight line, and the second phalanx moves to the object in a horizontal straight line.

This process is called the straight parallel pinch movement process, as shown in Figure 2.

2.2. Self-Adaptive Grasp Stage

In the process of straight-line parallel pinch movement, when a unit in the first phalanx first contacts the object and is blocked by the object and cannot continue to move, the motor will continue to rotate and drive the third link to rotate through the transmission mechanism, thus driving the fourth link, the first link, and the second link to rotate, and then driving the first phalanx and the second phalanx. The first phalanx is subject to the action of the second phalanx, the fourth linkage, and the object. The torsion spring connecting the two units of the first phalanx is opened, and the two units are toppled and enveloped in the direction of the object. The third shaft pushes the push block, and the push block compresses the spring compression part after being stressed. The spring compression part exerts force on the second phalanx, and the second phalanx rotates and tilts around the object in the fifth shaft. Finally, the first unit of the first phalanx, the second unit, and the second phalanx jointly achieve the envelope of the object, forming a stable grasp effect.

This envelope grasp process has adaptive characteristics for objects of different shapes and sizes. This grasp process is called self-adaptive grasp, as shown in Figure 2.

3. Design of the Hoecken Gripper

The traditional underactuated robot hand does not have the function of straight parallel pinch, which makes it necessary for the robot hand to rely on the lifting motion of the industrial mechanical arm when holding a thin object on the desktop, and the difficulty of grasp and controlling objects of different sizes is greatly increased.

To solve this problem, based on the Hoecken linkage mechanism, this paper proposes and implements a new type of robot gripper—the Hoecken gripper (as shown in Figure 3). Some parameters of the Hoecken gripper are shown in

Table 1. Parameters of the Hoecken gripper.

Parameter	Number
Number of motors	2
Degrees of freedom	4
Grasp range of parallel pinch/cm	14
Self-adaptive grasp range/cm	6
Output torque/Nm	0.5

.

3.1. Overall Composition

The self-adaptive underactuated robot gripper designed in this paper includes a base, a power input mechanism, the first and second unit of the first phalanx, the second phalanx, the near roller shaft, the roller shaft, first to fourth linkage, 1first to seventh shaft, first to second transmission shaft, the torsion spring, the compression spring, and the push block (as shown in Figure 4).

3.2. Connection

The motor is fixed to the base; the output end of the power input mechanism is connected to the input end of the power transmission mechanism; the output end of the transmission mechanism is connected to the sixth shaft; the first shaft, the sixth shaft, and the near roller shaft are sleeved in the base; the second shaft is sleeved in the first linkage; the third shaft is sleeved in the second linkage; the fourth shaft is sleeved in the fourth linkage; the fifth shaft is sleeved in the second unit of the first phalanx; the seventh shaft is sleeved in the third linkage; the roller shaft is sleeved in the first unit of the first phalanx.

The first linkage is movably sleeved on the first shaft. The two ends of the second linkage are movably sleeved on the second shaft and the third shaft, respectively. The third linkage is movably sleeved on the sixth shaft. The two ends of the fourth linkage are movably sleeved on the fourth shaft and the seventh shaft, respectively; the two ends of the first unit of the first phalanx are movably sleeved on the fourth shaft and roller shaft, respectively. The two ends of the second unit of the first phalanx are movably sleeved on the roller shaft and the fifth shaft, respectively. The two units of the first phalanx are connected by torsion springs.

The second linkage has a straight-line slot. The roller is sleeved on the near roller shaft and slides in the straight-line slot. The second phalanx and the third shaft are connected by push blocks and spring compression.

3.3. Parameter Relation

Set the center points from the first shaft to the seventh shaft and the near roller shaft as A, B, C, D, E, F, G, and H;

The lengths of AB and FG, AF and BG, BC and DE, and BD and CE are equal;

The length of AH and BC is 1.5-times and 6-times the length of AB, respectively;

The center line of the straight-line slot coincides with the line phalanx BC;

Point B and point C are located on both sides of the line AF.

3.4. Power Input Mechanism

The power input mechanism includes motor, reducer, worm, and worm gear. The motor provides torque. The reducer reduces the output speed through gear transmission and increases the torque obtained from the motor. The increased torque is transmitted to the worm and worm gear to provide power for the transmission mechanism. The power is supplied by the input mechanism to the first transmission shaft in the transmission mechanism.

The connection relationship is as follows: the motor is fixed to the base; the output shaft of the motor is connected to the input end of the reducer; the worm is sleeved on the output shaft of the reducer; the worm gear is meshed with the worm; the worm gear is sleeved on the first transmission shaft.

3.5. Power Transmission Mechanism

The power transmission mechanism comprises a first transmission shaft (number 6 in Figure 4) and a second transmission shaft (number 7 in Figure 4) as shown in Figure 5. The first transmission shaft transmits power to the second transmission shaft through gear transmission, and the second transmission shaft also transmits power to the sixth shaft through gear transmission, thus driving the whole mechanism to move. Through the force transmission of the first transmission shaft, the second transmission shaft, and the parallel four linkages, the power transmitted to the second phalanx always has a small pressure angle, so the transmission efficiency is high and the movement stability is good.

The connection relationship is that the first drive shaft and the second drive shaft are sleeved on the base; the worm gear is sleeved on the first transmission shaft; the first transmission shaft is sleeved with a gear, the second transmission shaft is sleeved with a gear, and the two gears are meshed; the second transmission shaft is sleeved with another gear, the sixth shaft is sleeved with a gear, and the two gears are meshed; the sixth shaft is sleeved in the third linkage.

3.6. Elastic Member

The elastic member includes a torsion spring connecting the two units of the first phalanx and a compression spring connecting the push block and the second phalanx. The main parameter is their stiffness. The determination of the stiffness of two springs follows certain principles.

For the torsion spring, if the stiffness is too large, it may result in the second unit of the first phalanx not being able to effectively tilt towards the object during the self-adaptive grasp stage, thereby failing to achieve the effect of expanding the grasp range of self-adaptive grasp. If the stiffness is too small, the torque provided by the torsion spring may be insufficient, leading to the second unit tilting before it should. For the compression spring, if the stiffness is too large, the push block cannot be pushed to the designated position, while if the stiffness is too small, the push block may be pushed in advance. These situations will greatly reduce the grasp effect of the gripper, so the stiffness of the two springs should be chosen to be moderate.

After theoretical analysis and experimental screening—using different stiffness torsion springs and compression springs for replacement experiments—it was ultimately determined that the more suitable torsion spring stiffness was chosen as 0.02 Nm/deg, and the compression spring stiffness was chosen as 50 N/m.

4. Motion Analysis of the Hoecken Finger

As the core mechanism of this paper, the end of the Hoecken linkage mechanism will move in a straight line under the drive of the mechanism. In order to verify the kinematic characteristics of the mechanism, we use the kinematics software (MATLAB R2021a) to model and simulate the mechanism.

In the simplified kinematics model of the finger mechanism, the length of BH, CH, and BC meets the following relationship:

$$l_{BH}+l_{CH}=BC$$

(1)

Let the included angle between BC and negative x-axis be α, the included angle between AB and the positive direction of the x-axis is θ, the length of AB is 2a, the length of AG is 3a, the length of BC is 12a, the length of BH is x, the abscissa of point C in the coordinate system is x_c, and the ordinate of point C in the coordinate system is y_c. According to the geometric relationship, there is an angular relationship between α and θ

$$\frac{3a+2a\mathrm{s}\mathrm{i}\mathrm{n}\theta }{x}=\mathrm{s}\mathrm{i}\mathrm{n}\alpha$$

(2)

The corresponding parameters can be obtained by using geometric relations:

$$\left\{\begin{array}{c}{(13+12\mathrm{s}\mathrm{i}\mathrm{n}\theta )}^{\frac{1}{2}}a=x(\theta )\\ [12-{\left(13+12\mathrm{s}\mathrm{i}\mathrm{n}\theta \right)}^{\frac{1}{2}}]\cos \alpha a=x_c(\theta )\\ (6sin\alpha -sin\theta )2a=y_c\left(\theta \right)\end{array}\right.$$

(3)

Calculate the first and second derivatives of x_c in Equation (3) for time to obtain the velocity and acceleration relationship of point C, whose expression is:

$$\left\{\begin{array}{c}\dot{x_c}\left(\theta \right)=v_c\left(\theta \right)\\ \ddot{x_c}\left(\theta \right)=a_c\left(\theta \right)\end{array}\right.$$

(4)

In the case of a = 1 cm, the curves shown in Figure 6 and Figure 7 are obtained through simulation.

From Figure 7, we can obtain the analysis results of approximate linear motion at the end of the Hoecken mechanism, and the linear motion error is:

$$\frac{10.015-9.985}{10}\times 100\mathrm{\%}=0.3\mathrm{\%}$$

(5)

The error of the Hoecken finger in the straight parallel pinch stage is small and negligible, and the straight-line motion property is good.

As can be seen from the curve shown in Figure 7, the fluctuation range of speed and acceleration of the Hoecken finger in the straight parallel pinch state are quite small, which makes the grasp effect stable, and is more advantageous than other fingers in grasping fragile objects.

5. Hoecken’s Finger Grasp Force and Grasp Range Analysis

In order to understand the mechanical characteristics of the finger during grasp, a mechanical analysis model is established, as shown in Figure 8a,b.

The process of straight parallel pinch is shown in Figure 8a.

In Figure 8a, the meaning and physical meaning of each symbol are:

F₀, F_1, and F₂ are the thrust of AB to BC, BC to point C, and the force that can be finally provided by the straight pinch of the device, respectively; ${\theta }_{},\alpha$ are the positive angle between AB and x-axis and the negative angle between BC and x-axis.

In Figure 8b, the meaning and physical meaning of each symbol are:

F₁–F₃ are the force acting on the two units of the first phalanx and the second phalanx, respectively.

P₁–P₃ are the action point of the force applied by the object to the two units and the second phalanx.

l₁–l₃ are the length of DI, EI, and EJ, respectively.

${\theta }_1–{{\theta }_3}_{}$ are the deviation angles of DI, EI, and EJ from y-axis.

T is the motor torque.

G₁ is the elastic coefficient of the torsion spring between two units.

$T_{G_1}$ is the torque of G₁ to the mechanism.

T₁–T₃ are the moment of F₁–F₃ to the mechanism, respectively.

In Figure 8a, from the relationship between geometry and force, order, M, is the torque that the motor can provide; the length of AB is 2a; the length of AB, BH, and CH is, respectively, $l_{AB}$, x₁, x₂.

We can obtain:

$$\frac{3a+2a\mathrm{s}\mathrm{i}\mathrm{n}\theta }{x_1}=\mathrm{s}\mathrm{i}\mathrm{n}\alpha$$

(6)

$$l_{AB}{F}_0=\mathrm{M}$$

(7)

$$\frac{F_0{x}_1\cos (\alpha -\theta )\mathrm{s}\mathrm{i}\mathrm{n}\alpha }{x_2}=F_2$$

(8)

The image shown in Figure 9 is obtained after MATLAB motion simulation. It can be seen that the force that the mechanism can provide in the parallel pinch stage is $\theta$ and steadily increases, indicating that the mechanism $\theta$. When the object is larger, it is more effective in grasping the object, the overall grasp force is larger, and the force change is relatively stable, which verifies the feasibility of the actual implementation of the mechanism in the straight parallel pinch stage.

In Figure 8b, the vector expression of four forces and force points is obtained through vector representation of the model

Order

$x_{p2}$ = $l_1\sin {\theta }_1+\frac{1}{2}{l}_2\sin {\theta }_2$;

$y_{p2}$ = $l_1\cos {\theta }_1+\frac{1}{2}{l}_2\cos {\theta }_2$;

$x_{p3}$ = $l_1\sin {\theta }_1+l_2\sin {\theta }_2+\frac{1}{2}{l}_3\sin {\theta }_3$;

$y_{p3}$ = $l_1\cos {\theta }_1+l_2\cos {\theta }_2+\frac{1}{2}{l}_3\cos {\theta }_3$;

$${\mathit{P}}_{\mathbf{1}}=(\frac{1}{2}{l}_1\sin {\theta }_1,\frac{1}{2}{l}_1\cos {\theta }_1)$$

(9)

$${\mathit{P}}_{\mathbf{2}}=(x_{p2},y_{p2})$$

(10)

$${\mathit{P}}_{\mathbf{3}}=(x_{p3},y_{p3})$$

(11)

$${\mathit{F}}_{\mathbf{1}}=(-F_1\cos {\theta }_1,F_1\sin {\theta }_1)$$

(12)

$${\mathit{F}}_{\mathbf{2}}=(-F_2\cos {\theta }_2,F_2\sin {\theta }_2)$$

(13)

$${\mathit{F}}_{\mathbf{3}}=(-F_3\cos {\theta }_3,F_3\sin {\theta }_3)$$

(14)

By taking partial differentiation of the vector at the corresponding point, the virtual displacement can be obtained as

$$\mathit{\delta }{\mathit{P}}_{\mathbf{1}}=(\frac{\mathbf{1}}{\mathbf{2}}{\mathit{l}}_{\mathbf{1}}\mathbf{cos}{\mathit{\theta }}_{\mathbf{1}},-\frac{\mathbf{1}}{\mathbf{2}}{\mathit{l}}_{\mathbf{1}}\mathbf{sin}{\mathit{\theta }}_{\mathbf{1}})\mathsf{\delta }{\theta }_1$$

(15)

$$\mathit{\delta }{\mathit{P}}_{\mathbf{2}}=(l_1\cos {\theta }_1\mathsf{\delta }{\theta }_1+\frac{1}{2}{l}_2\cos {\theta }_2\mathsf{\delta }{\theta }_2,-l_1\sin {\theta }_1\mathsf{\delta }{\theta }_1-\frac{1}{2}{l}_2\sin {\theta }_2\mathsf{\delta }{\theta }_2),$$

(16)

$$\begin{gathered} \mathit{\delta }{\mathit{P}}_{\mathbf{3}}=(-l_1\cos {\theta }_1\mathsf{\delta }{\theta }_1-l_2\cos {\theta }_2\mathsf{\delta }{\theta }_2-\frac{1}{2}{l}_3\cos {\theta }_3 \\ \mathsf{\delta }{\theta }_3,l_1\sin {\theta }_1\mathsf{\delta }{\theta }_1+l_2\sin {\theta }_2\mathsf{\delta }{\theta }_2+\frac{1}{2}{l}_3\sin {\theta }_3\mathsf{\delta }{\theta }_3) \end{gathered}$$

(17)

According to the principle of virtual work, it can be obtained as

$$\left[\begin{array}{ccc}T& -G{\theta }_2& -G{\theta }_3\end{array}\right]\left[\begin{array}{c}\mathit{\delta }{\mathit{\theta }}_{\mathbf{1}}\\ \mathit{\delta }{\mathit{\theta }}_{\mathbf{2}}\\ \mathit{\delta }{\mathit{\theta }}_{\mathbf{3}}\end{array}\right]=\left[\begin{array}{ccc}{\mathit{F}}_{\mathbf{1}}& {\mathit{F}}_{\mathbf{2}}& {\mathit{F}}_{\mathbf{3}}\end{array}\right]\left[\begin{array}{c}\mathit{\delta }{{\mathit{P}}_{\mathbf{1}}}^{\mathit{t}}\\ \mathit{\delta }{{\mathit{P}}_{\mathbf{2}}}^{\mathit{t}}\\ \mathit{\delta }{{\mathit{P}}_{\mathbf{3}}}^{\mathit{t}}\end{array}\right]$$

(18)

Among them, T provides torque for the motor, which is the required torque for the system.

The right end of Equation (18) can be represented as

$$\begin{gathered} \left[\begin{array}{ccc}{\mathit{F}}_{\mathbf{1}}& {\mathit{F}}_{\mathbf{2}}& {\mathit{F}}_{\mathbf{3}}\end{array}\right]\left[\begin{array}{c}\mathit{\delta }{{\mathit{P}}_{\mathbf{1}}}^{\mathit{t}}\\ \mathit{\delta }{{\mathit{P}}_{\mathbf{2}}}^{\mathit{t}}\\ \mathit{\delta }{{\mathit{P}}_{\mathbf{3}}}^{\mathit{t}}\end{array}\right]= \\ \left[\begin{array}{ccc}{\frac{1}{2}F}_1& \frac{1}{2}{F}_2& {\frac{1}{2}F}_3\end{array}\right]\left(\begin{array}{ccc}-l_1& 0& 0\\ -l_1\cos ({\theta }_2-{\theta }_1)& -l_2& 0\\ -l_1\cos ({\theta }_3-{\theta }_1)& -l_2\cos ({\theta }_3-{\theta }_2)& -l_3\end{array}\right)\left[\begin{array}{c}\mathit{\delta }{\mathit{\theta }}_1\\ \mathit{\delta }{\mathit{\theta }}_2\\ \mathit{\delta }{\mathit{\theta }}_3\end{array}\right] \end{gathered}$$

(19)

From (18) to (19), we obtain the torque expression required for the motor to stably grasp the object. The images are obtained after motion simulation, as shown in Figure 10.

Due to the functional separation of the first phalanx and the second phalanx, the self-adaptation is mainly completed by the first phalanx; it also improves the design position of the Hoecken linkage and parallel four-bar linkage in the robot’s hand. The torque required by this mechanism is relatively stable when the external force is small, and it basically decreases with the increase in ${\theta }_1$ when the external force is large. This indicates that the larger the torque required by this mechanism, that is, the greater the pressure it is subjected to, the greater the grasping force provided by the mechanism. Therefore, this mechanism can achieve stable grasping when facing objects of different weights. Meanwhile, the larger the rotation angle of the mechanism, the smaller the required torque, which is consistent with the actual situation and theoretical design. Therefore, this mechanism can effectively rely on smaller and more stable power motors to stably grasp objects, verifying the effectiveness of the actual implementation of the institution.

In order to understand the grasp range of the fingers in the process of grasp, the paper establishes the geometric analysis models of the first and second units of the first phalanx that are mainly used for self-adaptive grasp, as shown in Figure 11.

In Figure 11, the meaning and physical meaning of each symbol are:

l₁–l₂ are the length of DI and EI, respectively.

${\theta }_1–{{\theta }_3}_{}$ are the deviation angle of DI relative to the y-axis, the half angle of DI relative to x-axis, and the deviation angle of EI relative to DI, respectively.

R is the radius of the object to be grasped.

Order

$k_1=\tan {\theta }_2$;

$k_2=-\tan [\frac{\pi -{\theta }_3}{2}-({\theta }_1+{\theta }_2)]$;

$b=\tan [\frac{\pi -{\theta }_3}{2}-({\theta }_1+{\theta }_2)\left]\right(12a-l_2)\cos 2{\theta }_2$;

$a=(12a-l_2)\sin 2{\theta }_2$;

According to the geometric relationship:

$${\theta }_2=\frac{0.5\pi -{\theta }_1}{2}$$

(20)

$$y_1=k_1{x}_1$$

(21)

$$\frac{y_2-(12a-l_2)\sin 2{\theta }_2}{x_2-(12a-l_2)\cos 2{\theta }_2}=k_1$$

(22)

According to simultaneous Equations (21) and (22):

$$\left\{\begin{array}{c}{x}_R=\frac{b+\mathrm{a}}{k_1-k_2}\\ y_R=k_1{x}_R=R\end{array}\right.$$

(23)

In the formula, $x_1,y_1$ jointly express the angular bisector equation of DI and x-axis, $x_2,y_2$ jointly express the angular bisector equation of DI and EI, and $x_R{,y}_R$ are the horizontal and vertical coordinates of the object to be grasped.

As shown in Figure 12, the grasp range increases first and then decreases with ${\theta }_1$, with the largest grasp range at a certain angle; grasp range increases with ${\theta }_3-{\theta }_2$. The overall self-adaptive envelope grasp range is large, which can achieve reliable grasp. This corresponds to actual needs, because if self-adaptive grasp is carried out in the early stages of grasping, the captured object has a large size, and the grasp range should be large. Similarly, if self-adaptive grasp is carried out in the later stages of grasp, the captured object has a small size, and the grasp range could be small. At the same time, theoretical analysis shows that the length of the elastic element also affects the grasping range, and the length ratio of $l_1$ and $l_2$ should be changed to 1.5:1 within a reasonable range to increase the grasping range of the finger. In addition, after measurement, the grasp range of the Hoecken gripper’s straight parallel pinch reaches 130 mm, which means that the gripper can pinch most objects. The overall grasp range is large and can achieve reliable grasp.

6. Hoecken’s Gripper Grasp Experiment

In order to study the grasp performance of Hoecken’s gripper, a prototype of a Hoecken gripper was developed, and a grasp experiment of Hoecken’s gripper was carried out, as shown in Figure 13.

The grasp experiment includes three states: initial state (Figure 13a), straight parallel pinch (Figure 13b), and self-adaptive grasp (Figure 13c).

The remaining object grasp experiments are presented in

Table 2. Object grasp experiments.

Captured Object	Shape	Size/mm	Grasp Mode	Grasp Effect
Adhesive tape	circular ring	Φ100 × 47	Parallel pinch	Good
Phone	cuboid	160 × 75 × 10	Parallel pinch	Good
Water cup	cylinder	Φ60 × 170	Self-adaptive grasp	Good
Mouse	Approximate cuboid	120 × 50 × 30	Parallel pinch	Almost good
Plastic Block	cube	100 × 100 × 100	Parallel pinch	Good
Banana	Curved Spindle	Φ50 × 150	Parallel pinch	Good
Baseball	Sphere	Φ75	Self-adaptive grasp	Good
Towel roll	cylinder	Φ50	Self-adaptive grasp	Good

.

The “good” “Grasp effect” indicates that the gripper has a success rate of over 95% when grasping objects in the straight parallel pinch and self-adaptive grasp mode, and the object can be grasped without detaching from the gripper under the interference of at least 10 N external force applied to the object; “Almost good” means that the success rate of grasping objects by the gripper is greater than 90%, but applying a certain amount of external force interference may cause the grasped object to “eject”, that is, detach from the gripper for grasp.

The Hoecken gripper is controlled by Arduino and powered by an electric motor. In the experiment of straight parallel pinch, without the assistance of a robotic arm, the Hoecken gripper has a good gripping effect on objects on a plane, verifying its straight parallel gripping function and superiority over traditional robotic hands. The experimental results show that the Hoecken gripper can grasp objects of different softness, shapes, and sizes, and it has good grasping effects. The experiment shows that both states are stable grasping states, and the Hoecken gripper can meet daily and industrial grasping requirements.

7. Conclusions

Based on the Hoecken linkage mechanism, this paper designed and developed a kind of straight parallel pinch and self-adaptive gripper—Hoecken’s gripper. The structure design and working principle of the gripper are introduced in detail. The movement analysis of the Hoecken linkage mechanism is carried out, and the grasp force analysis and grasp range analysis under different parameters are carried out for the gripper. The analysis and experiment both show that the Hoecken gripper has great grasp performance and can meet the daily needs. The aim of future work will mainly be to conduct deeper analysis of the gripper, such as analyzing the difference of gripper’s grasping effect on objects of different shapes and optimizing parameters, such as optimizing parameters including unit length ratio to better grasp irregular objects.