2.1. Conceptual design

To stably grasp the edges of tilted objects, a gripper must apply different forces when pressing the surfaces adjacent to both edge sides (Figure 1). The gripper grasps the top right corner of an object, which is then held in place by the frictional forces triggered by the pressing forces $F_{1}$ and $F_{2}$ . The center of gravity of the object is to the left of the grasped edge; the weight of the object thus creates a counterclockwise moment. To ensure stable gripping, $F_{2}$ must be larger than $F_{1}$ ; this creates a reactive torque. To cope with this scenario, the gripper fingers operate independently to generate a torque equivalent to the object moment. The joint delivers various torques to the finger dictated by the slope of the grasped surface (Figure 2). When an object contacts the central gripper trigger, two fingers that were initially open close toward the object. When an object touches the contact trigger at the end of a finger, that finger stops releasing, and the joint applies a torque determined by the finger’s angular position.

2.2. Detailed design

Figure 3 shows the variable torque joint powered by a torsion spring. The joint has a cam, a cam guide, a spring pawl, and a pawl lock. The cam guide is fixed to the gripper body, and the other components rotate with the finger. When the gripper finger is initially open and locked by the central trigger, the torsion spring is compressed between the spring pawl and the finger and locked in place by the pawl lock, thus delivering no torque at this time. When the center trigger is activated, the fingers are released and close toward the object. When the object contacts the trigger at the end of the pawl lock, the lock is released to deploy the spring pawl (Figure 4). Then, a spring force acts between the finger and the pawl; the force is determined by the angle between the finger and the deployed pawl. The cam restricts the angle of the deployed spring pawl and thus determines the torque (Figure 5). The cam guide translates rotational finger motion into linear cam motion; the cam moves toward the pawl as the finger rotates. As this process evolves, spring expansion is limited, increasing the spring torque.

Figure 3.

Interior components of the variable torque joint.

Figure 4.

Operation of the variable torque joint.

Figure 5.

Spring restrictions at various finger angles.

At lower actuation angles, the extent of spring pawl release is greater, reducing the compression angle of the torsion spring and the torque delivered to the finger. At higher actuation angles, the cam further limits pawl release, and the spring thus delivers more torque at higher compression angles. Thus, the joint outputs different torques depending on the finger’s angular position. The detailed torque profile is determined by the cam and the cam guide.

Figure 6 shows a schematic in which a tilted object is gripped; $\theta _{1}$ and $\theta _{2}$ are the finger actuation angles, and $\phi$ is the angle at which the object is tilted. The top right corner of the object contacts the central gripper trigger. To create the maximum torque that counters object rotation, each finger must apply a different torque. For example, torques $T_{1}$ and $T_{2}$ are not the same; the difference ( $| T_{1}-T_{2}|$ ) is identical to the moment associated with object tilting. Figure 7 illustrates the designed output torque $T$ in terms of finger actuation angle $\theta$ . The friction at a rotating joint was not considered here. When the object is not tilted, or when $\phi$ = 45°, the finger actuation angle $\theta$ is about 54.3°. At that angle, all fingers apply the same torque to the object; $| T_{1}-T_{2}| =0$ . When $\theta$ exceeds 54.3°, one finger needs to apply more torque than the other fingers. For better gripping of tilted objects, the torque curve was drawn by reference to second-order polynomials (in forms presented in equations (1) and (2), both passing (54.3, 50)) with an inflection point at $\theta$ = 54.3° (Figure 7).

Figure 6.

Schematic of the gripper and object during gripping.

Figure 7.

Torque outputs of all fingers by the actuation angle.

(1) \begin{equation} y=f_{1}\left(x\right)=a_{1}x^{2} \end{equation}

(2) \begin{equation} y=f_{2}\left(x\right)=a_{2}\left(x-90\right)^{2}+100 \end{equation}

The output torque $| T_{1}-T_{2}|$ can then be written in terms of $\varphi$ (Figure 8). At $\varphi$ = 45°, all fingers apply the same torque, so there is no moment. When $\varphi$ is either larger or smaller than this value, a more actuated finger applies more torque and a less actuated finger applies less torque, thus creating a reactive moment that maintains the current object posture. To maximize $| T_{1}-T_{2}|$ , the torque output ( $T$ ) needs to increase faster when $\theta$ exceeds 54.3° and decrease faster when $\theta$ is less than 54.3°. Thus, employing second-order polynomials was more beneficial than employing a simple linear line.

Figure 8.

Torque difference (reactive moment) in terms of the object tilt angle.

Figure 9.

The designed cam images.

To implement a torque profile reflecting the finger actuation angles, the cam profile was derived based on Figure 7. To enhance torque performance, the cam surface has height steps of 0.6 mm, given that the surfaces of 3D-printed parts are rough. Figure 9 shows the cam with the various steps of different areas that create the required torque profile. As the contact (the spring pawl) moves counterclockwise, the finger actuation angle $\theta$ increases. When the angle of the horizontal dashed line is set to 0° (for an $\theta$ of 54.3°), Figure 10 shows the positions of each step when creating the torque output curve in Figure 7. After the spring pawl is released, its final position is determined by the specific cam step at that time, and the required torques are then sent to the fingers.

Table 2.

Torsion spring specifications.

The effective finger length (i.e., the grasping length) was 100 mm, based on the human hand. After the addition of the joint connection, the final finger length was 125 mm. The gripping force was about 4% of the average maximum force exerted by appression of the adult male thumb and index finger [Reference Nilsen, Hermann, Eriksen, Dagfinrud, Mowinckel and Kjeken25]. The distance between the fingers was 80 mm, and the fingertip force was 3.48 N. The torsion spring was selected by reference to these values. The maximum torque was the product of the finger length and the maximum fingertip force, i.e., 435 N-mm. Therefore, to deliver the required torque when a finger is fully actuated at 90°, the spring constant must exceed 4.83 N-mm/°. The spring radius was ca. 25 mm, i.e., the difference between the effective finger length (100 mm) and the overall length (125 mm). The torsion spring specifications are listed in Table 2.

Figure 10.

Angular positions at each step.

Figure 11 shows a CAD model and the fabricated gripper. Most components were 3D-printed using a material extrusion method using polylactic acid (PLA). To minimize the friction, commercial pin-socket joints were used. The cam guide, which must be highly accurate and have low surface friction, was printed via VAT resin photopolymerization. A polymer pad was attached to each fingertip to enhance gripping by increasing friction. Each pad was fabricated by curing a flexible PDMS polymer (Sylgard 184 silicone elastomer; Dow Silicone Corporation, USA), which is widely used for its ease of application and has a high frictional coefficient due to its strong adhesion [Reference Kim, Lee, Lee and Kim26].

Figure 11.

(a) A CAD model of the gripper and (b) the fabricated gripper.