A Compliant Hinge Joint Driven by the PneuNets Bending Actuator

Soft pneumatic robots have attracted much attention recently due to their safety, high adaptability, fast response, lightweight, and strong anti-interference ability [1]. As a result, a variety of soft pneumatic robots with different functions, such as climbing robots [2,3], walking robots [4,5], soft grippers [6,7], and soft pneumatic gloves [8,9], have been developed and applied in space exploration [10,11], medical field [12,13], and industrial field [14,15]. However, the inherent flexibility of soft pneumatic robots presents difficulties in carrying heavy loads and making precise motions, which restricts their further development.

Researchers propose two solutions to overcome this shortcoming. The first commonly used solution is the variable stiffness technique, which enables the soft robot to change stiffness through variable stiffness mechanisms or materials. For example, Kim et al. proposed a layer jamming mechanism, which uses negative pressure to change the friction between layers to achieve stiffness tuning [16]. Based on this work, Crowley et al. developed a positive layer jamming method to vary the stiffness of a soft pneumatic gripper [7]. Since the positive pressure has no upper limit, the positive layer jamming mechanism can change the stiffness in a more extensive range. In addition, some researchers have studied the inherent variable properties of smart materials for variable stiffness. Alambeigi et al. developed a continuum manipulator with a backbone made of phase-changing alloy [17], which changes the material’s stiffness by controlling the temperature. Yang et al. designed a robotic finger that can control stiffness using shape-memory polymer joints [18]. Although the variable stiffness technique can overcome the challenges by tuning the stiffness, additional devices and space are required, increasing the cost and complexity of robots [19].

Another solution is a hybrid robot system combining stiff backbone mechanisms with soft joints. An example is a terrain-capable hybrid soft/stiff myriapod robot designed by Ozkan-Aydin et al. [20]. The soft joint of the robot enables it to adapt to the complex terrain well, and the rigid-body mechanism supports the robot’s weight. However, rigid-body parts in this solution reduce the flexibility of the robots. Also, the robotic is hard to achieve high motion precision due to the compliance of the pure soft joint. Another example is a hybrid finger proposed by Lotifani et al. [21]. A rigid skeleton is embedded into a pneumatic soft finger to improve its torsional stiffness so that the gripper made of such fingers can grasp heavy objects. However, since the lateral bending stiffness has not been increased, it will still be unable to carry heavy objects and fail when the finger is extended to a certain length.

To improve this solution, we create a new hinge joint by combining compliant mechanisms (CM) [22] and PneuNets bending actuators. A typical compliant mechanism consists of a fixed and rigid moving stage. The two stages are connected by one or more flexure members. Compliant mechanisms offer many advantages over conventional rigid-body mechanisms, such as part-count reduction and reduced motion errors [22]. In addition, some CMs also have relatively high stiffness in a certain direction. Therefore, using CM as the backbone of the joint can improve the load-carrying capacity and motion precision. The PneuNets bending actuator shown in Fig. 1 consists of bellow-shape soft air chambers and an inextensible layer [23]. Pressuring the air chamber can inflate the lateral wall to contact the adjacent walls and generate moment. The moment will bend the actuator over a wide range. Compared with other bending actuators, the PneuNets bending actuator performs a high actuation rate, an extensive motion range, and a simple pressure–displacement relationship [24]. Those merits make it a good choice for joint actuation.

This paper proposes a new compliant hinge joint combining a compliant cross-axis flexural pivot with the PneuNets bending actuators. This joint can be in an active mode when pressurizing the PneuNets bending actuator or in a passive mode driven by external force/moment. We derive a pseudo-rigid-body (PRB) model to study the pressure–displacement relationship in active mode and validate the result with experiments and simulations. Experiments are implemented to evaluate the motion in both active and passive modes. Also, we experimentally measure the in-axis and off-axis stiffnesses of the joint and calculate the stiffness ratio.

The paper is organized as follows: Sec. 2 presents the design of the new joint. We derive the analytical models for the PneuNets bending actuator and the joint in active mode. Then, in Sec. 3, we detail the fabrication process, the experiments for the joint’s trajectory in both active and passive modes, and the experiments for the pressure–displacement relationship. We also develop finite element simulations to validate the theoretical model. A new design with higher motion precision is proposed based on the motion results. We also experimentally study the in-axis and off-axis stiffnesses and calculate the stiffness ratio. Finally, in Sec. 4, we conclude the paper.

In this section, we first detail the design. Then, we derive the theoretical model for the PneuNets bending actuator’s driving moment and a compliant four-bar mechanism based on the PRB model for the kinematics and statics. We finally derive the pressure–displacement model by combining the PRB and driving moment models.

The proposed novel joint is called PneuNets actuated cross-axis flexural pivot (PnACF pivot). Figure 2 shows its computer-aided design (CAD) model and prototype. A compliant cross-axis flexural pivot made of relatively stiff material is used as the backbone, and two soft PneuNets bending actuators are bonded on the two flexure strips for actuation. As the two actuators are attached to the strips, pressurizing one of them will bend the strip and rotate the PnACF pivot around the Z-axis clockwise or counterclockwise. The strip also works as the inextensible layer to limit the unnecessary expansion of the actuator’s bottom layer. Due to the high off-axis stiffness of the cross-axis flexural pivot, the PnACF pivot can carry a high payload in a non-motion direction while maintaining flexibility in the motion direction. Moreover, this high off-axis stiffness reduces the PnACF pivot’s parasitic motion error.

After finalizing the design and modeling of the PnACF pivot, we manufacture a prototype and evaluate its performance through experiments and simulations. We first fabricate a prototype by resin printing. The geometric parameters of the actuator and the cross-axis flexural pivot are shown in Tables 1 and 2, respectively. Then, we evaluate the motion of the PnACF pivot in both active and passive modes. Based on the results, we propose a new design with different parameters to improve motion performance in passive mode. Finally, we conduct experiments to study the in-axis and off-axis stiffnesses of both two designs.

Due to the complex shape of the PneuNets bending actuator and multi-material manufacturing requirements, the traditional casting method is not suitable for this project. Instead, this work uses the resin printing method to fabricate the actuator and the cross-axis flexural pivot with a DLP printer (AnyCubic Inc., Shenzhen, China). We first print two actuators using a commercial rubber-like resin (F69, Resione Inc., Dongguan City, China) and post-process them for 10 min with a curing machine (AnyCubic Inc.). Then, we print the cross-axis flexural pivot with a flexible and tough resin (UV Tough resin, AnyCubic Inc., Shenzhen, China) and do the post-processing for 3 min. Finally, we bond two actuators on the two strips of the pivot by super glue. We glue the actuator’s bottom to the strip’s top. A tensile test is implemented to determine the material properties of the rubber-like resin following the standard ASTM-D412 type C. Five dumbbell specimens are fabricated and stretched until they break on the tensile machine (Instron5542). The strain rate is 500 mm/min. The Neo–Hookean model is used to fit the testing data, and the mean value of Young’s Modulus from testing is 3.93 MPa. This value is used in FE simulations and analytical solutions.

A finite element model is built in abaqus. Young’s modulus of the cross-axis flexural pivot is set to 1 GPa, and Poisson’s ratio of the pivot is 0.35. Based on the tensile testing, we use the Neo–Hookean model with coefficients C₁ = 0.66 and D₁ = 0 as the constitutive model of the PneuNets bending actuator. Pressure up to 0.276 MPa (40 psi) is applied on the internal surfaces of chambers of one actuator for the PnACF pivot’s unidirectional rotation. The fixed stage of the pivot and the end of two actuators close to the fixed stage are fixed. Self-contact interaction is defined between adjacent chamber walls. The hybrid tetrahedron element C3D10H is assigned to the actuator, and the element C3D10 is used for the cross-axis flexural pivot.

The PnACF pivot has active and passive modes. In active mode, the PnACF pivot works as an active joint, meaning that the PneuNets bending actuator will drive the joint with air pressure. In passive mode, the PnACF pivot works as a passive joint, meaning that the external force will move the joint but not the actuator. Before the experiment starts, we define three translational directions (X, Y, Z) along the X-, Y-, and Z-axis and three rotational directions (RX, RY, RZ) around the X-, Y-, and Z-axis. The deformation axes are shown in Fig. 5(a). The experiment setup for the active mode is shown in Fig. 5(b). In which the PnACF pivot with an indicator is fixed on the table, and a camera is placed above it. One of the actuators is gradually pressurized from 0 MPa (0 psi) to 0.276 MPa (40 psi) to drive the pivot rotation. The camera captures the rotation angle corresponding to each pressure. Then, the imagej software is used to measure the angle in the pictures. To measure the angle, we draw the initial position of the indicator tip and rotation center on the grip paper. Then, we connect those two points and the center and current position of the tip in the picture. The angle can be automatically calculated by the tool in the software. We repeated the experiment thrice for each direction and calculated the average. Figure 5(c) shows the experiment setup for passive mode. The moving stage of the pivot is tied to the Mark 10 force sensor by a thin wire. The force sensor will move and rotate the pivot by turning the knob. The sensor records the driving forces, and the angle corresponding to each driving force is recorded by the camera above the pivot.

Our first design (Design 1) is pivot with the cross center in the strip’s middle. We define the length ratio from the moving stage to the center over the total length of the strip as a_l. Thus, for Design 1, a_l = 0.5. As shown in Fig. 6(a), given the dimensions in Tables 1 and 2, the ideal motion trajectory of the center point on the moving stage is a semi-circle with a radius of 22 mm relative to the cross center. We plot the trajectories in active and passive modes from the experiments in Figs. 6(b) and 6(c). We also calculate the absolute error of radius referring to the ideal trajectory for both modes and compared them in Fig. 6(d). From the result, the mean error in the passive mode is 0.1150, and that for the active mode is 0.1397. The main reason for the error is that the center axis of Design 1 will drift along the X and Y directions when rotating.

We compare the pressure–angle curve in the active mode from the analytical model with the curve from the experiment and FE simulation in Fig. 7. From the result, the error of the analytical model compared to the simulation is $1.15%$⁠, and it is $4.47%$ compared to the experiment. The reason is that the small amount of residual resin in the air chamber blocks the deformation of the actuator. Therefore, the PnACF pivot in the experiment cannot achieve as large as an angle in the simulation at the maximum air pressure. Within the pressure range of [0 MPa, 0.276 MPa], the analytical model and FEA show that the pivot can rotate in a range of [−43 deg, 43 deg]. And the experiment result shows a range of [−38 deg, 39 deg]. The result also shows a close to linear relationship, which means the PnACF pivot can be easily controlled to achieve a specific angle by air pressure.

To improve the PnACF pivot’s performance in the passive mode, we propose a new design (Design 2) with $al=0.5+(5)/6$ as shown in Fig. 8(a). The other geometric parameters are consistent with Design 1, except that the number of chambers n is reduced to 7. Based on Ref. [28], the center axis of Design 2 has zero error along the Y-axis and little error along the X-axis when rotating. The experiment results of trajectory in both active and passive modes are shown in Figs. 8(b) and 8(c), and errors are compared in Fig. 8(d). It shows that the mean error of Design 2 in passive mode is 0.0946, less than that of Design 1. The error in the active mode is 0.1161, which is also less than that of Design 1. However, the rotational stiffness of Design 2 is higher, so in the active mode, Design 2 can only achieve $9.2deg$ at 0.276 MPa.

We conducted four experiments to measure each design’s in-axis stiffness (RZ direction) and three off-axis stiffness (RX, Y, and Z direction).

Figures 9(a) and 9(b) show the experimental setup for measuring the rotational stiffness. We used a wire to connect the Mark 10 force sensor and the pivot’s moving stage. The length r_c between the connecting point and the cross center for Design 1 and Design 2 are 37 mm and 46.07 mm, respectively. By turning the knob clockwise, the force sensor can move away from the pivot and drive it to rotate. The force F can be read from the sensor, and the moment can be calculated by M = F × r_c. The camera above the pivot records the angle, which is measured by the imagej.

Figures 9(c) and 9(d) illustrate the experimental setup for measuring translational stiffness. When turning the knob counterclockwise, the tip of the force sensor will push the pivot to move along the direction to be measured. The traveling sensor and force sensor, respectively, can record the displacement and force data.

Based on the equivalent translational stiffness, we can calculate the stiffness ratio, defined as the ratio of off-axis stiffness and in-axis stiffness. The results are listed in Tables 3 and 4. For Design 1, the stiffness in the RX direction is 16.74 times the stiffness in the RZ direction, and the stiffness in the Y and Z directions is 471.75 times and 627.63 times the stiffness in the RZ direction, respectively. For Design 2, the stiffness in the RX direction is 9.37 times the stiffness in the RZ direction, and the stiffness in the Y and Z directions is 890.30 times and 479.97 times the stiffness in the RZ direction, respectively. A high stiffness ratio means the PnACF pivot has a large load-carry capacity off-axis while maintaining the compliance in-axis.

In this paper, we present the design and analysis of a PneuNets actuated cross-axis flexural pivot. Experiments and FEA simulations are developed to evaluate the motion of the proposed design in active and passive modes. The results show that the design has a relatively large motion range, and the pressure–displacement curve is close to linear. However, the motion error of this design is large in both passive and active modes. To address this shortcoming, we proposed an improved design that has a relatively small motion error at the cost of the limited motion range in the active mode due to high rotational stiffness. Experimental testings show that the two designs have a high stiffness ratio of off-axis to in-axis, which means they can carry heavy loads in the off-axis directions. Our design can be essential to soft robots with enhanced payload capacity and precise motion.

This material is based upon work supported by the National Science Foundation under Grant No. 2019648. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding agencies.

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.