A variable stiffness method for pneumatic flexible arms

Flexible robots have broad application prospects in the military, industrial, service, medical, and other fields because of their high flexibility and ability to adapt to unstructured environments; hence, they have been widely studied and various flexible mechanisms have been proposed.^1–4 For example, inspired by the elephant trunk, the German company Festo developed a series of bionic robots based on the McKibben pneumatic artificial muscle.^5–16 Jiang et al. were inspired by the origami process and developed a software robotic arm based on an origami structure.¹⁷ Jiang et al. developed a soft robot arm using a honeycomb pneumatic network structure that could easily complete various tasks, such as opening doors, twisting bottle caps, and pulling drawers in daily life.¹⁸ Surgical robots for medical applications have also been developed.^19–21 Flexible robots primarily use safe and flexible elastic materials that can continuously deform. However, the low stiffness and weak load capacity of flexible robots limit their applications.

Some studies have proposed variable stiffness and brake structures to improve the stiffness of flexible robots. The variable stiffness mechanism primarily includes particle blockage, layered interference, and functional materials.^22–26 In 2010, Brown et al. used the particle blockage principle to fabricate a sucker-type grasping device, which was simple in structure and easy to control; however, it could only grip small objects.²⁷ In 2016, Li et al. proposed a novel passive particle interference principle to achieve a variable stiffness that did not require a vacuum source or other control methods. When the inflation pressure increased from 20 to 80 kPa, the stiffness of the soft robotic arm could be increased by more than six times its original value.²⁸ Kim et al. proposed a new type of layered interference mechanism that could realize variable stiffness. The layered interference mechanism was achieved using the friction force between the layers of film materials and controlled by a negative pressure. The structure was hollow, compact, and lightweight; hence, it could be applied in minimally invasive surgery and other fields.²⁹ A soft gripper based on a shape memory alloy was designed, which could achieve a stiffness adjustment range of up to 55 times.³⁰ In 2017, Imamura et al. designed a soft variable stiffness gripper based on multi-layer dielectric elastomers. Stiffness adjustment was achieved by changes in the friction force between each layer of the dielectric elastomer under the attraction of static charges. The maximum stiffness change in the soft variable stiffness gripper could reach 39.2 times the original stiffness.³¹ Typical brakes are mostly used in automobiles and rigid robots;^32–34 flexible robot brakes have not been sufficiently investigated. Zheng et al. proposed an aerodynamic elastic spherical brake that could withstand a torque of 3 N m and solve the problem of multi-directional braking of flexible robot wrists.³⁵ Furthermore, Geng et al. designed a brake for maintaining the wrist posture, which compressed the middle brake ball through axial extension of the upper and lower brake airbags to achieve braking. The maximum axial thrust was 107 N when the brake airbags were filled with an air pressure of 0.14 MPa.³⁶ This method provides a large range of stiffness adjustments; however, the load-bearing capacity is insufficient. Thus, it is primarily used to adjust the stiffness of flexible arm-end effectors. However, it is not appropriate for the stiffness adjustment of flexible arms with large loads. Therefore, this study introduces a three-degrees-of-freedom (3DOF) pneumatic flexible arm with a brake that can adjust the braking force to achieve stiffness changes in the flexible arm, enabling the flexible arm to have anti-torsion and posture maintenance functions. The proposed flexible arm can be used in practical applications as it overcomes the shortcomings of existing flexible arms, such as insufficient rigidity and small bearing capacity.

Figure 1(a) shows a 3DOF pneumatic flexible arm that consists of a flexible actuator and flexible brake. The actuator primarily is composed of three groups of fan-shaped driving units uniformly distributed along the circumference. When the three sets of driving units have the same air pressure, the arm will elongate along the axis direction; the greater the air pressure, the greater the elongation. However, when they have different air pressures, the arm is bent within the three-dimensional (3D) space range. A composite deformation motion of axial elongation and spatial bending of the flexible arm can be achieved by controlling the air pressure in the three groups of fan-shaped driving units. The brake is connected to the actuator through an elastic cable and annular brake airbags are arranged inside the brake. After the pneumatic flexible arm reaches the set position, pressure is exerted on the brake airbags and they undergo a radial deformation. The axial and radial outer sides are limited by the brake shell, and the airbags can only expand to the radial inner side, promoting radial contraction of the fan-shaped brake pad installed inside the brake airbag. The fan-shaped brake pad contacts the brake shaft as the pressure increases, which generates a braking force. The greater the input pressure of the brake airbag, the greater the braking force. The stiffness of the flexible arm is adjusted in real-time based on the load or deformation of the flexible arm, and an arbitrary pose of the flexible arm is maintained. The effective carrying capacity of the flexible arm significantly improves, as shown in Fig. 1(b).

The stiffness of the flexible arm is anisotropic owing to the high tensile and low compressive strengths of the elastic cable. When the flexible arm is in its initial state, the brake is pressurized to hold the brake shaft, and the axial elongation and spatial bending deformation of the flexible arm are limited by the elastic cable. The flexible arm has greater stiffness in all directions because the initial state of the flexible arm is axially incompressible. When a flexible arm is axially elongated or spatially bent, the brake is pressurized to hold the brake shaft, and the elastic cables limit the axial elongation and spatial bending deformation of the flexible arm. However, the flexible arm is axially compressible at this time. Therefore, the flexible arm has the largest stiffness in the axial elongation direction, followed by the lateral stiffness, and the axial compression direction has the weakest stiffness. The braking pressure of the brake increases with an increase in the load or deformation displacement of the flexible arm to avoid a situation in which the flexible arm load exceeds the braking force of the brake and braking failure of the elastic cable.

The overall mass of the flexible arm is 4514.4 g, of which the mass of the actuator is 3313.3 g and that of the brake is 1201.1 g, as shown in Fig. 2(a). Table I presents the structural mass and material parameters of the flexible arm.

Figure 2(b) shows that the actuator is primarily composed of artificial muscles, elastic frameworks, elastic cables, nested restraint rings, and end covers. Several restraint rings are coaxially mounted without gaps to form three fan-shaped cavities. The nested structure of the restraint ring improves the torsional resistance of the flexible arm. Three artificial muscles are installed in each fan-shaped cavity to form a driving unit to achieve spatial bending and elongation of the flexible arm. The restraint ring limits the radial deformation of the artificial muscles without affecting the axial elongation and bending flexibility. Three elastic frameworks are evenly distributed around the circle between the three groups of fan-shaped driving units to improve the initial stiffness of the flexible arm and reset response speed. The inner ring is evenly distributed with three elastic cables connected to the brake.

The brake and upper end of the actuator are fixed by elastic cables, and three brake units of the same structure are formed by the brake airbag, fan-shaped brake pad, brake shaft, elastic cable, and brake shell, as shown in Fig. 2(c). The brake airbag is annular and made of Dragon Skin 30 liquid silicone material. A through-hole is arranged in the center to place the fan-shaped brake pad and brake shaft, and another through-hole is arranged on the side as a pressure–gas inlet. Two fan-shaped brake pads are used in each group. The brake shaft is fixed to the elastic cable in the actuator and assembled with a fan-shaped brake pad on the inner wall of the through-hole at the center of the annular airbag. The axial and radial rotations of the fan-shaped brake pad are limited by the coordination between the upper and lower shells of the brake.

The flexible-arm actuator and brake are operated independently. The brake operates when the driver is in the initial state, or undergoes a deformation displacement. The braking force is adjusted to change the stiffness of the flexible arm and maintain its posture by controlling the air pressure applied to the brake. Figure 4 shows that the external loads in the directions of F_n, F_φ, and F_t are applied to the upper end of the flexible arm. The stiffness of the flexible arm in different directions is analyzed. The F_n, F_φ, and F_t directions are the axis direction, bending direction, and direction perpendicular to the bending direction of the flexible arm, respectively. In order to simplify the analysis, it is assumed that the bending deformation of the flexible arm is assumed to be circular during the flexible arm motion, the spring skeleton, elastic cable, and artificial muscle deformations are coordinated and synchronized, and the deformation impedance force of the elastic cable and friction between the elastic cable and restraint ring are small and negligible.

After the initial state or deformation displacement of the flexible arm, the brake presses and locks the brake shaft to generate braking force Ff to maintain the position and posture of the flexible arm and guarantee the motion stability of the end effector. At this stage, axial force F_n is applied to the upper end of the flexible arm. Under the action of an external force, the flexible arm elongates along the axial direction, and the artificial muscle and elasticity framework generate impedance forces F_r and F_k that hinder the elongation of the flexible arm, as shown in Fig. 5.

The bending condition was the same in sub-regions 1–6 due to the central symmetry of the flexible arm. When the brake is not operating, the restraint layer is regarded as the central axis. After the brake is pressurized, because the elastic cable can withstand the tension but cannot withstand the pressure, the bending direction of Zone 1 is in the range 0° < φ ≤ 60°, and the restraint layer is set to pass through elastic cable No. 2. As shown in Fig. 6(a), artificial muscle P3 is elongated, P1 and P2 are compressed, elasticity framework T2 elongates, T1 and T3 are compressed, elastic cable H2 is pulled, and H1 and H3 are compressed [Fig. 6(a)].

When the bending angle of the flexible arm is θ, the brake operates and an external load F_t is applied to the upper end of the flexible arm. The flexible arm produces an additional bending deformation along the F_t direction, and the angle is θ″, as shown in Fig. 7(a). Under the action of an external load, the restraint layer changes and passes through elastic cable No. 3, as shown in Fig. 7(b).

The better the braking performance of the flexible arm, the higher the bearing capacity, and the better the output stability. The performance experiments were conducted using various experimental devices. The braking force and stiffness in the F_n, F_φ, and F_t directions were tested. The accuracy of the theoretical model was verified by comparing the experimental data with the theoretical analysis results, which provided a basis for the development of flexible arms.

Figure 8 shows the experimental principle and device used for the braking force test. The experimental device is composed of an air compressor, power, precision decompressing valve, air pressure sensor, dynamometer, and mobile slide. The dynamometer is fixed on the mobile slide, and the mobile slide pushes the dynamometer to slip along the axial direction of the brake to obtain the braking force under different air pressures. The three brake units have the same structure and only the braking force of one brake unit is tested. During the experiment, the pressure range of the brake airbag is in the range of 0–0.4 MPa, and the pressure interval increment is 0.05 MPa. The average value of five experiments is used to obtain the braking force of one brake unit under different pressures.

Figure 9 shows that the braking force linearly increases with an increase in the air pressure. When the air pressure is 0.4 MPa, the braking force of a single braking unit reaches 276 N. The experimental results were consistent with the theoretical analysis results. The error between the experimental data and theoretical analysis results was 1.73%, which verified the correctness of the theoretical model and feasibility of the brake structure.

Figure 10 shows the experimental principle and device used to test the stiffness characteristics of the proposed flexible arm. The flexible arm is fixed to a turning disk located on a fixed plate, and the fixed disk is fixed on a working table. The turning disk is equipped with locating and threaded connection holes; it is used to realize the rotation of the flexible arm on the fixed disk to conveniently measure the deformation displacement of the flexible arm in different bending directions. The experimental system uses a pulley group to guarantee the accuracy of deformation-displacement measurements. The rope is connected to the upper end of the flexible arm and dynamometer through the pulley group. During the experiment, the position of the pulley group is changed in real-time by moving the mobile slide to confirm that the pull direction of the dynamometer is always parallel to the upper end of the flexible arm. The dynamometer is controlled to apply a fixed external force to the flexible arm in different directions. The laser displacement sensor measures the deformation of the flexible arm under an external load. The acquisition card transmits the collected displacement data to the host computer. The experimental data are processed to obtain the stiffness values of the flexible arm in different directions under different states.

When the flexible arm is in the initial state, the brake is put into the air pressure in the range of 0–0.4 MPa, where the interval increment is 0.1 MPa. The relationship between the stiffness and braking pressure is obtained. Figure 11 shows that the experimental data and corresponding theoretical results are consistent, verifying the correctness of the proposed theoretical model. When the braking pressure is 0.4 MPa, the stiffness of the flexible arm in the F_n, F_φ, and F_t directions is 20, 6.7, and 6.3 kN/m, respectively; thus, it increases by 7.1, 9.3, and 8.8 times, respectively. In the initial state, the stiffness of the flexible arm nonlinearly increases with an increase in the braking pressure. Under the same braking pressure, the stiffness of the flexible arm in the F_n direction is higher than that in the F_φ and F_t directions, and the braking effect is better.

The flexible arm is spatially bent when pressurized. When the bending direction is 0° and the bending angle is 70°, the brake is forced into the air pressure in the range of 0–0.4 MPa, where the interval increment is 0.1 MPa. Figure 12 shows that when the braking pressure is 0.4 MPa, the stiffness of the flexible arm in the F_n, F_φ, and F_t directions is 8.6, 3.1, and 2.9 kN/m, respectively; thus, it increases by 6.1, 12.4, and 11.6 times, respectively. In the space-bending state, the stiffness of the flexible arm nonlinearly increases with an increase in the braking pressure. Under the same braking pressure, the stiffness of the flexible arm in the F_n direction is higher than that in the F_φ and F_t directions. However, the stiffness in the three directions is lower than that in the initial state, primarily because the stiffness of the flexible arm decreased with an increased deformation.

The stiffness of the flexible arm in different bending states is tested at a constant braking pressure. Seven marker points are set at equal angles in Zone 1 of the flexible arm; the bending direction is in the range 0°–60° and the air pressure of the three groups of driving units is controlled to bend the flexible arm in the corresponding direction to the seven marker points. Three external loads of F_n, F_φ, and F_t are applied to the upper end of the flexible arm to test its stiffness in different bending angles and directions. During the experiment, the braking pressure changes between 0 and 0.20 MPa, the bending angle is in the range of 0°–70°, the interval increment is 10°, and the average value is determined using five measurement results.

Figure 13 shows that when the braking pressure is constant, the bending direction slightly affects the stiffness of the flexible arm, whereas the bending angle exhibits a significant influence. The larger the bending angle, the smaller is the stiffness; thus, the stiffness of the flexible arm decreases with an increase in deformation.

The flexible arm is elongated when pressurized. At a phase angle of 0° and an elongation of 30 mm, the brake is introduced into the air pressure in the range of 0–0.4 MPa, where the interval increment is 0.1 MPa. Figure 14 shows that when the braking pressure is 0.4 MPa, the stiffness of the flexible arm in the F_n, F_φ, and F_t directions is 19.8, 3.3, and 3.3 kN/m, respectively; thus, it increases by 8.2, 8.3, and 8.3 times, respectively. In the elongation state, the stiffness of the flexible arm nonlinearly increases with an increase in the braking pressure. Under the same braking pressure, the stiffness of the flexible arm in the F_n direction is higher than that in the F_φ and F_t directions. The stiffness in the F_n direction in the elongation state of the flexible arm is not significantly different from that in the initial state. The stiffness in the F_φ and F_t directions is less than the corresponding stiffness in the initial state because the tensile strength of the elastic cable is high, and the compressive and flexural strengths are low. Therefore, the axial stiffness of the flexible arm in the initial and elongation states is high, and the lateral stiffness is low.

When the braking pressure is constant, the stiffness of the flexible arm in the three directions under the elongation state is tested. Over the entire circumferential range of the flexible arm, the marking points are set at equal angles and the angle increment is 10°. The three groups of driving units of the flexible arm are subjected to the same air pressure such that it is elongated along the axial direction, and the brake is pressurized. An external force is applied along the direction of the marker point to the flexible arm, and its stiffness in different directions under the elongation state is measured. During the experiment, the elongation is in the range of 0–40 mm, the interval increment is 20 mm, the braking pressure changes between 0 and 0.20 MPa, and the average value is determined using five measurement results.

Figure 15 shows the stiffness of the flexible arm in different directions in the elongation state. It can be observed that the stiffness is anisotropic. When the phase angles are 60°, 180°, and 300°, the stiffness change curve significantly fluctuates because the stiffness of the flexible arm is affected by the position distribution of the elastic cable. Under the same braking pressure, the larger the elongation of the flexible arm, the smaller the stiffness; thus, the larger the deformation, the smaller the stiffness.

The combined motion of the flexible arm in a 3D space is realized by adjusting the air pressure of the three driving units to verify the performance of the proposed flexible arm in the working process under the no-load state, as shown in Fig. 16.

In addition, a load experiment is conducted with other joints, including the flexible arm, wrist, and five-finger hand, to test the load capacity and motion characteristics under the load state. The flexible arm is used for dribbling, as shown in Fig. 17. The position of the ball is determined, the flexible arm is controlled to move to the position, the flexible five-finger hand is forced into the air pressure to grasp, the ball is transported to the set position after determining the stability of the grasp, and the five-finger hand releases the ball. The joint parameters are listed in Table II.

The flexible arm is used to pour water, as shown in Fig. 18. It grabs the water bottle and the control system controls it to pour water. During the pouring process, the flexible arm moves smoothly without any effect and no water spills out.

In this study, a 3DOF flexible pneumatic arm was designed. The brake was pressurized to hold the brake shaft. The axial elongation and spatial bending deformation of the designed flexible arm were limited by an elastic cable, and the stiffness of the flexible arm was changed to maintain its position and posture. A theoretical model of the braking force and stiffness was established, and a prototype was developed to test the braking force and stiffness. The results obtained using the theoretical model were consistent with the experimental data. The following conclusions can be drawn.

1. The braking force linearly increased with an increase in the braking pressure. When the braking pressure was 0.40 MPa, the braking force of a single brake unit reached 276 N.

2. In the initial state, the stiffness of the flexible arm nonlinearly increased with an increase in the braking pressure. When the braking pressure was 0.4 MPa, the stiffness in the F_n, F_φ, and F_t directions was 20, 6.7, and 6.3 kN/m, respectively; thus it increased by 7.1, 9.3, and 8.8 times, respectively. Under the same braking pressure, the stiffness of the flexible arm in the F_n direction was higher than that in the F_φ and F_t directions, and the braking effect was better.

3. In the spatial bending state, the stiffness of the flexible arm nonlinearly increased with an increase in the braking pressure. The stiffness in the F_n, F_φ, and F_t directions was 8.6, 3.1, and 2.9 kN/m, respectively; thus, it increased by 6.1, 12.4, and 11.6 times, respectively. The stiffness in the three directions was lower than the corresponding stiffness in the initial state. When the braking pressure was constant, the stiffness of the flexible arm decreased as the bending angle increased.

4. In the elongation state, the stiffness of the flexible arm nonlinearly increased with an increase in the braking pressure. The stiffness the F_n, F_φ, and F_t directions was 19.8, 3.3, and 3.3 kN/m, respectively; thus, it increased by 8.2, 8.3, and 8.3 times, respectively. The stiffness in the F_n direction was slightly different from that in the initial state, and the stiffness in the F_φ and F_t directions was less than the corresponding stiffness in the initial state. The stiffness of the flexible arm was anisotropic. Under the influence of the positional distribution of the elastic cable, when the phase angles were 60°, 180°, and 300°, the stiffness curve fluctuates significantly.

5. The load experiment with other joints revealed that the designed flexible arm exhibited a large load capacity and stable motion performance.

6. The proposed variable stiffness method for the designed pneumatic flexible arm can be used to maintain the posture of a flexible robot, which is of great significance for the development of flexible robots. The flexible arm has the largest stiffness in the axial elongation direction, followed by the lateral stiffness. The axial compression direction has the weakest stiffness, which needs to be improved in subsequent research.

This work was supported, in part, by the Jilin province development and Reform Commission industry independent innovation ability (Grant No. 2020C021-4) and, in part, by the Science and Technology Research Project of the Jilin Provincial Department of Education (Grant No. JJKH20210034KJ).

The authors have no conflicts to disclose.

Xia Wang: Conceptualization (equal); Writing – original draft (equal). Dexu Geng: Conceptualization (equal); Project administration (lead); Writing – review & editing (equal). He Peng: Data curation (equal); Formal analysis (equal); Software (equal). Wenzhi Xu: Data curation (equal); Methodology (equal); Software (equal); Writing – review & editing (equal). Dandan Wang: Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Writing – review & editing (equal). Lizhong Zhang: Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.