Design and gait research of pneumatic soft quadruped robot

Abstract

Soft robots possess flexibility and, in theory, an unlimited number of degrees of freedom. These characteristics give them superior flexibility, making them better suited for navigating complex terrain, such as steep or rugged environments, compared to rigid robots. The design of the quadruped robot is based on the deformation principles of a pneumatically driven honeycomb network, with leg deformation controlled through air pressure regulation. Furthermore, the mathematical model of the quadruped robot is developed, including both a vertical elongation model and a bending elongation model of the leg structure, established through geometric analysis. An experimental platform for the quadruped robot system is then constructed, and experimental analyses are performed to assess the elastic coefficient, hysteresis performance, elongation capacity, and bending performance of one of the robot’s legs, verifying the accuracy of the mathematical model. Based on control requirements, LabVIEW is used to design static and dynamic gait programs for the quadruped robot, ultimately enabling control of its walking motion.

Introduction

The biggest difference between soft robots and traditional rigid robots is the different materials. Traditional robots are composed of rigid parts and have poor flexibility, while soft robots are mostly made of soft materials that can achieve large deformations. Theoretically, due to the elasticity of soft materials, soft robots can bend, twist, and stretch. ¹ Such movements can be regarded as soft robots with unlimited degrees of freedom, ² so soft robots have strong flexibility. And soft robots have two deformation modes: active deformation and passive deformation, so soft robots have strong adaptability to unstructured environments, ³ and can complete tasks that are difficult for traditional robots in various complex environments. Due to the soft and variable characteristics of the material, the soft robot can be controlled to change shape or stretch to achieve the target requirements.

These materials give soft robots several advantages, including high flexibility, strong adaptability to various environments, and improved human-robot interaction. Consequently, soft robots are better suited than traditional rigid robots for exploring unknown and hazardous environments, significantly reducing the risk to human workers in dangerous conditions and enhancing both efficiency and economic benefits. ⁴

The earliest soft actuator, the pneumatic artificial muscle, was proposed by McKibben in the 1950s. Commonly used materials for McKibben’s artificial muscles include silicone rubber and nylon fibers, which are lightweight, flexible, simple in structure, and cost-effective. ⁵ These actuators have found practical applications, particularly in rehabilitation and corrective therapies for individuals with disabilities.

On the basis of the McKibben-type pneumatic muscle, scholars have continuously optimized and finally designed a variety of different pneumatic artificial muscle drivers, such as bending, spiral/winding, rotary and sheet multi-axis drivers, etc. ⁶ The team of Professor Leng Jinsong of Harbin Institute of Technology and Professor Norman Werely of the University of Maryland conducted research on elongation and contraction pneumatic artificial muscle actuators, ⁷ and inspired by the arrangement of elephant trunk muscle fibers, by introducing additional constraints. The structure realizes a series of elongation and contraction bending or helical pneumatic artificial muscles. ⁸

The University of Chicago has developed a universal clamping hand that uses the principle of “blocking” technology. The inside of the clamping hand is mainly composed of granular materials. When the particles are in a flowable state, the clamping hand can fully cover the surface according to the shape of the object to be clamped, and then the granular material can be quickly transformed from a flowable state to a blocked state by pumping air. Solid, to complete the gripping of the object. When the gas returns to the holding hand, the particles return to the fluid state and the object can be released. ³

Using soft lithography technology, Majidi et al. at Harvard University developed a pneumatic soft crawling robot capable of mimicking biological crawling movements. ⁹ Building on this, Katzschmann et al. designed a soft crawling robot with a body length of 0.65 m, utilizing air-driven mesh soft actuators. ¹⁰ In 2011, Harvard’s Ilievski et al. created a soft six-finger gripper with a pneumatic grid driver. Made from silicone, this gripper can grasp and release objects through inflation and deflation, providing good flexibility for handling fragile and irregularly shaped items. However, it faces limitations, including complex manufacturing, low gripping force, and a restricted diameter range for grasped objects. ¹¹

Inspired by the way an elephant’s trunk grasps objects, Uppalapati and Krishnan developed a spiral pneumatic soft gripper, ¹² which first wraps around an object and then grips it by controlling inflation and deflation. The researchers modeled the gripper using both a pure helical model and a spatial Cosserat rod model. The soft gripper can handle objects as small as a 20 mm fluorescent lamp and as large as a 60 mm PVC pipe, effectively grasping slender objects within this range.

Temple University developed ACSR, a soft robot capable of wall crawling, using innovative pneumatic bio-inspired soft adhesion technology. ¹³ This robot features a soft double-layer structure with an embedded helical aerodynamic channel and a cavity base layer. Unlike traditional wall-crawling robots, which rely on air extraction to adhere to surfaces, the ACSR injects air into the top spiral channel to deform it into a stable dome shape, creating a negative pressure state in the cavity. This allows the robot to adhere to walls regardless of surface conditions like humidity, smoothness, or incline. ACSR can carry loads over 200 g (more than five times its weight) while climbing vertically at a speed of 286 mm/min, making it a breakthrough for soft robots in tasks like window cleaning and underwater detection in harsh environments.^14–18

Tang et al. from Temple University proposed a gas-driven climbing soft robot that can operate both on land and underwater. The robot functions by utilizing a double-layer structure with a helical channel. When air pressure is introduced into the helical channel, it lifts to form a stable dome, allowing the robot to adhere to the climbing surface. ¹³ Meanwhile, Marchese et al. at MIT designed and produced a bionic fish soft robot. ¹⁹ The body of this bionic fish is made of silicone resin, and it features an internal air storage device. By controlling the deflation of this device, varying air pressures can be applied at different points within the fish’s body, enabling the robot to achieve diverse deformations and movements.

Inspired by the flexibility of an elephant’s trunk, Chen Xiaoping’s team at the University of Science and Technology of China developed a honeycomb pneumatic network structure for a soft manipulator. ²⁰ This manipulator is powered by multi-layer airbags, and its movement is controlled by inflating and deflating the air, allowing it to perform various tasks, such as opening doors, drawers, and twisting bottle caps. Even in the presence of human interference, the manipulator can complete tasks with precision. Additionally, researchers from Zhejiang University have created a worm robot using highly elastic materials, ²¹ mimicking the crawling motion of worms, with deformation capabilities allowing for end-to-end connection.

Li et al. designed a tetherless bionic quadruped robot with a highly flexible torso and a dual-cavity pre-inflated soft actuator. Although the robot’s stride length during walking is relatively short, the structure enables it to perform basic functions such as walking. ²² In contrast, the quadruped robot discussed in this paper has a longer stride length, giving it superior walking performance. While the robots designed by Muralidharan et al. are capable of performing multiple gaits, such as walking and jumping, they suffer from poor stabilization during walking. ²³ In comparison, the quadruped robot described here can maintain stability while walking without the need for additional assistive measures.

Currently, the integration of motors with soft structures is almost always achieved by using motor rotation to pull cables, driving the soft robot to perform bending and other actions. However, the motors typically used for such tasks are relatively large and bulky, which reduces the flexibility of the soft robot. ²⁴ By adopting smaller motors and combining them with pneumatically driven soft robots, it is believed that the precise rotational movements provided by the motors can complement the pneumatic actuation, resulting in an effective synergy between the two.

This paper mainly proposes a pneumatic soft quadruped robot based on the hexagonal structure of the honeycomb network. Its elasticity coefficient, hysteresis performance, single-leg elongation property, single-leg unidirectional bending performance, single-leg S-shaped bending performance, and hysteresis performance in the outrigger structure are investigated. By exploring these properties of the quadruped robot, it provides new possibilities for the creation of soft quadruped robots in the future.

Structural design and optimization of quadruped robot

The key component of the quadruped robot is its leg structure, which is mainly divided into two parts: the hexagonal honeycomb network cavity and the internal gas driver. The gas driver is folded in a specific manner and housed within the honeycomb network cavity. By controlling the inflation of the gas driver, the robot’s legs are able to deform. The model of a single leg is illustrated in Figure 1. The outer black frame represents the hexagonal honeycomb network cavity, while the blue section inside the cavity indicates the gas driver. To facilitate the connection between the legs and the robot’s torso in the subsequent structural design, a clamping mechanism is incorporated within the honeycomb network cavity.

Figure 1.

Single leg model.

Analysis of cavity structure parameters

The hexagonal honeycomb network cavity structure of the robot’s leg is made from TPU material, which is directly fabricated using 3D printing. TPU, or thermoplastic polyurethane, is a soft and elastic material with a foam rubber-like consistency. Its properties include a density of 1200 kg/m³, a hardness of 70 A°, a Poisson’s ratio of 0.38, and a Young’s modulus of 80 MPa.

We discovered that the wall thickness of the honeycomb cavity structure is a crucial parameter influencing the leg’s deformability. Wall thickness refers to the thickness of the walls of each chamber that comprises the quadruped robot’s leg. An optimal wall thickness enhances the robot’s walking performance by balancing flexibility and structural integrity.

To analyze the impact of wall thickness on the deformation performance of the robot legs, several honeycomb cavity models with varying wall thicknesses were created using SolidWorks and saved as x.t format files, with wall thickness as the only variable. These models were then imported into Ansys software for analysis. ²⁵ The material properties for the models were set with a density of 1200 kg/m³, a Young’s modulus of 100 MPa, and a Poisson’s ratio of 0.38. Figure 2(a) and (b) display the finite element simulation results, showing the deformation of the honeycomb cavity under identical conditions but with two different wall thickness parameters.

Figure 2.

Finite element analysis of honeycomb cavities with different wall thicknesses and opening depths. (a), (b) Deformation of the honeycomb cavity under different wall thickness conditions (c) and (d) show the deformations with opening depths of 1.5 mm and 0 mm, respectively.

Finally, deformation data for multiple sets of cavities with varying wall thicknesses were obtained. Based on this data, the relationship between wall thickness and the deformation of the honeycomb cavity under identical conditions of size and air pressure is plotted in Figure 3. The results indicate that as the wall thickness increases, the deformation of the leg structure decreases, leading to poorer deformation performance.

Figure 3.

Relationship between wall thickness and deformation of honeycomb cavity.

Additionally, to verify that the improved hexagonal honeycomb cavity design optimizes the deformation performance of a single leg, multiple cavity models with varying opening depths were created in SolidWorks for finite element analysis. In this case, the opening depth was the only variable, and deformation data were obtained accordingly.

Given that the gas actuator will be sealed during fabrication, it is essential to model each quadruped robot leg so that the gas actuator can be smoothly placed into its corresponding chamber without popping out during inflation. The opening depth refers to the small notches designed in each chamber to ensure that the gas actuator is properly seated and to enhance the cavity’s deformation capabilities. Figure 2(c) and (d) show finite element simulation results for models with different opening depths under identical conditions, with opening depths of 1.5 and 0 mm, respectively.

The relationship between various opening depths and leg deformation under the same conditions is plotted in Figure 4. Based on the simulation results and the corresponding relationship curve, it can be concluded that removing the internal opening enhances the cavity’s deformation performance.

Figure 4.

Relationship between opening depth and deformation of honeycomb cavity.

Optimal design of honeycomb cavity structure

TPU, known for its high softness and flexibility, was selected as the primary material for the design of the honeycomb cavity structure in the quadruped soft robot’s legs. The first step before 3D printing was to create a 3D model of the honeycomb network for a single leg using SolidWorks. The model was then saved in STL format and loaded into the 3D printer.

Each leg model consists of 64 hexagonal honeycomb cavities on both the front and back sides. The quasi-hexagonal honeycomb cavity design is based on a standard hexagonal honeycomb network structure, with an added rectangular notch measuring 1.5 mm × 0.43 mm on both vertical sides, as shown in Figure 5. The purpose of these rectangular notches is to facilitate the insertion of the gas driver into the cavity and enhance the deformation performance of the structure. The horizontal side length of the improved quasi-hexagonal honeycomb cavity is 10 mm, while the hypotenuse measures 15 mm.

Figure 5.

Improved honeycomb hexagonal network structure.

Gas driver design

The gas driver consists of a strip-shaped airbag made from two layers of plastic film of a specific thickness and a soft air tube. These soft air tubes are arranged and fixed between the two layers of strip-shaped plastic films in a specific order and position, serving the purpose of guiding air through the gas driver. The principle behind the gas driver is that when air pressure is introduced into it, the gas driver expands, which in turn causes the external honeycomb network cavity structure to deform. Figure 6 illustrates a comparison of the hexagonal honeycomb cavity containing the gas driver before and after inflation. Upon inflation, while the wall thickness and side length of the honeycomb network cavity remain unchanged, the height of the cavity increases due to the deformation.

Figure 6.

Effects of a honeycomb cavity before and after inflation.

The design of the gas driver is illustrated in Figure 8. Each gas driver is equipped with eight short flexible air pipes and seven long flexible air pipes. Additionally, a section of an air guide tube is fixed at the air inlet, which connects to an external inflatable device. After securing the air guide tube in place, the air inlet is sealed. By folding the gas driver along the dotted lines shown in Figure 7, the gas driver can be inserted into eight adjacent honeycomb cavities. This arrangement ensures that the gas driver can effectively expand and drive the deformation of the surrounding honeycomb cavities.

Figure 7.

Design drawing of gas driver.

The pressure-bearing capacity of the gas actuator is influenced primarily by two factors: the thickness of the plastic film and the width of the side seal of the gas actuator. During the research process, PE plastic films with thicknesses of 0.08, 0.24, and 0.32 mm were selected for testing. The specific parameters of PE are as follows: density 0.95 g/m³, hardness 50 A°, Young’s modulus 500 MPa, and Poisson’s ratio 0.41. A simple air pressure limit test was conducted to determine the maximum air pressure that each plastic film thickness could withstand. The results of this test are presented in Table 1.

Table 1.

The maximum withstand air pressure of plastic film with three thicknesses.

Plastic film thickness (mm)	0.08	0.24	0.32
Maximum withstand air pressure (bar)	0.04	0.26	0.34

According to the data in Table 1, the plastic film with a thickness of 0.32 mm demonstrates the highest pressure-bearing capacity. However, during experimentation, it was observed that this thicker plastic film posed challenges in sealing the edges, leading to loose seals and air leakage. Additionally, compared to the other two thicknesses, the 0.32 mm film exhibited higher hardness, making it more difficult to fold into the gas driver structure. As a result, a 0.24 mm thick plastic film was ultimately chosen for constructing the gas driver due to its balance between pressure resistance and ease of handling.

The laminator used for sealing the edges provides seals in three widths: 2, 3, and 5 mm. For each seal width, a set of gas actuators was fabricated and subjected to a pressure limit test. Table 2 presents the maximum air pressure that each gas driver could withstand, corresponding to the different seal widths.

Table 2.

Maximum withstand air pressure of drivers with different plastic package widths.

Edge sealing width (mm)	2	3	5
Maximum withstand air pressure (bar)	0.05	0.08	0.15

Based on the results shown in Table 2, it was found that the pressure-bearing capacities of the gas actuators with 2 and 3 mm plastic packaging widths were both less than 0.1 bar, and the sealing performance was poor, leading to frequent air leaks. Consequently, the 5 mm seal was chosen for the edge of the gas driver, as it provided better sealing and improved pressure resistance.

Structural deformation of the soft mechanical leg

To enable the soft quadruped robot proposed in this paper to perform basic movements such as walking in a straight line, turning, and other motions, the soft outriggers must be capable of stepping, lifting, and independently moving the legs. The outriggers are required to achieve different levels of elongation and perform multi-directional movements, including bending and forming an “S” shape. The mechanical outrigger is divided into two parts: the large and small leg segments. Each leg segment consists of an eight-layer hexagonal cavity structure. Eight pneumatic grid drivers are embedded in the corresponding regions of the soft mechanical legs, according to the design specifications. By applying different air pressures to the various actuators, the soft outriggers can deform accordingly, allowing for the desired leg movements. Figure 8(a) illustrates the side-bending effect of the outrigger, Figure 8(b) demonstrates forward and backward bending, and Figure 8(c) shows the “S” shape used during walking. Additionally, two long strips of soft pads made from silicone are installed at the bottom of the outriggers to increase friction between the legs and the ground, preventing the quadruped robot from slipping during motion. The pads are simple in structure and lightweight, adding negligible weight to the outrigger, ensuring they do not hinder the movement of the robot.

Figure 8.

Soft mechanical leg and deformation diagram: (a) lateral bending, (b) forward bending, and (c) “S” type.

Overall structure of quadruped robot

Considering material cost and time efficiency, the robot’s four legs are 3D-printed. The body torso, referred to as the base, is designed using acrylic transparent plastic, as shown in Figure 9. The connection between the four legs and the torso is achieved through a clamping structure located at the top of the legs. Specifically, the four corners of the base structure are embedded into the clamping mechanisms at the top of each leg and secured with screws. Furthermore, the four legs have been treated with an anti-skid design, with rubber pads added at the bottom of each leg to enhance grip and stability during movement. This ensures better traction and prevents the robot from slipping during operation.

Figure 9.

Overall body structure design of the robot.

Mathematical modeling of quadruped robot

When modeling and analyzing a leg, all cavities within the leg are inflated simultaneously, causing the leg to become upright and elongated, as illustrated in Figure 10. In this figure, L₀ represents the initial length of a single leg before inflation. Once inflated, the length of the leg extends to L. This inflation-induced elongation allows the leg to perform various movements and actions essential for the robot’s operation.

Figure 10.

Upright stretched state.

According to the relationship between pressure and force, the force F of a leg during upright extension should be:

$$F=\mathrm{PS}$$

(1)

In the formula (1), P refers to the input pressure, and S is the sum of the force-bearing areas in all the honeycomb structure cavities of a single leg during the inflation process.

After being inflated, the length of a single leg will elongate and change, which conforms to Hooke’s law. The elastic coefficient of a single leg structure is k, and the elongation change is the difference between L and L₀, then:

$$F=\mathrm{kx}$$

(2)

$$x=L-L_0$$

(3)

Combining formulas (1), (2), and (3), the air pressure elongation model of a single leg structure is finally obtained as:

$$L=L_0+\frac{1}{k}\mathrm{PS}$$

(4)

The bending state of a single leg can be categorized into two types: one-way bending and S-shaped bending. In one-way bending, the leg bends in a specific direction due to uneven inflation. Think of the leg as composed of two rows of honeycomb cavities: one row is fully inflated with air pressure, while the other row is either not inflated or filled with low air pressure. This difference causes the leg to bend toward the row with less air pressure, resulting in one-way bending.

S-shaped bending, as the name suggests, results in the leg taking on an “S” shape. This occurs when the upper or lower half of the honeycomb cavity structure in one row is filled with air pressure, while the opposite diagonal cavities in the other row are also filled with the same air pressure. The remaining cavities are either not inflated or filled with low air pressure, creating the distinctive S-shaped curve.

To better understand the bending state of a single leg, it helps to consider the leg as being divided into distinct zones. Figure 11 provides an explanatory diagram that shows the four zones of a single leg structure, clarifying how different sections of the leg contribute to these bending movements.

Figure 11.

Illustration of four partitions of a single leg.

One-way bending is to divide the four partitions into two groups: the first group 1, 3; the second group: 2, 4. When the two groups of partitions are filled with different air pressures, the single leg will bend to the side of the partition with the lower air pressure. As shown in Figure 12, the first and third partitions of the first group are filled with air pressure $P_1$ , and the second and fourth partitions of the second group are filled with air pressure $P_2$ , and the single leg is bent ( $P_1$  >  $P_2$ ).

Figure 12.

One-way bending of one leg.

The derivation and establishment process of the model for one-way bending and elongation of a single leg is as follows:

As shown in Figure 12, points A and B represent the two endpoints of the single-leg model, and O is the midpoint at the bottom of the leg. The midpoints of the two honeycomb structures are denoted as M and N, with OM represented by x. During one-way bending, when the air pressure is equally distributed into each cavity, the leg bends with equal degrees of curvature at both the top and bottom joints. The total bending angle of the single leg is recorded as 2θ.

In this case, the section of the leg filled with air pressure P₁ corresponds to an arc length L₁ with a radius R₁, while the section filled with air pressure P₂ corresponds to an arc length L₂ with a radius R₂. The overall arc length of the single leg is denoted by L_M, and the corresponding overall radius is R_M. The relationships between these parameters can be derived mathematically, connecting the bending angle, arc lengths, and radii, based on the air pressure applied to the leg’s different sections.

$$R_1-R_2=\mathrm{MN}=2x$$

(5)

According to the relationship between arc length and radius, we can get: R_M

$$L_1=\frac{2\theta \pi R_1}{180°}$$

(6)

$$L_2=\frac{2\theta \pi R_2}{180°}$$

(7)

$$L_M=\frac{2\theta \pi (R_1-x)}{180°}$$

(8)

Simultaneous formulas (5)–(7) can be used to obtain the bending angle θ of half of the bent single leg:

$$\theta =\frac{180°(L_1-L_2)}{4\pi x}$$

(9)

$$L_M=\frac{L_1+L_2}{2}$$

(10)

Analyze the front view of the cavity when all the partitions of a single leg are inflated, and simplify the quasi-hexagon to a hexagon. Figure 13 is a partial simplified front view after all the cavities are inflated, where d₁ is the hexagon of the honeycomb structure The length of the horizontal side, d₂ is the length of the hypotenuse of the hexagon of the honeycomb structure. No matter how much the air is inflated and the size of the inflated air, d₁ and d₂ will not change. After inflating, record the height in a single cavity as L_S, and record the stretching angle as α.

Figure 13.

Partial simplified front view of cavity inflation.

Through the geometric relationship in Figure 13, it can be obtained:

$$\mathrm{MN}=\mathrm{MP}+\mathrm{PQ}+\mathrm{QN}=2x=d_1+\mathrm{PQ}$$

(11)

$$\mathrm{PQ}=d_2·\cos \alpha =d_2·\sqrt{1-{\sin }^2\alpha }$$

(12)

$$\sin \alpha =\frac{L_S}{2d_2}$$

(13)

The number of cavities in a column of honeycomb network is recorded as n, and the wall thickness of each cavity is recorded as d, so the calculation of the height L_S of a cavity is as follows:

$$L_S=\frac{L_1-(n+1)d}{n}$$

(14)

From formula (11) to (14) can be obtained:

$$2x=d_1+d_2\sqrt{1-{\left[\frac{L_1-(n+1)d}{2n{d}_2}\right]}^2}$$

(15)

Then substitute formula (15) into formula (9) to get the relationship between θ and bending arc length L₁, L₂:

$$\theta =\frac{180°(L_1-L_2)}{2\pi *\{d_1+d_2\sqrt{1-{\left[\frac{L_1-(n+1)d}{2n{d}_2}\right]}^2}\}}$$

(16)

From the formula (6) and (16), the bending radius $R\prime$ of the air pressure $P\prime$ partition can be obtained:

$$R_1=\frac{L_1}{L_1-L_2}\{d_1+d_2\sqrt{1-{\left[\frac{L_1-(n+1)d}{2n{d}_2}\right]}^2}\}$$

(17)

Finally, the radius R_M of the overall single-leg bending can be calculated based on the difference between R₁ and x:

$$\begin{array}{c}\multicolumn{1}{c}{R_M=R_1-x}\\ =\left[\frac{L_1+L_2}{2(L_1-L_2)}\right]\{d_1+d_2\sqrt{1-{\left[\frac{L_1-(n+1)d}{2n{d}_2}\right]}^2}\}\hfill \end{array}$$

(18)

To cause S-shaped bending deformation of a single leg is to divide the four partitions of a single leg into two groups, and its specific grouping can be regarded as a group of diagonal lines, that is, the first group: 1, 4 partitions, the second group: 2, 3 partitions. Inflate the second and third zones with air pressure, and the first and fourth zones are inflated to 0. The deformation effect is shown in Figure 14. The deformation of a single leg is like an “S.” This S-shape is called “pre-S.” Sections 2 and 3 are not inflated, and the deformation of the inflated legs of 1 and 4 is a left-right symmetrical structure with the “front S,” which is called “back S.”

Figure 14.

S-shape deformation effect of single leg.

One leg is regarded as two parts of the upper and lower leg sections, the partition 2 of the upper leg section is filled with air pressure P_a, and the section 3 of the lower leg section is filled with air pressure P_b, so the bending of the upper and lower leg sections has different bending arc lengths and radii, angle. The radius, arc length, and angle of the upper femur are R_a, L_a, θ_a, and the radius, arc length, and angle of the lower femur are R_b, L_b, θ_b.

Figure 15 presents a simplified model of the S-shaped deformation of a single leg. The displacement change of the quadruped robot is influenced by the amount of deformation in the horizontal direction due to the S-shaped bending. In Figure 15, S represents the step length of the single leg in the horizontal direction during the S-shaped deformation, while H represents the height in the vertical direction after the S-shaped deformation. These parameters, S and H, are critical in determining how much the robot moves and lifts its leg during each step, playing a key role in the robot’s overall locomotion and stability.

Figure 15.

Simplified diagram of single-leg S-shape deformation model.

According to the geometric relationship in Figure 16, it can be obtained:

$$S_a=R_a(1-\cos {\theta }_a)$$

(19)

$$S_b=R_b(1-\cos {\theta }_b)$$

(20)

$$H_a=R_a\sin {\theta }_a$$

(21)

$$H_b=R_b\sin {\theta }_b$$

(22)

Figure 16.

Soft leg movement diagram.

Among them, $R_a$ , ${\theta }_a$ , $R_b$ , ${\theta }_b$ are obtained according to formulas (16) and (17).

And the chord length of the upper and lower leg joints can be obtained:

$$d_{\mathrm{k}}=R_k\sin \left(\frac{{\theta }_{\mathrm{k}}}{2}\right)(k=a,b)$$

(23)

Kinematics analysis of single leg model

The Piecewise Constant Curvature (PCC) model, also known as the equivalent D-H method, is widely used for kinematic analysis of soft-bodied robots. In this method, the soft leg is approximated as a series of circular arcs with varying curvatures, which are described using parameters such as arc lengths, bending angles, and rotation angles. These angular parameters act as virtual joint variables, enabling the kinematic analysis of the soft leg to be translated into a relationship between these virtual joint variables and the virtual joint space. The goal is to determine the maximum reachable range by the end of the soft leg, essentially defining the leg’s workspace. Therefore, only forward kinematic analysis is performed in this approach.

In this study, the soft leg structure is modeled as a cylindrical segment. A spatial coordinate system is established at the center of the plane on the side containing the connecting device, adhering to the right-hand rule, as illustrated in Figure 16. This coordinate system serves as the basis for analyzing the leg’s motion and deformation.

Taking the bending analysis as an example, the bending plane is indicated by the green dashed box. The angle between the fixed plane and the moving plane is the bending angle α, and the angle between the bending plane and the XOZ plane is the rotation angle β. During the bending process, the length changes from the initial l₀ to l₁, the bending arc radius is r, and the center point of the moving plane has the coordinates of O₁(x, y, z), which leads to the relevant equations.

$$r=\frac{l_1}{\alpha }$$

(24)

$$o_1=\left(\begin{array}{c}x\hfill \\ \multicolumn{1}{c}{y}\\ \multicolumn{1}{c}{z}\end{array}\right)=\left(\begin{array}{c}(1-\cos \alpha )·\cos \beta \hfill \\ \multicolumn{1}{c}{(1-\cos \alpha )·\sin \beta }\\ \multicolumn{1}{c}{\sin \alpha }\end{array}\right)·\frac{l_1}{\alpha }$$

(25)

Under the constant curvature model, the displacement of the moving surface in space can be obtained by translating the fixed surface center point O to the moving surface center point O₁, whose translation matrix is:

$$T_1=\left(\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & (1-\cos \alpha )·\cos \beta ·\frac{l_1}{\alpha }\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & (1-\cos \alpha )·\sin \beta ·\frac{l_1}{\alpha }\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & \sin \alpha ·\frac{l_1}{\alpha }\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right)$$

(26)

Rotating the coordinate system by β degrees around the Z-axis yields a rotation matrix R₁ of:

$$R_1=\mathrm{Rot}(Z,\beta )=\left(\begin{array}{cccc}\cos \beta \hfill & -\sin \beta \hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\sin \beta }& \cos \beta \hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right)$$

(27)

Rotating the coordinate system by α degrees around the obtained Y-axis gives the rotation matrix R₂ as:

$$R_2=\mathrm{Rot}(Y,\alpha )=\left(\begin{array}{cccc}\cos \alpha \hfill & 0\hfill & \sin \alpha \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin \alpha }& 0\hfill & \cos \alpha \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right)$$

(28)

Rotating the coordinate system by −β degrees around the newly obtained Z-axis yields a rotation matrix R₃ of:

$$R_3=\mathrm{Rot}(Z,-\beta )=\left(\begin{array}{cccc}\cos \beta \hfill & \sin \beta \hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin \beta }& \cos \beta \hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right)$$

(29)

The chi-square transformation matrix T for moving the coordinate system from the fixed plane to the moving plane can be obtained by associating equations (26)–(29) as:

$$\begin{array}{c}T=T_1{R}_1{R}_2{R}_3=\hfill \\ \left(\begin{array}{cccc}{C}_2^2{C}_1+S_2^2\hfill & C_2{C}_1{S}_2-C_2{S}_2\hfill & C_2{S}_1\hfill & (1-C_1)C_2·\frac{l_1}{\alpha }\hfill \\ \multicolumn{1}{c}{C_2{C}_1{S}_2-C_2{S}_2}& S_2^2{C}_1+C_2^2\hfill & S_2{S}_1\hfill & (1-C_1)S_2·\frac{l_1}{\alpha }\hfill \\ \multicolumn{1}{c}{-C_2{S}_1}& -S_2{S}_1\hfill & C_1\hfill & S_1·\frac{l_1}{\alpha }\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right)\hfill \end{array}$$

(30)

where S₁ = sinα, S₂ = sinβ, C₁ = cosα, C₂ = cosβ.

The position of the end of the floppy leg was determined from the single-leg positive kinematic expression and the variation of the virtual joint variables. Experimentally measured cylinder length variations ranged from 135 to 230 mm. Statistical modeling using MATLAB simplifies the study and reduces computational complexity by transforming complex problems into simulations and calculations with random numbers and their characteristics. Programming was done to implement the statistical method to derive the workspace for the foot end position and generate a 3D plot, as shown in Figure 17.

Figure 17.

Workspace of foot.

Experimental research on quadruped soft robot

The design purpose of the system experiment platform is mainly to be able to conduct simple open-loop experiments, obtain experimental data related to the single-leg model through the experimental platform, collect the input and output signals of the actual air pressure in the electric proportional valve, and then verify the Whether the model built in the previous chapter is valid, and realize the overall walking motion control of the quadruped robot.

Design of control scheme for quadruped robot system

According to the system’s control requirements, the design of the experimental platform involves coordination between the air path and the circuit design.

Air circuit design: The air compressor (air pump) provides the air source for the entire pneumatic control system, outputting high-pressure air. For safety, this high-pressure air is regulated to 1 atmosphere via a pressure-reducing valve. After regulation, the air is directed to the quadruped robot through an electric proportional valve, which adjusts the air pressure as needed.

Circuit design: A constant voltage power supply is used to convert the 220-V AC power into 24-V DC power, which is required to operate the electric proportional valve. Additionally, a data acquisition card, PCI-1724U, is installed in the control computer to receive control signals from the LabVIEW program. These signals are then sent as inputs to the electric proportional valve to regulate the air pressure directed into the quadruped robot. A second acquisition card, USB-6216, is used to monitor the voltage signal and provide feedback to the control program, ensuring that the air pressure reaches the desired set value. This feedback loop enables the pneumatic control system to accurately control the soft quadruped robot’s walking motion.

The overall control system design is illustrated in the block diagram shown in Figure 18.

Figure 18.

Overall control schematic diagram.

The entire pneumatic software quadruped robot control system mainly includes one air compressor, one pressure reducing valve, one constant voltage power supply, one control computer, eight electrical proportional valves, and two data acquisition cards.

Selection of hardware equipment

According to the working principle of the air pressure control system, the experimental platform architecture is designed. According to the signal types involved in the whole system working process and the specific control process, the instruments required for the summary experiment platform mainly include:

1. Controller: one computer running LabVIEW control program;

2. Voltage signal: one piece of data acquisition card that can output 0–10 V voltage signal; one piece of data acquisition card that can input 0–10 V voltage signal;

3. Power supply: one constant voltage power supply that can convert 220 V AC into 24 V DC;

4. Air source signal: air compressor, one set; pressure reducing valve, one set;

5. Air pressure signal: electric proportional valve, eight pieces;

The hardware equipment selection information of the whole experimental platform is shown in Table 3:

Table 3.

Hardware equipment selection table.

Name	Model	Function
PC	Founder 3011	Execute the LabVIEW control program
Data acquisition card	PCI-1724U	Multi-channel analog output
Data acquisition card	USB-6216	Signal monitoring feedback
Air compressor	OTS-1100	Provide compressed air
Pressure reducing valve	Rexroth KIE 1404	Compressed air decompression
Electric proportional valve	ITV0030-3BS	Control air pressure

The setup of the whole experimental system is shown in Figure 19. Figure 19(a) is the connection diagram of the hardware equipment, and Figure 19(b) is the actual quadruped robot, and the connection between the robot and the electric proportional valve is through the air duct.

Figure 19.

Experimental system establishment. (a) Hardware equipment system (b) Actual quadruped robot model.

Experimental research on the performance of quadruped soft robot

Elastic coefficient test experiment

The elastic coefficient of the single-leg structure was measured using a dynamometer and a vernier caliper. A leg structure without a pneumatic actuator was placed on a smooth table surface, with the clamping structure end of the leg fixed and the bottom end connected to the dynamometer. The bottom end of the leg was then pulled to displacements of 45, 80, 105, 125, and 130 mm, respectively, and the pulling force was recorded. Each displacement measurement was conducted 10 times, and the final result was the average of the 10 experimental data points. Based on equations (31)–(34), the displacement-stiffness curve (Figure 20), strain-stress curve (Figure 21), and strain-elastic modulus curve (Figure 22) were plotted.

$$K=\frac{F_{i+1}-F_i}{x_{i+1}-x_i}(i=0,1,2,3,...)$$

(31)

Figure 20.

Displacement-stiffness relationship curve.

Figure 21.

Strain-stress relationship curve.

Figure 22.

Strain-elastic modulus relationship curve.

In the formulas: K represents the stiffness, F_i+1 and F_i denote the forces at two adjacent positions X_i+1 and X_i, respectively, while X_i+1 and X_i represent the displacements of the single-leg structure at these two positions.

$$\sigma =\frac{F_i}{A}(i=0,1,2,3,...)$$

(32)

In the equation: σ represents the stress, F_i is the force measured at each displacement for the single-leg structure, and A = 0.4653 m² is the cross-sectional area.

$$\epsilon =\frac{\mathrm{\Delta }{L}_i}{L_0}(i=0,1,2,3,...)$$

(33)

In the equation: ε represents the strain, L_i is the displacement of the single-leg structure, and L₀ = 0.13 m is the initial length of the single-leg structure.

$$E=\frac{\mathrm{\Delta }\sigma }{\mathrm{\Delta }\epsilon }=\frac{{\sigma }_{i+1}-{\sigma }_i}{{\epsilon }_{i+1}-{\epsilon }_i}(i=1,2,3,...)$$

(34)

In the equation: E represents the elastic modulus, σ _i and σ_i+1 represent the stresses of the single-leg structure at two adjacent positions (strain states), and ϵ _i and ϵ_i+1 represent the strains of the single-leg structure at the two adjacent positions.

Hysteresis performance test experiment

Since the leg structure of the robot is a nonlinear structure, the process of inflation and deflation is a one-dimensional deformation. Based on this, the hysteresis performance of the leg structure is explored, that is, the deformation effect of a single leg from inflation to deflation. Figure 23 plots the elongation of the single-leg structure of the robot during the inflation process and the deflation process under different air pressures with a gradient of 0.05 bar. The two curves in the figure show that the hysteresis of the single-leg structure is greater at low pressure and less hysteresis at high pressure.

Figure 23.

Hysteresis characteristic curve.

Single leg elongation characteristic test

The experimental test of the elongation characteristics of a single leg is to uniformly inflate all the partitions of the single leg at the same time, and the single leg produces an upright elongation change. In Figure 24, the comparison between the non-inflated state of a single leg and the saturated state of elongation of a single leg after being inflated with air pressure is shown. The length of a single leg in the uninflated initial state is 130 mm, and the saturated length after inflation is about 260 mm.

Figure 24.

Elongation property experiment.

In this experiment, the elongation length of a single leg was tested under different air pressures. In order to visually express the change of the upright elongation length L of a single leg obtained in the experiment with the air pressure P, all partitions of a single leg were filled with different air pressures in Figure 25. The experimental results and theoretical model calculation curves between the length of a single leg upright extension.

Figure 25.

Experimental results and theoretical model.

The following conclusions can be drawn from Figure 25:

1. When the air pressure is <0.05 bar, the elongation and deformation of the single leg will not occur. Only when the air pressure reaches 0.05 bar, the elongation and deformation of the single leg will increase, and obvious elongation will occur. This is because the cavity structure itself has a certain height, and when the air pressure is less than 0.05 bar, the deformation of the internal gas driver cannot reach the original cavity height, which is not enough to deform the single-leg structure.

2. The relationship between the length of the upright elongation of a single leg and the air pressure basically shows a monotonically increasing change. As the air pressure increases, the length of the single leg increases gradually;

3. As the air pressure increases, the increase rate of the length of the single leg decreases. When the air pressure reaches 0.3 bar, the vertical elongation of the single leg gradually reaches the saturation value of 260 mm.

4. The framework load and ground reaction force are closely related to the elongation of a single leg. The framework load refers to the vertical downward force applied to the single leg. At lower air pressures, the effect of the framework load is more significant, limiting the elongation of the single leg and causing the experimental results to fall below the theoretical model curve. As the framework load is relatively small, with the increase of air pressure to 0.1 bar, the air pressure pushes the single leg to elongate further, and the effect of the framework load becomes negligible. Additionally, due to material constraints in the actual model, the length of the single leg structure remains unchanged after the air pressure increases to a certain level. The ground reaction force is the upward force generated when the single leg comes into contact with the ground. In the early stage, it can balance the framework load, allowing the single leg to quickly surpass the low air pressure phase. In the later stage, it ensures the single leg structure quickly reaches its maximum length and provides greater load capacity for the robot. The interaction of both forces together determines the elongation of the single leg.

According to the single-leg upright elongation model, the elongation length L and the air pressure P should have a linear relationship with a slope of 1/k. The main reasons for the error between the experimental test curve and the model calculation curve are:

1. There is minimum deformation of the leg structure to start the air pressure;

2. The actual single-leg structure is not linear, and the elastic coefficient k of the single-leg structure is an increased value;

3. During the inflation process, the stress area S increases, but the increment is extremely small, and the increase speed is far less than the increase speed of k;

4. After 0.25 bar, the discrepancy between the theoretical and actual results gradually increases. This is due to the limitations of the materials used in the single-leg structure and the pneumatic actuator, where deformation stops once the air pressure reaches a certain level.

End-effector output force experiment

The output force at the leg end is an important indicator for measuring the anti-interference performance of a quadruped robot’s single leg or overall system, reflecting the actuator’s working intensity. The measurement of the output force at the leg end uses the stiffness test method, as shown in Figure 26. The robot’s leg is laid flat with the top fixed to the test platform, and a force gauge is used to pull the end to the elongation corresponding to the given air pressure. When the force gauge reading stabilizes, the data is recorded as the output force corresponding to the input air pressure. Due to the minimal deformation pressure, the output force reading only starts when the input pressure exceeds 3 kPa. As the air pressure increases, the output force increases correspondingly. When the pressure reaches 35 kPa, the output force reaches a maximum of 73 N, and the maximum air consumption of the single leg during testing is 0.8 L.

Figure 26.

(a) End longitudinal output force experiment and (b) relationship between air pressure and longitudinal output force of the leg.

One-way bending performance test of one leg

Divide the first and third divisions of the single leg into a group, and the second and fourth divisions into a group. The two groups of partitions are filled with unequal air pressure to cause bending and elongation deformation of the single leg. Like the upright elongation deformation of a single leg, the bending deformation of a single leg also has a saturation value. Figure 27 shows the saturated state where the second and fourth partitions are not inflated, and the first and third partitions are filled with air pressure. The bending angle of the single-leg bending saturation state is 136°, and the radius is 82 mm.

Figure 27.

Bending elongation saturation state.

In this experiment, the bending angle θ and bending radius r of the single leg were measured when the second and fourth partitions were inflated to 0, and the first and third partitions were filled with different air pressure values P, and will be drawn in Figure 28.

Figure 28.

(a) The bending angle θ and (b) bending radius r of a single leg under different air pressures.

From Figure 28 it can be concluded that:

1. The bending performance and elongation characteristics of a single leg are the same, and there is a minimum deformation air pressure. The air pressure charged is too small to make the single leg bend and deform, and the bending angle of the single leg is very small;

2. There is a monotonous increase between the bending angle of one leg and the air pressure. With the increase of air pressure, the bending angle of one leg increases gradually;

3. When the air pressure reaches 0.3 bar, the bending deformation of the single leg enters a saturated state, and the saturated bending angle is about 135°;

4. When the single leg is not inflated, there is no bending radius, and when it starts to inflate to reach the bending deformation air pressure of the single leg, it will produce a greatly abrupt bending radius value. After that, as the air pressure increases, the bending deformation of the single leg increases, and the bending radius gradually decreases until the bending deformation of the single leg reaches a saturated state, and the bending radius approaches a stable value.

The theoretical calculation curve and the experimental test result curve of the relationship between air pressure and bending angle are plotted in Figure 29 for comparison, which verifies the validity of the bending model derived above. It can be seen from the figure that in the low pressure section, the experimental curve conforms to the linear law. In the high-pressure section, there is a large difference between the model calculation results and the experimental test results. The main reason is that the single-leg structure does not conform to the linear structure in the high-pressure section, which leads to larger errors when the air pressure is higher. Here, Elongation refers to L_M.

Figure 29.

Theoretical model calculation and experimental test curves.

Single-leg S-shaped bending performance test

In addition to testing the unidirectional bending performance of one leg, it is also tested for the performance of S-shaped bending. As shown in Figure 30, it is a single-leg S-shaped bending deformation performance test experiment, with 1 and 4 partitions as a group, and 2 and 3 partitions as a group, respectively filling the two groups of partitions with different air pressures P₁ and P₂, and measuring at this time Single leg stride length and height.

Figure 30.

S-shaped bending performance test experiment diagram.

As shown in Figure 31, for the inflation and deflation process of a single leg of a quadruped robot with time variation, it can be seen that the maximum amount of deformation is reached within 10 s of inflation. The deflation process is completed in 8 s. Thus, the maximum amount of deformation is reached within 10 s of inflation, and the deflation process is completed in 8 s.

Figure 31.

Single leg gas charging and discharging sequence diagram.

Both P₁ and P₂ take 0 air pressure as the initial value and increase to 0.4 bar with a gradient of 0.05 bar. Multiple groups of different P₁ and P₂ combination experiments are carried out. Figure 32(a) plots the height data of multiple groups of P₁ and P₂ under the single leg. In Figure 32(b), the step length data of multiple sets of P₁ and P₂ for single leg are plotted.

Figure 32.

The relationship between air pressure and deformation in the diagonal region of a single leg: (a) the height of one leg under multiple sets of P₁ and P₂ and (b) the step length of a single leg under multiple sets of P₁ and P₂.

As shown in Figure 32, it can be observed that as the pressure in regions 1 and 4 increases, the step length gradually increases, while the height first increases and then decreases. This is because, during the initial deformation, the expansion in region 1 causes an increase in height. However, as the pressure continues to rise and the bending angle of region 1 reaches its maximum, further increases in pressure reduce the bending radius, causing the height to decrease. When regions 2 and 3 along the diagonal start inflating, the obvious phenomenon is an increase in height, but the step length decreases. When the pressure in all four regions is equal, the height and elongation characteristics are the same as the experimental data. However, due to the different bottom structures of the side chambers, the end will still tilt slightly, even in the elongated state.

Figure 33 illustrates the nonlinear relationship between pressure (kPa) and displacement (step length, mm), revealing a trend of gradually increasing step length with rising pressure. This study provides a basis for optimizing robotic motion control strategies, enabling different step lengths and gait variations through pressure adjustments. It contributes to the design of more flexible gait control algorithms that can adapt to various terrains, thereby enhancing the robot’s motion efficiency and flexibility in complex environments.

Figure 33.

Relationship between pressure and displacement.

Quadruped robot movement gait experiment

Static gait walking experiment of quadruped robot

The static walking motion design of the quadruped robot imitates the triangular gait walking principle of quadruped animals. The overall walking of the quadruped robot can be regarded as a periodic cyclical movement. In one cycle, the quadruped robot takes a triangular ground-supporting step, that is, at any time, there are always three legs supporting the body of the quadruped robot. The experiment is roughly The process of the quadruped robot is as follows: the initial state of the quadruped robot is that the four legs are all filled with air pressure to make the robot have a certain upright height, and then the four legs are moved one by one in the order of the left front leg, right rear leg, right front leg, and left rear leg. The completion of all the steps of the four legs means that the quadruped robot has completed a cycle of motion.

According to the static motion gait planning of the quadruped robot introduced above, the triangular gait walking test experiment of the robot is carried out. Figure 34 shows the stepping action of the four legs of the quadruped robot according to the triangular gait sequence in one cycle.

Figure 34.

Triangular gait sequence with four legs in one cycle: (a) left front leg step, (b) right hind leg step, (c) right front leg step, and (d) left hind leg step.

The stepping action of each leg is divided into several frames. Since the initial state of each leg has a certain height, the first frame of all single legs to realize the stepping action is shortened, that is, the single leg that needs to step Do a lift and hang in the air, and then land after lifting. Landing not only requires the legs to complete the elongation of the height, but also the displacement of the legs in the horizontal direction, so the landing of the legs means that a single leg needs to complete a “front S” deformation action, so that a single leg can be realized. The step of the leg.

Figure 35 is a sequence diagram of the steps of the quadruped robot in one cycle in the static triangular gait walking experiment.

Figure 35.

Step sequence diagram of static walking experiment.

Dynamic gait walking experiment of quadruped robot

To complete the dynamic walking motion of the quadruped robot, it is crucial to maintain balance and stability throughout the movement. The quadruped robot’s dynamic walking plan is tested using a diagonal gait, a gait where one diagonal pair of legs supports the body while the other pair is lifted to take strides.

In the diagonal gait, as shown in Figure 36, the robot alternates between two diagonal leg sets. During the movement, the left front leg and right rear leg lift simultaneously, performing an S-shaped stepping action. Once this movement is completed, the left rear leg and right front leg lift and perform the same action. This sequence ensures that the robot maintains dynamic balance as it moves, with one set of legs always supporting the body while the other set steps forward.

Figure 36.

Diagonal gait walking: (a) initial, (b) left front leg, right back leg step, and (c) left hind leg, right front leg step.

As shown in Figure 37, the step sequence diagram of the quadruped robot in two cycles in the dynamic diagonal walking experiment.

Figure 37.

The sequence diagram of the dynamic walking experiment.

In the design process of the soft quadruped robot control system, there is a certain difference between the actual system and the theoretical model due to signal transmission lag and the delay brought by the actuator, which may lead to system instability. To address these issues, the control system can be improved by adding feed-forward compensation and adaptive control algorithms to alleviate the possible instability of the system.

Conclusion

The performance of a quadruped soft robot can be divided into at least five parameters: (1) elastic coefficient, (2) hysteresis performance, (3) elongation characteristics of a single leg, (4) unidirectional bending performance of a single leg, and (5) S-shaped bending performance of a single leg. The hysteresis performance of the leg structure is explored, that is, the deformation effect of the single leg during the process from inflation to deflation. The results show that the hysteresis of the single leg structure is greater at low pressure, and less hysteresis at high pressure. When carrying out the experimental test of the elongation characteristics of a single leg, only when the air pressure reaches 0.05 bar, the elongation and deformation of the single leg will increase, and obvious elongation will occur. The relationship between the elongated length and the air pressure is basically a monotonically increasing change. As the air pressure increases, the length of a single leg increases gradually, and as the air pressure increases, the rate of increase in the length of a single leg decreases. The bending performance and elongation characteristics of a single leg are the same, there is a minimum deformation air pressure, and the bending angle and air pressure change monotonously, and the bending angle of a single leg increases gradually as the air pressure increases. When the single leg is not inflated, there is no bending radius, and when it starts to inflate to reach the bending deformation air pressure of the single leg, it will produce a greatly abrupt bending radius value. After that, as the air pressure increases, the bending deformation of the single leg increases, and the bending radius gradually decreases until the bending deformation of the single leg reaches a saturated state, and the bending radius approaches a stable value.

In static walking, the quadruped robot adopts the triangular gait principle, as shown in Figure 34. A single leg completes a “forward S” shape deformation to achieve the stepping motion. Following the order of the left front leg, right rear leg, right front leg, and left rear leg, the robot can accomplish overall walking. In dynamic walking, as shown in Figure 36, the quadruped robot uses the diagonal gait principle. First, the left front leg and right rear leg form a pair, lifting simultaneously and performing the “forward S” shape stepping motion; then, the left rear leg and right front leg form another pair and complete the same movement, achieving a diagonal walking pattern. Through experiments validating both static and dynamic walking modes of the quadruped robot, the basic walking functionality has been confirmed, and the deformation performance of the single-leg structure in the robot’s motion process has been demonstrated in terms of its role and reliability.

In the future, we will consider combining soft links with electric motors to enhance the capability of quadrupedal robots in complex environments. This combination will leverage the precise and flexible control of electric motors along with the flexibility and large deformation capability of soft actuators. In addition to this, we will implement the following improvement measures: 1. Improve gait design by optimizing algorithms to enhance balance and stability, thus shortening the gait cycle; 2. Adopt advanced control optimization algorithms to reduce decision-making time and improve response speed; 3. Refine the mechanical design to enhance joint flexibility and improve the drive system, further reducing the gait cycle.