Development of a Pneumatically Actuated Quadruped Robot 2 Using Soft-Rigid Hybrid Rotary Joints

: Inspired by the musculoskeletal systems in nature, this paper presents a pneumatically 9 actuated quadruped robot consisting of a rectangular torso and four 2-degree of freedom (DoF) 10 planar robot legs with soft-rigid hybrid rotary joints. We first introduce the mechanical design of 11 the rotary joint and integrated quadruped robot with minimized onboard electronic components. 12 Based on the unique design of the rotary joint, a joint-level PID-based controller is adopted to con-13 trol the angular displacement of the hip and knee joints of the quadruped robot. Typical gait pat-14 terns for legged locomotion, including the walking and Trot gaits, are investigated and designed. 15 Proof-of-concept prototypes of the rotary joint and quadruped robot are built and tested. Experi-16


Introduction
Working in extreme conditions, including explosive, nuclear, high-voltage and 27 magnetic-resonance environments, is a big challenge for robots.Electric motor-driven 28 robots have been extensively studied due to their efficiency, precision, versatility and 29 adaptability [1], including the wheeled, tracked and legged robots that use electric mo-30 tors as the primary power source to perform various tasks and functions.For example, 31 the fully sealed wheeled robots [2][3] use electric motors to drive the wheels for opera-32 tions in chemical, radiological and nuclear missions.However, the degree of confidence 33 decreases over time as the sealed components show different responses to vibrations, 34 temperature, etc. Besides, the tracked robots [4] relying on electrical motor-driven tracks 35 have advantages over their wheeled counterparts in load tolerances and flexible mobili-36 ty on soft, slippery and rough terrains without sinking.They have been developed for 37 use in radioactive and hazardous environments [5] and offshore oil and gas platforms 38 [6].However, the operating time of these robots in radioactive conditions was limited by 39 the maximum dose their weakest components could sustain.Compared to wheeled and 40 tracked robots, electric motor-driven legged robots [1,7] have the ability to make and [10][11] and underground mines [12], but their electronic components are at high risk of being damaged in radiation-contaminated spaces.They could cause electrical sparks and disastrous ignition in explosive environments.
Soft-legged robots [13] powered by pneumatic actuators can be fabricated using flexible materials with minimized electronic components and exhibit excellent characteristics such as inherent compliance, good impact resistance, high energy-to-weight ratio, safety interaction with humans and adaptability to a variety of hostile environments.
For example, the untethered soft robot [14] fabricated with silicone rubbers had the adaptability to harsh environmental conditions, including snowstorms, fires and water, although the locomotion speed is slow, only 0.0077 body length per second (BL/s).The soft quadruped robot with three degrees of freedom (DoFs) per leg [15] uses simple pneumatic oscillator circuits without any electronic components to generate walking gaits for operations in environments where electronics are not suitable, but these circuits are acted on additional control elements or based on manual input.The untethered hexapod robot in [16] with soft fluid-driven actuators composed of elastomer bladders enables complex deformations by leveraging viscous flows to produce non-uniform pressure between bladders to drive the robot to move in one direction at 0.05 BL/s.Further, soft-legged robots [17][18][19] are highly flexible, and their legs may exhibit other unwanted forms of deformation, such as radial and axial expansion due to the fabrication using soft materials, bringing challenges in kinematic model and motion control [20][21].
Aiming at addressing the challenges of electric motor-driven robots and soft-legged robots, pneumatically actuated soft-rigid hybrid-legged robots, also known as musculoskeletal robots [22][23] inspired by the musculoskeletal systems in nature [24][25], demonstrated promising capabilities in both building a precise kinematic model like electric motor-driven robots and adapting to various environments like soft-legged robots.For example, with the rigid exoskeleton providing structural support and the flexible pneumatic joints providing actuation and inherent mechanical compliance to absorb impact and improve safety in interactions with humans, soft-rigid hybrid bipedal robots [26], quadruped robots [27], hexapod robots [24] and arthropod-like robots [22] were proposed.These robots can achieve movement with simple gaits by manual input.Besides, despite the actuation delay and the decrease in actuator accuracy, the quadruped robots with antagonistic pneumatic actuators developed by Tsujita [28][29] achieved stable locomotion in walking and trot patterns by adopting an oscillator network controller and adjusting the stiffness at the trunk.The quadruped robots built by Fukuoka [30][31] can adapt to speed variation and stabilize the pace of running using a neuromorphic locomotion controller with leg loading feedback.However, human assistance was needed during experiments to avoid the possibility of falling over.Despite having great potential, soft-rigid hybrid-legged robots are a new trend in robotics and have not been thoroughly investigated [32].One of the main limitations is that the development of legged robots using pneumatic actuators is complicated.
To further explore the potential of soft-rigid hybrid robots for operations in special environments, This paper proposes a soft-rigid hybrid rotary joint and develops a pneumatically actuated quadruped robot integrating a rectangular torso and four 2-DoF planar robot legs.The main contributions of this work include the mechanical design and experimental evaluation of a soft-rigid hybrid rotary joint, and the development of an integrated quadruped robot and its feasibility validation in typical gait control.
In the following sections, we first introduce the rotary joint design, quadruped robot integration, and controller development.The foot trajectories and typical gaits of the quadruped robot are also investigated.Following this process, section 3 presents the torque test of a single rotary joint and demonstrates the trot and walking gaits of the quadruped robot.Section 4 provides a detailed discussion of this work, and section 5 concludes the paper.

Design of the soft-rigid hybrid rotary joint
The variable-stiffness actuators for soft robots have been developed by using antagonistic fluidic actuation [33].For example, the theoretical models of the antagonistic rotary joints in [34][35] indicated that given an angular displacement/a torque, the torque/angular displacement is linear with the pressure difference between the two muscles, and the stiffness is related to the sum of pressures of the two muscles.However, the hinge function of the rotary joint in [34] may decline when the chambers are inflated to the fully deployed state for a long time.The maximum contraction ratio of the McKibben artificial muscle in [35] is only between 20% to 30%.
Inspired by the antagonistic fluidic actuation, we propose a soft-rigid hybrid rotary joint based on the twisting actuator developed in our previous work [36], as illustrated in Fig. 1.Two twisting skeletons are connected to the skeleton connector, where one performs a clockwise helical motion and another produces an anticlockwise helical motion.A soft bellows muscle is coupled to the twisting skeleton with a bearing and a bearing connector.The bellows muscle can be vacuumed and inflated, thereby generating linear driving force to actuate the twisting skeleton.The left end of twisting skeleton 1 is fixed to the end plate, while the right end of twisting skeleton 2 is connected to the output shaft.The outer ring of bearing 3 is fixed to the housing via a bearing cover.The linear motion of the output shaft is restricted by the inner ring of bearing 3 and the shaft sleeve.Therefore, the rotary joint generates pure rotation and torque at the output shaft by adjusting the pressures supplied to the two bellows muscles.The two twisting skeletons of the rotary joint are illustrated in Fig. 2(a).The kinematic structure is shown in Fig. 2(b).The base, the middle platform and the upper platform are parallel and denoted by identical squares□A1B1C1D1, □A2B2C2D2 and □ A3B3C3D3 with a radius of r, respectively.The angular displacement of the middle platform corresponding to the base of twisting skeleton 1 is denoted by θ1.It can be measured between lines O1B1 and O1Q1 where Q1 is the projection of vertex B2 on the base.Besides, the link connecting the revolute joints R112 and R113 is denoted as L11 and its length is defined by the distance, l, between the two parallel joint axes.The length of the projection of L11 in the direction of O1O3 is denoted by h1.Similarly, the angular displacement of the upper platform of twisting skeleton 2 is denoted by θ2.It is measured between lines O3B3 and O3Q3 where Q3 is the projection of vertex B2 on the upper platform.The length of the projection of the link L21 connecting the revolute joints R212 and R213 in the direction of O1O3 is denoted by h2.The angular displacement of the rotary joint defined as θ is calculated by where and the sum of h1 and h2 is constant.

Integration of the quadruped robot
Using two rotary joints as the hip and keen joint, respectively, a planar 2-DoF robot leg is designed.The thigh consists of a beam and two plates where one plate connects the output shaft of the hip joint and the housing of the knee joint; another plate rotating freely around the hip joint provides structural support for the knee joint.The shank has a similar design to the thigh to support and drive the foot.
The quadruped robot consists of a torso and four 2-DoF robot legs, as shown in Fig. 3 . The torso of the robot is a rigid rectangular plate.Four legs are fixed to the torso symmetrically.The design specification of the quadruped robot is listed in Tab. 1.

Controller for the soft-rigid hybrid rotary joint
The angular displacement of the rotary joint mainly depends on the pressure difference between the two bellows muscles due to the compliance of bellows muscles and flexure hinges of twisting skeletons.The larger the pressure difference is, the larger the angular displacement of the rotary joint is.Based on this characteristic, a joint-level PID-based controller is adopted to achieve the active angular displacement control, as shown in Fig.
where t is the current time; ti,0 is the start time of the current gait cycle of the i-th leg; T is the period of one gait cycle.For higher velocity, a smaller gait period T is more suitable.
With a constant gait period T, a smaller duty cycle d would result in each leg having an increased aerial time and create a more dynamic gait.Note that the four legs may have different values of ti,0 and .
Besides, the phase offset [ ] is defined to coordinate the phase of the i-th leg with respect to the phase 1 of the leading leg to produce different gait patterns through the relationship: ii (5) Table 2 lists the duty factor and desired phase offset for defining typical gaits, including the crawl, walking, trot, pace and bounding gaits.Based on the parameters given in Tab. 2, Fig. 5 lists the sequences of the leg movement with the trot and walking gaits.The solid blue color bars indicate the stance phase of the corresponding leg, while the white color bars represent the swing phase of the corresponding leg.In the trot gait, two diagonally opposite legs (e.g., the right front and left hind legs) are in contact with the ground while the other two legs are lifted and move forward.In the walking gait, a leg in the air is set down at the same instant as another leg is lifted, and three legs contact the ground at all times.

Foot trajectory
As illustrated in Fig. 6(a), a coordinate frame O1−XYZ is set at the leg where the origin is located at the center of the hip joint O1, the X-axis is horizontal, and the Y-axis is vertical.Based on the Denavit-Hartenberg method, the position of the foot O3 3 , 3 , 3 expressed in the O1−XYZ can be derived as where 1 , 2 ,  1 and  2 are the length of the thigh, the length of the shank, the angle of the hip joint and the angle of the knee joint, respectively.The inverse kinematics can be solved based on the conventional geometric approach.For a given position of the foot, the angles of the hip and knee joints are derived as where One gait cycle of the foot can be divided into a stance phase and a swing phase.To minimize the impact between the ground and the foot, the foot trajectory should meet the demand that the velocity and acceleration of the foot along the direction of the Y-axis (Fig. 6(a)) become zero at the time of touchdown, liftoff and maximum foot height [37].Besides, the legs support the torso to move forward during the stance phase.To make the torso move steadily, the acceleration of the foot along the direction of the X-axis shown in Fig. 6(a) needs to be as small as possible.Therefore, the foot trajectory in the swing phase can be composed of a cubic curve along the X-direction and a cosine curve along the Y-direction, while the foot trajectory in the stance phase can be a straight line along the X-direction [38].The equations for defining the foot trajectory with respect to the coordinate frame O1−XYZ set at the center of the hip joint are: where Ls denotes the stride length; 1 represents the remainder of the real-time divided by the gait cycle T; Lt denotes the distance between the axis of symmetry of the foot trajectory and the hip joint; Ht and Hf represent the height of the hip joint and the maximum height of the foot in the Y-axis with respect to the ground, respectively.Equations ( 10) and (11) represent the position variation of the foot along the X-and Y-axes during the stance phase, respectively, while Eqs.( 12) and ( 13) represent its position variation along the Xand Y-axes during the swing phase, respectively.
To evaluate the dynamic performance of the quadruped robot, trot and walking

Experimental evaluation of the rotary joint and integrated quadruped robot
To evaluate the performances, prototypes of the rotary joint and quadruped robot are fabricated using 3D printing and CNC approaches to achieve rapid and low-cost manufacturing.TPU 95A filament is selected as the material for printing the soft bellows muscles of the rotary joint due to its exceptional wear and tear resistance and rubber-like flexibility.The thighs, shanks and housings of the rotary joints are 3D-printed with the PLA material.The twisting skeletons of the rotary joints are CNC-machined using multi-layered aluminium composite panels (HYLITE) with a polypropylene core and aluminium cover layers, which have good fatigue resistance and can provide a compliant hinge function to withstand repeated bending without damage.The feet are fabricated by injecting Dragon Skin 30 into prefabricated molds.The torso is cut with a carbon fiber plate.The quadruped robot prototype is obtained by assembling the modularized rotary joints and aforementioned components.

System integration
The quadruped robot is tethered, and the pneumatic and control systems are off-board to reduce the weight of the quadruped robot.The PID-based controller is adopted for the rotary joints to achieve the active angular displacement control, which was developed using Matlab ® on a desktop computer.The bellows muscles of the rotary joints are capable of being both inflated and vacuumed, and their pressures are adjusted by the pneumatic system which is composed of an air compressor, pneumatic

Trot gait test of the quadruped robot
To prevent the quadruped robot from falling sideways during tests, one end of a linkage connects the torso of the quadruped robot with a revolute joint, and another end of the linkage connects a guide carriage with a revolute joint.The guide carriage has one DoF and is capable of moving on the guide rail.Besides, to prevent the quadruped robot from radically rolling forward and backwards, the torso of the quadruped robot is also connected to strings.Another end of the strings is tied to a linear guide block, which can move along the linear slide rail.Using the theoretical trajectories shown in Figs.This is because only a set of PID parameters is used to control the angular displacements of the hip and knee joints of the quadruped robot.(The robot in [32] was able to move without assistance by using two sets of PID parameters to control the leg in the swing phase and the stance phase, respectively, but its movement was clumsy.)Besides, to minimize the impact between the ground and the foot, the velocity and acceleration of the foot along the direction of the Y-axis are set to zero at the time of liftoff, so the foot does not generate a high force to kick the ground.muscles needs to be specifically designed; and the robot's initial configuration is complex to be adjusted.By contrast, using the rotary joint with embedded pneumatic muscles as a module, the quadruped robot developed in this paper has a concise and compact structure and can be quickly adjusted.

Discussions
In this paper, a gait generator is used to generate typical gaits to coordinate the movement sequences of different legs, and a joint-level PID-based controller is adopted to control the angular displacements of the hip and knee joints of the quadruped robot.
The feasibility of the quadruped robot in gait control is demonstrated, although preliminary experiments illustrate that the quadruped robot slips sometimes, exhibits large tracking errors in the swing phase and needs assistance during movement, including the linkage and strings presented in section 3.3.To eliminate the undesirable oscillatory behavior and realize stable movements of the quadruped robot without any assistance, control strategies such as the active model-based control [39], the adaptive fuzzy sliding mode control [40] and the reinforcement learning-based control [41] for predicting the unknown disturbance and improving the tracking performance of pneumatic artificial muscles with uncertainty and a considerable delay in characteristics will be further investigated.Besides, the advanced controllers used in electrical motor-driven robots like the torque control in adjusting the ground reaction force of supporting legs [42], and the closed-loop central pattern generator in leveraging compliance of elastic legs [43] could be considered for controlling the pneumatically actuated quadruped robot in the future.
Besides, variable stiffness is an essential characteristic of robots for safe physical human-robot interaction and adaptation to various environments and applications [44].
The variable-stiffness actuators for rigid-bodied robots have been developed by connecting motors to adjustable springs [45][46] or by building virtual controllers for motors [47][48] or by combining both [49].In contrast, the variable-stiffness actuators for soft robots can be developed by using a pair of antagonistic muscles [33].For example, the rotary joints reported in [34][35] have variable stiffness, which can be adjusted by controlling the internal pressures of the two antagonistic muscles without changing their position.The proposed rotary joint with two antagonistic bellows muscles has a similar arrangement to [34][35], and our previous work in [50] demonstrated that the bellows muscle is equivalent to a non-linear spring and the muscle can adjust its stiffness by changing its internal pressure.Hence, the proposed rotary joint has the potential of variable stiffness.How to derive the theoretical model of the output torque and stiffness of the proposed rotary joint and make use of it for highly dynamic motions of the quadruped robot will be explored in our future work.
It is worth to mention that pneumatic actuating systems composed of the air compressor, regulators and solenoid valves are normally heavy and bulky.Like most pneumatically actuated robots [27][28][29][30][31][32], in this work the pneumatic actuation system is off-board and the robot is tethered to make the robot lightweight.Though pneumatic robots without any external power source have recently been developed as reported in the references [14,15], the lightweight mini air compressors and valves bring in additional challenges, including the small volume of compressed gas, low flow rate, limited operating pressure and imprecise pressure control.Besides, the maximum output torque of the rotary joint used for the quadruped robot depends on its maximum operating pressure of 250 kPa, which is determined by the material selection and fabrication ap-

Figure 1 .
Figure 1.Exploded view of the soft-rigid hybrid rotary joint.

Figure 2 .
Figure 2. (a) 3D model and (b) Kinematic structure of the twisting skeletons of the rotary joint.

4 .
Using the reference angular displacement θref as the controller's input and the real-time angular displacement θ of the rotary joint as the feedback signal, the error between the reference angular displacement and the real-time angular displacement is calculated and transmitted to a PID module.The desired pressure difference ∆P is set as the output of the PID module.Combining the minimum pressure Pmin which is directly set from an external port by users, the theoretical pressures P1 and P2 of the two bellows muscles are determined by Eqn.(3) and transferred to the pressure control units, which control the pressure supplied to the two bellows muscles of the rotary joint by using the pneumatic regulators and solenoid valves.In addition, the values of PID parameters are tuned manually by lifting the quadruped robot and evaluating the response of each joint to a step input, while Pmin is set to 30 kPa for the robot's experiments.

Figure 4 .
Figure 4.The PID-based controller of the rotary joint for the angular displacement control.

Figure 5 .
Figure 5. Sequences of the leg movement with typical gaits in one gait cycle.(a) Trot gait.(b) Walking gait.The solid blue color bars indicate the stance phase of the corresponding leg while the white color bars represent the swing phase of the corresponding leg.(LH: the left hind leg; LF: the left front leg; RF: the right front leg; RH: the right hind leg.) gaits are chosen for experiments.The parameters T = 0.8 s, Ht = 297 mm, Hf = 70 mm, Ls = quadruped robot with the trot gait, while Figs.6(c) and 6(d) illustrate the theoretical angular displacements of the hip and knee joints of the quadruped robot with the trot gait, respectively.(The angular displacements of 0° in Figs.6(c) and 6(d) are corresponding to  1 5° and  2 7 ° shown in Fig. 6(a), respectively.)By contrast, the parameters T = 7.84 s, Ht = 285 mm, Hf = 80 mm, Ls = 100 mm and Lt = 50 mm are set for the walking gait.Figs.7(a) and 7(b) illustrate the theoretical angular displacements of the hip and knee joints of the quadruped robot with the walking gait, respectively.

Figure 6 .Figure 7 .
Figure 6.Leg movement of the quadruped robot with the trot gait.(a) Schematic diagram of the 2-DoF robot leg.(b) Foot trajectory of the quadruped robot with the trot gait.(c) Angular displacement of the hip joints of the quadruped robot with the trot gait.(d) Angular displacement of the knee joints of the quadruped robot with the trot gait.(LH: the left hind leg; LF: the left front leg; RF: the right front leg; RH: the right hind leg.)

Figure 8 (
Figure 8(a) illustrates the prototype of the rotary joint and experimental settings for evaluating its output torque.The base of the rotary joint is mounted on the load cell of the Instron machine E5967 by using two clamps, while the output shaft of the rotary joint is fixed to the gripper of the Instron machine.A 6-axis torque-force sensor (RO-BOTOUS RFT40-SA01) is installed between the twisting skeletons and the base of the rotary joint to measure its output torque.

Figure 8 .
Figure 8. Torque evaluation of the soft-rigid hybrid rotary joint.(a) Testing platform.(b) Output torque of the rotary joint.The output torque of the rotary joint is tested by fixing the rotary joint and maintaining the bellows muscles at constant pressures.As the rotary joint is symmetric, the case of P1≥0 and P2≤0 is evaluated in this work.The experimental results in Fig.8(b)reveal that without actuating the two bellows muscles (P1 = P2 = 0 kPa), the absolute value of the output torque increases when the rotary joint deviates from its initial position θ = 0°, which results from the compliance of the bellows muscles and flexure hinges of the twisting skeletons.Given a certain angular displacement, increasing the pressure of bellows muscle 1 from 0 to 200 kPa and decreasing the pressure of bellows muscle 2 from 0 to -80 kPa leads to increased torque of the rotary joint, but the torque change of the rotary joint at the angular displacement θ > 0° is lower than that at the angular displacement θ < 0°.The reason is when the angular displacement increases from 0° (θ > 0°), bellows muscle 1 is extended, and its contact area with twisting skeleton 1 decreases; bellows muscle 2 is contracted, and its inner space decreases; inflating bellows muscle 1 and vacuuming bellow muscle 2 at the angular displacement θ > 0° generates a lower force than that the at the angular displacement θ < 0° to driving twisting skeletons.Further, The rotary joint is capable of generating a maximum torque of 5.83 Nm under the conditions of θ = −60°, P1 =200 kPa and P2 = −80 kPa.
regulators (SMC ITV-212BL4), three-port solenoid valves (SMC VDW350-5G-4-02F-Q), two-port solenoid valves (SMC VX220AGA) and vacuum generators (SMC ZH07DSA-06-06-06), as shown in Fig. 9.The data acquisition device (NI USB-6343) is connected to the computer for transmitting control commands from the controller to corresponding valves.The data acquisition device directly controls the pneumatic regulators via analogue output, while the two-port and three-port solenoid valves are driven by motor driver controllers L298N.(When receiving digital signals from the data acquisition device, the motor driver controller L298N generates analogue signals to control the solenoid valves.)Markers are attached to the thigh and the shank of each leg to measure the angular displacements of the hip and knee joints via the OptiTrack motion capture system, respectively.This information is fed back to the controller to realize the closed-loop control of the posture of the quadruped robot.

Figure 9 .
Figure 9. Pneumatic and control systems of the quadruped robot.
Fig. 10, the moving sequences of the quadruped robot with the trot gait are left front and right hind legs lift synchronously -left hind and right front legs lift synchronously -left front and right hind legs lift synchronously, which matches the sequences depicted in Tab. 2 and Fig. 5(a).As we control the position instead of the torque of supporting legs in the stance phase in the experiments, the elastic properties of robot legs are not fully explored and the strings connecting the quadruped robot are constantly under tension.

Figure 10 .
Figure 10.Moving sequences of the four legs of the quadruped robot with the trot gait.Solid circles denote the feet in ground contact.White circles denote lifting feet.

Figure 11 .
Figure 11.Movement of the left hind leg of the quadruped robot with the trot gait.(a) Angular displacement of the hip joint of the left hind leg.(b) Angular displacement of the knee joint of the left hind leg.(c) Errors of the angular displacements of the hip and knee joints of the left hind leg.(ST: stance phase; SW: swing phase.)

Figure 12 .
Figure 12.Moving sequences of the four legs of the quadruped robot with the walking gait.Solid circles denote the feet in ground contact.White circles denote lifting legs.

Figure 13 .
Figure 13.Movement of the left hind leg of the quadruped robot with the walking gait.(a) Angular displacement of the hip joint of the left hind leg.(b) Angular displacement of the knee joint of the left hind leg.(c) Errors of the angular displacements of the hip and knee joints of the left hind leg.(ST: stance phase; SW: swing phase.)

Table 1 .
Design specification of the quadruped robot to represent the percentage of the gait cycle during which the leg is in contact with the ground.At the start of the gait cycle, each leg starts in stance with a phase of The leg switches from the stance state to the swing state when increases to .Once the phase increases to the maximum value of 1, it wraps around to zero, and the leg switches from the swing state back to the stance state, starting the next gait cycle.The phase of the i-th leg can be calculated by 2.4.Gait analysisQuadruped robots can implement various gaits, such as crawling, walking, trot, pace, and bounding gaits, determining how the robot moves and interacts with the environment.The typical gaits are different in sequences, as is the duration that each leg is in contact with the ground.To generate periodic gaits, the phase [ ] is used to depict the state of each leg in a gait cycle, and the duty cycle is denoted by [ ]

Table 2 .
The duty factor and desired phase offset of the typical gaits, including the crawl, walking, trot, pace and bounding gaits for the left hind leg (LH), left front leg (LF), right front leg (RF) and right hind leg (RH).

Table 3 .
[31]arison of the quadruped robot with existing soft and soft-rigid hybrid robots.Tab.3 reveals that the proposed quadruped robot has a more simplified kinematic model for motion control and is capable of higher locomotion speed than soft-legged robots.Besides, compared to existing soft-rigid hybrid legged robots, the locomotion speed of the proposed quadruped robot is competitive, only slower than the soft-rigid hybrid-legged robot driven by McKibben-type pneumatic artificial muscles[31].However, the McKibben-based robot is complicated since one leg has three joints with two external muscles actuating each joint; the distribution of McKibben artificial