Design and Dynamic Model of a Frog-inspired Swimming Robot Powered by Pneumatic Muscles
Design and Dynamic Model of a Frog-inspired Swimming Robot Powered by Pneumatic Muscles
Ji-Zhuang Fan 0 1 2
Wei Zhang 0 1 2
Peng-Cheng Kong 0 1 2
He-Gao Cai 0 1 2
Gang-Feng Liu 0 1 2
0 State Key Laboratory of Robotics and System, Harbin Institute of Technology , Harbin 150080 , China
1 & Gang-Feng Liu
2 Supported by National Natural Science Foundation of China , Grant No. 51675124
Pneumatic muscles with similar characteristics to biological muscles have been widely used in robots, and thus are promising drivers for frog inspired robots. However, the application and nonlinearity of the pneumatic system limit the advance. On the basis of the swimming mechanism of the frog, a frog-inspired robot based on pneumatic muscles is developed. To realize the independent tasks by the robot, a pneumatic system with internal chambers, micro air pump, and valves is implemented. The micro pump is used to maintain the pressure difference between the source and exhaust chambers. The pneumatic muscles are controlled by high-speed switch valves which can reduce the robot cost, volume, and mass. A dynamic model of the pneumatic system is established for the simulation to estimate the system, including the chamber, muscle, and pneumatic circuit models. The robot design is verified by the robot swimming experiments and the dynamic model is verified through the experiments and simulations of the pneumatic system. The simulation results are compared to analyze the functions of the source pressure, internal volume of the muscle, and circuit flow rate which is proved the main factor that limits the response of muscle pressure. The proposed research provides the application of the pneumatic muscles in the frog inspired robot and the pneumatic model to study muscle controller.
Frog-inspired robot; Pneumatic muscle; High- speed switch valve; Pneumatic model
The increasing demand for underwater exploration and
development, water detection, and other works in the
scientific and military fields [
] has led to the rapid
development of bionic underwater robots [
]. These robots
mimic animal locomotion mechanisms, such as biological
motion and musculoskeletal characteristics, which have
been the focus of several studies [
]. With regard to
underwater swimming methods, most research has focused
on the waving and oscillating propulsion of fishes, jet
propulsion in squids, multi-legged crawling propulsion,
hydrofoil flapping propulsion in turtles, and swimming
motions of frogs [
Frogs brilliantly possess land jumping and underwater
swimming abilities [
], and thus greatly inspired relevant
robot designs in bionic research. The amphibious robot can
serve in an extended wide terrain to complete tasks.
Therefore, frog inspired robot is a promising topic to
mimic its swimming and jumping movements which are
accomplished by the same hind leg. However, current
research focuses on the jumping prototype and mechanism
of swimming, while the development of frog-inspired
swimming prototype is less studied.
Swimming mechanisms have been widely investigated
through the analysis of motion characteristics [
propulsive force [
], and flow field structures [
experimental observations, Pandey et al. established a
CAD model of a bionic frog swimming robot to mimic
biological frog motions [
Frog swimming is intermittent and shows explosive
movements. Thus, the driver must have high output power
and fast response. Pneumatic artificial muscles have
highpower mass ratio and flexible characteristics, and are
similar with biological muscles in terms of driving the
skeletal system [
]. In the present study, the frog-inspired
robot is driven by pneumatic muscles that simulate the
performance of biological frog muscles. The similarities
between the pneumatic and biological muscles must be
determined and analyzed to understand the relationship
between the amphibious movements and musculoskeletal
system of frogs.
Traditional drivers, such as motors, run through
complex mechanisms [
]. Meanwhile pneumatic
muscles have been widely used because of its simple
] and are mainly controlled by tracking
control methods [
]. Several studies have performed
on pneumatic muscles, mainly focusing on the modeling
and control of pneumatic valves [
]. The precise
pressure proportional valve is typically used to achieve
output accuracy, but the cost is expensive and the flow
rate of the pressure proportional valve is relatively small.
Further, such setup is not suited for situations demanding
large flow inputs. In this paper, pneumatic muscles are
controlled by high-speed switch valves, which have high
flow rate and fast response, thereby simplifying the
design of the frog-inspired robot. In addition, high-speed
switch valves are light weight, occupy small volume,
and incur low cost, and is thus preferable in the robot
To mimic frog swimming performance, a frog inspired
swimming robot is developed with pneumatic muscles and
high-speed switch valves to study pneumatic
characteristics during leg extension.
2 Design of Frog-inspired Robot
2.1 Modeling of the Frog-inspired Robot
Figure 1 shows the model of the frog-inspired robot, which
is based on the morphology of real frogs. The palms were
modeled as flexible planes with embedded spokes. The
forelegs were relatively small compared with the thick and
strong hind legs, which contribute slightly to the swimming
propulsion. Therefore, forelegs were disregarded in the
design. Considering that all the limbs moved in planar
space, frog joints were modeled as planar revolute joints.
The palms are main propellers for swimming, and thus, the
multi DOF hind legs could be simplified as a three-DOF
link mechanism for each hind leg. The entire frog model
consists of the body, thigh, crus, and palm connected by the
hip, knee, and ankle joints, respectively.
2.2 Structure of the Frog-inspired Robot
2.2.1 Robot System
The major pneumatic system is designed inside the robot.
Figure 2 shows that the pneumatic units are integrated in
the robot body, but several pneumatic muscles are in the
hind legs. A small upper shell and large bottom shell are
separated by a metal baffle plate on which two cylinder
Micro air pump
tanks of 2 L each are placed as source chambers. The
switch valves are installed inside the exhaust chamber and
connect muscles and the source and exhaust chambers by
air pipes. The body and legs are sealed by the shells, while
the joints are dynamically sealed according to the joint
design in the following section.
A micro air pump is placed in the rear trunk.
Meanwhile, given the size of the air pump, we make it pass
through the baffle plate connected to the cooling fin of the
pump. The source chamber is connected to the outlet of the
pump and filled with high-pressure compressed air. The
high-speed switch valves are mounted in the middle of the
baffle plate on the bottom face. The electrical unit is
located in the front head of the exhaust chamber, which is
also connected to the inlet of the pump. Therefore, the
switch valve controls the flow from the source chamber to
the muscle for pressurizing and from the muscle to the
exhaust chamber for depressurizing.
2.2.2 Design of the Hind Legs
The robot is designed according to the model in this paper.
Each leg (Figure 3) is framed by two upper and lower
Y-shaped skeletons connected to one end of the pneumatic
muscle. The other end is connected to the shaft via a crank to
enable the muscle to rotate on the joint. To realize the seal for
the leg and dynamic seal for the joint, the thigh and crus are
covered by the shells and separately laminated on the shaft.
The crus is then fixed with the shaft to enable the crus to
rotate relative to the thigh through the thigh muscle drives.
Each joint has two muscles mounted in an antagonistic
way to control the angle position and joint stiffness. For the
knee and ankle joints, we selected the pneumatic muscle of
DMSP-20-150N-RM-CM, whose length is 150 mm and
has a maximum contraction rate of 25% and maximum
contractile force of 1500 N. We selected those with lengths
of 180 mm for the hip joints.
Owing to the water seal and drag reduction in water, the
leg is sealed with the shells, which are made by 3D printing
with tough resin as material. Figure 4 shows the joint
dynamic seal structure, where a V-shaped seal ring is
firmly set in the joint axis. The upper part of the seal ring
contacts and presses the bottom face of the thigh shell
during joint rotation to prevent water from entering the
interior of the shell. Meanwhile, a cavity is formed along
the shaft between the bearing and the shell, which forms an
oil chamber, which is filled with waterproof grease that
could further prevent water from entering the shell.
In the propulsive phase, frog palms are completely open to
increase the area for water repulsion. This action generates
the reactive force for propulsion. In the recovery phase, the
palms shrink to a minimum area by concentrating the palm
spokes to reduce fluid drag. To realize those functions, the
design shown in Figure 5 is used. The design consists of
palm spokes, steering motor, membrane, and connector for
the joint shaft. One distal spoke is fixed, while the other distal
spoke is rotated by the steering motor.
2.2.3 Design of Pneumatic Circuit
Given that the pressure proportional valve has the
disadvantages of large volume and mass, high cost, slow
pressurizing, and depressurizing speed, we use pneumatic
highspeed switch valves to control the pressure process of the
pneumatic muscles. The pneumatic circuit of the hind leg
of the robot is shown in Figure 6.
The three revolute joints of each hind leg use six
pneumatic muscles in total, and each pneumatic muscle is
controlled by two high-speed switch valves. Thus, the
entire hind limb requires 12 pneumatic high-speed switch
valves. We consider the hip joint as an example to explain
the working principle of the pneumatic circuit. The body
and the thigh crank are connected with a joint shaft. Each
Crus recovery muscle
end of the crank is hinged to the muscles, and the other
ends are connected to the body frame. During muscle
contraction and stretching, the crank (fixed with the thigh
frame) rotates relative to the body. To reach the initial
position, the charging valve 1 and discharging valve 2 are
open, and the discharging valve 1 and charging valve 2 are
closed. The recovery muscle then starts to contract, and the
driving muscle stretches until the joint rotates clockwise to
the preset position (Figure 6). During the propulsive phase,
the hip joint must rotate counterclockwise quickly, and thus
the charging valve 1 and discharging valve 2 are close, and
charging valve 1 and the discharging valve 2 are open. The
recovery muscle of the hip joint then starts to stretch and
the driving muscle contracts. The hip joint rotates
counterclockwise quickly to a specific position.
3 Model of the Pneumatic System
The dynamic characteristics of the pneumatic system must
be analyzed to determine the function of the
musculoskeletal system of the frog-inspired robot and lay the
foundation for its control system design. The pneumatic
muscles used in the robot have apparent nonlinearity and
hysteresis characteristics. The previous modeling of the
pressure dynamic process is based on the ideal conditions,
which can lead to different response results. The dynamic
models of the source chamber, muscle volume, exhaust
chamber, and switch valves are established to simulate the
pressure process of the pneumatic system.
3.1 Dynamic Model of the Source Chamber and Exhaust Chamber
The source chamber is regarded as a variable mass system.
The equation of the thermodynamic process can be written
according to the first law of thermodynamics [
dQs þ isdMs ¼ dUs þ dWs þ idM;
where dQs—Heat gained in the source chamber caused by
the heat exchange between the inner gas and the outside
world through wall, is—Specific enthalpy of the gas flow
into the source chamber, i—Specific enthalpy of the gas
flow to the pneumatic muscle from the source chamber,
dMs—Gas mass flow into the source chamber,
dUs—Internal energy change of the gas in the source chamber, dWs
—Expansion work done by the gas in the source chamber,
dM—Gas mass flow into the pneumatic muscle.
According to thermodynamics, dUs = CvMsdTs,
dWs = PsdVs and idM = (Cv ? R)TsdM, where Cv is the
constant volume specific heat of air, R denotes the gas
constant of air, and dTs is the temperature differential in the
Given that the gas flow in the air pipe is faster than the
heat exchange rate between gas and external environment,
energy loss in the air pipe is much lesser than total gas
energy. Hence, heat exchange can be ignored. Therefore,
rapid pressurizing and depressurizing processes of
pneumatic muscles can be regarded as adiabatic processes, such
that dQs = 0. The volume of the source chamber is
constant, such that dVs = 0. If the air supply to the source
chamber is disregard, then dMs = 0. Therefore, the gas
thermal process in the source chamber can be simplified as
where Ts—Temperature in the source chamber,
ps—Pressure in the source chamber, R—Gas constant of air,
287 J kg/K, K—Specific heat ratio of air, K = 1.4.
The relationship between temperature and pressure in
the adiabatic process is represented as
where T0—Initial temperature in the source chamber, p0—
Initial pressure in the source chamber.
Similarly, the thermal process of the pressurizing
process in the exhaust chamber is derived as
kRT dm ¼ Vedpe;
where T—Initial temperature in the pneumatic muscle,
pe—Pressure in the exhaust chamber, Ve—Volume of the
exhaust chamber, dm—Exhausted air from the pneumatic
3.2 Dynamic Model of the Pneumatic Muscle
When the pneumatic muscle is assumed as a cylinder and
the noncylindrical joint at both ends of the muscle is
disregarded, the pneumatic muscle structure can be modeled
as shown in Figure 7. In the figure, h is the angle between
the braided thread and central axis, and D is the pneumatic
muscle diameter. Muscle volume can be expressed as
where L—Pneumatic muscle length, n—Number of loops
of braided thread in the muscle, b0—Length of the
nonretractable braided thread, V—Internal volume of the
According to Eq. (1), the pressurizing process of the
muscle can be simplified as
kRTSdM ¼ kPdV þ VdP:
The thermal process of muscle depressurizing is
kRTdm ¼ kPdV þ V dP;
where P—Pressure in pneumatic muscle, V—Volume of
3.3 Dynamic Model of the Pneumatic System
The pneumatic system in this paper uses the switch valve,
which has a simple structure and digital control signal. The
flow characteristics of the pneumatic system mainly
depend on the valve [
]. Thus, we use the high-speed
switch valve MHJ9-LF manufactured by FESTO. The
valve Sonic conductance (C) is 0.4 L/s bar, and the critical
pressure ratio (b) is 0.38. The volume flow through the
valve can be solved by the following equation:
Q ¼ pr2v;
Qm ¼ CðP1 þ P0Þ
:> 1 1 b
xðr; bÞ ¼
P2 þ P0 ;
r ¼ P1 þ P0
where P0—Atmospheric pressure in standard condition,
P1—Upstream pressure, P2—Downstream pressure, T0—
Gas temperature in the standard state.
As the gas cylinder connects six branch pipes for the
muscles from one main pipe simultaneously, the gas flow at
saturation, and the gas flow in the main pipe can be
obtained through the continuity equation:
where Q—Main pipe flow rate, r—Pipe radius, v—Flow
velocity in the pipe.
On the basis of Eqs. (2), (3), (5), (6), and (8), a
simulation model of the pneumatic system of the robot is
established to simulate the dynamic characteristics of the
process of pneumatic muscle pressurizing, as shown in
The saturation term is added in the simulation process
because of the calculated maximum flow rate in the main
pipe. The muscle length is derived from the joint angle.
With muscle length and initial pressure from the source
chamber, the pressurizing processes in the chambers and
muscles can be simulated, and the rationality of the
established model can be verified by comparing the results
to lay the foundation for future designs of robot control
4 Simulations and Experiments
The prototype of the frog-inspired robot is shown in
Figure 9. This prototype is based on previously introduced
designs. In the experiments, the source chamber was
connected to a 6 mm pipe (with inner diameter of 4 mm) to
serve as the high pressure source. Then, the pipe was
divided into six branches to connect to the switch valves
and muscles with 4 mm pipes (with inner diameter of
2.5 mm). The parameters in the robot are shown in
Flow rate model
Arduino Mega 2560
The swimming cycle is divided into the propulsive and
recovery phases according to body velocity. The leg
motions are powerful and rapid in the propulsive phase,
while the legs recover slowly during the recovery phase.
Therefore, the experiments and simulations on the hind leg
extensions were conducted to test the muscle capacity to
drive the system. The pressures in the pneumatic muscles
were controlled by the high-speed switch valves with open
4.1 Swimming Experiments
For the validation of the designed frog-inspired robot
system and experimental data collection, an experiment
wherein the leg extension had no load and robot swimming
experiment were conducted. In the experiments, the
recovery muscle was removed and replaced with a light
spring for recovery to eliminate the interaction of the
recovery muscle which response differently according to
control strategy. The source chamber was pressurized to six
bars in advance, and the muscle was aerated through the
switch valves. The open loop control was applied to the
muscles, which were pressurized for one second and then
released for one second. The swimming experiment for one
period was conducted to verify the robot design, as shown
in Figure 11. The results of the joint angle are shown in
Each joint had a delay of about 0.1 s, because of the
influence of the delay in the pneumatic system and
mechanical clearance in the joints. The joint data would be
used in the subsequent simulations to compute the pressure
dynamic process according to previously established
4.2 Simulation of the Pneumatic System
The experiments in the air were conducted, and the frog
body was fixed on a base. The joint angles were measured
by the sensors in the robot, and the pressures were
measured by the sensors in the external circuits. With the
pneumatic muscle volume data derived from the joint angle
results, the dynamic model built in MATLAB/Simulink
was able to simulate the pressurizing processes in the
muscle volume and source chamber. The pressurizing
process from the experiments and that of the simulations
were compared (Figure 13) to validate the model of the
pneumatic system. The pressurizing processes in the hip
muscle from simulation matched well with the experiment.
Figure 14 shows the results in the source chamber.
Compared with the small muscle volume, which rendered the
noise evident, the pressure drop is smaller because of its
large chamber capacity. Finally, simulation results are
consistent with the experiment and converged to the same
end. These results verified the feasibility of the pneumatic
model built in this paper.
According to Eq. (8), the aeration volumes of each joint
can be obtained by integrating the flow rate during muscle
pressurizing. The aeration volumes at standard state were
computed in the simulation. The aeration volume at the hip
muscle is 0.455 L and that at the knee and ankle muscle is
0.401 L. The aeration volumes are the key factors that
verify the design of the pneumatic system.
4.3 Analysis on the Pressurizing Process
The response speed of the pneumatic muscle reflects the
performance of the driving capacity and is a key factor to
simulate the musculoskeletal system [
]. The response
speed is mainly restricted by the pressurizing process. The
simulations with different source pressures, muscle
volumes, and flow saturations in the pneumatic circuit were
calculated to analyze the factors of muscle performance.
Therefore, the pressure response in the pneumatic muscle
can be obtained.
The simulation results of the pressurizing process in the
hip-driving muscle was influenced by different initial
source pressures (Figure 15). The branches from the main
pipe were considered during the calculation of the
maximum flow rate in the main pipe through Eq. (9). The
saturation for each branch pipe connected with switch valves
and muscles were set at 45 L/min after the average
distribution of the flow rate. The final steady muscle pressures
varied with source pressures (Figure 15(a)). However, the
pressurizing process was identical before the balanced
state. The pipe saturation limited the flow rate through the
valves and muscles, as normal sources of pressures in the
simulations generated a flow rate higher than 45 L/min.
Therefore, at pipe saturation, the initial source pressure has
minimal influence on the muscle pressurizing process.
Figure 15(b) shows the results without the pipe saturation.
After removing the saturation, the pressure response
shortened and fell within 0.2 s, thus illustrating that the
source pressure promotes the flow in the pipe.
Figure 16 shows the simulation results of the
pressurizing process in the hip joint muscle at an initial source
pressure of 6 bar and 100% muscle volume constant. To
estimate the influence of each muscle volume, the 100%
muscle volume constant was set as the maximum of the
muscle volume during contraction, and 0% muscle volume
constant was set as the original muscle volume before
pressurizing. Figure 16(a) shows the results with saturation
and without saturation. Although a small volume required
low-pressure air to receive faster response, the difference
between the maximum and minimum volumes during
contraction was relatively minimal at small volumes
compared with those at the large initial muscle volume.
Therefore, the results reflected the minimal influence of the
variation of muscle volumes.
Figure 17 shows the simulation results of the
pressurizing process in the hip joint muscle at initial source
pressure of 6 bar and 100% muscle volume constant. At
0.2 0.4 0.6 0.8
(a) Hip muscle pressure with saturation
(b) Hip muscle pressure without saturation
different saturations in the pipe, the pressurizing processes
showed different response speeds.
The flow rate limitation in the pneumatic circuits is the
main factor that controls the pressure response in the
muscle. Therefore, in the robot design, extending the main
A frog-inspired robot is designed to mimic the frog
swimming based on its three DOFs in the leg and the
pneumatic muscles serving as the joint driver. The
prototype can perform untethered swimming which
are supported by the integrated pneumatic, control,
and communication systems in the body.
The pneumatic system is modeled based on flow rate
characteristics during the pressurizing and
depressurizing of the muscles to analyze the nonlinearity of
the driver. The pneumatic model is proved
reasonable by the simulations, and the findings is the base
for the controller design in the near future.
The experiments and simulation of the robot indicate
that the robots driven by pneumatic muscles are
feasible and the flow rate is the main factor for the
quick response of the pneumatic system.
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Ji-Zhuang Fan , born in 1976, is currently an associate professor and a PhD candidate supervisor at State Key Laboratory of Robotics and System, Harbin Institute of Technology, China. He received his PhD degree from Harbin Institute of Technology, China , in 2008 . His main research interests include mechachonics engineering and bionic robotics. E-mail: Wei Zhang, born in 1988, is currently a PhD candidate at State Key Laboratory of Robotics and System, Harbin Institute of Technology, China. He received his master degree from Harbin Institute of Technology, China in 2012. His research interests include mechanical design and bionic robotics. E-mail: Peng-Cheng Kong, born in 1992, received his master degree from Harbin Institute of Technology, China , in 2016 . E-mail: He-Gao Cai, is currently a professor and a PhD candidate supervisor at State Key Laboratory of Robotics and System, Harbin Institute of Technology, China. E-mail: Gang-Feng Liu, born in 1980, is currently a lecturer at State Key Laboratory of Robotics and System, Harbin Institute of Technology, China. He received his PhD degree from Harbin Institute of Technology, China , in 2010 . His main research interests include robotics, teleoperation and space manipulator technology .